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The History Of Mathematics In Terms Of Awareness A Seminar Conducted By Caleb Gattegno At Bristol in 1984

Transcribed and Edited By:

Dick Tahta

Educational Solutions Worldwide Inc.


First published in 1991. Reprinted in 2009. Copyright Š 1991-2009 Educational Solutions Worldwide Inc. Presenter: Caleb Gattegno Transcriber/Editor: Dick Tahta All rights reserved ISBN 978-0-9517077-0-8 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com


Table of Contents Introduction ..................................................................1 Opening Discussion .......................................................7 The Square Root Of Two ................................................35 Subdivision Of A Line ....................................................49 Numeration ...................................................................61 Counting Infinity .......................................................................75 A Lesson On Integration ................................................89 Some Feedback ..........................................................................99 Lesson On Quadratic Equations.....................................115 Discussion ..................................................................................135 Equivalence ...................................................................149 Geometric Images..........................................................165 Culture And Civilisation ............................................................173 Final Feedback ..............................................................199


Glossary Of Mathematicians Mentioned In The Seminar ........................................................................ 209


Introduction

Caleb Gattegno convened a seminar on the history of mathematics over two days in October 1984 with the following participants: David Berington Davies, Phil Boorman, David Cain, Diana Chatley, Joy Davis, Cos Harnaz, Dave Hewitt, Michael Hollyfield, Maurice Laurent, John Mason, David Pimm, Alan Roselle, David Sturgess, Dick Tahta, Jo Waddingham. The original invitation had been to consider the history of mathematics in terms of awareness. But - as was pointed out by one or two people during the seminar - no seminar given by Gattegno ever stayed exclusively within the terms of reference of a given title. On the other hand, it could be said that every seminar given by Gattegno involved some aspects of awareness. In this case, the seminar was a fascinating - and sometimes disconcerting - mixture of history, mathematics and pedagogy in terms of awareness. The following brief overview of the proceedings takes the place of an annotated table of contents. It may be of some help to 1


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readers who were not at the seminar. Those who were there will know that continuity over the various sessions was to be found in the attention of the participants and not in the sequence of topics discussed. The seminar opened with a general discussion (p6) which lasted for some time as people searched for some common interest. There was some discussion, taken up later in the seminar, of the ways in which mathematics contained its own history. A related issue was the place of fashion in determining what was eventually held to be significant in the mathematical activity of a given time. Noting the presence of some people who were not specialists, Gattegno then offered what he called stories - interpretations which “we are embellishing and stressing” - about some early Greek mathematics. There was some discussion about the historical truth of his account of the Pythagorean discovery of the irrationality of root-two. (p20) As if to emphasize his preference for ‘narrative truth’, he then described a lesson -with some children usually taught by a member of the seminar -that had involved an imaginative reconstruction of a ruler and compass construction attributed to Thales. (p27) The next example of the way in which the historical record enshrines awarenesses was numeration. (p33) In this case, the recasting of the material in terms of awareness meant that number could be treated - to begin with - as a matter of language usage. Numerals are noises which may then be written down in words or signs. This then suggests an alternative

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Introduction

approach to the foundations of mathematics - in the pedagogic as well as the philosophical sense. A brief mention of this then led to an exposition of some of Cantor’s work on infinite sets, (p40) In particular, some imagery was invoked to yield a dynamic approach to the enumeration of some infinite sets of infinite sets. Some further examples of the recasting of history were worked out in the form of lessons, this time specifically addressed to the non-specialists. First, they were introduced to the problem of finding the area under a curve, using a dynamic approach suggested by Newton’s fluxions. (p47) This was a tour de force which ended with the integral as a Riemann sum. Later, the same group was given a lesson on quadratic equations (p60), notable for Gattegno’s use of what he called the saucepan strategy -”If I have solved a problem, I don’t have to solve it again.” It was also notable for the spontaneous applause when one of the ‘virgin’ students effected a particularly sophisticated transformation. The next session concentrated on the notion of equivalence (p76) - a particularly goad example of an awareness which immediately demands a radical recasting of the traditional treatment of arithmetic and algebra. “The thing which we have to examine carefully is the revolution it brings to our thinking to our work - to give equivalence its rightful place... I am speaking of something tremendous.” The way in which algebraic transformations can create ‘another name for’ an expression was later also illustrated in a description of an approach to the teaching of division of fractions. (p96)

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Between the last two examples, there was a brief session that involved some geometry. (p85) Some work on mental imagery led to the construction of various curves. Implicit in this account was the notion that the history of geometry reveals a succession of topics that have been thoroughly worked on at one time, but have not then been retained as necessary items in the curriculum. Such examples - of what Gattegno has elsewhere called ‘the folklore of mathematics’ - emphasize an underlying theme of the seminar, that the historical record is at our disposal for whatever pedagogic purposes we choose. Cassini ovals are not now particularly important. But we can invoke a version of these curves when we need some suitable geometrical material for students to work on. Typically, this was an occasion for ‘intuition’ - which in Gattegno’s work means invoking both parts of one’s brain, using the ‘whole of the self’ Anyone interested in the style of Gattegno’s seminars will be intrigued to note that the session on geometry then moved - in what may seem a mysterious way - into a brief, but challenging, discussion of culture and civilization. (p89) Unfortunately there was a break in the recording at this point; there was a pause while a few people left to catch trains, and it was some time before it was noticed that a new tape was required. During this time, Gattegno had spoken most movingly of the way that we could now evolve beyond the bounds of nationalism, and of the part mathematics might play in the education of genuinely cosmopolitan and open-minded world citizens. Those who are familiar with Gattengo’s seminars will know that they always involve regular demands for ‘feedback’. There were a lot of these during this particular seminar; some of them 4


Introduction

involved a succession of statements from everyone, others involved some dialogues in which particular points were made and elucidated. Some of these have been heavily cut, but enough has been retained to give a flavor of what were - unusually -some of the most fruitful parts of this seminar. The various omissions that have been made in editing the full transcript have not been indicated by the customary dots. This is partly in order not to break the narrative flow - there are such a large number of omissions when any recorded speech is transcribed to make some sense to a reader. Anyone who has tried to transcribe audiotaped discussions will know that it is difficult to reproduce these in any form that is comprehensible to the participants let alone others. A small tape-recorder picks up lots of extraneous noise and it sometimes fails to catch the softly spoken word. Moreover, it is not until one tries to transcribe a recording of a discussion that one realizes just how redundant, and at the same time elliptical, the most accomplished speakers can be. In a sympathetic group it seems we rely on gesture and tone as much as words; we impose on what we hear a logical and grammatical form, which as the tape clearly shows, is not always in the spoken word. It is almost impossible not to begin some sort of editing even when attempting a verbatim transcript. And it is almost impossible to edit without changing nuances of meaning and intention; even punctuation can become misleading. Even if it were possible, a completely faithful transcription would be confusing, if not tedious, to read - especially for people who were not at the original event. So in preparing this version of the seminar - mainly with such people in mind - I have been 5


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unashamedly ruthless in cutting, condensing and occasionally modifying, a first verbatim draft. I had originally sent a copy of this to Gattegno; he had himself suggested that it. needed to be drastically cut and edited. Unfortunately, I was not able to complete such an edited version while he was still alive. So the responsibility for this final version is now solely my own. Finally, I should explain why I have included a glossary at the end. (p108) There may be some readers who will not immediately recognize some of the names of mathematicians that were mentioned by various people during the course of the seminar. It becomes a tedious convention in written accounts to introduce all names with some such phrase as ‘the great nineteenth-century Patagonian mathematician, So-and-so’. To avoid this, and yet not to assume that everyone will know in every case who So-and-so is, and what he or she did, I have listed in the glossary all the mathematicians mentioned in the text. I have confined the brief notes on their work to just those aspects that were raised in the seminar. Dick Tahta

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Opening Discussion

CG: The attempt is to work as a group of people in the field to see whether we can write a new history of mathematics. This may require a different kind of historical research. I proposed the same task to the International Commission in 1957. But this didn’t get taken up, so it has been postponed twenty-seven years. Nobody has yet written that new history of mathematics. We can perhaps try to see what is needed in order to be able to produce a history of mathematics in terms of awareness. If we can do it for mathematics then we would be able to ask also for the history of art in terms of awareness, and so on. What is it that we need to start with, David, in order to say we have something to offer that is worthwhile? DP: Some common feeling for whatever we are trying to account for. Also that the history of mathematics runs up to the present day. CG:

And tomorrow.

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DP: And continues tomorrow. In fact, when we’re studying mathematics we’re actually studying the history of mathematics. CG: When we are studying mathematics we are studying the history of mathematics - are you sure of that? DP: I don’t see a distinction between studying mathematics and studying the history of mathematics. CG: You don’t see the distinction? Perhaps you are Italian? In the field of mathematics education that is what they thought: if we teach mathematics we teach the history of mathematics. This was supported by Castelnuovo and Enriques and all those people at the time; I had to work hard to dissuade them! DP: There is a difference between teaching and studying. When I’m studying, I’m studying the history of mathematics. CG: No you’re not, you’re studying perhaps the history of the topic on which you are working in order to know who’s done what. Is that what you mean - bibliography? DP: No, it’s more a reaction to what is the prevailing view that mathematics is timeless, and the feeling that the present is not a privileged position from which to look at the past -it is already part of the past. CG: What then is the privileged position from which to look at the past?

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Opening Discussion

DP: There are no privileged positions - there are different positions. Therefore, if it makes sense to look at mathematics in terms of awareness, one starting place might be to look at present mathematics in terms of awareness to see if we can identify what we would say to the future historian about our current awarenesses. CG: So you are taking the position of a mathematician talking about his work? DP:

I am proposing that as a possible starting point.

DT: I take it that, in one sense, mathematicians are consulting and describing the ways in which they think. CG: That’s a very modern thing. Put that so that everybody may know it: Pythagoras was a mathematician, Pappus was a mathematician - where they doing that? DT: I’m not sure; I’m trying to be tentative about this, because I realize there’s all sorts of complications in what people may have said they were doing. We may feel that they were doing something else as well. The question of whether people put what they think out into the world, or whether they put what is out there into their thought, is a complicated one. But, yes, it’s a modern vantage point that mathematicians are in a sense introspective describers of their own thought processes. CG:

A handful of them.

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DT: And that one way of thinking of the increasing tendency to abstraction and structural thinking that we have seen particularly in this century - might be to suggest that mathematicians were getting - I won’t say closer, but trying to get more deeply - into very fundamental mental structures. To express the history of mathematics in terms of awareness suggests that my own history may be relevant so that to look at the history of my mathematics in terms of my awarenesses may be a relevant starting point for this sort of study. CG:

Inevitable - without that you can’t do it.

PB: I am interested in the notion that when you study mathematics you are studying the history of mathematics. Would David say that if you were doing mathematics you were studying the history of mathematics? CG: Why do you use ‘to do’ which is an action word: to do mathematics? PB: Make mathematics; if you are making mathematics, or creating it. CG: If you have made mathematics, you know that this is not true. When you make mathematics you work on a pinpointed item, trying to sort this thing out, and then it’s relation to other things to see where it’s relevant and has significance. And then you submit your paper - if it isn’t accepted you keep it in your drawer. You don’t work on other developments of this, unless you are concerned with bibliography. You may be struck by a 10


Opening Discussion

property of the numbers you are playing with and wonder whether it can be extended. Then if you want to know whether anyone has solved the problem before you, you go to a bibliography. That is one aspect of the history of mathematics. But if you are carried away by your enthusiasm, you say well I’m going to pursue that myself. After that you look into the archives to see if anyone has done this work before you, better than you. PB: I didn’t make myself sufficiently clear. The thing that interests me is that looking back over a certain amount of the history of mathematics it astonishes me how often - and this is not a peculiarity of mathematics - how often it is that the same sort of ideas and awarenesses occur to different people in different places. CG: This is a slogan that has been repeated many times. If I ask you to give examples you will find at most two. You know who started it? Perhaps it will frighten some of you - it was Lenin. Lenin wanted this to be widely believed. PB: There does seem to be a climate which is receptive to certain sorts of new awarenesses, new ideas. CG: But, you see, that’s a strange thought - that if I create, it is likely that someone will create the same thing somewhere else. PB: Oh, no, not the same thing. But it will be part of the general climate.

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CG: Maybe it will be one of the things that we shall have to dispel in our study here - this notion that because Newton and Leibniz were concerned with the calculus at almost the same time, the same idea appeared in two minds. Their ideas were not at all of the calculus. It is what people did after them that became the calculus. The inspiration of Newton and Leibniz, and their relationship to it, came from directions that were totally unrelated. You may find two cases where something of the kind that you mention has taken place. I am a little vehement about all this because I know the uniqueness of each contribution. Whereas you see two things looking alike; you take the likeness as being the same - you start with likeness and from then on you see sameness. We have to be concerned with these ways of distorting history because we need to work in a certain way. Perhaps we could make clear from the start that when we talk of history we are concerned with a stretch of time, a duration, which is long enough for us to look at - that we can talk about it now since we are here - that a certain awareness of a certain thing has been altered by reflection, by work on it, by investigation, by discussion, by exercise, and so on. Whatever you want to put into it - if it is history - has to be concerned with a stretch of time. Thus, when you use the word history, I conceive of it as referring to what has gone on for some time. This will be so for whatever I am working on. So that you could say I am putting myself in the stream of history by my contribution -that would be understandable to me. So I could say I was making history of mathematics by my production. That I understand - as a 12


Opening Discussion

meaning that you (to DP) perhaps had intended to suggest, but it wasn’t in your words. DP: No, what I meant was that by studying mathematics, by reading other people’s papers, even if they were written last week or the week before, I am studying the history of mathematics. CG: Well, there are millions and millions of people who live and who have no impact at all on history. So it’s not because you read some papers that the papers are part of history. Neither yours nor the ones you read. In order to be part of the history of mathematics, or the history of physics, they have to be significant and we haven’t considered what makes things significant. There are thousands of papers appearing every month in the United States. If you sift all these, it would be a great thing if every month three remained to belong to history. DP: From the way you are talking it sounds as if history is something external to us. CG: To a certain extent it is - you mustn’t make it too personal, it isn’t yours. DP: It depends. The question of significance is a very interesting one, because in reading histories of mathematics the notion of significance changes quite remarkably. CG: That’s part of our study - we don’t start with it - we shall end with that, possibly because we shall understand then what 13


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we are talking about. We shall see why something lasts. When I was a working mathematician before the second world war, I was astonished to find that so many people were so excited by the things that excited me. Ten years later, nobody was interested in them. And I started asking myself: are there fashions in mathematics as there are in women’s clothes? If there are fashions would the history of mathematics not be a history of fashions - which is a different subject. We are going to refine our sense of history, our sense of mathematics, our sense of significance, by working on all these matters with an open mind and without any preconceived commitment that this is what history is, that this is what mathematics is. This is what we should hold in our minds to begin with. DP: By calling it fashion, you’re drawing attention to particular aspects of change in mathematics. I don’t think it is so much history of mathematics versus history of fashions: it could be history of fashions in mathematics. CG: Yes, sure, a history of fashion - not of skirts, but of mathematics. DP:

But fashion seems to me quite a loaded term

CG: But if it is, what can I do about it? It means that you have to throw away my suggestion on the basis that it is not true DP: No, metaphors don’t have to be true or false to be helpful. It is more a question of looking at the particular aspects of

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Opening Discussion

mathematics that are highlighted by the use of the word fashion and finding whether a more appropriate term is available. CG: When I was at school there was a big stress in the last years of mathematics studies on descriptive geometry. There were several generations of people who had done descriptive geometry. Nowadays, it’s out of fashion, it’s not in the curriculum anymore. What does it mean to say it’s out of fashion? You see, someone has decided that it’s irrelevant -why waste time doing this? - and they have convinced the authorities that it shouldn’t be taught. When nobody learns about descriptive geometry it is as if it doesn’t exist, you will find it only in the history books. AR:

Or in Switzerland! (where AR teachers)

CG: Or in Switzerland. . . You see, perhaps I like it - it’s a nice subject and shouldn’t be taken away. But it has been taken away. You can look now at the history of mathematics in terms of the content of the maths curricula, either in college or school, and you will see how many things have been dropped by successive generations. That means that the notion of a fashion should remain at the back of our mind. Maybe there is a sense in which if we like it, we cultivate it - that’s one attribute. But if we don’t like it, what do we do with it? We can’t convince everybody to drop it. But it happens again and again that there have been certain topics that were once extremely important - like elliptic functions, say, where lots and lots of mathematicians gave all

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their time, all their being, to produce theorems, yet hardly any mathematician is concerned with them today. We don’t know yet what we have to look for in order to be able to rewrite the history of mathematics in terms of awareness. But one beginning has been taken, one step has been taken already by saying that somewhere there must be criteria for us to decide on why some things last. We are not the ones who choose Pappus’ theorem, but it’s still in many of the books. I did not publish these books, I didn’t write these books, and it is there. One aspect of our study of history in terms of awareness, is to define our approach to it - rather than saying history of mathematics is the set of history books of mathematics that are in our library. You take these books for granted when you read them. Yet, Bell, who wrote such lovely and lively history of mathematics, is being found again and again to be misleading. I have been misled by him and by others. Can I shake my head and get rid of these things? That is one of the incentives for trying to rewrite the history of mathematics in terms that make significance significant, rather than stories one after the other. You see the mathematical work that I sweated over is worth nothing. Nobody’s interested, nobody wants to know what I found; the theorems with my name - nobody knows about them. DP:

Does that concern you?

CG: It doesn’t at all. It helps me understand the history of mathematics. Maybe it helps you to understand that it’s not because someone works hard on something that makes it

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Opening Discussion

significant. I had a drawer with so many calculations, and I found a beautiful relationship between Liouville’s differential equation and Bernoulli’s. It was an illumination - who cares about that? So, if I know this, I know that you make yourself obsolete by working on things that are not significant sufficiently significant. DT: Can we rehearse that in terms of the example of elliptic functions? Because I am still not quite clear about this notion of significance - whether it’s a social phenomenon, or whether there is some other criterion being invoked. The sort of questions in my mind are: have elliptic functions gone in and out of mathematics? Is it still mathematics? CG: It is part of mathematics, but of no interest to mathematicians - or at least to most working mathematicians. DT:

In a sense it is then history.

CG: Well, it is history - in the sense of a depository where you put things. PB: Would you be the same man if you hadn’t found this lovely relationship? CG: Oh, surely a better man. (laughter) I could have spent my time on other, much more significant, things. But I thought - I fooled myself into thinking - that little thing which looked beautiful to me was important. I switched from ‘I like it’ to ‘you will like it’. 17


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DC: I still don’t understand what significance is about. It has always puzzled me. CG:

It’s a beginning - we are finding stepping stones.

DC: One of the things that puzzles me very much about mathematics is that what is significant depends on whom you talk to. CG: Sure, and that is also part of our study of awareness -that you can have two members of the Royal Society and they don’t have anything to tell each other - they have nothing of significance to say to each other. Because mathematics is not mathematics - it is something else; and at the end, when you know what it is, you say its mathematics. DT:

You’ll have to repeat that last sentence!

CG: When I was concerned with these matters, I discovered that what we have to be concerned with is mathematization not mathematics. Because mathematics leads to this situation where you think it’s all the same thing and it ain’t. So that two mathematicians - they call themselves mathematicians, everybody says they are mathematicians - when they meet each other, they don’t understand each other. We have to study why this is so. What does it mean that I don’t understand, that I am not understood? We have to work on this in terms of awareness. It seems to be unimportant for mathematicians, because by 18


Opening Discussion

definition a mathematician is schizophrenic. In order to work on mathematics you have to develop in you a capacity for leaving everything else out. Musicians are like that - so when you are working, you have to be cut off from everything, other than whatever occupies you. This is not a condemnation of them; it is a description, using a word that has been put into circulation about ordinary people. When people close themselves in and don’t see the rest, we say they are schizophrenic. And that applies eminently to mathematicians and musicians - and often both come together in one person. If you have met a number of working mathematicians, you cannot escape noticing this. Not all of them are like this, of course; but there are so many who are. Some of you may have known, or known of, Littlewood. Among mathematicians, he was considered a man who was entirely by himself, never with other people. He was entirely cut off from the others in what he did. He had a great reputation for the papers he wrote, he held a high position at the University of Cambridge, he was a delightful person for everybody outside mathematics. But within mathematics he was on his own, cut off. You can say this to a certain extent of Weil, one of the founders of Bourbaki, who - in addition - had a tremendous contempt for others. Not only was he by himself, but he looked down on everybody. Let us come back to what I wanted to say a moment ago. If mathematics is a confusing word we have to change it into something that is less confusing - so, mathematization. Mathematization is the making of mathematics - not of one mathematics, but of the mathematics that belongs to this act of 19


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mathematization. In the field where you are working, you mathematise something that is not mathematics and you obtain mathematics. Now once you have produced the mathematics, it may have its place - if it’s significant. It may have its place in history because it was significant, because it had consequences and lots and lots of people produced further things and went on producing them. You see, one exercise for you is to imagine a great mathematician of antiquity coming among us today. What would he feel? Would he feel anything other than that he is a stranger and had better run away? How is it that he couldn’t come back and say: oh, you are a mathematician, I’m one too. And we would reply: you’re not; what you were doing, working on, is trivial - we are not interested, shut up. Even Pythagoras or Eratosthenes, great founders of chapters of mathematics, would have no concept of what we consider to be mathematics. DP: And, equally, I doubt whether we have much concept of what they considered to be mathematics. CG: What we don’t know is how they worked. That’s why the history of mathematics can be put back onto the agenda for us to see what sort of awareness is required in order to be Pythagoras, as against being a Cauchy or whoever. If history is not of mathematics, but of mathematicians, then the mathematics changes from chapter to chapter. We can quickly dismiss this bias - that there is a history of something called mathematics. We are concerned with a human activity, with the activity of some human beings. If we could understand this, we would be

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able to say: they are mathematicians because so-and-so. And if we found a similar thing in others of whatever period, then we could say they were mathematicians as well. If you said that Pavlova was a dancer, everyone would agree. She wasn’t the only dancer but she was a dancer. There will be some attributes - we have some films of her or we know my grandfather saw her, or I saw her in flesh and blood on the stage - so we can say Pavlova was a dancer, because grandfather says so. Now we have to find whether there is an attribute of mathematics which will tell us that when you list mathematicians you don’t include Pavlova, or Le Corbusier. Once I worked with quite a sizable mathematician - bodily and otherwise. He would never fail to publish the smallest thing he could find; he would know it’s new and it would appear in half a page. He had a sense that whatever he did produced mathematics. If I had written that little paper for him he would have told me to take it further. But as for him, it was just as in an exhibition in the Royal Academy I saw many, many years ago. There was a piece of paper this size, well protected, under glass, with a circle on it. . . almost a circle: it was by Leonardo da Vinci, he hadn’t even completed it. DT: This man - this is a type of man; it doesn’t tell us anything about mathematics, does it? CG: No, but that was in passing - so that people who think mathematics should be democratized, that everything is of equal significance, may have a moment to reflect on that.

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PB: I don’t think everything is equal in significance, but I do think that it is very difficult to know whether something is going to be significant, and therefore that it may be that small steps made by a hundred people are more significant than the big step made by one. CG; There was a meeting in Vienna around 1890 between Gorky and another novelist. The whole conversation was about when Zola’s next novel would appear. To expect a novel by Zola meant to expect something significant in the field of literature. If it appeared, if it came out, it was worth waiting with trepidation for it. Well, its the same for some mathematicians. There was a French mathematician called Gaston Darboux, who published a beautiful series of books that nobody reads any more. Darboux was like a steamroller; he had such power, mental and otherwise, that if people working in and around the Ecole Superieure or the Sorbonne heard a rumor that he was onto something they were working on, they would drop their research - they would abandon the field at once. To know that Darboux was putting his mind to it was sufficient to discourage them. That’s part of the history, because through it you learn that mathematicians may have some qualities but also have some other qualities - some other aspects of themselves - that don’t contribute to mathematics. We don’t know whether all these people who were discouraged, might not have done something valuable or not; all we know is that they didn’t. DP: There is a nice example of a similar thing, only it’s where the problem itself - rather than the individual - put people off. It’s an account of three generations of mathematicians working on the Poincaré conjecture talking about their attitudes to the 22


Opening Discussion

problem, the methods and techniques which were used, and so on. A number of them said: I warn graduate students off this problem. It was not that the problem wasn’t significance - it had great significance - but it was considered too hard. You would waste your life working on it. CG: Do we read these things in the usual history of mathematics? The books tell us about the things that have been achieved. Then they turn in the next chapter to other things that have been achieved. They don’t tell us how a field vanished because the remaining problems were either insoluble or too difficult for the mathematics of their time. Now that’s part of the history of mathematics, and it is salutary to know that there are people with tremendous technical capability who have to give up. So when we see our children, our students, working on something, we think it is easy because we know the answer. We don’t realize there is a mental development to a certain stage where I know what I am doing and I know whether I can continue or whether I can stop. I would like to put all this within mathematization. This has two meanings - it is a process, but also a result. It’s often like that: in order to work out, say, seven add seven, you ‘sum’ the numbers to get their ‘sum’. Can we have some views now on whether we have started, or whether we prefer that we change course and do something different. DC: Could I come back to significance. You mentioned this mathematician that you knew. Did he think that all those little pieces were significant?

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CG: He would say you would never know who needs what, perhaps there are people who are waiting for something new, in order to continue their work. DC: If you think you have discovered something then there are times when you may say there is something there of significance. At other times it needs other people to say it... DT: One example to consider might be Cauchy. It may be that the pressure for someone like him was not towards significance, but more like the pressure on say a certain type of hostess to feed people. I mean he just had to pour it all out and he appeared to be such an opposite character in this respect to Gauss who had to keep it all in. I don’t know whether these two illustrate a distinction; perhaps significance was more important a question for one than the other. In response to your first question, I think this would be an excellent direction to continue with. DC: I have written music and sometimes when I have written a piece I’ve said yes; and at other times I’ve written music and I’ve said: well, OK, it goes like that, it does its purpose. Is significance to do with some awareness of what it is that you are at that moment feeling? CG: I don’t know enough about music to decide about significance there, other than looking to see whether it establishes a kind of lasting fashion for people to do a similar thing - as with Bach or Mozart who had repercussions and who 24


Opening Discussion

inspired generations of people by this way of composing. To the extent I know music, I would I would say Bach is significant, not only because his pieces are beautiful and attractive, and so on, but because of what he started. There may be another whose music is delightful, but it remains an option, it didn’t create another tree. In mathematics, it’s easier, because although mathematicians have the same mental make-up as the musician, what they produce can be worked on by other mathematicians. No musician would say: I am going to improve upon this by adding a few notes here, a few notes there. This is excluded in that realm - music remains what it is. In mathematics you can take the work of anybody and improve upon it DP: There are minor changes - I mean orchestrations, changing instruments, and so on. CG: It’s compatible. But you can’t say I am playing Hindemith or I am playing Elgar. On the other hand, you can say: there is the theorem of So-and-so and So-and-so. PB: ‘Pictures at an Exhibition’ is the classic example of a piece that was really worked by two musicians, isn’t it? I think there are some other examples of these DT: Oh, come on, these are minor counter examples. Isn’t the point to stress that in mathematics theorems can be worked on by other people and accumulate a whole series of re-writes. JM: Name any theorem that you know and I will predict that theorem as it is presented currently would not be recognized by 25


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the man or the woman whose name is attached to it. In fact, you’ll find that it’s been through at least a transformation per generation. So if you take, for example, Riemann’s theorem - the fundamental theorem of the integral calculus - that the area function is the same as the derivative, I think you would find that Riemann wouldn’t recognize it and neither would any of the people of the intervening generations. Lagrange and Cayley didn’t know the theorems now known by their name. You can’t find their theorems anywhere in their writings. Certainly Cayley’s theorem, a fundamental theorem in group theory, doesn’t exist in his writings. Unless you dig and dig and dig and recognize, because of your current awareness of what is a significant result, a sort of little kernel, a little something or other which you want to hold on to and say that is a theorem. It seems to me that in mathematics we transform every time we present a theorem. It’s been transformed every time you walk into a classroom and say: right, now we’re going to do Blah’s theorem. It’s not Blah’s theorem at all, it is something that has been derived from the work of Blah, and it has been transformed through many, many generations of expositors. It’s quite, quite different from what’s going on in music. There are a few minor examples, say where people have taken a theme of Thomas Tall is and mucked around with it and presented something else, and somebody will then perhaps muck around with it yet again. DC: I’m not sure about that, John. If you go back into the Renaissance and the Baroque, then there was music around that

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Opening Discussion

didn’t belong to anybody. It was worked upon by all kinds of people and turned up in different forms in different ways. But when you come to the nineteenth century you do get things in their own right, which are not to be tampered with. You also get much more definite statements from composers: I have sweated blood for so many years and have just completed a work which I feel is great. Many composers haven’t needed anybody else to say this, they’ve said it themselves. Do mathematicians make these statements? I ask as a pretty ignorant person in this area - I just wondered whether such statements are made. CG: There can’t be a group in which people will not say: I speak as an ignorant person. We haven’t yet come to a consensus on whether what we are doing seems acceptable, encouraging, exciting, to those who are here. JM: It seems to me clear that the question of significance generates energy if nothing else. I don’t think we are ready to discuss significance yet, There are a lot of concomitant notions which we haven’t considered - for example, awareness. I think we’ve laid out some of the questions; we now need an organized means of attack. CG:

A priori or a posteriori?

JM: I suspect there are certain principles that are going to be invoked and I want to see them invoked. 27


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CS:

Now?

JM:

Yes.

CG: I always try to prepare my lessons after I have given them. Do you feel engaged with what we are doing? JW:

Yes.

CG: Enough to go on; but not enough to put yourself into it, not yet? Allright. David? DS: I would certainly like to go on with significance. It is something that has always puzzled me very much. CH: I was also very interested in the idea of there being a social aspect like fashion. The introduction of music generated energy into the discussion. I would like to go with the flow a little more for the moment. JD: I’ve been thinking about using questions as instruments and what you have raised previously about what makes good questions. At the moment there is a lot of material that would be attractive to pursue, but I don’t feel the enunciation of a question. ML: I have a question about this issue of significance. Since maths is an activity of the mind, is it possible to look at the results that mathematicians obtain and see if these come from a

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new awareness which includes all that was valuable before then as well as giving a new start for the other mathematicians? CG: If you want to keep these two questions together, you will get into difficulty. Because what goes on may not be at all a new start. A new start may be cutting off entirely from what was going on. ‘A bas 1es triangles’, said a group of French mathematicians in the fifties - down with triangles, so don’t lets keep any Euclid in the curriculum. DP: I have a new start, which picks up on John’s question about awareness. I have an example that I would like to offer: conics sections. At first they were classified by words amblytome, orthotome, oxytome - that described the different cones being cut. Then there was a shift to words - ellipse, parabola, hyperbola - that reflected the way different cuts were made in the one cone. This seems to me an example of shift in terminology that might carry with it a shift in awareness. CG: Do you want to give this change of name as an example of awareness? DP: I was trying to focus on the question of words carrying with them traces of awareness. A change of name might then mean a change of awareness. I was proposing this as a possible line of attack, following up John’s suggestion that awareness might be something that we should focus on before looking at significance.

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CG: Well, that’s what he said. But maybe he is prepared to say he doesn’t know whether one comes after the other. Perhaps they come together. If we come across significance first, we are not going to disregard it because it doesn’t come in a certain sequence. You are prepared for that, John, aren’t you? JM: Yes and no; my experience with you is that you use words that I think I understand, but you use them in ways that are slightly different than the ways in which I use them. So I would like an opportunity to come to appreciate more fully how you are using awareness and history and mathematization in this context. CG:

Yes, but if I say the word ‘reading’ do you understand?

JM:

No

CG: With good reason, for there are at least twenty meanings for that word. But in mathematics, when something has more than one meaning it has no meaning. We tend to protect ourselves by keeping to single meanings; if there is need for a new meaning, then you use a new word. We may end up with any number of mathematizations, with any number of chapters of mathematics. We need not object to that, or protest at the way of working which is being exemplified here. I have wanted to work on this problem for years and years. It is not worked out and this is an opportunity for us to do something together, which I have felt for so long is worth trying. The truth of our research will be measured by the extent to which we end 30


Opening Discussion

up having arrived at a similar point of view. Some of what we see will be full of details, some less. One part will be colored in green, another will be colorless - and so on, according to our temperament, according to whom we are. We are going to make our understanding of this vague thing called the history of mathematics becomes more precise. Have we consensus? Do we have your support or shall we choose another program, another example, another alternative, another start? DP: We have at least two possible projects; I am quite happy which we start on. CG: I don’t understand yours. Your example has confused me. Are you concerned with labeling realities of the mind so that one label triggers a certain meaning? If there is a richness of meaning behind that label, are we making sure that everybody that hears that label reaches more or less an equal amount of richness in it? Is that what you had in mind with your example? DP: No, what I had in mind in my example was a suggestion about how you might try to look for - how you might try and gain access to the awarenesses mathematicians might have had. CG: We can’t go and consult them. We have to take responsibility ourselves. If I say that so-and-so had this awareness and someone doesn’t agree with me, we should have a quarrel between us, not with him.

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DP: But our quarrel won’t be - at least I hope it won’t be -just us quarrelling we might be able to appeal to others. CG: Behind your question there is a whole universe - I don’t know whether we can get into it. How do I enter into the mind of someone else, particularly if that person is not accessible? Well, that is not what we can handle here; it is too big a task, although it is an important one. But we do carry something we call mathematics in our own sensitivity, our memory, or some other place. Can we relate to that amount of furniture of our mind and say something about it which throws light on the title that brought you here - the history of mathematics in terms of awareness? If we are not too ambitious, too greedy, we shall end up with a lot. If a lot is too little we shall have to come again and try to get a bigger banquet. DT: There’s one aspect of what David is saying that interests me and I think is probably fertile - but it may not be what he is concentrating on. It seems to me that many of us might well be at the moment looking for a closer understanding of the shifts of awareness in history, and one clue might be the renaming of words, or the shifts of language that get made. CG: What is a shift of language? Do you mean speaking of filters rather than limits? DT: Yes, that would be a typical example. I think it is a very big issue, and that possibly it’s also geared to the significance issue. So if there’s still a question of which direction we probe in, 32


Opening Discussion

I would like to probe into significance first, because that’s what we’ve started on. I don’t feel that would be ignoring awareness, which is very much at the front of my mind. CG: If what is needed is that we be concerned with awareness, then we are not in the theme of our seminar. If it’s absolutely necessary to consider awareness today and then start on the history, then we should do so. But it seems to me that it is possible to have a growing awareness of what we call awareness as we go on; we don’t have to have all these things out together from the start.

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The Square Root Of Two

CG: For me, the challenge is new in spite of all the time I have spent working on all this. It is new because we are here, and also because I’m aware that there are three people here who don’t know enough mathematics. I don’t know enough, but I know a lot more than they do. I would like to keep them awake bytelling stories which are true, and which are relevant to what we are doing. It’s a process by which we eliminate technicalities. If we go on speaking of elliptic functions, I will have to send them out, because we are not entitled to spend so much time on something so narrow - even if it was the field of mathematics for fifty years and the greatest names were into that field. There are beautiful stories that we can start with. They belong to history. But they are stories, because we are not reading the original manuscripts - we are interpreting, we are summarizing, we are embellishing and stressing, and so on. One of these stories - a historically remarkable one - is about the fear that the Greek mathematicians experienced when the Pythagorean system of mathematics collapsed because root-two appeared on the scene. Now, how could lofty minds - such competent people

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- have such fear? You can only understand it in human terms. You can only understand them as having done a number of things with themselves to come to the point of saying: we have to turn our sights somewhere else. And they did; they changed the thrust of Greek mathematics completely by not allowing the best among them to think of what became later known as algebra. We shall be concerned with human beings working on a challenge, a professional challenge in their field, and not understanding it. In the Pythagorean school of mathematics they used to study lots of things; and they were able to find important relationships within what they were looking at. But what was the premise on which they worked? Whole numbers - and a connection between two numbers as found in music, a ratio. If you choose one note, you can describe the other notes in relation to this one. The ratios are pairs of whole numbers; one note is three-fourths of another, or eight-ninths of another, and so on. They convinced themselves that the whole numbers, and the pairs of numbers which we now call fractions, were sufficient to describe the universe. But one of them, working on another problem, put marks on a straight line . . . (drawing on the board). . . If you take this length, and you take half of it, or a third of it, and so on, you get lots and lots of points that you can label in terms of pairs. So you use a kind of measurement to give a name to a point on the straight line. Now, you make a square on the line and you draw a diagonal In order to measure it with the measures you already have, you rotate it and let it drop onto the original line. But they

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The Square Root Of Two

could not give a label to the point on which the end drops. It’s impossible - and they had a proof that it was impossible.

Their philosophy, their metaphysics, their religion, all seemed to hinge on the belief that the universe was harmonious. But it was only harmonious when it had the whole number properties they had selected in advance. Now, they had already proved that in a right triangle, the square on the hypotenuse is equivalent in area to the sum of the squares on the two sides. So when - coming back to my picture - you take the side of the square to have length 1, the square on it will have area 1, and so the square on the diagonal will have area 2. Hence the diagonal has a length which when multiplied by itself gives 2. This length was later called by a new name, ‘root-two’. It can be proved that this cannot be represented by a fraction. There were then two possibilities. One was to suspend judgment and ask what had happened. Can we trace our steps back and see how we started? Was there any necessity in the way we

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started? The other possibility was to say: everything has been working so well and now it isn’t anymore - well, why not shift to doing something else? Which is what they chose to do. And it took a long time - centuries - before there was a new start with respect to all this. The person who helped most to make a new start just said: why not begin by saying every point on the straight line has a label, and that label - which is the measure of the distance to the origin - is a number. You can then define the operations which were valid in the field of integers and fractions on these numbers. The unscrupulous person who helped us was Descartes. He was the first person who realized that you have a new start when you put the question the other way round by saying: to every point I give a number. Number is now the name of the measure of every segment that you can make. That measure became much clearer at the end of the 19th century when the axioms of Dedekind, Cantor and so on, were put into perspective. Why did the Greek mathematicians decide that this area was doomed and that they had better not get into it? There is some indication that they were at ease with it when Plato considered in some dialogue that the roots of numbers have an existence, that you can say root-three and mean something. He did not show it - he was a philosopher not a mathematician - but he suggested it. Although this lingered in the psyche for generations, nobody really developed it because the new start they made was so successful.

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The Square Root Of Two

That new start was to get into geometry, to abandon the field of these numbers. So the Greeks became geometers; they got into sculpture, rather than music - into geometry rather than algebra. They contemplated space. One of the things they did for us was to discover the conics as conic sections, not as second degree equations in the plane. Well, that’s my story. How do we handle it as a starting point to be concerned with people in the past, with whatever awarenesses that we can grant them and that we must deny them. DT: I think it’s a useful story to work on, because it seems to raise a lot of the questions we were asking earlier. I’m not sure, when we make these accounts, how much we are putting into them. I understand parts of the story - that the Greeks did, as it were, shift to geometry. It seems to me that they were still handling the root-two lengths but not in the same way as we do. There is this book 10 which is full of extraordinary stuff -clearly some sort of classification of these supposedly terrible things. So there must have been a long tradition, and Plato must have known of it because -of people like Theatetus. CG: Let me tell you that North of the Mediterranean is not South. When you come to Alexandria, which is my city, there wasn’t only Euclid, there was also Eratosthenes and Eudoxus. These people had different views of the world. They shifted away from what had been so vital for the Pythagorean school. You can say that the earlier ideas were still important, were significant, because you find them again and again all through the ages,

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changing but always there because of the effect of that thinking of the 6th century BC. But if you say that the later Greeks did not abandon them, then you have the responsibility as a historian to say who did what. Plato was still not satisfied that they should have done that, but he couldn’t do anything else but say a few things in a few pages. Then we come to an example where we find that the preoccupation was not to save Pythagoras. Euclid’s preoccupation was to create the edifice of the geometry that the Greeks had found -to give it an order, and to make it into an axiomatic theory. You see, measurements are not mathematical. Measurement is the impact of physics on mathematics. Today we can say that in the consciousness of mathematicians there is no time or anything connected with time. There is no measure. Measure is a physical activity and time is available as a notion that you can use as a frame of reference, as Kant wanted us to do. Space and time were brought together by the Alexandrians although time had not for them the reality of space. They were unscrupulous. I shall use this notion again and again; they were unscrupulous: that means they didn’t mind turning their back on the traditions to bring us onto a path of their own. The fertility of the work of Eratosthenes, of Eudoxus, of Euclid, is the justification for maintaining this. DP: The more I read about the enterprise of the people we are calling the Greeks - predominantly a group of perhaps ten mathematicians working in Athens in about 350 to 300BC - the more opaque I find it.

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The Square Root Of Two

My reading of the Elements is that it is not about geometry. It is written in the language of geometry, and geometry was their language of proof in exactly the same way as algebra is our language of proof. Book 10 - which is in terms of number of pages a third of the entire corpus - is this massive classification program of incommensurables. Geometry is the technique of proof for arguing about magnitudes. There are also some books on number theory as well as books on geometrical objects treated geometrically. It seems to me that it’s much more a whole approach, that is continuous from Pythagoras, of working geometrically on mathematical problems. DT: Translators in the 17th and 18th centuries didn’t bother to put Book 10 into their school texts. They made a judgment of significance - or difficulty. They chose to pick up all that stuff about triangles and circles and so on, which then became the popular view of Euclid in Europe. But book 10 did worry a few people though it wasn’t actually put into school texts. CG: Let us remain in this room with what we have in us and how we interpret whatever comes to our mind. We cannot enter the mind of the dead, we cannot speak on behalf of the Greeks. But we can discipline our minds to work in a certain way. For example, you can’t deny time when you say: here are two triangles - they are going to be called congruent, say, if one can be put on top of the other to cover it exactly. But the language of congruence ignores the? taking and putting on top. So I look at these two triangles and through measurements -which are actions which are not mathematical - I say: these two triangles

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have side AB and side PQ equal, and so on. Then I say that the two triangles are congruent and I produce something for people to agree with and to play some games with me. The awareness which is directing me is that I can now look at this process at the various levels of my human condition - not of the discipline itself. In the discipline itself, there is no motion. It is a development of the 19th century to have studied transformations and put these as axioms at the basis of geometry. This created a multitude of geometries, whereas there was only one geometry for Euclid. When Euclid did his work he didn’t think of the critics who would turn up centuries later -he was his own critic. He wanted to be rigorous in his own terms. That is where the history of mathematics as awareness comes in. So that you understand that Euclid’s dialogue is with himself, and he is going to find whatever he can find. In the 19th century, the tendency - which came from the middle ages - was towards generalizing all these things, making them universal and eternal, and so on. Today, we are bringing back the human into the geometer, into the algebraist. True history happens because men evolve - they change, they entertain different things. There is no further obstacle to saying we can recast history in terms of awareness. We can now ask the question: what was in the awareness of these people to produce these things - which awarenesses were needed? An awareness which is in every one of us is that there are two modes of working. One is to make chains of arguments that support each other, that are consistent with each other, that are

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The Square Root Of Two

equivalent, so that we see that there is progress by a movement which we call transitivity - as we pass from one to the other. But there is another movement, which is what we do in the beginning when we provide definitions. This is the creative moment in mathematics - it’s when we introduce a new entity. If we become aware that definitions are new starts, every definition that I make opens up something - if it is a good one. What the Bourbaki did - without telling us that they were doing it - was to consider a very fruitful, a very rich, theorem and call it a definition. That means it is something that allows you to start: I collect all these things; now you have them all see what you can draw out. When the Greeks abandoned a field because of the panic - which is an important notion here - they did not find in their makeup that which exists in all children. They did not find in their awareness of themselves that they could take advantage of the shock and continue with a new definition. They didn’t have the definition, they couldn’t produce it. They said: lets do something else. But more than this, what they did was an expression of themselves. They provided us with a multitude of statements for those that want to hear - that images are important. So, in terms of awareness, the Greeks were aware of imagery and of the dynamics of imagery - and they exploited it beautifully. They could do all sorts of things with their imagery. It became an isolated universe - but one worth investigating - to know what sort of thing the mind can do with images. Now if you give a prominence to images rather than operations and that’s what you are concerned with, then you become a Greek. Then you can recognize that you see anew.

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When I animate geometry - in making films after Nicolet - that is precisely what is in my awareness. I am aware that I am not going to talk of time, but that I am going to use time - by making a film. You don’t have to know that I use time and I don’t use it explicitly. Because I know the techniques of animating a diagram, I can generate geometrical entities, and these are related in my memory when the film is over. I flood my mind with images, and when I start working on them what I obtain are geometric results. So animation is the subterfuge that inserts time into geometric situations - where time does not belong. We can, of course, see that the Greeks used it implicitly, without telling us. Whenever they used motion, they used time. But they didn’t use motion to study kinematics, they used motion to establish relationships which they brought back to use for another geometrical situation. DT: That is very illuminating. You’ve reminded us that geometry is a far richer notion than the sort of thing that is usually described. It’s an interesting re-evaluation of what the Greeks did. I need to think a bit more about it. And about time I hope that you will work a bit more on that. CG: I know some business people. They think all the time of business. What’s wrong with that? There is nothing wrong; that’s their pre-occupation, that’s their occupation. They find some distractions - they have lunch and some drinks and they go to the theatre and so on, but this is not essential - their life is business. After you meet such people, can’t you say that the Greek geometers were pre-occupied with space and its properties. They were compiling so much; why should they do anything else? Just as the businessman makes more money and 44


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then feels justified, so the geometer finds more theorems and that’s a satisfaction to him. I don’t have to give him thoughts he doesn’t have. He was entitled to pursue his thing in his way, just as you are entitled to pursue yours. This is how to respect history. You don’t want to distort history by projecting where you are today onto where they should be. You don’t quite understand everything they do, but at least you are disciplined not to say they should be thinking like you. AR: But when you’ve come to a point where there is a shock like this, rather than accept the discovery and use it, you flee. Or instead of accepting something, you burn the library at Alexandria. Or because of a disease like the bubonic plague, you burn the Jews - as in the middle ages. Is that not to be viewed not necessarily judged, but viewed - as a bias which prevents the taking of one path? CG: But we are not concerned with that. We are not moralists at this moment, we are not historians of civilization. We are trying to find the awarenesses that we can reach when we consider what happened in mathematics and there are vast numbers of such moments which you can find within mathematics without having to call on the invasion of Alexandria by the Arabs or . . . AR:

Or by the Christians.

JM: When I make an assertion of the form ‘somebody was preoccupied with something’, I am trying to make deductions 45


The History Of Mathematics In Terms Of Awareness

from a few surviving material objects. I don’t know the state of mind of those people. I can look inside myself, I can discover the presence of images, I can weave a coherent story which suggests that something similar was happening to them, but all the time I am still making an interpretation of what it was that actually preoccupied them. I, myself, cannot assert that the Greeks were pre-occupied with space. I can’t even assert that Plato was pre-occupied by space. I have not read any other Greek geometers - except in many, many layers of later translation, so for example I cannot assert that there was a shock. It seems a very plausible story and it seems to make some sense. It allows me to be then invited to agree that they were concerned with properties of lines and circles when it might be that they were actually concerned with measurement, but that their mode of thinking was through the imagery of points and lines as the only way they could think about measurement. CG: There is evidence, because there is Euclid. Euclid summarized the heritage that he had received from many other people who had found the various theorems. He put them together in a systematic way. JM: I’ve heard lots of people attribute systematization to Euclid. I can enter that because in my experience of myself -and in observation of my fellows - I notice an inner desire to systemize which seems to me manifested frequently. So that seems a plausible story. Then I’m invited to agree that Euclid

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was systematizing geometry and that he was actually beginning the axiomatic method. But that isn’t obvious to me. CG: I would like to know if you are saying history is an impossible task? History is not writing about one thing after another, but finding a way in our psyche that allows us to reach the psyche of previous generations. If we don’t reach it, what we write can be disproved by counter-example. If we do, it convinces all of us, because we all inherit that past. I have to be what I call disciplined. I have to place barriers here and there, so that I do not put in things that were found much later. I have to understand that history exists. I have to know that some people have found in themselves that there is sense there is a meaning - in looking at events that are no longer there and taking note of them. Historians are not chroniclers, they are not novelists. They have put more and more constraints on what they are going to say. There’s a scientific approach - you can’t say things that can’t be found by others. Now, I heard you say you can’t. I can’t either - I can’t say such and such a mathematician thought such and such a thing - I can’t say that. But I can do something to myself, so that I find that if I were in their circumstances - not with my experience, not with the heritage of history - I would probably find what they did, because probably they did something not so different from what I would be doing. JM: I am very happy about that as a method. But I notice, for example, that it was very easy for me to be convinced for a long

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time that the Greeks must have been shocked - and to have sacrificed many bulls - when they discovered root-two was irrational. I want to be careful, because I think the notion of impasse - and possibly of shock, though I’m not sure about that is a critical one in the development of mathematics. Perhaps I agree with that and I want to pursue it. But I observe that it is very easy to talk myself into it, yet what I’m talking myself into is, in fact, a story that has been handed down to me. CG: If you stop there, you will just be saying: that is the challenge. You will be missing an opportunity of seeing whether there is another example which will indicate that I am justified in making statements about Greek geometry.

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CG:

Are there other examples that I can be connected with?

It occurs to me that Maurice asked me two years ago to be with a course of four one-and-a-half hour geometry lessons to his students. He gave me the task of teaching Thales theorem. I said: allright, let me try. What I tried provides an example of my way of working on this kind of challenge - where the discipline that I give myself yields results. If I remember rightly, I started by asking them what they knew - which was that they had used ruler and compasses and knew perfectly well how to divide a segment into two parts, and how you can then divide it into four, and eight, and all the powers of two, because you will be iterating the same process. So I said: right - we got excited just now because we could find the fourths, we could find the eights, we could find all these. But can we divide it in three? Wouldn’t that question come to a Greek who had been working on these things and can say to himself: since three comes after two, can’t I ask the question how to divide into three parts. So the Greek fellow went to bed and fell asleep thinking: can I divide?

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They had answers. They said: fold it and fold it. How do I know it’s one third? How could I know? Can you divide a segment into three equal parts? You can’t by remaining on the straight line and using the compasses - you can’t. So what - in his despair did Thales do? He fell asleep, he slept on it. And having fallen asleep he saw himself preoccupied ... ah well, I want three, I want three. The next day, he went to the beach as he used to do every day. And on the beach he started walking, asking himself how to divide a line into three. Then he shifted his awareness from the problem to the fact that he was walking. He became aware that he was walking, Can you imagine that? And he said: oh well, if I stop after three steps - one, two, three - and I put this on the line, then I have divided it into three parts. Then a critic said: but what if my line is like this? Well - he says - it’s quite easy; I join this endpoint to that endpoint, I draw the parallels - here are the three points. And what - I asked my class - do you think he did after he found the answer to one third? It was obvious to them: can I divide it into five, or into seven? Why do you say five, seven - can’t you say something more reasonable? Oh yes, I can divide it in any number of parts; in particular, the four, eight, and so on. Thales became a person who had problems, who pondered on them, who found himself caught in the tradition, and who got out of the tradition and tried something else. All this was done as if it was a Greek discussion on the beaches of Mi1etus. Well, this is the kind of thing we can try to do - is there something else we can do?

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Subdivision Of A Line

DT: I want to work on the implications of that sort of imaginative reconstruction. I’m quite happy with the notion that history contains not only what might have been supposed to have happened but also what everybody else has ever said since about it. I see it as a myth in which I am participating. So I think that account of Thales is very powerful. I see Thales walking on the southern shores of Anatolia on the sand. I like that idea. I can see that it is possible to work on what I know and come up with intuitions of that sort. Would you say that history is also the accumulation of what we say about it? CG: This is another problem. What I would like to do is to see to what extent it has been made obvious that you can recast the history of mathematics in terms of awareness by taking one or other of the examples. PB: I’m not sure whether one is recasting history. There are people who would take the view that this is just fanciful nonsense and that what we are concerned with is just facts. But I get far more revealing insights out of ‘truth stories’ - as I would call them - rather than the factual account. I accept this sort of story straightaway as revealing insights into the nature of what has happened. Whether it is possible that these accounts really do have a close parallel with something that happened so many thousand years ago doesn’t strike me as being an important question, particularly as I don’t think it will ever be possible to answer it. AR: The seminar is on the history of mathematics in terms of awareness and not the history of mathematics in terms of social

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dress, or towns that were lived in, or the invaders. If the history is coherent then it is the history of mathematics in terms of awareness. The other point I wanted to make is that there is no history of Thales, because that went up in smoke with everything else in Alexandria. DS: If it is to have a point for me, it must be to make me more aware of what mathematics is. Therefore, if that happens, it doesn’t seem to me to matter terribly whether this is historically accurate or not, if it checks out in my experience and it rings true. In fact, if a history of Thales was discovered, and there was a bit in this that said something totally different about the way that theorem was actually found, it wouldn’t seem to matter. CG: It doesn’t actually matter; but I say it is a recasting of history because it makes sense to see a man being challenged within the Greek system of geometry that uses compasses and rules and has created so many constructions of so many things. You can’t attack the problem of dividing a segment into three equal parts. His solution is irrational - it is true, it is valid, but where did it come from out of the blue? He was walking because he was preoccupied and nervous. He wanted to know how this thing could be done. He walked on the sands and shifted awareness. This is what I did in my film on the foundation of mathematics when I presented Thales as a theorem. I actually produce the steps. Then everybody will benefit from this awareness that you abandon a line of conduct and generate a new one - but a new one that is justified by the result that now I can actually solve

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the problem of dividing any segment into any number of equal parts. This has a cardinal place in geometry and it yields lots of further results. It is of the greatest significance. That’s why we still call it a theorem and give it the name of the inventor. In the science of geometry, this is a way of worshipping the gods - the heroes, the discoverers, the people who make lasting gifts forever. John, what do you say? JM: I’m very happy with the Thales example. I’m tempted to conjecture that imaginative reconstruction is a powerful tool when looking at local results, and it is one that I have used with students - perhaps not with the same attention or with the same care. But I remain doubtful about the extent to which I can apply this method and not mislead myself. CG: Would you not know yourself better, at the same time as understanding mathematics and the history of mathematics better, if, instead of taking this position of expectancy, you were to take an aggressive position and say: I have seen in the example of Thales someone who had to abandon tradition and who took an innovative approach which was so fruitful that we can talk of it here. Isn’t this the opposite of what happened in the other case -about the Pythagoreans? That was an example of how when we find something we can’t do, instead of entering into an innovative way of handling the matter, we go into another field. This allows me to say there was a shock. This doesn’t come from me. It

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comes from them that they had a shock, that there was something which they called irrational. How could there be something beyond reason, since they had just found it? What is irrational is the relation of the new to the old. JM: I don’t know when the word irrational was first used. My suspicion is that it doesn’t appear in Eudoxus. CG: Why do you quarrel with me, as if you were a better historian than I am? Both of us are bad historians. You don’t know any more than I do. You don’t control all your statements about history any more or less than I do. This is not at stake between us; what is at stake is whether it is possible through our exercise to know ourselves better, to know humanity at work better, to get some lighting in a field in which we earn our living, in which we are engaged and have a lot of experience. If I had asked you to do that about archaeology, both of us would perhaps stop talking after five minutes. (end of tape - some missing contributions) CG: I remember once giving a lecture to the London branch of the Mathematical Association. The topic was algebra and my first question was: what is algebra for you? One person said it was arithmetic with letters and other people said other things. I wrote all these things on the board because I knew if I collected all these treasures - ones I wouldn’t have been able to invent we would have an exciting afternoon. There were a dozen impossible definitions. None of them was concerned with awareness. None of them could bring out what algebra actually

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was, where we are going to find algebra, and why this algebra appeared so late in history. These are topics for us here; if we recast history we can ask about the things that don’t appear, not just the things that do. I would like for a moment to be concerned with the contrast between Thales and Pythagoras. PB: I am a little worried about the assumption that because we’ve got a pun on the word irrational, the Greeks must have had one too. The fact that there was no ratio does not mean that it was particular shacking and against reason. I don’t know enough Greek to know if there was a similar thing in the original. CG: Well it doesn’t seem to be crucial to decide this matter now. I don’t know who invented the word irrational; I don’t know why they didn’t choose another word. Who invented it, when it was invented, is immaterial. But with the word appears the fact that I couldn’t think of a possibility. It says to me that the shock translated itself into an interpretation that these things are odd ones out. There was no idea that there were many more irrationals than there are rationals. We have to wait for Cantor and his successors to become aware of these things. The Greeks had no such notion. But they had the notion that the irrational an exception. It was like finding a leper among your sheep. You recognize at once that it’s not a sheep. What do you want to do when you have a leper? You just get rid of it.

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We only have a few minutes; can we try to say something about the first long session this morning? David? DC: I’m still struggling with significance. I’m still struggling with the story on the beach. I’m obviously reminded of Archimedes and the bath. When you make that story and say that’s what may have happened, it’s still doesn’t answer the question for me of what were his awarenesses at that time. CG: Let’s not remain only with the story but try to see if you found any significance in the work we did this morning. JW: I still don’t think I have any feeling what awarenesses mathematicians in the past had. My view of, say, Pythagoras’ theorem is very different from what Newton’s view must have been, and what somebody else’s way back had been. I’m interested in how this changes. CH: I’ve been trying to get an awareness of the universe that the Greeks were in, so that I can locate some of the people who were exploring that universe and uncovered certain truths. I have resonances with significance because the truths that they were discovering might be picked up by the milieu that they were in. I’m also puzzled a bit by how something can be picked up in later times that isn’t significant at the moment of its discovery. CG:

That may be a red herring, but it came to you.

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JD: In terms of acceptance of what we’ve been saying this morning, I’ve been experiencing a sequence of ebbs and flows, and I find that quite significant. There do seem to be certain statements which have emerged which are acceptable to me. I liked your remark that the Greeks were into the dynamics of imagery. Other items have been far more difficult and I’ll have to think more about them DC: I feel very bombarded by a lot of information which I’ve heard. So I need time to think about that. I think probably the firmest image has been the one of the sand. DP: I’ve been thinking off and on, ever since you gave the example of Thales, and trying to see what components there are to the story. It seemed to me there were two: the first is that you provided a context in which the question itself makes sense, by talking about halvings as things that you were able to do, and so to making it a more general problem. The second part seemed to be in providing a context in which a construction in the solution could be made to have a human sense. I was wondering whether both those elements are necessary, and whether this is an example of what seeing the history of mathematics in terms of awareness might produce. In other words, whether it’s directed towards the teaching of mathematics, as opposed to an activity in its own right. CG: I offer you a beautiful rose and you ask yourself how it would look when it fades. There is a rose in front of you – a beautiful rose. Is there not a moment of contemplation of the beautiful rose?

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DP: I don’t see this morning as a disparate series of events and stories that don’t have links with one another. I’m trying to create some of those links. CG: There are pedagogical consequences of some findings which are not necessarily pedagogical. The finding is that engaging in the history of mathematics you want to understand these things - not as of a recitation of events - but in such a way that you feel the movement within the mind of the finder. Even when this is artificially imposed, it’s at least an effort at making the mind at work appear to those who look at it. If you give yourself this bias, you look at whatever parts of history that interest you, and you look in terms of awareness. What does it do to my awareness? Not to my memory, not to my information - I’m not concerned with information at all, I’m concerned with illumination. Is it possible to find in those things that remain - that are talked about, that are stored in history with reference to the people and the place and so on -an indication that someone at a certain moment had a certain awareness of something that became significant after it was uttered - that had consequences. We could perhaps go back to what we know is in the history books and examine whether we can give it a new life by injecting into it the awareness of the people who thought of it. We don’t aim to be perfect, complete or authoritaritative, in these matters. We shall offer it as a gift, like a beautiful rose. Let’s relate to this gift; and then when we are on our own - in our own situation - see if we are in a position to use it, to take

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advantage of it, on behalf of our charges. The pedagogical aspect is not essential for our study. You will use what we do in your teaching, once it becomes available to you as an instrument which you like and which inspires you. ML: Another question for me: is what can we say that is valid has a certain sense of truth? Before the break, I was like others with two questions which were co-present: what is significant and can we recognize the shift of awareness? I was working on the possibility of the equivalence of these two questions. We have had two examples and I have another one in my mind - not from maths but from physics. We spoke of Copernicus in the previous seminar. We can imagine a story that Copernicus sat on the Sun and looked at the Solar System from another point of view. Well, the story is a story; but I am sure that here is a shift of awareness - to look at the universe from the Earth, to place myself in another area, to have another point of view. In the case of the story about Thales, I also have to recognize that to treat this question it was important to change the view not to concentrate on the segment that had to be divided. But I have a little problem with the question of the Greeks being shocked, and deciding to work on geometry when they saw that certain numbers were not rational. For me, that’s an interpretation and I don’t see in here a shift of awareness. I don’t understand this example and I can’t recognize it as true, as I recognize it’s true in the other two cases. CG: The job we have is not to find which is the awareness, but the fact that it is possible to force awareness. For instance,

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suppose I offered a new example now to force your awareness, for you to come to say: yes, I see the truth. I wouldn’t say I had convinced you, I wouldn’t say I had explained it well. I would say I forced your awareness. This is a new aspect of the job of teachers: the main function of teachers is to force awareness. If they know how to do it, it happens. After that, you give people time to obtain the facility that is required in that field. First you force awareness, and then you give time to obtain facility. Once students have both together, they don’t depend on you any more. This applies not only to what you do in the classroom - you can do it anywhere.

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CG:

Let me try to force an awareness here.

I play a game with four-year-olds. Suppose you have managed for whatever reason, from wherever you come from - to have the rigmarole from 1 to 20. Now, the game between you - Phil and Dick - is the following: both of you have given proof that you can go from 1 to 2o, so one of you starts and then the other follows. When the other makes a noise, you don’t. . . (DT and PB count alternately to 20) That’s as far as you can go. Now can you play it on your own without him, it’s the same game . . . (DT counts odds, PB counts evens) Allright - now I have forced awareness that the set of integers can be divided into two subsets which have certain properties. Now imagine what happened in Pythagoras’ ashram. It’s usually called a school, but it was more than a school, it was a community in which people lived and were passionately involved with their studies. Imagine what happened in that context when someone brought the bad news to the community. We don’t know whether they responded by fasting for five days

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to see if it could be put right. This is not in our knowledge. It was a catastrophe which we don’t understand. Because we refuse to get into their mind - into their having invested so much of themselves, and having found that it works so well. For instance, I don’t know whether they did this, but we can. Take the Pythagorean table in base ten, from one to ninety-nine, (writing the table on the board) Now, I produce the table of numeration base nine, (masking off the last column and last row which include the digit 9) What’s at the end? ... it’s 83 base nine. And when you do base eight? ... It goes from 11 base two, to 22 base three, up to 99 base ten. You stop before 100 in all of them. The Pythagoreans could have found these things; there is a pattern, there is harmony.

10 20 30 40 50 60 70 80 90

1 11 21 31 41 51 61 71 81 91

2 12 22 32 42 52 62 72 82 92

3 13 23 33 43 53 63 73 83 93

4 14 24 34 44 54 64 74 84 94

5 15 25 35 45 54 65 75 85 95

6 16 26 36 46 56 66 76 86 96

7 17 27 37 47 57 67 77 87 97

8 18 28 38 48 58 68 78 88 98

9 19 29 39 49 59 69 79 89 99

Here is another example - they didn’t work on this but it is striking. When I make the multiplication table for base two, I have 1 × 1 = 1. If you develop this for all the other bases you will get 2 × 2 = 11 base three up to 9 × 9 = 81 base ten. But whatever the base you can always say 10 × 10 = 100 and - with the exception of base two – 11 × 11 = 121.

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These are the sort of questions you can put in an ashram. You play with each other. You develop some insights; for example, if you add all the odd numbers up to a certain point, you will always find the square of the number that you started. This was part of Pythagorean arithmetic. We can say that they were engaged in producing mathematics of a certain quality. Because they were getting such remarkable results, the realm of integers seemed endless. In fact, it is endless. You can get so many results in the theory of numbers - even today - that you can say the field of the integers is very rich. So it’s not as if you go from one to nine and then stop. If you play for a little while with the rigmarole - one, two, three, and so on, this process of counting from one in the field of the integers you recognize that by going on you begin to lose grip. Although it’s so easy, as you go on you find that it becomes to be too much. You don’t know how to handle that. You haven’t asked a new question, but arriving at that point you can feel what is a large number. Now, that is not a mathematical problem. But it is about the numbers. Suppose I were to say: can you conceive of a number that has one million digits? The only thing that you can say is what has in fact been said. You can say: yes, I can conceive it, because- I can make the statement. But there is no way in which you can be in contact with this challenge. You have learned, you have become aware, that you can talk of these things without their actual presence. If I tell you that I can find a shorthand, that I can say a number that has ten million digits, can you give me an example of such a number ? It won’t take long at all: the

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number is ten to the ten to the ten. Here the answer is given in the notation. (end of tape - working at board and describing a demonstration lesson by the next tape) I wrote ten to the ten to the ten and I said how many zeros are there? There are three zeros for anybody with eyes to see. The boy sat and sat and eventually said: ten billion. You can imagine those adults, those teachers, seeing the little boy there. I said: how do you know? He said: well, ten to the ten is ten million, and ten to the ten-to-the-ten is ten to the ten-million, and therefore there are ten million zeros in that number. He had worked from the first ten up to the third, and he had seen that this will provide the zeros. It wasn’t a mathematical problem. It was a mathematical awareness, concerned with the specific problem of relating something he could perceive to something which has been defined and putting them together to answer the question. After that, I foolishly asked: how many minutes will it take to count that, if you can count four of them every minute? And of course he got lost -and I lost face! DT: Could I ask for a pause there, because I would like to try to see where I am. It seemed as if you were saying there are ways of looking at things like notation in which I may be able to catch something that notation did for people. This is, in a sense, a matter of historical record. So in doing mathematics I might at the same time be doing the history of mathematics, if I allow the mathematics I am doing to tell me about the shifts that were

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made. There could be a very large chapter of Renaissance algebra in ten to the ten to the ten. CG: I don’t understand a thing; but that’s allright, it’s my turn not to understand. We are trying to work on a number of things, one of them being what is the place of awareness. Since awareness is still a mystery to so many people, we are trying to bring everybody in contact with themselves, to the point of saying: oh, I but of course I have always known it, I have never lacked awareness. It is nothing exceptional. What is exceptional is that people don’t see that’s it’s universal. The reality is that awareness is your endowment as much as mine. Without it, you wouldn’t be anywhere. You need only to have the awareness forced and then you will recognize its place and its role in the field of mathematics. An example is when I recognize that I can count as a rigmarole of repeated sound that I hear around me. Do I recognize that from the first to the second, to the third, and so, on I’ve added one? What is the one I’ve added? When I says one, two, three, four, five - (putting out a finger at each count) - I’ve added a finger every time to the set. But what have I added when I only say it as little children do? A breath - one, huh, two, huh, three, huh. Without making it strong, I knew that - I can put a silence in between. The day I can do this rigmarole in order, I have recognized that. From one to two to three, the sliding, the passing from one to the other, is as concrete as putting the finger out. So I have this awareness - that by going from one to

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the next I have done something, a transformation, a definite transformation on arbitrary noises. I can transfer this to a movement on the board. . . (making a diagram on the board with arrows from one number to the next) . . . I make it visible and I can force awareness by focusing only on what I’m pointing at. Look: become aware of your sight and say something about what you see. You don’t have to go very far to find awareness. It is not a mystery. But it hasn’t attracted your attention to make it into an instrument for your work. +1 1 —→— 2 —→— 3 —→— 4 —→— 5 —→— … After a while, I insert - like Peano did - that it is plus one... (writing + 1 over the arrow). . . Peano was not aware that when he gave his axioms he was not giving us the axioms for the integers. He was giving us the structure of an arithmetic progression. For those who don’t know, the Peano axioms arose from the attempts at the end of the last century to know which is the smallest number of assumptions that you can make to generate mathematical systems - in this case, the whole numbers. If I say there is a one, and I give the figure 1, that’s one axiom. You have to agree with me that there is a 1. Then I say there is a next, and you have to agree with this. And so on. The same axioms apply to any arithmetic progression. But I was diverted by the Peano axioms. What I wanted to indicate was that you can recognize the next step as an addition. You make it be an addition. You keep the previous one and the

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next one. So you create aggregates. You can say ‘one’, and then in order to say ‘two’, you have to keep this 1 there and put this 1 with it (writing 1 + 1). And then you have to keep these to put this 1 with it (writing 1 + 1 + 1). and so on. This is obviously true of a set, but it is not obviously true of the writing - the figure 3 doesn’t contain the figures 1 or 2. You have to lift yourself away from the notation and ask what is it all about. Then you will see grains of rice, or peas, or whatever you counted. You then become aware of the set, and aware that you are adding one. But there was no adding when you said or wrote the integers in sequence. Peano’s first axiom says there is only one 1. This has been a hangup for so many logicians, that there is only one 1. For then I don’t dare put another finger up. Because if this is 1, that is not 1, it is another - but not another 1. This doesn’t matter when you are making the sounds that you make when counting aloud. These produce the illusion that you are concerned with numbers. But you are only concerned with labels. And you are not obliged to do the same thing going from ‘one’ to ‘two’ as from ‘two’ to ‘three’. Look - my fingers are all different. To become aware that the things you’re adding are different and that, in fact, you can’t add them in this way, because you can’t add different things, you have to be both with it and not with it. This is part of being aware, this is forcing awareness, as I’m doing here for some of you. The fact is that we made the assumption, that one (1)) is one. When we use it in counting things, one (1) is not one. There are as many Is as you want. If you say in your first axiom that there is one 1, and then you say there is a next - which we write n + 1 - is the 1 here the same 1 as

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before? It must be because there is only one 1. But if there is only one 1 how is it that there is another? So there is another awareness - that as soon as you utter a word for the numeral you shift from the realm of objects to the realm of words. Words apply to indefinite classes; this is an awareness of the universe of language - I could have said infinite classes, but until we are clearer about that I prefer to call them indefinite. AR: What is the awareness that was required to move away from a numeration system that did make this more obvious such as those used by the Babylonians or the Egyptians, or even the Greeks and the Romans - but which were extremely clumsy. CG:

How could they be clumsy if they were what they were?

AR: Our system is a step away from clarity in terms of adding the ones. CG: If you stay with your question you will get no answer. If you lift yourself a little and be concerned with what is behind it, then you will understand that while you are making sounds, you are looking at things. You are relating all this in your mind; they don’t relate in themselves. Mathematics is of the mind, language is of the mind. You have to find inside your mind what it is you are doing. As long as I have invented the zero, I don’t have the zero. As long as I haven’t seen Neptune, there is no Neptune. The day I 68


Numeration

have zero, there is zero. The day I know that my ones are all different in counting, I will not worry. So I can apply it to different apples, or to apples and pears, and all sorts of things that are in my basket, because I know what I am doing. I am only getting a noise at the end - twenty-seven. And now I’m going to say twenty-seven is a label that expresses something , an attribute of this set which I can perceive through the process I just used. Because it exists now in my consciousness, I’m going to give it a label; I will call it the cardinal of the set. So what are the cardinals of sets? They are the last noises you make in using the counting rigmarole while touching the various things in order. But you may say you’ll count them as well -you don’t believe me. What will you do? Will you remember which ones I touched? You wouldn’t. Therefore, as part of your awareness, when you start somewhere else and still get twentyseven, you’ve found a property of the set. It’s an attribute of the set which is independent of the order in which you count it, provided you don’t miss any object, and provided you don’t skip any of the noises you make. What if you now eat one of the fruits you are counting? You could count again. But since you are smart, you don’t. You remember something about enumeration and you take one back. This is where we concern ourselves with awareness, and with awareness of the awarenesses. Is it clear for you, John, or are you still struggling? JM:

It’s clear in these instances, yes.

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CG: If you want every example to be worked out, you will never know what awareness is. If two or three examples are worked out by you and you are not concerned with the thing, but with what happened in you, then you will not need anybody to say it again. AR: While you were doing your exercise, I created an image in my mind of a set of 27. When you counted in a different way, I did something in my mind - in the image but without actually going through each of the things. Is that type of imagery common to this type of exercise? Is that a criterion of knowing that you function when you count something? CG: Who is the you? If it’s me, I may say yes; but if the ‘you’ is anyone, I don’t know. If I am too young or senile, I wouldn’t know. JM: Could I just rehearse the one example - counting back? First, suppose I’ve set that up with somebody - and they don’t know how to count back. Then suppose I performed the same activity with someone and they count back. Then - a third instance - suppose they spontaneously say: you just count back. Now in the third case, I have independent evidence, without probing, without forcing. Is that the awareness, or the awareness of the awareness? I’m sorry, but I’m hung up on that level. It seems to me important when I am trying to move on to other topics. In one sense I could speak of, I know inside me -I just have to count back. I perform it; but I may not be articulately aware. This may not be a question of being able to say it; it may not even be part of my inner world that you count

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back. In general, though, when I’m put in front of the instance rather like riding a bicycle - I can do it. It’s a hackneyed example, like the case of the telephone number which only comes to me when I confront the telephone with the image of the person and I recall the number. CG: You are creating obstacles for yourself. If you ask me to count back and I say I don’t know how to, then you have one of my awarenesses - that I don’t know how to. You can make me count back, and through your guidance I learn to count back not necessarily from twenty-seven, perhaps from nine. Then you ask me to count forward and to count back. I demonstrate that I really know it - I’m aware of the words you said, I’m aware of what I do, what I have to do, and I do it. You then ask me whether I can take this further and count back from something you haven’t given me. Suppose I can manage this and you ask me how I did it. I says: well it’s the same. I am aware of my awareness, I am aware of what there was to do. It’s the transfer which is the criterion for you. I have done the transfer to demonstrate to you that I am on top of it, that I am aware of what your question is, and how to do it. Once, in West Bromwich, I had a child with an IQ of 43. This child was 11 years old. He had all- the symptoms of a very retarded person. I’d been invited by the educational psychologist to demonstrate to people whether I could do something with the rods in their classes. This child had been drilled over years to count from one to ten, and he could do that perfectly - he could even go beyond. When I made the staircase, he could go one, two, three, up to ten. When I put his finger back one, the machinery took him on, and he said eleven. I asked what he had 71


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said before. ‘I don’t know’, he said. Say it. ‘One, two, three . . .’ Stop, what did you say here? ‘Nine’, he said. It was a new exercise for him altogether, to discover what he was doing in order not to be carried by the habit, like so many we know do. I worked exclusively with ten and nine until he could do it. I never said anything, I only asked him to do things. So when he produced ‘nine’ I knew that he had now abandoned the habit. He may not have known how to take the next one back, so I did the exercise again. After a few minutes, he was counting back to eight. He was constructing this new - for him - system, creating a new habit with all the elements which were in his mind. They had been loosened. At the and I asked him to do it with his eyes shut. He had already been able to count up to ten? but in counting back you could see that he felt that he needed the support of the imagery - or rather the objects. I asked him to try again with open eyes. Then he did it with his eyes shut. In forty minutes this child had conquered so many things. It was clear that the approach was based on letting him be aware of what was asked of him, letting him know that he was working on his own awareness, on something which he can relate to. When he made the transference to the next problem, how could I deny that he was aware of which awareness he had. DC: Is this what happens when a child goes to twenty and then says: I can only count to twenty. They also know that there are other noises that seem to tie up somewhere. But are there are also holes in between - gaps in the noises? Presumably for the Greeks there was a similar lack of awareness that there were

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gaps in their numbers. If they were labeling these points on a line, there were a lot of points which didn’t seem to be label led. CG: They were not concerned with points and lines. They were concerned with something else, which created the problem of having in their mind the one, the two, the three, and recognizing that you can reverse some of these operations and get halves and so on. But you can’t yet say there are an infinite number of points between zero and one. You can’t say there are so many rationals and so many non-rationals - all this is beyond perception. DC:

Yes, this is what I find hard to get hold of.

CG: That requires Liouville’s, Cantor’s and Dedekind’s - and lots of other people’s - awarenesses to be brought in. You see, you never take steps on the things that don’t matter to you; you leave them alone. It is when they begin to matter that you get involved - that’s common sense. DC: Except that in the Greek story, when something did impinge upon them, they didn’t take the steps. CG: No, they took another step. They were able to says there is no ratio which represents root-two. They knew that was not possible, because of the properties of multiples and divisibility, and so on. They had all these concepts at their disposal. What they didn’t know is what else they could do? We can only use what we know, and if you examine in which way you move

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towards what happens in your life that is new, this may happen several times a day. Have you taken the trouble to say to yourself: what do I do with myself to allow the unknown to get known. If you don’t know the answer to this one - well, it’s a lovely problem for the rest of your life. You can work on this: how does the unknown get known? Is there one way of doing that? If you notice when you give explanations to your children, to your students, to your colleagues, to your friends, whether you always aim at replacing what is new by the old - which they already have - to make it acceptable to them. You use the basis of their experience to make it likely that this thing may happen. Every time you involve yourself in explanations, that’s what you do. You reduce the unknown to the known. Well, that’s not possible. But we use it constantly. So we get the shock, some day or other. I thought I understood that - I even explained it, I even taught it - and I didn’t know it. I didn’t understand it, because it is not the same thing, to allow the unknown to penetrate and to inhabit your mind, as to reduce the unknown by a combination of the known. This is one of the ways in which we get so many misunderstandings between us, that we have little concept of how the unknown gets known. We get hung up on it having to be in agreement with what I know. It should be reducible to what I know. When it can, well and good; but if it can’t, you abandon you don’t recreate your self to meet the unknown. This is the cause of stagnation - when we are indulgent and we allow things to continue without taking the trouble to see what we are missing.

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Counting Infinity CG: At the beginning of this century there were tremendous, extremely interesting discussions which are recorded in the literature. I wasn’t there, but I have read the accounts of the shock of the awareness like that we came across with ten to the ten to the ten, and that type of thing. Do we have any idea of this set of integers represented by the three dots? (representing the ‘ad infinitum’. . .) Do we have any idea of the mysteries and the complexities and the demands it makes upon us? None - it’s far beyond our capability. With a hundred billion neurons, we don’t have enough for such questions. So what is our attitude towards this? We say we are too low, too small, to handle this matter. Let us aggrandize ourselves like gods and say: I know the whole set. And I put the three dots to prove my superiority It’s a matter of joking with something that is serious. But it’s exactly what you do. We have no idea what it is, yet we say that we have a clear idea. We don’t know at all what goes on — after a certain number we are completely lost. We can’t handle numbers of more than a hundred digits or so, yet they are only the beginning. What do we do? In the end, nothing. One awareness of the history of mathematics has been the avoidance of being caught. We either leave the field or create something else to replace it. For instance, the exponentials have suggested themselves to allow us to handle something beyond. It’s a way of demonstrating to ourselves that in order to be concerned with matters beyond what we can actually do, we need a notation

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which is going to help us. Then we can generate lots of things to talk about - things that are firm, and lots of things which are suspected to be allright, and lots of things about which we have absolutely no idea. When we become aware of this - at that moment - you can say there is place for awareness in the history of mathematics. Because what I discovered about myself must have been true of all my predecessors. It is not due to me. With this awareness, you could say: how do we handle transfinite numbers? How do we get into this maze? We don’t know what infinity is, yet we are already into transfinites. (end of tape - transit ion to work at blackboard) As for the diagonal process, I teach it to anybody. Those of you who don’t know it can know it at once. Suppose I write 1, 2, 3, 4, 5, and so on. You can see it can go on and on and on -it doesn’t create any trouble. But if I take these two rows (the odd numbers and the evens), in order to have them all, I have to do this... What am I doing? I am actually symbolizing threading them. It’s a string; I’m threading them so that I don’t lose them. I don’t lose them at all. 1 2

3 4

5 6

7 8

9 10

... ...

The three dots are the greatest creation of mankind - the nothing which allows you to hold everything. This is part of becoming aware of what you are doing in mathematics. All 76


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mathematicians put three dots. They know - without knowing what it means - that it is a way of handling infinity. For infinity, you need eternity - and you don’t have eternity, you have now. So what do you do? You put three dots. Now I start again with the odd numbers. Suppose I double this one. What do I get? Now what do I get when I double the rest? Don’t get lost with the dot dot dot. What do I get if I double again? And again? . . . 1 2 4 8 . . .

3 6 12 24 . . .

5 10 20 40 . . .

7 14 28 56 . . .

9 18 36 72 . . .

. . . . . . .

. . . . . . .

. . . . . . .

You see, I am very clever. I have only worked with the beginning - with the little that I control. When I put three dots there, what impression do I give you? That what I was doing, I can do again. Even worse - that I have done it. This is where you change from being a mathematician into being a god. You give yourself the power of generating, not only one infinity, but an infinity of infinities. What did Cantor tell us? I am going to thread them all - like this. And then I pull the thread - like this - quickly. I have got them all. Could I get them all? Are they all there? Well, tell me which one you want. If you give me a number that I can handle, I’ll give you two numbers - the address, which line and which column it

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is in. There isn’t one you can name that I haven’t got - all are potential. That is an awareness that makes us mathematicians. The difference between mathematicians and others is that this awareness is operative - it is present in the mind - and it works. When Cantor was looking for a proof (that the rationals mere denumerable), nobody was telling him what to do, because nobody knew what he was doing. He did not know how to solve the problem, but one thing he knew for sure was that if he went on and on the straight line he would never come back. So he knew: I have to avoid going straight, I have to run crooked. All these are at an elementary level. You may have some difficulty in getting into these things when we take notions that are on the boundary of the perceptions of our sensibility. Can we have some minutes on this section before we continue on something else? Is it of any interest to you? DP: Some interest; less interest than this morning. I feel in a way some responsibility, because I made some suggestions about awareness. DT: I feel I have moved since this morning; you’ve charged me with poetry and passion. But at the back of my mind is another enterprise, in the sense that I have no idea of the instruments that enables me to gain what I think is a tremendous insight in this particular case. I sometimes find when I look at some bits of mathematics that I know the essence - but not often. This one shocked me when I first met it.

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CG: I didn’t give this one properly - because the threads get enmeshed. Cantor’s original way is better. It was a shock for me too. DT: This seems to be a good example of what I think of as your historical method. Because ‘diagonalisation’ is usually specifically attached to the attempt to count the irrationals. What you set up was a version of another Cantor discussion, which is about counting the rationals. That is done by threading a two-dimensional array of number pairs - the addresses in your example. It seems to me that what the poet is doing - the poet in you in this case - is bringing together two entirely different things in Cantor’s life. Cantor himself did not draw attention and nobody else normally does - to the fact that he once used a diagonal pattern and then repeated it a few years later in an entirely different context. I know of another person who does that sort of thing and that’s Robert Graves. He interprets the Greek myths and claims the right of the poet to state what the Greek myths were about. He ignores totally the innumerable charges of critics that actually that’s not what the Greeks thought it was, and it’s not what it’s about, and all the rest of it. But the notion that one can actually get at the core of the poetic process -that’s what I thrilled to today. I think that the juxtaposition of two processes in Cantor’s work is very illuminating. The other thing that was obviously brought up very strongly is the economy of access to the mathematics in this particular story. Clearly it was also around in the computer program we

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saw the other night for five- or six-year-olds. Perhaps the more intense the poetry, the more accessible it becomes. CG: You see, even in this, there is no mathematics - no mathematics at all, unless I wanted you to tell me where a certain number is exactly on the thread. You’d have a problem then; it is of no interest to anybody but it is a mathematical problem. As far as forcing awareness of what the field is all about, I think this is an excellent example; it doesn’t require anything else. DT:

I would like to know what it does require.

CG:

It requires only that you can go from 1 to N.

DT: I was thinking about what is required to put some of the pieces together. CG:

In the case of Cantor.

DT:

No - in the case of Gattegno.

CG: Why do I put it there and why did I put it into my computer? DT: No - why did you bring certain pieces in the record together? How did that happen? When you rehearsed what you knew of Cantor’s life and discoveries, how did certain aspects of these things happen to come together?

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CG: The experience that came to me is the one I stated. I say it is out of the question - if I want to know these things -that 1 go on a straight line. I just can’t manage it, because I get lost in the infinity and I wouldn’t be in contact with the others. So how can I get them all in? I can’t get them all in. But I can get the process which allows me to put three dots after a certain number on the horizontal row, to show a process which can be repeated. Then they are entitled to say: do you know the address for that particular one? Is it certain that everyone is there? Then I have a secondary problem - which is now a mathematical problem. Will there be a place for any number? How do you tackle that? To tackle this problem, I have to find a power of 2 say 2^m - and an odd number - say (2n-1) - which will together give me the address. So I have to get the exponents m and n which express the given number in the form 2^m, (2n-1). if I have a device for doing this I can find the address of any number that you can think of. DT:

I think you are still telling me the poem ... (laughter)

CG: When I look at the number you give me, I don’t know what to do in order to find which power of two is a factor of it. If the number is large, this may take many hours on a large computer. But as a mathematician I have finished if I know the process. That is one of the awarenesses about which we should spend sometime tomorrow - how we become aware of algebra, which is what is behind all this. It is one thing to get answers within the field of arithmetic. And quite another thing to see it as done, once I know how to do it.

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This is one of the awarenesses that mathematicians don’t have, because they don’t need to talk about it. But when I talk to them, they say: of course. I remember when Dieudonne was in a seminar of mine - held in 1952, near Paris. On that occasion we were talking of a number of things. I gave my definition of algebra; for my pupils algebra is operations upon operations. He agreed: yes, that’s what it is. He had a thousand times more algebra in him than I had, but he had had no need to look at it in this way. His algebra was about rings and fields and putting them together and so on. He was taken by what he was doing to get new results because the content mattered to him. I can generate awareness of algebra with very young children, and from there make them do more and more algebra, because they understand what it is. It’s as much theirs as we believe arithmetic can be theirs. We shall come to that tomorrow with more precision. Coming back to our theme, is it possible to recast history of mathematics in terms of awareness. JW: I am willing to work at it, but I don’t know whether it is possible yet. DC: I flounder and then you do something like that and I think: yes of course it is. What I wanted to know was what prompts a mathematician at any moment to decide to work at whatever it is they have decided to work at?

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CG: It doesn’t work like that. Mathematicians work on specific problems. It’s after they invent something, and someone asks about it, they say: yes, well it must have happened like this to me. I once went to Qstrowski with my work on the very difficult question of how it is that the area under the curve is the inverse of the gradient of the tangent. They don’t seem to have anything to do with each other. When I told him my solution, he said: but we have always known that. DC: Had anybody tried to work at the infinity problem before? CG: Nobody took a boat from Spain and Portugal to go west to the West Indies before someone did it. Why didn’t the others do it? DC: Why did it only happen then? Why do these things which are so blindingly obvious or beautiful happen when they happen, with whom they happen? DP: That it is partly obvious to us might be a product of our education. It may not be obvious in itself. CG:

But this is common sense.

DC:

Yes - exactly - it’s so obvious that it’s common sense.

CG: You can’t do otherwise. You started with these odds and you doubled them, and you doubled them again, and so on. if

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you want to say that they are all there, you have to show that they are all there. But you can’t go straight - it’s obvious, you can’t go straight. So what else can you do? When you look for that, you say: there is only one way I can go, because I know 1,2,3, and so on, and therefore I am going to string them in order. When I string them in order, I will have something that I can pull. That’s my imagination - I have the string and I pull it. Then I have a denumerable set and so a denumerable set of denumerable sets is denumerable - a beautiful theorem of Cantor’s. JM: I have two points. One I know you won’t want is that I would love to take some topic that somebody names at random say solution of quadratic equations - for a group of us to work on trying to get at what the awareness is, so that we could look at the method of uncovering the poetry. A second point - to do with Cantor - is that I’ve frequently failed to get a response of the kind that I would have liked from students when showing that the set of real numbers on the interval (0, 1) is not denumerable. It’s one of those occasions where everybody assents to each of the details. You are trying to show that no matter how you string it, it’s not going to string. CG: The proof is one of those to which you can object because it’s a reductio ad absurdum - it shows a contradiction. Your sensitivity may tell you that there is an excluded middle. But I don’t think that.

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CG: It is an obligation for the person who wants to prove something that is extremely difficult, to find a way of making it simple. That is what was demanded of Cantor when he had to demonstrate that the set of transfinite numbers is transfinite. Then he had to find the diagonal processes for the irrationals and for the rationals. He had to find these two in order to make it acceptable to those who didn’t believe him. Gauss and Kronecker - great names in mathematics - had refused to consider the concept of infinity. There was no place in their work for it. DT: We have to put that in a different way - because they did deal, of course, with infinity. The issue is - and this is one of the problems about the squeezing of the poetry into one single image - that actually you have to rely on the fact of being able to quickly say: I pick up the string and I ... Now, that is very dramatic - it works, it tells us what to do. But when do we say: oh, can we pick up the string? CG: When I abuse language and start speaking poetry, you would bring me back to my senses! DT: No, no - I want to say there are two poetries here. It’s perfectly possibly to say: yes, that’s delightful and that’s the argument. But it seems to me that Gauss and the others were perfectly aware of infinity, but were saying: there is no way in which you can have a string. You can put dot dot dot, but that doesn’t mean a string. And other people since then have said that, as well. I mean it’s the same thing as the students rejecting the diagonal argument or the excluded middle - that people are

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still uneasy about the story, however excitingly we can actually produce it. CG: This you can produce, the other you can’t. The other diagonal process is entirely in the mind and if you resist letting the penny drop, it doesn’t drop and you will say I don’t understand. I have met so many times this resistance in people, until I have brought them to the realization that you can prove something by disproving the contrary and that they use this in their life. But they don’t think this legitimate in mathematics. Brouwer and his disciples are of that kind. That’s not a proof, they say. It doesn’t convince them. What can I do if the curvature of their brain is such that it doesn’t convince them? . . . (laughter) . . . I didn’t say that, it was Zermelo who said it. PB: What is so surprising is that some things wait for centuries before they flower again. Zeno put a whole lot of questions which you would have thought everybody would have wanted to chase up. Apparently they didn’t. What is that - all of a sudden after all those centuries failing to grapple with the argument at all - there is a Cantor. And when Cantor comes along, his contemporaries, to a very large extent, just wouldn’t believe the evidence of their own eyes, although - as you say - it seems so simple, so obvious. CG: This is not only true of Cantor. It occurs repeatedly in history. Newton was recognized to be a great man in England in the 1670s. But across the Channel, he wasn’t recognized until 1741. His contemporaries didn’t accept him for all those years. Leibniz is still not understood. Cantor is still not understood.

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There are many papers of his that most people don’t know how to read. The work of Leibniz was collected many years ago, but it has not yet been fully catalogued - there are lots of things he wrote that nobody knows about. JM:

This brings us back to significance again.

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CS: Something is significant when it has consequences. There is an elementary example of great depth in the study of Newton’s fluxions. This involves a simple idea which has been accepted immediately, but which is philosophically ungraspable. How is it acceptable in mathematics? I take a segment of a line - say, part of a road - and I let a car pass over it. Imagine the wheels of the car going over the line. Now, reduce the car to a point. Start here and move towards there, (drawing a line-segment one the board) Will it go through all the points in between? Well, then, invert the image and says this segment is generated by this dot moving in a certain way and leaving a track behind it. This way of producing a line out of a point is going from zero dimension to one dimension. Now take this line and move it like this to produce a square on the plane (by a parallel displacement). You have generated one thing out of another and that is not permissible. But if I christen it, if I give it a name, it becomes knowable, recognizable, acceptable.

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DBD: Give what a name - the line? CG: No - the process by which I take all the points of this line - or all the segments of that square. So that if you see a square, you are allowed to think of it as made by the sweeping of one of its sides - whichever you want - in a number of ways. With this process, I am generating one thing out of another. Both exist in this case, and you can pass from one image to the other. When you carry out the process with things on which there is no agreement, I say it’s the same process. So, I now take a segment that is defined from this line to a curve -whatever curve you put there. (The figure below shows a vertical segment from a horizontal line to a curve.) The segment is going to change. It becomes bigger and smaller according to what curve I have there. But I still have the same sweeping process to produce the area underneath the curve.

The image I am creating depends on which curve I’ve drawn. I am at liberty to draw the curve like this, or like that. What I am telling you is independent of the curve, isn’t it? Here is the diagram that has to be in your mind. It’s in the mind because

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you can construct such things - a segment that has to go from here to there, moving like this - only it doesn’t move. DBD: So it gets longer and shorter as it moves along? CG: It doesn’t matter which. All I’m asking you to do is to let this segment cover the area underneath this curve. When I convince you - have conned you into accepting that - I have generated new instruments. They were extremely important historically, though here, for you, they are nothing. Before they were created, nobody could do what I did with you. But once they are created they are available for anybody to use. This sweep - this way of generating what wasn’t there before belongs to a branch of mathematics called the calculus. DBD: I thought you were talking about Newton’s fluxions. CG:

Yes, I was.

AR:

A way to calculate functions.

CG: The awareness that we don’t give our students is that these processes that were invented at that time by Newton and Leibniz - though in very different ways - were a source of new definitions. They generate new entities. We teach it as finding the area, instead revealing to our students that these are a new set of machine tools that didn’t exist before.

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Now that we have an example of another kind to work on, can we try to see Newton’s awareness and then the awarenesses of others, including the awareness of people in this room. DP: It’s an example where motion and time re-enter mathematics. CG: The explanation of what you do is not mathematics. In mathematics, there is no time - you don’t need time. DP:

I thought the fluxion was rate of change in time.

CG: You may use this idea to begin with because you study a problem in which time is one of the variables. But then you are studying applied mathematics. That’s what Newton was all about; he wanted a tool to solve a problem of natural history. When you give a lecture with epsilons and so on, you never see time appear there. But it is there, because you talk, because you write, because you continue to exist. So it is your humanity your humanness - that brings all these variables into the situation. But it is not part of mathematics. In mathematics, you don’t study time. It is such a dull thing! What is there mathematically in time? Nothing! You replace it by a continuous variable - a single continuous variable. DC: I am still with the imagery. When you did the sweep, then I actually saw. I could see this thing sticking on top of the line sweeping along. If you don’t allow a process in time to take place, then it seems to me that that image doesn’t exist. 92


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CG: You are right in anthropomorphic terms; you are not right in mathematical terms. In mathematics, you look for something which implies time - but not explicitly. It is there because you talk, because you draw, because you think one thought after another. That’s where you can’t escape the presence of time. But it is not a mathematical entity. This is a fine distinction between pure and applied mathematics. DS: I think that you are here using a different instrument. We have had the Thales story. But now - coming closer in time - it seems to me that one is actually sayings this is what the originator thought. I am not quite sure what the instrument is in this case - for the awareness, I mean. CG: If you are interested in the awareness, you will find it. But if you are interested in results, you will find results. The purpose of the example was to show awareness at work when those who have not studied the field are presented with it. Since it is not yet a living thing for them their first thought is: shall I remember all this? AR:

So everything comes from one thing?

CG: There is nothing to remember - it is so simple. The first question could be: shall I know this? Do I know the meaning of it? Can I use it in any way? You can use it because of the consequences that can be drawn from it. Historically, it has been extremely powerful and has changed the world in which we 1ive.

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If you are a pure mathematician you would never present it in the way I did. You would be working in a way it that leaves no room for intuition. When I give you images, when I moved my hands and so on, I force awareness. These things are part of what I did. There was something arbitrary - and that was the curve. If you know that when I did this curve I did any curve, what is the freedom that you feel in your self? If it is one, it is any number. I can replace this one by another one. If I know how to get this one, I will know how to get the other one - or the other, or the other, or the other. The top point is being guided by the curves themselves. There are infinite possibilities for what the top is going to do. It could move on a parallel straight line, in which case it would form a rectangle. Well, how dull! A rectangle! It’s a special case - I could go very far and come back, or I could come as close as I want to the bottom line. There is your mathematical imagination. But it is imagination - that means images - with a dynamic which respects that it is on the plane, that it is continuous, and so on. You can generate any number of such curves. You could say: if I prove for one of them that I can get the area underneath, and if you select a curve, then I will give you the area underneath that. This solves in one go an infinite number of problems, without giving you the particular answer to any particular example. So I can now lift myself from the example I have given and introduce a notation which indicates to you that this curve is going to be called f of x. What does this mean? It’s a function of this point, because the other point depends on it. The imagery creates a mapping - a correspondence – between these two points. In the process of going from there to there, I choose a point 94


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anywhere. And because of this choice, I go there. So this is an x. I represent the liberty I have in choosing it by this letter. So I write fx - with a bracket if you wish, f(x) - and we can read it as giving the second point. As the first point moves from there to there, fx will describe this. I am putting into the notation what I said before. If the images are in your mind you will understand what this is. So if I take this particular point x - meaning it is one of all possible examples - then fx is the name for this point. Any one of these points can be called x; x is a length DBD: Is fx a length? CG:

Sure it is a length - from here to the point that defines it.

DBD: So fx is a lot of lengths. AR:

Can we give the x’s different number of names?

CG: If I asked for a glass would you show me all the glasses in the world? It’s a word. I have shown how to generate a line from a point. I put a point on the line. It is one of the points that have generated the line any one. I put it here - anywhere. I put it so that there is room on both sides - elbow room. Now I tell you that this fellow that’s why I put f . . . (laughter) - goes from there to there. For this x, I call this fellow fx. Why? Because had I chosen another x there would be another fellow. At the same time as this x gets there, this fx will be doing the dance up there, going from there to there. Do you see them going at the same time? 95


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DBD: There’s an infinite number of them. CG: Yes - but we are not concerned with that. We are only concerned with the fact that for each x, there corresponds fx, and when x goes from here to here - from a to b, fx goes from fa to fb, by passing through all the possible positions - which depend on your choice of curve. I will always use the same process, whatever the curve. It doesn’t matter which curve I take - another one, or another in between. Do you have enough imagination to say all these will work?

When you do the job of going from here to here, we call what is generated underneath the area under that curve. This is represented by a beautiful A - for area. As we don’t know how to find it in general, we can talk about it without saying what it is. So I am going to write it this way - and because tradition demands it, I could write something else here, but I won’t now... (laughter). This says x will go from a to b along the curve, which is represented by the f, and when the A is elongated - into something like an S - this says that you have found the area. Now you know as much as I do.

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AR: What is the thing that you wanted to add that you didn’t? (Leibniz’ ‘differential’, dx) X:

Monsieur Leibniz has just walked into the room.

CG: It was simply a concession to tradition - don’t worry about that. AR: I wasn’t worried; I just wanted to know what it was.... (laughter) CG: What matters is that, by arranging things in a certain way, I can force awareness in you that corresponds in some sense to what Newton saw as one of the problems he had to solve separately in his mind before he applied it to other problems which were in nature. To teach you a little more I would choose an easy fx. For instance, I could choose fx like this -obviously making two rectangles. You know what the area of a rectangle. But I can also demonstrate how it can be obtained by the previous process, (works example on board) DBD: What I see now is a way of describing the problem. I can work out the two rectangles, I have a notation for describing the area with a wavy line, but I don’t see how it is going to help if I want to actually calculate the area under the wavy line. CG: I would teach you that later. The only thing you need to know is how to calculate and that’s not hard. For instance, you can make a trapezium, then another trapezium, and so on. Then you add them all up. 97


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DBD: You have to divide it up so that you cover the irregular bits. AR: What if this were three-dimensional - why can’t you do three dimensions? CG:

I can, but you can’t... (laughter)

AR:

I bet I can ... (at the board)

CG: It is not too different. You see, in this intuitive way, you don’t consider that Riemann’s formulation is essential, there is no need for that rigor. When Newton started he knew so much mathematics about little things he didn’t need. What he did need was to know how to meet the new, and he couldn’t do that with what existed before. He had to take another step. That step appears to be such an intuitive one - one that we can expound in a few minutes to people who have not done it before. When I worked this with the girls in Putney High School many years ago, they devised an approach similar to Riemann’s. They 98


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could say that to find the area of this potato, why not find an area that I know is smaller and an area that I know is bigger. Then I can say it’s between these two. With a system of grids, they produced polygonal approximations from inside and outside. They saw that if they went on and on, they would get closer and closer. They didn’t take the step that we can take to say that - instead of three dots - we have done it. Instead of all these additions, we are going to do that as one operation. Can we have a little feedback on this?

Some Feedback CH: I have been going up and down in my feelings a bit like fx and I wonder what area I’ve now swept out. I’ve been very moved by the poetry of it. I loved maths at school when a teacher came into my universe for a while and uncovered the poetry of calculus for me. He left when I was about 15 and I couldn’t manage it by myself. Now that I’m coming back to being a teacher of maths myself, I want to transmit those awarenesses to the young people I teach. . . . At the moment there’s a dot dot dot - but not a frightening one. DC: Who was Cantor’s forcer? For me it comes back to this. Who forced the Cantors or the Newtons? How did it happen? You have given an impression that once that initial opening is made lots of people jump in. One lot - the mathematicians - then get hold of it, close down on it, push it as hard as they can and as

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far as they can, and then use it. But it seems that there are other lessons to be learned from this - ones that Cantor and Newton went through. We’ve reacted to the beauty and the simplicity and the blinding obviousness of it; that has opened up something in us which is perhaps not mathematics. CG: Mathematics is of the mind and the mind is more than mathematics - it’s human. So you can have mathematics as a byproduct. DC: The forcing process does seem to be different in these different cases. Going back to the Greek problem, what could have been a forcing of awareness actually shut it down. CG: That is inevitable; because it depends on how you see it. We have to look at me playing the piano with a bow. DC:

Will you tell me what it sounded like? . . . (laughter)

CG: It’s a fanciful idea - but it doesn’t work, it’s not viable. There are things that are viable, they serve some purpose, you can do various things. Then you can ask whether you can do more. Can I tackle the third dimension? But these are not the same movement as finding how to tackle what nobody has seen before. DC: When Pythagoras put forward this theory that the sum of the squares on the two sides was equal to the square on the hypotenuse . . .

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CG: It was done empirically. Then it was recognized that what was being talked about were relationships. I have a triangle which is a kind of relationship. It has a right angle - which is another kind of relationship. If I put squares on the hypotenuse and the two sides, then I find a relationship -when I have the areas of two of them, I get the third one. DC:

And you said that for any triangle with a right angle.

CG:

You also tried to prove it

DC: This is what I find extraordinary. He knew he was creating squares with irrational sides - he must have done. CG: Of course if you do it in geometry, you don’t ask the question. DC:

That’s right, you can see root-two when you draw the line.

CG: You don’t even have to see it. It was an accident in the work that one of the disciples did. But he didn’t have to connect with this - because he could dismantle pieces and rearrange them - so that what covered the two squares would cover the big square. You can also get the Pythagorean theorem independently of geometry. DP: Provided you are happy about assigning numbers to lengths.

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CG:

Letters to lengths.

DP:

Which are now generalized numbers.

CG: I don’t know. That’s where I can hide the fact that the square root of two isn’t there. When I say ‘a minus b’ , I haven’t said whether a and b are rational. I only ask you to believe that they can be related by subtraction. If I have a length and make a square, there is the area. The square is the one number multiplied by itself. PB: So are you saying Pythagoras could get away with not operating with numbers, provided he used the calculus. CG:

No, I am not saying that.

PB: Surely if you use geometrical methods to square the line, you are sweeping out. CG: No, I’m not squaring the line. I’m generating a square on a certain line. It is a totally different thing. PB: But you are getting the area by precisely the same sweeping out process. CG: Not at all. You see, the notion of area has only become really rigorous in this century.

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DC: You can sweep out any shape - any simple shape - and it still works. CG:

Provided they are similar.

PB: And, of course, it will work in a field in which there are no geometrical squares. It will work equally well on an isometric grid. But this idea of sweeping out must have been implicit in the thinking of the Greeks about areas of squares, even though it never became explicit. The idea must have been around without awareness, as it were. CG:

Sponges can integrate - but they don’t know the calculus.

We only have a few minutes before five, Can we see where we’ve gone as far as the day is concerned. DC: I have become aware that I can relate to numbers and pictures better than words. So I found this morning’s session very difficult to follow, in that I automatically cut out some parts of it because I couldn’t take in the information quick enough. Whereas this afternoon’s session seemed to me much slower - so I could understand. JW: I think I have been charmed by the things this afternoon. The Cantor theorem was lovely. I am not sure about the explanation of the calculus. I don’t know whether it gives me an inkling - or picture - of how Newton was working when he came to it. I think I would quite like to follow those up a bit. I don’t think Newton started with what you gave us, did he? 103


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CG: I didn’t have in mind the sequence he did things in. That’s allright for those who want to ascertain that. But I want to work on the awareness behind it all, and on why he presented his work in a certain way because he was addressing the Royal Society - people he knew personally. JW: Yes, I think I probably want to know a little more about why he came to think in those ways. CG: I don’t know. This is precisely what I call his genius - to have thought of it, when nobody thought of it before. DS: When the name of your seminar was announced, I said I’d come expecting you wouldn’t be doing the history of mathematics anyway. I had very little investment in that, and never have had much. So it was a bit of a surprise to find that you actually were doing. . . (laughter) . . . I very soon became convinced by the awareness story. It ties in very much with things I wanted to look for, and I’m beginning to see very clearly why the history is important. I asked earlier about the instrument. That seems to me to be where I need to look. It’s been very rich for me. PB: I’m very happy about the general thesis. I keep coming back to your saying that time is not a mathematical entity. But then you start some sort of continuous process - with a continuous variable. It seems to me that a rose by any other name smells just as sweet.

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ML: For me - during this afternoon - it’s as if we are interpreting the subject. My reflections during the afternoon were intuitive. The use of intuition can give students some access in different ways - which are not the ways used by mathematicians. CG: I don’t know what you mean at the end - what is it to use ways that are not used by mathematicians? ML:

You said intuition doesn’t have any place in mathematics.

CG: Did I say that it has no place? Or that when the product is finished it doesn’t require that you show your working? ML: We accept that first of all we need intuition, and then when a result is obtained the main objective is to remove intuition. It seems a bit dishonest. CG: No its simply being a member if the club. If you are a member of the club, you behave according to the rules. MH: I can only say that this afternoon was much better than the morning. It’s funny, isn’t it? When he gets the chalk out, and starts drawing and behaving like a teacher, it makes me relax. One thing that interested me was the question about what is happening to these mathematicians at the point where they make these great discoveries.

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CG: When Hadamard made an inquiry about this to write his book, he found that nobody knew. JM: I can add that when I made a similar investigation in this country they knew even less. CG: They are entitled to say: if I am a beautiful dancer, you don’t want me to tell you about how I dance - you want to see me dance. Mathematicians want to produce their theorems that’s their job, they have to create new mathematics. DP: I’m now confused about the relationship between mathematicians. Newton One might be the historical Newton, and Newton Two might be the awareness that you attributed to Newton, by calling it Newton’s use of fluxions. I am puzzled about the force of your saying this is just an awareness and inviting us to consider it. There seem to be two claims. One is that, as a result of historical work, you are proposing that this is how Newton looked at it. So the force of us looking at it has to do with its relationship to Newton’s work. This would be specifically a historical activity. Another claim - in the context of this seminar - would be that you are you are inviting us to look inside ourselves in relation to awarenesses that we may have. In which case, I don’t see the force of the historical side of things. So I think I’m puzzled as to the relationship between the two. CG: Well I think I have an answer, but I will go on with the others and then we’ll come back to it. 106


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JD: My attention during the last hour has been very much with a session I conducted with students earlier this year on calculus. I’ve been thinking about what this afternoon’s experience has to tell me about these students and their awareness of calculus. I feel I have got a view of some extra tools to bring to bear in future - some ways of looking at what I offer students to talk to each other about, and at what they understand by a derived function. DT: There are one or two things I wanted to say. One is that this was a most interesting session. I was grateful that we had people with us who prompted you to do what you did. I was thinking at one stage of how many times I had given that particular lesson. If I had got to where David and Alan got in that time, I would have been very satisfied indeed. There is also something about charging, which may be a better word than poetry which has other connotations. I am not clear about this - is the charging required to permit the process of forcing awareness? I am reminded - if I can digress a little -of a story about Freud reflecting on a visit to the Acropolis with his brother. He had been depressed in Trieste as he waited for the boat to take them to Greece. Later, he thought that this was something to do with the fact that he and his brother were surpassing their father who had never been able to travel. He recalled that when they got to the Acropolis, he had told the story about Napoleon at his coronation, turning round to his brother and sayings and what would monsieur notre pere make of this?

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Freud’s reference to Napoleon is moving, because we immediately have some feeling what it is like to be setting yourself up and surpassing your father. Whether that is true in the historical record or not, becomes unimportant besides the use that Freud made of it. The story has a charge. JM: My attention has been drawn to forcing of awareness. What is the force of the word forcing, when you use that term? I suspect that somewhere in there is where the power lies. Because I feel that I recognize in the various examples we’ve had some elementary attempts of my own to try and pick out essences - or awarenesses perhaps - of mathematical ideas and to communicate those in some way to students. My attention is frequently on trying to get hold of what the essence is, and to try to communicate that. Because I am not interested in detai1s. I want to look more at the business of forcing and charging. Because I think it is there that I am not drawing on the power that is available. CG: When David Pimm speaks of the true Newton, he assumes that he can define the true Newton better than I could. I don’t claim that I am the true Newton. No-one cab be the true Newton. What Newton did will remain forever in history. If I associate Newton with what I did, it’s because I did not present the calculus in the way Leibniz did. I presented it in a manner which included the intuitive notion of sweeping under the curve. On balance, what I was doing goes to Newton, rather than to Leibniz. I could have done it the other way. But that is less intuitive and it would require a little more work.

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I have made myself vulnerable to the workings of the mind. When I read something, I read it at two levels. I will be concerned with what there is to learn. I am concerned with what the person who is there is saying telling me about how things are done. It is not always clear. But it is often there, and not picked up - because we are concerned with the knowledge that they have passed on. I am concerned with how they worked. Therefore, I can get to something and make some suggestion that this seems to be the way they work. So Newton Three has something of me, just as I have something of Newton. If I have learned his work and I have assimilated it, I have something of Newton. Nobody can deny that, because I can present a number of his procedures - his experiment -repeating these as he did. So I know something of Newton the physicist. I’ve read some parts of his book, and some books on his books, and I came to the conclusion that when he was with the pressure, he the spent the time idling about - he wrote nothing. Then, suddenly, he was driven to produce something. He was in a hurry and in competition - so he came out with it. These parts of his history are available to me; they can be ignored they can taken in. So, when I enter into this field, I say: Riemann wasn’t there. What did Riemann, Cauchy and Weierstrass do? They worked on something long after it was the new. But these virgins that we have here (the three to whom the calculus lesson was specially directed) learned heuristically - this is the name for the work of midwives, Socrates’ mother was a midwife. They learned

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heuristically to gat their minds come to the point where they say something spontaneously. David Cain was jumping in his chair, recognizing that it was connected with the thing - not out of erudition, not out of knowledge, but out of understanding. So I was confirmed that what I was doing is proper. Just as the little girls of Putney confirmed for me that the only difficulty in learning the calculus at eleven is because the teaching is so bad - not because the girls can’t do it. We need to recast this work that others did. We recast it in such a way that it makes sense to the goodwills of new minds. I have not yet written my treatise on the calculus. But I have thought it out for many, many years and I know that if I live long enough, and I am strong enough to put it on paper, then it will stand the test of being read by people who don’t know the calculus, but will end up knowing it, and rejected by all those who do know the calculus - because it wouldn’t look like the calculus to them. I don’t mind, because I don’t feel the pressure to write it. I am sure it is going to be found later, when more people know how to work on awareness. If Newton didn’t worked on his awareness, what did he work on? His bottom? . . . (laughter) . . . Sitting on his bottom - yes - he had to work on his awareness. I can ask myself how clever of Newton to have made something into an entity that could be labeled - something which is not part of mathematics, but part of everyday intuition, that the continuum can be generated by a point. We know how difficult it

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is to make it mathematical. If you try to be philosophical, you have the definite challenge: how do new entities spring to mind? You’ve all been asking this. You want to know how entities arise in the mind? It is a creation. It is a proposal for something that someone has seen. This sweeping process which generates a new dimension is very clever. It can’t easily be axiomatised, but it’s certainly vital, it certainly has changed the face of things. Let those who use only histological studies make a million cuts into the living things to find what they can’t find, because they cut everything up. If you start with the unit - the whole - you can say that imagery is undoubtedly part of the working of the mind. There is no doubt that there is a support from the grasp of the whole, from the intuition of the whole. There is no doubt that one can be helped by recognizing that something that one has done could be done differently - like drawing a curve. So I know that whatever particular I was concerned with, I’m now concerned with that which is no longer particular. If I remain with all these things - things I know little babies have in their minds when they learn to speak - if I work on the powers of speech in everyone, then you can say: a word is a class. And you can say: a curve is a class. Questions which looked obvious to Euler were covered up and became mathematical under Cauchy. Now Euler was the intuitive mathematician and found many more things than Cauchy. But Cauchy has a greater reputation, because he was the goldsmith. There are people who admire the finish of things, and there are people who admire the inner wealth - the content. Now, I have always said 8 I work with the wealth of my students, not with their poverty. The poverty is in their mathematical 111


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knowledge. Suppose I started asking them: what did Weierstrass do, what did Riemann add? I would get blank looks. I have no right to work with their poverty. You see, Newton was particularly fortunate the day he saw that something totally unrigorous, yet that was apparently true, could serve as a basis for a new departure. If you make it more rigorous - that means satisfying people of a later generation that’s your right, you belong to the later generation. But when I start with people, and I say I want to give them an entry into a momentous historical contribution, am I not working on the history of mathematics? I would like there to be a lesson in awareness for people, so that I can say they have understood in a few minutes things that I took years to learn. I learned through Goursat and other books that required pages and pages, and thousands of exercises, to be understood. I had rules at my disposal, but I hadn’t grasped the subject. When I spoke of Thales’ theorem, I gave a dramatic content to bring together that the Greeks worked a lot with the ruler and the compasses, and that they could not solve the problem of one third with these. I was concerned with their problem, which was solved by an entirely different approach. So my students know a story, they know the theorem. But they know more than this they know that it is possible to use one method for one thing and one method for something else. It’s part of the curriculum to teach constructions. Well, let’s teach constructions as devices for solving some problem in a certain frame of reference. Let’s show not only that this solves these problems, but that it doesn’t solve others. This is a mathematical education.

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It took hundreds of years of reflection to understand that the axiom of Euclid doesn’t need to be proved. Many famous people tried, because there was a misconception that comes from the fact that nobody understood the meaning of Euclid. They learned it by heart, they used it in the machinery of deduction, but they didn’t heed what he said in his text: I beg you to agree. Well, you can say: I don’t want to agree. That wasn’t a possibility for a long, long time, though he knew when he put it there, that he had other choices. He didn’t pursue the other choices. He didn’t have enough years to live to be able to say: now let’s start again, and let’s start with this axiom and see what it does. He might have done that, because he was capable of putting together a multitude of thoughts which were found by others at various times. Suppose I had been there, and had knocked at his door and said: look, I’ve proved your axiom. He would have said: but you can’t prove it. Don’t you see that I beg you to accept that this is so? How could you prove anything? Otherwise I would have called it a theorem, and I would have proved it myself.

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CG: I often give a test called the saucepan test. If you know it, keep quiet. I will use David as the subject. David, in your kitchen there is a stove that can be lit, water in the tap and a saucepan on the wall. Tell me exactly what to do to get some boiling water. DBD: I take the saucepan off the wall. I fill it with as much water as I think will meet your requirements, put it on the front left burner, turn on the gas tap and press the little red button that ignites the gas; and then I sit down and wait -reading the newsletter the while. CG:

Now the next morning, the saucepan is on the table.

(end of tape - various solutions offered) JM: I’ll try. I put the saucepan back on the wall and tell myself I knew how to do it in the first place.

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CG: You see you already solved the problem when it was on the wall, didn’t you? This reduces the problem to a problem already solved. If you do that you are a mathematician. If you don’t, you are a cook or whatever you want, but you are not a mathematician. It’s a very good test. I don’t know who invented it - I heard it in Zurich in 1936, so that it could very well have been Polya. This is an awareness you have to come to - this conclusion that if I have solved the problem I don’t have to solve it again. There are two ways of working as a mathematician. One is as in this kind of test. The other is to ask: if it works in this way can it work in other circumstances? Which other circumstances? Someone asked yesterday about quadratic equations. Well, we are going to work on quadratic equations - as mathematicians. I am going to play a game with you, Alan. I think of a number, I square it and I get 9; can you guess what my number is? AR:

3.

CG:

How did you know that?

AR:

Because I worked backwards

CG:

What did you do?

AR:

I took the square root of 9.

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CG:

And you found?

AR:

3.

CG: So, allright, you know that. Well, it wasn’t 3 . . . Of course it was 3 . . . (laughter) . . . But I have the possibility of saying it wasn’t 3. What else can it be if it isn’t 3? What is x when x squared is 9? AR:

3 or minus 3.

CG: So you are as aware of that as I am. I can’t say more about this than you. If you know a few things, it doesn’t require more than common sense to get this answer. Now - think carefully - I think of a number, I take away 1, I square and I get 9; what is my number. AR:

Well I’m not sure.

CG:

Saucepan, sir, saucepan!

AR:

Minus 3, but I don’t see why.

CG: Saucepan doesn’t mean 3. (laughter) What was the problem you solved before? You found a solution of an equation in which one term squared is another number. Which number did I square? AR:

The number after subtracting 1. 117


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CG: Then is it not still the square of a number equal to something else? Can’t you do what you did before? Why do you get paralyzed? AR: I don’t know. That’s what I’m trying to figure out - the fact that there was an additional operation beforehand. CG: Because you are undisciplined, sir ! Saucepan, first ! What can you say because you did it before? You see what happens? People are used to writing everything down, and then they paralyze their capacity to visualize the problem. Work on this problem in the way I suggested to you. Now, I think of a number. So what do you automatically? AR:

I wait for the next step.

CG: You put a question mark - because you don’t know that. Now I say I subtract 1 - so you do it. You don’t know what’s coming next - I have to tell you what it is because I’m the one who has the question mark in him. So then I say 9. What do you get when you unsquare? AR:

3.

CG:

No, sir, you forgot what you did earlier.

AR:

3 or minus 3.

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CG: We say plus or minus 3 and write it like this, (writes ± 3 on board). The ‘or’ has disappeared when you write one on top of the other, but when you read it you restore it. Say it again. AR:

Plus or minus 3.

CG:

And that stands for what?

AR:

The square root of 9.

CG: It stands for the process of going backwards. What’s my number now in your mind? AR:

I have no idea.

CG: It’s a question mark in your mind. But did it remain a question mark or did I do something to it? AR:

For the moment nothing has been done to it.

CG:

What did I do to it before I asked you the question?

AR:

You subtracted 1.

CG: Well, why don’t you say so, since you know it? So, what is in your mind? AR:

Question mark minus 1.

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CG: And that equals what you said you had from the first problem. AR:

Plus or minus 3?

CG: Allright - but why the question mark in your voice? . . . (laughter) . . . So what have you now? AR: Question mark - but that has nothing to do with your problem, (laughter) CG: Discipline, sir ! You have a certain number of atoms in your mind; will you gather them so that you know what you are looking at? AR:

I know exactly what I am looking at.

CG:

What are you looking at?

AR:

I’m looking at the panic.

CG:

That doesn’t solve the problem.

AR: I don’t care if it solves the problem or not - I’m looking at it. That’s what I was looking at. CG: Stop looking at panic and look at the problem. Write it in front of you, and speak it out loud so that we can hear what you are seeing. 120


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AR:

Question mark minus 1 squared equals 9.

CG:

But you have already worked out this one?

AR:

Which one do you want me to work out?

CG:

The saucepan now tells you what?

AR:

4 minus 1?

CG: Not yet. Tell us what is in front of you, before you do any calculation, so that you have an image of the problem on which you will work in a minute. AR:

9 equals . . .

CG: That’s finished. That was when it was hooked on the wall. Do you know what he should have in his mind, David? DBD: Question mark equals. AR:

Question mark minus one

CG: That’s what he had in his mind. You should have it too because that’s where we are with the saucepan. And what do you have on the other side? DBD: 9.

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CG: No - it can’t be 9, unless it’s a square. You didn’t say square - I don’t want you to say square. AR:

You don’t want me to say square?

CG: Because you have the saucepan. This is an example of the saucepan. You know that you don’t have to go to the tap and do all these things. You have the same problem as before. So what do you think he should have in his mind? DBD: If I follow your instructions, question mark minus 1 equals plus or minus 3. CG: That is what he had in his mind. But his panic produced a curtain - he couldn’t see. AR:

I didn’t know I had it in my mind.

CG:

Well, that’s an awareness.

AR:

That I didn’t have it in my mind?

CG: Both an awareness that you didn’t have it in your mind, and that you did have it in your mind. That you didn’t have it in your mind was because you concentrated on the wrong thing like the others who didn’t want to put the saucepan on the wall. They concentrated on the wrong thing - they thought they had to give me water, but they had already given me water. Now, take over, David, you are not much better, but take over.

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DBD: I often see in classes that the panic in one student enables another student to do the work. CG: Then don’t panic now so that he has to do the work for you. Do the work to the end. DBD: Do the work again? CG:

NO - to the end. You haven’t finished, have you?

AR:

It’s 4 or minus 2.

CG:

How did you get that?

AR:

Minus 2 minus 1 is minus 3.

CG:

Are you sure? You could have answered that?

AR:

Not necessarily

CG:

Why not

AR:

I say not necessarily no

CG: Well from now on - necessarily, yes. From now on; it’s as easy as that. You understand how you got all these results. So you can tell me what I thought of.

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DBD: The answer’s 4 or minus 2. CG:

No - this is no answer. I asked you what’s my number?

DBD: Your number could be 4 or minus 2. CG: Why are you so timid that you think you may be wrong? Why do you not think you may be right? Was there anything else to do in what you did? DBD: I didn’t want to be so inflexible as to be . . . CG: Be flexible when it is needed, rather than now! The game is nice and simple, the rules are all given. So we don’t have to panic. Let’s start again. I start with you this time. No calculation this time. I think of a number, I add 2 to it, I square, I double it, and I get 18; what was I thinking of? MH: Sorry - you thought of a number? CG:

Why didn’t you listen to me? Saucepan, sir !

MH: Plus or minus 3. CG:

Why do you say that?

MH: Because that’s what the saucepan is. That’s putting the saucepan back on the hook to get half of 18 is 9.

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CG: Well, why don’t you say so? So that we’ll know that you did it. How could we guess you did it? Perhaps you were thinking of the other problem - tell us, please be explicit. MH: The half of 18 is 9. CG:

Why do you take half of 18?

MH: Because you doubled it CG:

Well, why don’t you tell us?

MH: Because you doubled it, I have to halve it to get 9. Then because you squared it, I have to find the square root of it -plus or minus 3. Then because you added 2, I have to figure out as quickly as I can - if you add 2 and end up with plus or minus 3, you must have had either plus 1 or . . . shsh . . . or minus 5. CG:

How do you get that?

MH: Because you had plus or minus 3. If you added 2 to minus 5 you get minus 3, and if you added 2 to 1 you get plus 3. So I say you get plus or minus 3. CG:

That’s what he got - plus or minus 3 - what did you get?

MH: I’ve just told you - plus 1 or minus 5.

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CG: Are you sure? Are you sure? Well, if you accept, that I bow to you and say you are right. It is. Be firm. MH: I am being firm CG: This is my way of forcing awareness. A lot of force. Allright; now I go back to you. I think of a number, I square it, I double it, I subtract 5, and I get 7; what did I think of? AR:

7 plus 5 is 12, divided by 2 is 6 - square root of 6.

CG: Yes, but you forgot something from the saucepan - from the first one. AR:

Plus or minus the square root of 6.

CG: Yes, and now he is confused. Let’s go over it to see if it is true. Square plus or minus the square root of 6. AR:

6.

CG:

Double it

AR:

12.

CG:

Subtract 5

AR:

7.

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CG: So you can verify for yourself - doing what to get the right answer? AR:

Doing the opposite - going backwards.

CG: In the reverse order. What else beside the reverse order? The saucepan is a state of mind, you see. We go it when we want to be whole. AR:

But there comes a point when you can’t go any further.

CG: Well, don’t. Be aware that there are two things - the reverse order and something else, the one you did yourself. MH: The right order? The way you went back over it in the right order. CG: But you didn’t do only that you did something else as well. AR: Was it that I have to change the operations, I have to reverse the operations? CG:

Well, why don’t you say so?

AR:

That’s what I meant by the reverse order.

CG:

The reverse order is only the reverse order.

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AR: That’s true. So I have to reverse the order and reverse the operations. CG: Each operation - that’s a correction of language which makes your thought a little more precise. Do you see that it doesn’t take long to teach quadratics? I can make it a little more complicated. But it doesn’t serves any purpose. I can make it more complicated to test my students, so that they can take one more operation into account. But for the moment I’m going to summarize what you know. I think of a number, I do something to it which can be (writing on board) either add or subtract alpha, or multiply by capital A, the number I have in my head. Then I can square this, to get something called T. I can also do something else - I can multiply this by B, which could be divide, and I could add or subtract C, and the result is T. I could do all thus with you or with my class. T = (A × ? ± α)2 × B ± C Now what would you do to tell me what my question mark is, what I have in my mind? AR:

To find the question mark?

CG: To find what I have thought of. Say it with letters so that everyone can follow.

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AR: I would start with the T. Then minus or plus C, divide by B. Take the square root of A, times question mark plus or minus alpha. CG:

I’ll be your scribe - say it.

AR: Take the square root of A times question mark plus or minus alpha CG: No - I want to write the steps you have already in your mind. What shall I write on this side that you have not yet done? AR:

A times question mark plus or minus alpha.

CG: Yes - and on the other side what do I have? You said it before. AR:

T minus or plus C.

CG:

And then?

AR:

Divided by B

CG:

But you forgot something, sir. What did he forget?

MH: Just the fact of the square.

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CG: Because of the saucepan. The first one led to a doubt, didn’t it? AR:

I’m with something else.

CG:

Don’t be with something else be with this.

AR:

I’m with that.

CG:

Yes, but we haven’t finished yet.

AR:

Plus or minus.

CG: And you know why - because there is doubt. Now this one is the consequence of doing all the things you know on a concentrated problem. Can you take the next step? What will the next step be? Instead of solving this problem, you have replaced it by an easier one. Can you solve this now to get the number I have in my head? AR: I’ll do it, if you will promise that I will understand why you have taken the square root of T minus or plus C multiplied by B CG:

Because you told me to do that.

AR: I didn’t tell you to do that. I told you to divide T minus or plus C by B.

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CG:

That’s what I wrote.

AR:

Well, what’s the square root doing there?

CG: You told me there was a square root - because there was a square there. AR:

I told you to take the square root of the other one.

CG:

Which one?

AR: Oh well they’re equal I guess . . . (laughter - applause). . . OK. Well then - if you will subtract or add alpha. CG:

What will you be left with?

AR:

A question mark.

CG:

Not a question mark.

AR: That’s what I said - ‘ay’ question mark otherwise I would have said ‘uh’ question mark . . . (laughter) CG:

What shall I write?

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AR: Plus or minus the square root of 7, minus or plus C, divided by S minus or plus alpha. So question mark is everything on the right divided by A.

CG: For which you get 20 out of 20. The notation only represents the content of your mind, not what the teacher told you to do. When you get an answer, you know it. AR: Why did I have to use the saucepan before I got to the end? CG: There is no saucepan here - this is an interference of yours. I said saucepan when you put the square root of this. All this in your mind can be called my number, my thought. This can be question mark squared. So if I give you this problem which equals V, can you solve this problem. Well, the pattern is the same. It only requires this awareness - that from a certain point n you know what to do. For instance, we can play with this adding and subtracting long before the squares appear. So that it becomes part of second nature in you to reverse the order and invert each operation. This is applied to a new situation in which there is doubt - therefore plus or minus appears. Of course, I would take more time if I worked with young children, because we have to agree on the notation. It expresses the content of the awareness. They have to know exactly what they have in their mind. Now, in order to complete the theory of quadratic equations - which is the other aspect of the problem, 132


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no longer the saucepan, but the saucepan with extension - which other cases can I handle? I think of a number, I square it, and I get minus 9; what is my number? AR:

The square root of minus 9.

CG: Well, you see, it took a century of discussion to allow the square root of minus one to appear in mathematics. In terms of the dynamics, you can say that it doesn’t mean anything if you think of squares as being shapes, and square root as being the side. But it will mean some thing if you can give a meaning to it. That’s another story a nice story. I can play another game with you. I think of a number, I subtract 3, I get minus 1; what’s my number? AR:

Two.

CG: You can do that because you went to school. But it wasn’t the same when people were first confronted by all this. How could they subtract more than they had? It was a great heresy to think of negative numbers. But not for the people who didn’t care about such restrictions and could say: we can generate this extension of number to allow us to solve an impossible problem. It was impossible in the concrete sense, not mathematically. Because in mathematics you are concerned with relationships, with statements about what is in your mind - not what is in the pebbles. You are concerned with dynamics of the mind; so why shouldn’t I subtract? I have in mind a number -and I subtract.

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This extension has been accepted historically - it has been a landmark. But it isn’t a landmark for our children. It’s only our understanding that it can be done, that if we work on the number line we can get to the other side, and say what I was saying for the other. Similarly, you can recognize that fractions make certain divisions possible - divisions that could not be done in a restricted field. And we restrict students without telling them the rules of the game. PB: Can we briefly have a moment on a pedantic point though it is a very real one. When we write the square root sign, we mean the positive square root. But when we’re talking, this ambiguity is always there. So when he says the square root of 6, he’s really quite right. We shouldn’t want him to put plus or minus in front of the root sign. It’s an ambiguity that exists in speech, but we’ve ironed out on the paper. CG: This links us back to the Pythagoreans who were afraid when they found that the length of the diagonal didn’t fall on one of the fractions. People later invented a new notation. They said: we can see it, we know what we’ve done, we know it’s a name. The notation allows you to talk about the square root of 2. It doesn’t tell you what it is. We can present our understanding of the progress of history without thinking of a sequence of shocks. That’s why I could not say accept, David, when you said you taught history of mathematics when you taught mathematics. I don’t. I’m guided by the history of mathematics to avoid the shocks which came from prejudice - not from mathematical thinking, but from

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another source in people’s minds, from their habits of thinking and from what had been passed onto them. CG: Can we have a little more on this, so that we know how reflection on the history of mathematics in terms of awareness can improve our teaching as well as our understanding of the past?

Discussion DS: I’m not sure that I’ve completely understood what you’ve said, because there seems to be a contradiction with your story about Thales, where you were, in fact, presenting a shocks you can’t divide it into three, so you go for a walk. Are you saying that you shouldn’t present this to children as a shock? CG: If there is a shock, that is educational. I have learned from history that people recognized that the construction of the point that divides a line in a certain way was not feasible with ruler and compasses. Why would they say that everything could be done done that way? They can only say: I am going to investigate which are the things that can be done by this operation. It is clear that you can do a lot with ruler and compasses. But something as simple as dividing by three shocked them. I can shock my students for a little while. But Thales salved the problem, and I wanted to make history of mathematics part of my teaching, because I can in this way force awareness that something new is needed. I can cultivate in them

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the sense that the adventure into the new is as much mathematics, as much as mathematics education, as passing on the successful work of the others. DS: I took you as saying that we shouldn’t regard the negative numbers as being an adventure into the new; or have I misunderstood? CG: It has been historically. Our ancestors had some inhibition - perhaps a bias towards moving always in the same direction. But the rest of the line is there, and if you play the game of subtraction on it, you can get to the other side. I can find a characteristic of all these numbers on the left, and then I have a handle on directed numbers. But this will not be a new chapter or a new awareness. It may become so at the moment when I introduce something which is not legitimate on the straight line - namely multiplication of directed numbers. Multiplication is an extension of addition, in the sense that it is repeated addition. We teach that to our children by giving them the tables. By saying ‘add 2, add 2, add 2, add 2’, we have introduced multiplication as something to do with addition. The impression we always give children is that multiplication magnifies. One day, we’re going to teach them that multiplication also miniaturizes, because one half of something, one tenth of something, will make it smaller. That’s where epistemology is required. We have to be very close to how we operate this, how people operate it and why they got into trouble. We don’t want our children to get into trouble

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because our ancestors did. It is part of our teaching not to smooth things out by making it all clear and simple, but by making them recognize that, when they get to a certain level, they can put a question which allows them to widen the field. That’s what our ancestors did. I think it was Vieta who wrote somewhere that only the infidels could think of the unknown as if it were known. I think of a number - and you work on it with peace and tranquility. Well, at the time he was writing, people could not do that. There was an orthodoxy, and you had to be infidel like the Arabs, to be concerned with the fact that you can work on things that are not known. This is an awareness the orthodox could not have. The role of preconception and prejudice in mathematics is equivalent to that in any other field of thought. But it’s more subtle. Once it is eliminated, it seems it wasn’t there - you don’t think of it, you don’t talk of it anymore. The fact that we can’t do division forces us to consider fractions as an entity that replaces the division you can’t do. If you divide 6 by 2, you get 3. But if you divide 6 by 5, you can’t get something that you can do. There will be a remainder - that will be one form in which you do it. But another one is when you divide and get 6 fifths. Then you have developed a facility with fifths that you had with integers. As a self-taught person, I had lots of misconceptions about lots and lots of things. One day I was writing something on fractions. I had to explain why you can’t add and . Why did I have to do something about it? Children don’t have the problem, they write naturally. Because addition tells them what to do, they do it. But why am I inhibited? I am inhibited, not because they 137


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are fractions, but because of the plus. What do I know about the plus? That it only works on entities that can be merged together. So it’s the plus that requires the denominators, not the fractions. I went back and I said: let’s add 2 cows and 3 horses. You can only add animals - if you want to, you have 5 animals. So you already have a common denominator when you are working with objects that have different names and you change the name. You change the name of the objects to have a class that includes both. Therefore, to add and - in order to be able to do it - you have to find the same name. This is in the classes of equivalence - sixths. You say plus - listen to my voice TWO sixths and THREE sixths - and sixths play the roles of pears and apples. The numeral adjectives - and they are numeral adjectives - are called numerators. And the denominator is what gives them their name. In 1957, I was already an old man when I had this shock of finding that for all these years I had lived without this awareness - that it’s the plus that asks me to change the denominators. I don’t know whether any historian of mathematics has been concerned with these matters and has seen that when mathematicians come to something they can’t do, they introduce a definition - a new definition - which allows them to go ahead. In our curriculum, we make a great deal of the appearance of the negatives. It’s a turning point. We make another big deal with fractions, and so on. Well, there is no reason for this except our clumsiness, our not having thought of how to generate the proper awarenesses. When you work on decimals, you discover one day that the decimals have the property of allowing you to use the saucepan. All the things that you knew how to do with 138


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integers, you can do now, provided you know how to rename numbers and then put the decimal points underneath each other. Or - in a certain context - that decimal is only another name for fraction. This is not a new mathematical instrument, it is another language. Once you have established that all you can do with fractions, you can do with decimals, then you find why you should have decimals. I have decimals because I can do new things that I couldn’t do with fractions. I can do lots of new things, because I have this notation at my disposal. That’s the second part of the awareness - the extension to asking what else I can do with what I have. I can ask questions about the recurrence or non-recurrence of decimals, and ask what sort of things these generate. Thus, I can use decimals to encounter an extension of numbers. I can make people aware that there are things that are neither whole numbers nor fractions. The Greeks didn’t have this at their disposal. Have I gone too fast? I don’t ask the virgins - they have to remain virgins until the end of the day, but after that they too can become bores if they want . . . (laughter) It is possible to express in terms of awarenesses in the classroom. So it is possible to do it in the history of mathematics. I know how to do it in the classroom. I haven’t done it in the history of mathematics. But the work can be done, because history can be shown to be a succession of awarenesses. We put them together in our teaching and we force awareness that every child can handle mathematics in his mind. It is not a privilege, it is not a special gift of the brain for those of us who

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take steps to become good at mathematics. There is nothing exceptional in mathematics. It’s like all other things - a matter of becoming aware of the content of your mind and making it explicit. Education can shorten history - so that the new generation doesn’t have to waste centuries to get something. There is a way to do this - you increase the power of the students, you give them the tools that have been developed by the generations one after the other. You give them a tool functionally, not in the historical context. They use it, and they work with it. They have a future, because they start where others leave off. Let’s have some other voices than mine; and don’t ask questions - please make statements. DT: You said you could do it in the classroom and asked whether it could be done in the history of mathematics. I can see very vividly the force of it in the classroom. I was reminded of the fact that you wrote about quadratic equations - it must be nearly forty years ago - and that it was one of the earliest papers that I read of yours. I didn’t believe it when I read it, and it wasn’t until I taught quadratic equations in that way that I realized the power of it. It remains difficult to communicate except through practical example. You have to hear the matterof-factness in which Alan says ‘plus or minus alpha’ or whatever. I am still very intrigued by the discussion that David set off about the matter-of-factness and the shock.

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I am interested in the challenge of what can be done for the history of mathematics. Take the story about Vieta. The insight that you offered about Vieta seems to be an intuitive reading between the lines, impossible to interpret from the text directly. I feel the force of what you say. I think of people just coming out of mediaeval times daring to speak the name of God for themselves. To grasp the as-yet-unknown, and to hold it and to be able to deal with it, is indeed something that mediaeval Christendom couldn’t do. Vieta was in the process of doing that. But he did not think the Arabs had done it - he was scathing about them. Are we just cultivating our own insights? CG:

Yes.

DT:

Or trying to communicate them?

CG: I have attempted to do this - to rewrite history - in the realm of the social sciences. I could perhaps take some time later to show you how we can rewrite history in terms of awareness. The history of mathematics would be a special case. Then you would understand that there is no need to subject our young generation to the errors of our predecessors, our ancestors - that education is not the transmission of culture as it is, but the benefits of culture. If we can extract these benefits, our children are entitled to them. But if you just repeat the unfolding of history, you will teach first arithmetic and then algebra. You will teach first addition and then subtraction. You will teach first integers and then fractions. All these things are prejudices. We don’t have to

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be tied and enclosed and held by this type of choice. We have learned some things since. We don’t have to be connected to everything that has been uncovered. We have the right, and the duty, to recast those things according to our lights. If our lights are true -justifiable then though we can be rejected if we do fantastic things, and we can be rejected if we do the right things, the fantastic things will dissolve and the right things will re-appear - will re-emerge somewhere in another form. Since we are searching we are searching for truth, we are not spending our time caressing and inflating our egos. DT: The point I’m trying to make, though, is that it is the actual experience of working with Alan on holding the unknown, or the experience of working in the classroom on working on transformations, that is what is convincing, rather than the recasting as such. CG: It would not necessarily be convincing if it were not true. So there is a truth that precedes the example. The example only convinces people who are at the level of the example. DT: When Alan said very quietly that you reverse the order and you invert each operation, he said it so matter-of-factly. There is a sense in which he may not know what he said. CG: No, on the contrary - he knows what he is saying I agree what he says. It is as simple as that. Percy Nunn - or perhaps someone before him - introduced the balance for solving 142


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equations. All the students who were introduced to it thought equations could be solved by the balance. Nonsense, nonsense! You don’t get the reversal of operations you don’t need to take something from one side and put it on the other. This is true in the model of the balance, but if I think of a number and I make an operation and I give you the result, you will think of the reversal. When I was at the Institute, I had to fight with all Nunn’s disciples. I couldn’t make a dent, because I was considered the odd one out. I didn’t mind, because I had other things to do. But what was interesting is how prejudice enters the mind. I don’t want this to happen to me. I don’t want it to happen to the children. I don’t see why they should be with the prejudice of someone they have never met, why they should be conditioned by people they have never encountered - people who don’t know how to give them a chance repeatedly every day to prove that they can use their minds. These children used their minds so well when they were one or two years old, when they learned to speak. Then they come to do futile small things in the classroom, and they fail. These same giants are dwarfed - and more and more dwarfed everyday - by our approach to these things. But we wouldn’t have an alternative if we didn’t have awareness at our disposal. This is the other thing I have said to you for so long, awareness of the awareness exists. It is what you all have, what they all have. Let’s use it. I had the job of recasting mathematics in terms of awareness, and I did this - it’s in my books. The history of mathematics is another such exercise.

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DH: I realize that I personally have become aware of what is required for quadratics - not being colored by the way I was taught, just what is required minimally. An exercise like you’ve done now concentrates on the only things you need to concentrate on to give you the ability to handle quadratics. Also - listening to you talking about fractions, increasingly made me realize that I need to look much more at mathematics myself, to get rid of lots of cloud around it. I’m particularly interested in this, because I’ve actually been working in a similar way with some kids in my classroom. I’m interested in the way you worked because to me it reminds me of what I felt about the way you worked over the weekend. I remember that at one point John said: I want to go out for a walk now, I want to be with what I’ve got. You were really quite insistent that he stayed and made use of you in the time that you were here. Then he was bombarded, in a sense, with more and more and more - he could go out for his walk on another day. I felt that something similar was happening with Alan. I felt at times as if Alan actually wanted to go for his walk - he wanted to mull things over - and I felt you were pushing him on and on. AR: Well, that’s not the way I felt. As far as I was concerned, I did go for my walk, because I went for my walk when you switched to Michael. As David so correctly pointed out, when panic sets in you shift, and then the panic often shifts as well. It didn’t in this case, but it can. The panic disappeared for me - or I made the panic disappear. So, in fact, there was a moment when I could be alone and try to see what on earth happened between something that seems so obvious and something that should

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seem so obvious - simply because it was only the addition or subtraction of one. Why the panic set in, I still have no idea. I know I’ve acquired enormous biases and prejudices, at least in this domain - in other domains as well, but especially in this domain. But I did feel that I went for a walk during the lesson, and when I came back the panic wasn’t there. There were things I couldn’t understand as we went along. But there was no panic. DH: The things that you didn’t understand - do you understand them now? CG:

He doesn’t have to understand it like us.

AR: Yes, there are certain things that I didn’t understand and which seems to me terribly obvious at the moment. For example, for a certain period of time I didn’t understand why he insisted on putting the square root over one side of the equation and not the one it was supposed to be - the other one. Until it clicked that it didn’t make any difference since they were equal. MH: Isn’t it funny? I was thinking that once it was written there I found it strange and kept wondering why. What’s the difference between doing it verbally and then looking at something written down and not knowing what I am doing with all these symbols? AR:

That’s the panic I’m talking about.

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CG: You have not been taken through a whole course of exercises. You have been thrown into the pool and set swimming - and the pool was deep. We have done so many things. What no one is saying at this moment - though perhaps everybody has it in mind - is that there was not the slightest panic at the moment I shifted to letters. He has understood it as long as he can handle numbers. He doesn’t know what number it is - but it’s potentially something he can know. He recognized something which was recognized yesterday in the calculus lesson - that if I draw a line, I have drawn all sorts of lines. AR: To come back to one more thing. I feel very comfortable with what I know of this. It doesn’t present me a challenge any greater than the challenge should be. I’m not intimidated by this. CG: It will be greater when you become aware of your awareness. You have become aware of something. We’ve worked on it, but you are not yet aware of your awareness. So you can’t work on the awareness which contains some complicated things. Letters can stand for something, because all we’re doing is to be concerned with operating with sequences of operations - not with letters, not with values. Therefore, when you become aware of this, you will say: well, what was I doing? There was no number, there was no result, what was I doing? And someone will tell you that you were doing algebra. The one who says that will be concerned perhaps with a definition - that algebra is

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arithmetic with letters. But if I force awareness in that person that it is operations upon operations, then he will be aware of the reality of this, as you became aware. You have not been introduced to numbers - you’ve been introduced to operations. There is something to hang on the operations - hooks for recognition. Notation is just a hook -so that you don’t get lost. You know that there are so many and not more, that they are connected in this order and not another. We use notation to ease the thinking of our students. I would like to take another topic - another awareness - which came quite late in my life, because I had met it first within mathematics and I had not allowed myself to touch it, to do anything else but what was presented in it’s own ritual. It is so important that it revolutionizes altogether the thinking of those who allow it in. It puts a lot of order in what you do. I will tell you later how I put it into the elementary school curriculum.

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CG: Every time I wrote the equals sign there, I felt that I shouldn’t have done, that I was perhaps misleading some of you. Because the equals sign is connected with lots of previous experiences, some of them perhaps not totally digested. It has no place in mathematics. We still use it - I use it - but the equals sign can be used without danger only by people like me. Percy Nunn’s balance is there (writing on the board), but I don’t want this to be in people’s minds. I want them to enter into the dynamics. This thing becomes something which I can name. So what is on the other side is an entity that has another name. We are used to saying it’s the answer, which is a bias - a distortion. Even if it were an answer - a final one -there should be an arrow to indicate that all this leads to that. So that we take responsibility that the movement is in that direction. Is it possible to reverse the direction? If I start here, will I get that? That is much richer - so I’m already misleading people if I put an arrow back.

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At the level of language, you will know that if you forget a word as I do so often - you have to find a whole statement which generates the meaning. Then someone suggests a word that telescopes that meaning. What do we call this in language? We say: but that’s an equivalent expression. The statement ‘Alan is on the right of Maurice’ is equivalent to ‘Maurice is on the left of Alan’. When you look at these two statements, which is preferable? Neither. When you start with Maurice, you have to say ‘on the left’, when you start with Alan you have to say ‘on the right’. That’s where the obligation is; but as far as the description of the situation is concerned, they are equivalent - of equal value, equi-valent. Equivalence is the notion to generate in our students. We have it implicitly when we become mathematicians, because we have a store of knowledge. For our children, the notion that things are equivalent is more descriptive of the truth. You keep them in contact with the truth, rather than mislead them. Infant school teachers - who may have trained with some of you - have been told things that are so terrible and so cramping that their sense of truth is distorted. If I say 2+3 is the same as 5, then I don’t know the meaning of ‘same’. This pen is the same whether in my pocket, between my legs, or on the chair. That is what the language says. How could 2+3 be the same as 5? They obviously look different, so you are forcing me not to trust what I see by saying they are the same. They are not the same - ever. They are both present there, because there is a movement of the mind. I need sometimes to be with 5, sometimes with 2+3. So I need to be in contact with classes of equivalence from the start. I prefer the continental ‘class of equivalence’ rather than 150


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‘equivalence class’; because I’m concerned that you emphasize the equivalence. So what is one? One is a class of equivalence. Not ‘one and only one’, as the song says. We’re going to be concerned with classes of equivalence. Your students will then be in contact with reality. What is 1 + 1? You want another name? I’ll give you a name - 2. What’s 2? Another name for 1 + 1. So it’s not as Russell says, ‘the twoness of two’ an impossible thought that everyone goes on repeating. What is in order, is to re-establish the notions that Russell concerned himself with. What I want to convey is that the cardinals are also words. And words belong to classes. There is no sixness of six. All sets that have this name are called six. It’s a word - ‘the twoness of two’ means that ‘two’ is a word. The twoness of two is an example that keeps you alert. You say what’s the meaning of that and for years you may handle it, you may repeat it in a lecture, as if you were a reader of Russell, and so on. Then you come to the point where you see the damage that it does. (end of tape) DP: I have often found myself resisting referring to mathematical symbols as names. What I’m thinking about at the moment is why am I resisting that. Because at one level it is quite apparent that they are names for things. I’ve found the expression ‘the twoness of two’, or whatever, illuminating when thinking about different number notations, and situations in notation where the relation between the sign and the meaning is not arbitrary.

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CG: So does the ten-ness of two make sense to you? That’s what two is in base two. There is no twoness of two in binary. DP: There won’t be a twoness of two, because two won’t be around. It will be around as ten, so it won’t be the ten-ness of two - it will be the ten-ness of ten. CG: I can go from one system of numeration to another in the way I can also go from one language to another. I can put these things together. DP: Surely when you were talking about the ten-ness of two you’ve got one thing in one language and one thing in another. CG: And why not? Par excellence. You don’t mind if I say that, because it’s been adopted in English. You go to another language - it says what you want to say. But I didn’t want to get into a discussion of this matter. The twoness of two was an absurd notion that I swallowed for a long time, but I have vomited it completely - it is no longer in my blood. There is no need for such things - it’s misleading, it’s impoverishing. It was right for the game that Russell wanted to play in his constructions. Outside that it fails. One has an infinite number of representations. If you allow subtraction, you can say 1 = 2-1 = 3-2 . . . and so on, for an infinite number of times. How do you present this to students? It is a fact about 1, that it is the name for anyone of these differences. That’s really the discovery, the important thing that mathematicians do - as Hardy used to acknowledge - that you 152


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can talk of something being the same, yet it’s always different. That’s how mathematics arises, that’s why the virgins call it magic. Because you start with something, and you transform it with another light, from another point of view. You start with 1+1, it suddenly becomes 2; suddenly 4 becomes 2x2. There is a context for the language. If you are careful, nobody is confused. If you are confused, how could the others not be? PB: Aren’t you worried by using tens to refer to the squiggle rather than the number? That worries me tremendously. CG:

Why worry because I make noises?

PB:

But some of the noises frighten me.

CG: If you are one of my students, you will not be frightened by noises. Accept it. It would be frightening if it was thunder, or the collapse of your building. But the noises we make in classrooms are normally not so frightening. We should be a Lot more concerned with the role of language in mathematics. Bourbaki claims language as a foundation. Babies enter into language so easily. We are losing the help they can get from something they have mastered and which they-use so beautifully. It only requires - among other things - that we drop the equals sign, that we don’t use the word ‘same’, that we say deliberately that it is another way of saying things -that all these things are equivalent.

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The example which I prefer to all others is this, (writing 3 + 3 ~ 6 on board) Once you have learned 3 + 3 is another name for 6, you become aware, through a separation from it, that there are two 3s, and you say there are two 3s in this. So you can say two 3s is another name for 6. (writing 2 × 3 ~ 6 ) If you have learned to read in Arabic you can just read it backwards to say 6 is another name for three 2s. (writing 6 ~ 3 × 2) I’ve gained the possibility of saying: I think of a number, I multiply by 3 and I get a number which I call 6; what’s my number? (writing 3 × ? ~ 6 ). They put a box where I cover it up or ask the question. That’s an invention that came long after my work. I didn’t use boxes - I use boxes now since everyone does. How many things can I say when I know this? I get another relation here - there are two 3s, which I can write in this way. (writing 3 [6 ~ 2) This is linguistic - since you know how to read English, I say ‘how many?’. This is the way in which I write it in the Anglo-Saxon world, because that’s how you say it. After a while you accept my notation. This is an extension on the linguistic level - nothing beyond that. Then I ask you what else I could say. You will invent how many 2s in 6? You know it’s 3. Then I’m going to tell you - and this is arbitrary, I impose it upon you just as I imposed this notation which I borrowed from history - that from now on you have another language at your disposal. You can say 6 divided by 3 is another name for 2. (writing 6 ÷ 3 ~ 2) What else can I say?

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So I have the division notation, the goes-into notation, the equation notation, the multiplication notation with commmutativity, the repeated addition notation - and it’s not finished. Because another of the horrors of education is that if I am twice as heavy as you are it stops there. Nobody asks you how heavy you are with respect to me. You have to wait two years for that question. How unjust it is! We are here together, talking of the two of us. Why should they be so sexist as to only say that I’m twice as heavy as you? Can’t I start with her, instead? How heavy is she compared to me? So at the same time as I have all this. I will have one-half-of, and one-third-of, and so on. (writing × 6 ~ 3, × 6 ~ 2) If I recognize that this is connected with one plus one plus - with the right number of ones - I have another notation, (writing 1 +1 + 1 + 1 + 1 + 1). How many is? So I put six is. How many 6s? One 6. (writing 6 ~ 6 × 1 ~ 1 × 6) Therefore I can say one oneth ( ) of 6 is what? One sixth is what? (writing × 6 and × 6) All this is linguistic. Once it is established it will be second nature every time they see one of these they will say I know all this paraphernalia that goes with it. They won’t need to tell you because they know it. All this is because we have taken equivalence - ‘is another name for’ - as the most important notion. To take one thing and look at it in another way is to do mathematics. JM: Can I ask a technical question? One of the things I have difficulty in dealing with back where the 3+3 is the same as 6.

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CG:

Not the same, it can’t be the same.

JM: Sorry - is equivalent to. I’m asking how many 3s are in 6 and somebody replies 4. I look at them with a quizzical look and they says yes, 3 + 3 + 3 - 3 gives me 6 so there are four 3s in 6. I find it hard to know what to do with that. CG: If you had worked on classes of equivalence you wouldn’t be troubled. But you did not confine the question precisely. You could say: why do you say there are 4? You could have said there are 24, you could have said 701; why do you say 4? Since the answer is totally ambiguous, it is of no use. Can we make my question ‘how many 3s in 6?’ mean only one thing. Of course, if they want to pull your leg they can do all sorts of things. JM: I take the message that when someone comes up with another way of seeing through the language that I’m using, then I’ve not been precise enough in my language and I have to cut their language down. CG: You can make them see that there is advantage in having a more precise statement. If you say there is an infinite number of equivalences once you use subtractions, then they can’t say there are only four 3s in 6. There are also whatever number of 3s that I want. But you hadn’t asked for all that, you were asking in a very precise way. What you had in mind was the reversal of multiplication. JM: The reason I asked the question is that it is my experience that almost always one or more people interpret what I’m saying 156


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in a way unexpected by me. It takes me quite a long time to work through that. CG: You must welcome that. It tells you that there is need for them and you to agree on the rules of the game. If they don’t keep to the rules, you can say: you’re not playing my game. JM: The reason that the answer 4 arises in this particular context is that prior to pushing on to the multiplication I have been working on different ways of making up 6. So that is more present in the mind because it is recent. CG: If you do that, then you are at fault. You shouldn’t have expected an answer of this kind. JM: Yes, I know I am at fault - that is why I am asking the question. My first assumption is that I am at fault - that is my axiom. DT: In this context, there was a 3+3 written down as equivalent to 6. The question ‘how many 3s’ is then a very concrete question. JM: But I didn’t hear the question ‘how many 3s are there in this way of writing 6?’ If its haw many 3s in this way of writing, then I’m not sure what I’m writing is a member of the class of equivalence of 6. CG:

Of course.

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JM:

But that language didn’t come out in your presentation.

CG: It did - because having written 3 + 3 ~ 6, which was the only thing written on that part of the board, I say how many 3s do you see? JM:

You didn’t, actually. I can see you could ask that.

CG:

I did - you have it on tape.

JM: I don’t recall it. One of the things I’ve observed is that even though I may write spartanly on the blackboard, so that there is only one thing that I think everyone is attending to, it is also my experience that there are at least two people in the class who are attending to something quite different. CG: Then you test this. Every time you make a statement, you go over it and you see whether these two can join the flock. JM: Sometimes I don’t discover it until twenty, or even five, minutes before the end - or even in the next lesson. Then I have a lot of work to do. DS: I would like some help on the question one should ask, because I’m stuck on that. John said: how many 3s are in 6? I find it very difficult to phrase a question that asks for the equivalence of the opposite to multiplication - I am getting myself linguistically tied up.

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CG: There are three ways - three notations for that. They come together. One is the fraction notation, the other is ‘goesinto’, the other is ‘divide by’. You can say: from now on, you have three ways of writing this. DS: The question I have to ask is what is 6 divided by 3? I mustn’t change this into how many 3s are in 6. CG: In English, you can also say: 3 goes into 6 twice. You say that to your students. DS:

I don’t - because I think that it is very misleading.

CG: But it does - it goes into 6 twice. The ‘goes into’ will develop into an awareness that whereas multiplication is repeated addition, division is repeated subtraction. If you teach long division, that’s how you teach it - as repeated subtraction. You have to choose sometime which is the most telling of the different ways that are available for them to use. It helps them as it is an answer to their problem. If I introduce these three notations at the same time as the inverse of the multiplication, I could also say that the inverse of each of these ways is the multiplication. But they are notationally and conceptually different. The division sign is arbitrary - on the continent there is no bar between the two dots. (; rather than -) . The fact that the British, and then the Americans after them, chose to represent division by the two dots with the bar says that they want to remind the students that division is attached to subtraction. 159


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PB: Surely most of this trouble arises from the idea that one should avoid linguistic ambiguities. But you can’t; and so surely the thing one has to do is to welcome them. CG:

When I hear ‘surely’, I hear doubt.

PB: Possibly - it may be that all one needs to do is to welcome linguistic ambiguities and discuss them, and then agree on what we are going to accept for the moment. CG: All these things represent turning points in my life, in my reflections, in my contact with what I do. What is the content of this in terms of awareness? If you want examples, you will find thousands of them in my writings. But I don’t want to set aside the fact that you need more than one instance for awareness, and the awareness. If you don’t have it in one example, doing another one will be like any teaching of arithmetic which goes through all these problems in turn. You don’t gain anything by going through them except a waste of your time. We are not going to have a break this morning - it is going to be a continuous thing. But we can have a kind of break when you speak - and don’t make it too short, Dick . . . (laughter) AR:

You mean let’s have a long break.

CG: It is a requirement of working in a group that one hears what the others think about the things that have been presented. If I go on, it will be as bad as lecturing.

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PB: To come back to ‘how many 3s in 6?’ again just for a moment. In school, where the children have done a lot of playing with things, you may well get 6C2 or 6P2, 6C3 or 6P3 (i.e. combinations or permutations of 6, 2 or 3 at a time). Suppose they have had 6 children out at the front. The teacher says how many 3s are there? They have thought: well, we can have John and Jim and Jack, and so on. There will be combinations - or permutations even - and these are valid ways of thinking of the question in a certain context, aren’t they? CG: Do you know what it is I invite you to add? Something you know? Or are you going to talk about other things? What is the central issue that I brought now, that I think is helpful to everybody - students, teachers, professors, mathematicians and philosophers, and so on. What is it? DT: For the moment, I see the answer to that question as the economy that is gained when what is offered to the student has been carefully chosen. CG: It ain’t the question. Don’t start rambling about something which is less important. The thing which we have to examine carefully is the revolution it brings to our thinking - to our work - to give equivalence its rightful place. This is what I have been trying to illustrate. I did so in two ways - one was to use the phrase ‘is another name for’, and the other was to ask what can be done with the same situation in a different light. Now I am not mincing my words and I am not saying something trivial. I am speaking of something tremendous. We have to cultivate equivalence. We do cultivate it subconsciously as

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mathematicians, but we have to make it explicit when we arm working with students - with students of any age, with teachers. It is because it is such an important thing that we need to spend some time on it. AR: I realized that myself in working this way. Whereas I am quite willing to look for an equivalent expression in language, I didn’t give myself that same liberty. That’s why I was so shocked to discover that as a matter of fact they were equivalent expressions on the board. I could decide to concentrate either on one side of it or the other. I felt the idea in this work as a liberation. PB: I’m not misinterpreting you, am I, in suggesting that what happened up there was that you expected Caleb to take the square root of one side? In fact, he took the square root of both sides. The squiggle (the square root sign) only appeared because he couldn’t do anything to that side. AR: What I’m speaking of is the squiggle I expected there. I now realize there could be a step in between and this is why I spent such a long time looking at the board. CG: Orally you can just string the things - you can just say them one after the other. I didn’t write all you said. AR: Yes - you certainly didn’t write one that I was expecting, and it was that one which forced me. CG:

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AR: No, I wanted something to say: A times question mark plus or minus alpha squared equals and then the square root of, and so on. You didn’t do it that way. CG:

I didn’t write it, but you did it.

AR: It was that which surprised me. I know that this is something I have difficulty in doing - at least in mathematics. Because when I work with teachers in school, when I take part in lessons with children who are 9 or 10 years old, I realize to what extent I am paralyzed in this field. I don’t have this flexibility. I was surprised to have to say that I don’t remember whether it was the answer. I don’t remember what it was in one of the exercises we did. The only thing I could say was that the answer had to be the square root of 6. It wasn’t. I couldn’t give you a number in the way I wanted. CG: Those of us who don’t know the content of your mind at once put this and the 6 inside it. Because that’s what we would do. Would you do that step? Would you do that? Well, then, why are you astonished by the squiggles? It’s a saucepan problem. The 6 has been replaced by something else. If you had it in your mind when it appeared, you could say: oh yeah, it’s an old friend. AR: It’s an old friend. But I wanted to do something to the thing that was inside. CG: When you left 6 by itself, you didn’t do anything to it. You didn’t touch 6 - you only touched the environment of 6. 163


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AR: But there is a prejudice in me that says it’s not solved until there is only a number. PB:

And if you can’t put a number?

AR: No, I recognize that when that’s forced. It’s not something that comes freely.

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CG: What is important in this seminar is not that we treat the whole of mathematics, taut that we realize that having treated a certain number of examples in terms of the awareness of the awareness, people who are interested can recast the rest. When I worked with Choquet, I would work with him on what I did with the awareness of the awareness at my level. He went back to teach his students and he said! I teach them in your way, I subordinate teaching to learning at the university. He was the only professor who was doing that. People who worked with him enjoyed what they were studying and became researchers. Perhaps the question I put to you is not one that can be handled here and now. It represents a revolution in the whole of mathematics. In my chapter in the Sillito memorial book, I showed that there is a whole field of study for mathematicians that they haven’t ever got into - the mathematics of language. The mathematics present in every living language hasn’t been tackled. But since there is equivalence in language already, there is an entry for mathematicians. When I talk with them, they say: do it, since you understand what it is I say. But I don’t have the

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equivalent knowledge. If I were to learn al1 the mathematics that exists and then found it wanting, what would I have achieved? I would have given my time to learn things per se, not for my problem. The problem- has to be the source of the challenge. Which are the structures that mathematicians can find in living languages? There are lots of them - I’ve found a few, some of them. The revolutionary notion of the importance of equivalence offers an opening for a new field for mathematics. Since physics, biology and so on, have been used as fields for creating new methods in mathematics, there could be some development which comes from language. JM: Would it be appropriate to ask a question? I’m thinking of a class which I have conducted several times - on projective geometry - in which, despite students’ protestations that they want to know how to do the assignment questions, I invite them to work on imagery. So that if they can simply enter the geometry of the question, it will solve itself. I’m fairly confident that a good number of them don’t go away and work on what has happened in the way that Alan was saying he had done when the pause took place. I think I can assert that, because when there are pauses, or when one person is being addressed and the rest of them are free to attend to whatever they wish, I don’t later on find the proficiency that you appear to get every time you do it. CG: The field is wide open - but has very few workers in it -to make each lesson an experiment in subordination of teaching to

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learning, via awareness of the awareness. Before I prepare myself to go into a class, I have to know how that topic can be put into these terms. If you have a class of projective geometry - which I don’t have, which you have - you must ask yourself: why does this topic belong to projective geometry? What is appropriate for this? I remember that Poncelet - in his prison in Odessa - was on his back. He had no paper, nothing. He could find all sorts of projective properties by projecting onto the ceiling. I have the same gift as Poncelet - only I don’t do it with projective geometry. I lie on my back, and the ceiling is enough for me to know what to work on. My lesson is prepared in a few minutes. I know that when I go back to my class I will give some exercises on imagery which will give them the theorem as their own discovery. They won’t be able to prove it, because that’s not compelling for them. It’s just in passing - they get something. I need to do another thing and that is to find the limiting case. They are shocked that there is a different aspect when it is a limit case. For example, I say the diameters of a circle are all hypocycloids with two cusps. Well, of course, it’s invisible until you put the diameter within the situation. If I start with this, they might say: what are you talking about? How could a diameter be a hypocycloid? Then I bring them to realize it’s what happens when a class becomes closed - is complete.

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JM: Taking up that point, I’ve found tremendous resistance when I’ve tried starting with the assertion that a straight line is a circle. My difficulty is that after twenty minutes working they still have the shock - they won’t buy it, even though I’ve gone through mental image sequences with them. CG: One shock I create in my classes is to put a dot on the board. Do you see this straight line? Where? Here - and I point my pencil and ask: what do you see when you shut one eye? A point. What about the rest? It’s behind the point. So there is a forcing of awareness that a straight line can be a point. And a forcing of awareness that a circle can be a straight line -by asking what will happen to the curvature of this circle when I increase the diameter, or push the centre outwards indefinitely. JM: What is the force that, through you, breaks down those biases with people you work with - but when I try, it doesn’t? AR: This crazy guy is trying to tell me that the straight line is a circle. Well, it isn’t. A circle is a circle, and a line’s a line.

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CG: I saw a film for a national test in the United States. It gave five figures and asked which was a trapezium. The figures were a general quadrilateral, a trapezium, a parallelogram, a rectangle and a rhombus. They asked which one was a trapezium. I could have answered which one was not a trapezium. But not which one was a trapezium, for there are four of them. When I was working on these things - about 40 years ago, in the schools of London - there was the same resistance. I was having lunch with some colleagues - one of them was a teacher of French at the University. I said: if I have two pounds, do I have one pound. She said: no, you have two. I said: but can’t I spend one? She said that was a different problem -I’d asked her if she had two pounds. I said: Well, you have two pounds, but I say I have also one. She disagreed: no, not at all. And it was an endless metaphysical discussion. CG: I had students who failed in three-dimensional geometry because they couldn’t get it from the book. I lapped it up - it was so simple for me. It wasn’t because of my brain. It was because I was a bad student at school; I was able to keep the connection between my two hemispheres intact. So my brain works as a whole - whereas many people are dominant on one side or the other. Well, I’m not dominant. So I had worked through some things which I realized weren’t present in the minds of my students. I had to create a new exercise. Perhaps you’ve read the study of three-dimensional geometry that I did with teachers, college students, and schoolchildren. I

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found that there was no difference between the ability of 11year-olds and honors-course students. It wasn’t part of the curriculum to make people think in three dimensions so they didn’t cultivate it. I tried to see how long it would take to give it them. It is an awareness that one can image in three dimensions. Of course, every geometrical figure on paper - or on the board - is two-dimensional. The problem is whether I can look at a two-dimensional figure and see it as threedimensional. Of course, I can - because I have two eyes. Since most people two eyes, they can do something. I ask them first to shut their eyes, and I give lots of exercises. They manage to tell me things - they are perpendicular to each other in three dimensions, they are equal in length, and so on. You don’t ‘see’ as vision, you ‘see’ as an ingredient of discourse I put it there, it’s that length, it goes from there to there, and this one goes to there. I tell you to take two points on the sides of a triangle. Then you find strangely that another person took two different sides. So you have to account now for his saying the same thing as you, although he has different points. You learn by proxy that this property can be shifted from one pair of sides to another pair. That is geometrical education - not that they know a theorem and they can recite a proof. DT: I wanted to ask John’s question again: how do I work with rigidities? CG: It is not easy to answer that, because rigidity is manifested in so many forms, in so many people. I can’t work on rigidity, it can’t be dissolved.

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DT: But you said something very useful to John when he asked that question. Then you immediately left it just as I was getting interested. The question re-surfaced in my mind because you have taught us over the years, in one way or another, to work with imagery. The problem lies in the difficulty of conveying what it’s all about except by example - as you did just now, when you were working with the virgins. The difficulties that people have about the thing is not I think the insight - the sort of thing that says: look, when I’ve got projective geometry classes, I must seek what it is that is in projective geometry, and what is it that’s in my students - or feeling into the life of the baby to ask the question: why am I not frightened when my mother gets smaller. The difficulty seems to be not in appreciating such insights, but rather in acquiring the techniques that are appropriate to teaching in these ways. CG: I don’t find that I have anything to offer that will satisfy. It’s too loose for me as a question. I may enter into this and then talk of something entirely different. I may not have experience that is valid for the context. I renew myself in every lesson. The only way of learning is to go into classrooms and learn what they are doing, how they are doing it. You see, when I came to the Institute, I’d not taught in any English school. I had no idea what an English textbook was like. When I looked at them - Durell and the other ones - they were disgusting. What could I do with these streams of exercises. So I started from scratch. When I returned to the Institute after a visit, and Daltry asked me what had happened that day, I would

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say I learned a lot. He asked whether they had learned something. I said: well, I don’t know - for the moment, you’re talking to me and I can tell you I learned a lot. He wanted to know how I would know what they learned or didn’t learn. I said: I will tell you next time - I will pay attention to that and I will bring some evidence to you - but if I learned a 1ot why are you complaining? Why do you change the question? I learned a lot and I can tell you what I learned. DH: I think what you said is very important. I’m also trying to tie other things that have been around the last few days and feeling that perhaps it’s really a question of making people get rid of things - to open their eyes. It’s no good dealing with it intellectually. There’s got to be a gut thing. I can’t see there’s something ever to be passed on. I feel that perhaps there’s something to be blown away. This can offer opportunity for people to open their eyes a bit more. CG: Had I been English, I wouldn’t have learned so much. I’ll tell you why. When I got into London classrooms, I didn’t have the inhibitions - the restraints - that all of you have. Once, there was a boy in the front row who could not see something in the figure. I asked him to come out. I turned him round, I held him and said: now look now and I pushed his head down. My students said: we never do these things in this country. The teacher, who heard from my students that I did that, said: we don’t want this man again. What’s that? You see, this boy got the answer at once. He had forgotten that when he was a little child he had separated his legs and had looked at the world through them to see how the world looked upside down. But in geometry, if the triangle is ABC, it can’t be PQR. If it’s PQR, it’s 172


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no longer ABC. So because I was a foreigner, I took steps. I didn’t have the same inhibitions. I could have had more inhibitions, being a foreigner, but as it happens I had fewer and therefore I could learn.

Culture And Civilization CG: During the previous weekend, we spoke of the possibility -because of the instrument of the psyche, the awareness of awareness - of tackling questions that had been prohibited before, such as the origins of species and similar matters concerning our prehistory. I also said that the scientific method which had been in use in physics and chemistry was not a priori. It does not necessarily apply to other fields. We can widely ignore the demands made on us that we should in all fields apply the so-called scientific method - observe, hypothesize, experiment, verify and so on. It’s also possible to start at a certain point in an inquiry and work on the models that you want to create. Out of the models you may find new consequences which are verifiable like those the scientific model requires. This allows you to say that there may be science or sciences which don’t operate in the same way. One of the questions which I have entertained for most of the last 40 years has been: is it possible to take mankind as a whole and consider how civilizations and cultures are generated? Why do cultures and civilizations disappear? We know that quite a number of brilliant civilizations have disappeared, even if the descendants of the ancient Egyptians are still in Port Said, and 173


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the ancient Greeks have become modern Greeks, and the ancient Romans are Italian. We can see that there may be a biological continuity, but no longer an adherence to ways of thinking - in particular, religions. The challenge I had doesn’t belong to mathematics, but mathematics must enter into all these things. Which comes first, civilization or culture? Do we know how to consider this item? By looking around, we know that the Spaniards, the Portuguese, the French, the Italians, the Germans, the English, Welsh, Scots and Irish, they all call themselves Christian. Then, within this unity, they call themselves Breton or Alsatian, Yorkshire or Lancashire, and so on. They put themselves into smaller boxes. Is it possible to relate a civilization to an awareness, and see how this awareness is affected by an environment in order to produce varieties within the civilization? I have found one answer, that satisfied me sufficiently for me to entertain it and work on it. I would say that civilization begins with one person. Civilization begins by one person becoming aware of one or more ways of being human and asserting it in action, thought, teaching and so on. From then on, you get its passage to other humans who adopt it. There is the Buddhist religion, which started 2,500 years ago with one man called Gautama Buddha. You can see Christianity starting with one man, Jesus Christ, who is still recognized as Buddha is recognized. You can start Judaism with Abraham. There are individuals who begin things. Islam with Mohammed, and Communism with Lenin - Marx and Lenin if you prefer.

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The world today is subdivided into a number of civilizations, which are things that relate individuals to the large cosmic manifestation in certain ways. (end of tape; the rest of this passage is missing - as is the final feedback from ML and AR who had to leave early to catch a plane) When we work on the history of mathematics for the purpose of expressing it in terms of awareness, we can recast it. We are not changing the history of mathematics, we are recasting those things that matter for the next generation. We can say the Greeks worked in this way, they contributed this to us - but they didn’t work in that way, and they didn’t have a contribution to make there. The Italians of the Renaissance, who followed the Arabs, wanted to work on equations. Poor things, they wasted so much time solving particular equations and then they produced a theory that is so slim - though they have fat books with lots of things in them. What remains? Only for a few specialist mathematics does this type of thing exist. That’s why there are fashions, Because there are experiments. People get into these experiments and they find what they find, If some of it stays, it is useful. You don’t have to teach linear equations and then equations of the second degree and so on. This is a false view of the impact of history on mathematics. The inheritance is what matters, not the history. It’s what contributes to our evolution.

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We have to stress that, because there are so many people writing so many papers nowadays. Are we going to learn a hundred thousand papers before we can continue. It’s absurd -that’s not the way to do research. You don’t have to assimilate literature before you take the next step. You have to come with your sensitivity and your sense of a problem that you want to work on, and what is relevant to it. I spent a long time in my youth studying treatises that were absolutely irrelevant to my education. But they were part of the curriculum, they were on the shelves. I did lots of exercises and things which will only go to dust when I die. Some stays in my brain and may come back with that ... that’s a good point to pause for those of you who have to leave. Well can we have a little feedback on all this? CH:

I would just like to say that 2+2 is equivalent to 4.

CG:

But not in all systems of numeration.

CH:

I was in base ten.

JM: I found the switch to the history, a change of gear. Trying to account for civilization and culture is not for me a question at the moment, although I’m aware there is a question. The parallel with the notion of evolution that was developing on Sunday is attractive. But my attention at the moment still wants to come back to questions of being in classrooms. CG:

You can’t, because we’re still in time. 176


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JM: I’m trying to reconstruct some of the remarks at the end. There were one or two sentences about significance and fashion. They seemed so completely appropriate that I am sure I can reconstruct them when the time comes. DP: Can I offer a couple of things? Right at the end, I found a possible link with the first half of this afternoon. One of the remarks you made was about there being biological continuity, but not necessarily cultural continuity. One thing you were referring to - certainly one that struck me - was us sifting through the mathematics of the past, looking at things that are of significance to us. Yet it’s the inheritance that matters, not the history of mathematics - mathematics starts again and again. The image of mathematics forever restarting prompted a question - it’s the question of cultural continuity. Why are our concerns now in mathematics what they are, and how do they relate to what they were? I am getting back to the question with which I started yesterday morning. It’s the question of restarting mathematics anew or afresh, with particular problems. Yet at the same time - having had a mathematical education which has been influenced by past awarenesses -looking into past mathematics for things that can help us with our current problems. I find myself sliding between those two. DT: I have also been trying to slide between those two. I thought the change of view this afternoon was very helpful. It is easy to respond to the notion - the vision - that there is a possibility that the differences that one can see in history could be flipped away. That one can in the same day be talking

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meaningfully about mathematical shifts - from awareness of actual historical record and awareness of inheritance – and then be able to speak of that as the metaphor for teaching Chinese and French children, seems very powerful. CG:

I thought words went up - they seem to go down.

DH: My mind is very much in the classroom. My mind is very much with neutrality, it is also very much with common sense. I know that today I am actually itching to start teaching again, because I want to explore. I am looking forward to Monday. DC: I feel exactly the same thing. It started yesterday afternoon. It’s very strange - that’s never happened before -the first day of the holiday. CG:

Wait until Monday - today’s only Tuesday.

DC: How far these truths - these strange neutral truths - go in terms of development. I have been very aware in the last day and a half of the way you’ve drawn this line between that which appears to have relevance - no, that’s not the right word - and that which touches us, and can touch everyone. As with equivalence, or with the inversions that were just drawn out suddenly. There’s obviously some point where you say: but after that, that’s for the mathematicians, it’s for the small group of mathematicians who want to have big books full of things. What happens to everyone else, what do they do?

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CG: Those who want to be mathematicians join the group. Those who want to be skiers join another group. Those who want to be painters join another group. Who am I to tell them they should all become professional mathematicians? DC: No, you’re not. But it does seem to be that if you’re tapping these things . . . CG:

Are you tapping in singing or writing, as well?

DC: Right; I’m a bit overwhelmed by this - it’s all starting to get a bit big. CG: It’s not too big, because besides all these things in one life which is mine, there are are lots of other things. Therefore it’s not so much. It can’t be so much just as I say languages can’t be difficult if babies learn it. It must be that you are doing the wrong things, that they are doing the right things. Let’s all do the right things and see what happens. There are right things. I gave seminars called doing the right things. There are right things to do. DH:

Whatever the titles of your seminars ... (laughter)

CG:

Why?

DH: I always feel I’m dealing with the same things. It’s because they’re very powerful things - and they are about wherever I look. So, in a sense, I just say that the title for me is

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looking at these powerful things in a different way. But I am still dealing with the same things. CG: I once had an MA student at London University. He was the head of a school in Isleworth. He wanted to write a thesis and had mistakenly chosen me as his supervisor. He would come to me and talk about what he intended to do. We would work for an hour, say, and he would leave with lots of copious notes. Next time he came, I would talk of something entirely different. After three such visits, he went to Vernon and said: it’s impossible to work with him. I had the reputation of being so poor at leading people to their MA that they didn’t send me any more. It was a nice thing for me, because I had more time available. How could this person come to work on something and expect that when he comes again I will repeat myself. I have said already what I said lado I need to tell you the same thing again? That’s impossible. DC: But this goes back to what Dave said which is that he wants to go and do something. I want to go and do something now. CG: Doing consumes time. When you take your doing step by step, in the end you have only made a small journey. But inwardly you may have made a big one. You’ll only have the evidence that you can get from the little thing. You asked where does the truth go. I am going to tell you what I’ve said so many times. My father taught me that truth has very short legs and goes very slowly.

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Well, if you can’t join in, I can only throw more at you. Is that what you want? Because I have finished with this. I have something else to do, and another thing and another thing and another thing. Is that what you want - to be drowned? DH:

I drowned a long time ago

CG:

So, please yourselves, I am here with you.

DH:

What actually does worry me is that I’m sitting here.

CG:

Doing nothing.

DH: Yes, that’s right. I expressed this to David as I was taking him to the station. What worries me is that I’m just sitting here. If something you say strikes a chord with me - it seems true then why aren’t I working on that and seeing where that leads me. At the moment, I notice I’m actually not doing that. I find that disturbing. I’ve noticed this morning that I haven’t taken the reins in my own hands. I’m reminded of that by what you just said. DC: That implies, somehow, that there is something wrong with agreeing. CG: Well, don’t make it a cafe discussion. He started saying what he felt. On another occasion he will say something else. What he suggested to me was an exercise to give you all, in which you will recognize the work of awareness of the

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awareness, the production of mathematical things, and at the same time see that it begins by being elementary. It doesn’t remain elementary - because you work on it. Mathematical imagination tells me how to start with a very simple problem - an extremely simple problem. Then - suddenly -it becomes a world. I can take it a certain way; after that I am incompetent or lazy. If I wanted to go on with it, I would need to stop all the rest of my work, go into the library and read lots of books. I never did that - so I have been accused of being only a starter. Perhaps that’s what I am - I can only begin things. Although I work out all the details of the things I do, I stop somewhere and I leave it undone. There’s plenty of work to do for anyone who is in sympathy with this way of working. I will take an exercise - and make a children’s story out of it. You have a bucket in front of you. You observe what comes into it with the rain. What do you get with the rain? Chaos. You get all sorts of thing and you can’t handle that. Well, the rain has stopped. You put the bucket under a tap and you open the tap. You get one drop at a time. There is something to do there. You can see something and you can describe it. A circle is generated that moves away towards the walls. When it meets the walls it doesn’t come back as a circle. During that time there have been other drops that fell. They produced a system of circles - to begin with they look like that. So you can stop in time, and you can ask what sort of thing you have there. You have generated a family of concentric circles. But if you put the wall in, the mathematics becomes incomparably more difficult,

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because now you have reflections and the interference of the two systems. You are going to abandon that, because the instruments to work on that are not available. You take another example. Now there are two taps dripping at the same time. What will each do? What they were doing before? But when the two are together, there is going to be a system of interference. Now I have new problems. I can talk about these. And I can go to the board and do the same thing with compasses. I no longer have a disturbance, no longer any water, no longer any of these things that are going to distract me, and take me into difficulties. What I am going to work with is a stationary system of interfering circles. Do I see the conies that are there? They are all there? Do I see them? What do I need to do to see them? Where are all the intersection points of two equal circles that are big enough to intersect? I obtain what the Greeks knew -the perpendicular bisector. Now is there only this one perpendicular? Can’t I force myself to say: I am going to take the circles, pair them up so that the sum of the radii is constant? Where are all the intersections now? Or when I take the difference of the radii? Or the product of the radii? You see, I started with so little. Of course, it is making them aware that they can work on their eyes, they can work on their mind.

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But I can start the whole thing again, taking lines from the centre, rays from the centre. What happens when I have rays instead of circles. The radii that I thought of before are now objectified; I can say: when these rays go round and they intersect, I can see that I can choose the points where the lines intersect. I generate another language for the same reality. It may not be historically important, it may not be preparing for any exam, but it gives you the sense that you can use yourself. You can use yourself fruitfully, you can synthesize, you can get Cassini ovals at the same time as you have conies, and quite a number of other things that are there. PB:

If your two taps are dripping at different speeds?

CG:

You will get into something else.

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CH: I felt like the others who said they wanted to go out and be in a classroom. It would be another way of expressing what I’ve gained this weekend. The last few days have put a spotlight on for me - it’s a way of expressing that, other than with words, which I don’t always find easy, especially in this sort of context where I am trying to talk to you and forget there is anyone else here. CG: Why do you want them not to be here? You have some social investment in the community. CH: One of the things I work on is getting rid of certain attachments, not just of material things, but other things. CG: There is no greater ownership than that of an idea. Giordano Bruno is an example - he went to the stake because he believed in Copernicus. Idiot! Why should you be killed to prove a truth that can be proved neutrally - without any passion? (end of tape; transition to description of some work with fractions)

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A LESSON ON FRACTIONS CG: How many thirds in 1? That was a harder question. One said 3. Why 3? Because one third means that there are 3. Then I changed numbers. How many sevenths in 1? After we did two or three of these here, I said: be careful, I am going to ask you one over - but I am not going to tell you what I have in my mind, I’ll call it a - how many one-a-ths in 1. At once, a. How many one-bths? This they liked very much - how many one-b-ths. Then I said: I’m going to change language, I’m going to speak Greek to you. How many one-lambda in 1? I am going to change language again, I am going to say it in Hindi. How many onemers in 1? So it was fun - it was play. Having done this, I told them that I was going to keep to one-a-th, because it was in all of these. What do you prefer? Shall I keep this, or shall I give you a larger number? They were unanimous: this is much easier to remember, much easier to hold in the head. Even if I say a is a billion? You can make it as long, or as small, as you want. We spent a very short time on this. I asked the same question all the time - how many in 1 - that was all. But now I’m going to play the game differently. Listen to me, boys and girls. How many one-thirds in 2? You know the answer -be careful - how many one-thirds in 2? How many one-thirds in 5? You already know the answer. Now - in ? What’s the answer to that? If I cover this, what will you say? What did I ask? How many one-thirds in FIVE-sevenths - I make a difference in the two sounds. So what do you say? Then we can change to this. How many one-a-ths in b c-ths.

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Allright - that was fun all the time. Before I go into the final step for the division of a fraction by a fraction, which is slightly more delicate, I say: now, since you are so good, why shouldn’t I teach you a new language? Instead of asking ‘how many in 1?’, I am going to say: divide 1 by . But don’t solve that problem change it back to the one you know. What we are doing is linguistic translation. I give you a problem - you solve an equivalent one, with which you are familiar: how many in 1? This makes sense whereas the other doesn’t. Having established in only a few minutes that this can be done, we obtain the next frame - how to divide by . How would you work this out? You see - the saucepan appears again and again and again instead of rules. How do you go from this to this? Make c to be 1. It’s more difficult - but it’s an opportunity to tell them that division by 1 has another notation. Another way of saying ‘divide by 1’ is to say it as it is. I can have this in the three notations. One way of saying it is ‘ ab ‘ - another way is ‘a times b’, another way is ‘a dot b’. The equals sign is not necessary, because it’s language - all language. b÷

~ab~a × b~a.b

In a lesson, you meet a few paints that you refine. They take two or three minutes, but you are sure that by doing this, you are sweeping away lots of cobwebs that stay in so many minds. Now that I have this, I ask the final question - how many in 1? That’s another type of difficulty. The 2 is going to diminish the answer - there are fewer two-thirds in 1 than one-thirds. This requires a discussion in common sense, not in mathematics. How many quarters in 1? Why are there more quarters than thirds? Why are there more thirds than halves? 187


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I make them sense a physical interpretation. They have to sense that if the things get bigger, there are fewer of them in the same quantity. If is this, will be that. It is not a matter of triggering them to say half. The awareness that you are insisting upon is that they see in their mind that this is - and here there are . (drawing on the board) Such a relationship - that the third becomes half - has an importance in itself. With the rods, we show this by taking three rods and showing that the same rods change their name constantly.

Here this one (shaded in the above diagram) is of the whole. But we are not comparing it with the whole. We are comparing it with the other two. So it has to be of them. And the whole will be one-and-a-half times them. If I only give the reversal of the fraction as an observation, they will not be connected with the reality behind this - they will learn it by rote. I can extend this one to be the other one - by working with this sense that if you increase the element that goes into the unit, then you must have fewer. So this will be one here, and one-and-a-half here. PB: In doing this with children I’ve oscillated between ‘ 1’ and ‘ into 2’ to start with.

into

CG: The only thing is that you give enough time. Until they can talk about it and realize they are saying what they

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understand, not what you said. Since I don’t say anything - but only ask questions - whatever you hear comes from them. So you have to grant that they are working in the proper manner. PB:

You would have no preference for one or the other?

CG: If I had some doubt, I would make an experiment. The first time this appeared for me, was in a class with a boy that used to jump off one end of the table. They were all like him they all refused systematically to do any mathematics. But when I worked with this, it was beautiful. It was the first time I saw it myself - this refinement that you don’t ask for 1 divided by until you make it a linguistic thing. To solve a problem you always refer to what you know - the saucepan. DP: I’m asking myself about contacting the reality about which this language speaks. It’s a hypothetical question because I do not have a class of children in front of me. It’s the difference between dividing fractions, and being able to talk about dividing fractions. CG: There is no division of fractions. It ends up by being something which you mathematise - you call it division of fractions, because there is a division sign and on both sides there are fractions. But you are not dividing fractions - you are learning to divide fractions by playing my game. JD: How did they deal with your initial question: how many in 1?

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CG: Where are we if I can’t grant in 10- or 11-year—olds that they know how many halves there are in 1? DH: I take from that that there’s this doubling thing again, that kids double. Half of 1 is only that. CG: That’s right - and it’s the name you see. If I’m twice you, you are half me. If they don’t know that I will introduce it. When I started, I did not know what they were going to say. When they said it unanimously, I could not deny that they said it, that they know that there are 2 halves in 1. Are there 2 halves in 1? It is a definition of half that there are 2 of them. PB: Whether this is a useful activity or not, it has been a pleasant game to play. CG:

It was a solution of a pedagogical problem.

PB: Whether you can use it later on doesn’t really matter, does it? The use of division of fractions is another question. CG: I’m not concerned with that. I teach mathematics, not the application of mathematics. If it’s not needed it shouldn’t be in the curriculum. If it is in the curriculum, here is how we do it. I didn’t create the curriculum in Bristol - or anywhere else. PB: But who cares about the curriculum - if it is fun to do this, let’s do it.

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CG: Masturbation is fun for them in puberty. I don’t ask them to do it. (laughter) I expected - perhaps wrongly - that you were going to give a professional post-mortem of the lesson. That you were going to say that we had taken care of all the elements that are in your sensitivity, your vision and in your experience - that by handling it in this way, we have eliminated obstacles that are generated by I don’t know what -that there were no difficulties, because when it is language it’s language, and when it is experience it’s experience - and so on. JM: I can’t be absolutely sure that if I tried that in a classroom, exactly as you did it, that I would get all the responses you got, and that I wouldn’t come a cropper at or or wherever. But the program, as you outlined it, copes with all the blockages that I am aware of In the facility of doing computations and making scribblings on pieces of paper in order to satisfy examiners. CG: I may have a group of students who know that you reverse and multiply. They may be teachers - they all know reverse and multiply, but they don’t know why. DH: For me, that was purely a language game. There was no sense of a mathematical feeling - that that’s right or wrong. I was enticed into saying 1 - nothing more than that. CG:

Thaty4 right.

DH: Well in that case, why don’t I say: turn upside down and multiply.

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CG: I will tell you a story. I was once put into a conference program by Polya, who wanted to show me off. There were hundreds of Californian teachers at this conference. He asked me to give a lesson. They brought me five or six children. There was a 13-year-old boy who showed me that he could say that if you add 2 and 3 you get 5. When I asked him to add 200 and 300, he would collapse. Well, I forgot Polya and I forgot everybody else. I worked with him on his own. I said: now listen to me, you are going to work with what you know. Do you know what 2 plus 3 makes? Allright, now listen to me: two pears and three pears. Then after a few other fruits he seemed to know perfectly well, so I said: two million and three million -what is it? Five million. He had never realized that it’s a noise. He thought million was a trap, that million was something you had to know inside out. Then he worked through it all. When he finished, everyone was disgusted with me - except for one woman who said to me: it was beautiful to see this boy blossom. Polya was so nervous: you had a chance to make a place for yourself among the American educators, and you made that mess! From Polya, that was a little too much. DH:

I feel there is something different here.

CG: You can go on thinking of number - but it is also a noise. It’s a number in a certain context. But when I add two hundred and three hundred, it is not a number that I add, it’s two and three - hundred. By the distributive law, it is 2+3 hundred.

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DT: It is interesting that the professionals have such difficulty about the issues that you are raising. Because Alan spoke very clearly of what he thought about it. I don’t think the triggering of the action here is anything like that of ‘turn upside down and multiply’. There is an internal logic - an internal mathematics which I suppose is the mathematics of the language. I would be delighted to be able to say that I had kids who were operational with the question: how many in . Then I could do all sorts of things with them. That seems to be quite different than giving them a rule and having them make noises. The whole of that notion is such a startling idea to us professionally. We’ve taken it on board, yet we are very hooked up on understanding and meaning and reality. We could actually begin to look at language, going to a lower level first - to what can be done. CG: I haven’t heard from you - or from anybody for that matter - that I have taken some liberties with the mathematics because I’m a teacher. I did not want them to be involved in 1 divided by . I said when you have a question like this, I change it into another that makes sense. Only when I am sure that all the background is there will anyone know that when I work the division I work it through the understanding I have of how many halves in 1. Then I can say it’s equivalent. It’s a different writing - it’s not the same. That’s why we have this second kind of equivalence coming into this. It’s another level. There is one kind of equivalence - of things like 1 ~ 2 – 1 ~ 3 - 2 and so on. Then there is this other,

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which is that there are two ways of saying things. French and German are equivalent, they are not the same they differ by agreement - this is German, this French. This is the writing and reading of a certain kind, and this is the writing and reading of another kind, (comparing equivalent arithmetical expressions on the board) We are saying the same thing once we know what we are saying. PB: When children have done this by rule - ‘turn upside down and multiply’ - you would never be able to ask the children about the division of fractions. It’s the sort of rule that was invented by someone who never understood what they were doing. When you ask children who have done it - a over b divided by c over d - they’ll tell you straight away they won’t fiddle about, turning things upside down and rubbish like that. They go straight to the heart of the matter, they understand what they’ve been doing. CG: There’s no point in giving the answer. The answer has been given to children as the most important part of their studies. Their understanding is more important; when there is understanding, there will be an answer. If there is an answer, there wouldn’t necessarily be understanding. So I’m not concerned with ab over c. But I’m concerned with ‘another way of saying’. When I have done it all I will then make an observation: oh, you’ve crossed and multiplied. So what will you do next time? You will cross and multiply like everybody else. You found a short way of handling the problem. How many times goes into . Does it mean anything? You can’t do it, you can’t work on it. But you may make a noise at the end of it.

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DBD: I found myself doing that sum in my head not by rule. You tickled my awareness by going through those? things. There was no routine about it - I’ve not been so clear before how the teacher works on awareness. CG: The treatment was given simply as an example of how you can take a complicated chapter that teachers dread teaching and make it into fun. Some things are linguistic, some are notational, and some are physical. You have certified your lesson in terms of which awareness is going to be called on at each moment. JM: Even though I have used the language pattern for hundreds and thousands and millions, I have been making the mistake of equating awareness with that which lies behind. If somebody had asked me about the awarenesses connected with fractions, I would have thought in terms of fractional parts, of pictures or whatever, and not thought of the semantic level. CG: But in your own thinking, and in your own use, you use fractions in two meanings - you use them as operators. JM: No - here I’m not sure I’m going to buy it as an operator. At the moment, it’s just part of the linguistic pattern. CG: But of course it’s an operator. Half of 1 - you operate on 1 with the half.

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JM: Yes but you are inviting me to see the stroke as merely a chalk mark. I wasn’t being invited to conjure up all sorts of references to rods and so on. CG:

There are no rods in what I am doing.

JM:

Not quite in the one-third of five-sevenths.

CG: When I say I am twice as heavy as Jo - which is not true, but I say it now - twice is an operator. It is not the numeral adjective, two. JM: But when you were doing it on the board, I didn’t feel half was being used as an operator. CG: I did not take apples - there were no objects ... (various interjections) When I wrote ‘2 × is another name for 1’, 2 is an operator that applies to something that is not an operator. When I read it in Arabic it will be × 2, and now 2 is the object and is the operator. So halves and twos become operators. Moreover, the cross is another operator, and I have two operators working together. So you have different interpretations - different lights - which enriches this. But it doesn’t mean that it is not there - either as an operator or as an object - when you need it. I can always work on an algebra of operators. But I can also work on an algebra of operators that act on a set of objects. I can do both things. According to the moment, I will draw attention to which is the sensitivity that is at work. But you can shift from one to the other. Because you can start from the left or start from the right. You have this attribute that you can change - you are not tied to the nature of

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the object. We should turn out the light there, and turn it on here.

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CG: Let’s start to pull threads together so that we can say what has been our experience. And I would like some precision. I ask this in order to know if the time we’ve spent together represents some shifting from me to others, so that it can be shifted from others to others, so that it can serve the community at large. I don’t want it to be an investment on my part. It is simply that I am an elder in the community of teachers of mathematics. I meet juniors - and half-juniors or twice juniors among the people I work with. I want to know if the oral tradition of this type is passed on from the elders to the next generation. JM: I for one can’t speak about what is passed on. Because that requires that I have the perspective of knowing both sides. I don’t feel that I want to rehearse the various things that I want to think about. And I want to avoid the language in which I say there were three or four things I want to work on, because I get tired of hearing that. For me, that is just fancy talk. So I am not sure that I can met your request. One thing that I’m pretty sure of is that if you were present suddenly at some event that I was

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participating in, or if you looked at something I was involved in writing or co-writing, you might not recognize what was there. But that doesn’t follow that there hasn’t been some influence. CG:

Why shouldn’t I recognize it, if there is some?

JM: That is up to you - that is your problem. I’m not in the game of trying to preserve a tradition. I am trying to understand for myself. CG: Which tradition is the one that is on the side of the rebels? Which tradition represents a revolution? JM:

Every tradition represents a revolution at some time.

CG:

But does not represent a revolution that is in the making.

JM: I don’t see the task of mathematics - or my task in mathematics education - as being one of revolution, of fighting. CG:

But it’s needed.

JM: Perhaps. But I don’t see it through that metaphor. It’s a metaphor which invokes for me unusefu.1 tensions and conflict. For me, the task is to understand. That’s my task. And, when possible, to express that understanding. In some sense - but in a smaller version of what you’re doing - there may be one or two people who would themselves come to an understanding, through that expression. That’s as much as I could hope for.

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DH: There is a lot of things I want to say. One thing that has been very clear to me today - which I’ve mentioned before – is the direct challenge to go ahead and work further myself. I remember being asked once why - if I am involved in my own personal work - why I was not completely autonomous, why did I bother coming to a seminar with you. I said that it was like sitting in a room where someone comes in and switches on television. I keep on looking up at the television. After half an hour, someone comes along and switches the television off and I am very annoyed. I feel that sometimes in my day-to-day working within school. Without realizing it, I’m letting a lot of fog come in. I’m getting involved in that fog and it actually starts becoming important to me. I just need someone to come along and remind me that it is just fog. I think that has been done a bit today. So what’s nice is that I feel that I’m perhaps clicking back into work. As for the revolution, well, I’ve had one over the last five years. Yes, I think a revolution is needed. But, first and foremost, I’ve got to make it happen in my classroom. I will also be involved with in-service as well, on the way. But I’m finding - particularly over the last year - a very direct challenge to me to make that revolution happen in my classroom. If I can’t make that happen in my classroom, then I don’t know what I’m doing. But I feel a revolution is very much required, I want to take on that challenge - I know that today has helped me work further on that. PB: I’m sure that the revolution is necessary. But I feel that revolution is perhaps the wrong word. A lot of teachers feel they cannot do a lot of things they would like to do - because of 201


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parents, examinations, school organization, and so on. Although some of those things don’t need to bother us very much, some of them must to some extent. I don’t think that this revolution causes any difficulty with examinations, because although they maybe quite useless and an awful waste of time, if the children have understood mathematics, they’ll eat the examinations. So we don’t have to bother about that. And we shan’t be bothered about covering the curriculum. But it will shrink to be only a small part of what we are involved in. In that sense, I don’t think the revolution is a necessary description. It doesn’t have to destroy the shibboleths that other people hold on to. They can keep them, until they don’t want them any more. JD: I’ve been asking myself what will be helpful to me in the questions I address in the classroom. I have a feeling of scrabbling around like a mouse too much in that. Because the moments that are most vivid are moments when you’ve been quite startling in the way that you have looked at a particularquestion. That seems to be a task - to learn how to be startling in some way. Maybe you express that when you talked about looking at the fractions and recognizing the awarenesses that needed to be forced. The problem is that I find it easy when I watch you do it - but very very hard when I try to do it myself. JW: I share Joy’s feeling of being startled at times. When I saw a film of you working with a little girl at the board, I thought, you were quite forceful - and I was quite shocked. I was quite shocked this morning when you were working with Alan you pushed so hard. But what I was reading as being difficult wasn’t difficult at all. That’s quite a surprise.

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CG: A young boy was sent to me once by a clinic in New York. His tutor came with him to watch me work with the boy. While I was working - for an hour or more - she sat beside us. Afterwards, she said: why did I bring you here - he is destroying all the work we have been doing for six months, he is doing it in a few minutes? As she took the boy down, he said: this man loves me and understands me. That for her was a shock. She called at once when she got home to say she had to tell me she had been in despair all through the lesson. But this boy had found just the opposite. A few months later, he was top of his class in a district full of people who are aggressively competitive. He was the son of a famous theatre manager, and the grandson of a famous television star. He had shrunk and had become smaller and smaller. He was treated as someone who was the son and the grandson of these people. There he was in front of someone who trusted him. In fact, I had given him such a simple exercise. Instead of requiring him to be with me and my voice and my gesture - all these things that made her nervous - I had asked him: tell me what twice two was. What is twice two hundred? Twice two thousand? Then I put these together. What is twice two thousand and two? What is twice two million two thousand two hundred and two? I kept him within such a small thing. Everything he said was right and acceptable. He became ten feet tall. I knew that all he was doing was being with the challenge - sorting it out, taking the next one, knowing it is harder but not much harder. He ended up doing it with 8, which meant that he had to say 16, and so on.

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The appearance is not the reality, my dear. From now on, when the man shakes you, that’s the appearance not the reality. Do you understand? JW:

I didn’t understand the last bit.

CG: Most people find me too hard on teachers, and too lenient on students. Well, I am truly with the children. They have the right to all their mistakes. And the teachers don’t have- the right to their nonsense. Therefore, I object if they come to me to be handled gently. It doesn’t mean I will remember the next day, whether I did anything with them. Because it was all acting, it was all proper for the moment, and only for the moment. There is so much talk of the psychological background of the learners, that we only think of them as handicapped, as needing help. We are all terribly resilient and we have a lot of power. Why don’t we work with this aspect of people instead. I would like to do that with teachers but they are on the defensive and I can only get somewhere by being brutal - going round and clubbing them one after the other - virtually, of course. DH: I am struck by the personal energy need to go in and do a day’s intensive teaching on this sort of level. CG: The difference between us is that I have a small quantum of energy - but in reserve is the whole cosmos. I will never be short of energy with the very small quantum - I can’t, wear it out, it’s too small for anyone to find. Why is it at the end of a certain number of days of working one is less tired than when 204


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one started? There must be a way of going to the store of energy that is there for one to use oneself. DH: Yes, that’s right. After several weeks I notice 1 have low energy, something is going wrong somewhere, I notice that over a period of time something begins to go. Maybe it’s this idea that I need a holiday - I get that so firmly in my mind that I actually convince myself I do need a holiday. CG:

My last holiday was in ‘57.

DH:

I hope you enjoyed this nice one in Bristol.

CG: I found that if my work is my rest I don’t need to go out. I work several days a week - sometimes twenty hours a day. I don’t feel the fatigue that young men like you do. They give me energy, the ones I work with, and they have energy. DH: I think it could have something to do with the fact that I still have some investments around CG: Yes, drop them - drop them and you will see the difference. The French have a saying: the most beautiful woman on earth can’t give more than she’s got. (end of tape) DP:

I think I will decline to say something.

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CG: Decline? I don’t mind criticism. I thrive on opposition. What you said yesterday about Newton was the most welcome moment of the whole seminar for me. Because then I could see something I hadn’t formulated before - that T could tell you why I have a sense that 1 can speak on behalf of someone else. 1 wouldn’t have said that had you not spoken. CH: I look forward to teaching again - the guiding light for that is to be sure to seek truth in what I do — to shrug saying I have to do this thing because the external situation forces me to. That idea was mentioned earlier - I do catch myself doing that. MH: I don’t feel I have much to say. I wanted to let you know that I feel I was very caringly treated by you in the mathematics seminar. To my astonishment, it has been illuminating. CG:

Can we close our meeting? (to DT)

DT: I have had (great difficulty with feedback today. I am not. sure what I can do to close. But I would like to make a personal statement of where I’ve been for four days. It has been like floating on a magic: carpet. I’ve been moved and I’ve been very stirred - specifically this afternoon. It seems it doesn’t matter whether one is talking about the four realms, or about the history of mathematics - or about teaching, civilization, languages or religion - it’s as if they can all be woven on the same thread. I was invited to shift from one thread to the next - with, I might say, enormous resistance at various times. It was very remarkable for me when - in a seminar on the history 206


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of mathematics - you offered the suggestion that we have? to transcend nationalities. Of course we need a revolution. I don’t mean that in a dramatic sense. I mean it in the sense that the vigilance - the watchfulness - that you are asking for throughout - seems to be something that one has to work at in a revolutionary way. I feel that I have to learn to be able to work with nothing - to be able to take it on a lower key than I am tempted into sometimes. There does seem to be an enormous amount of matter-of-factness available when one gets rid of distraction. I appreciated something Cos said about attachment and it seemed as if there were one or two occasions when as a result of quite startling shifts, I was temporarily detached from my attachments and could feel just, a little glimmer of the simplicity of what there is to be done. In that sense, it seems to me, there is a need for revolution. I don’t want to put emotional stress on this, but I know that continually being with that simplicity is very difficult, and involves for me a continual attempt to turn around. You raised the question of the oral tradition and I am intrigued by something to do with charging - the charging that is associated with neutrality that you have given examples of this weekend. It is often difficult to capture that. My last thought is about how your writings come alive in your seminars. I hope there will be more seminars - I think this has been a very remarkable one.

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CG: May I add that what is on the board has shown that you can teach multiplication and division of fractions before addition. You don’t, have a hierarchy. Thank you for your attention.

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ARCHIMEDES (Greek, C3 BC): the greatest mathematician of classical times. He is mentioned here for the story of his discovery about the displacement of a body in water. His cry ‘eureka’ - now commonly refers to the sudden intuitive solution of a problem. BELL, Eric Temple (Canadian, C20): wrote several popular books on the history of mathematics. His biographies have been criticized for their unsupported anecdotes and prejudices, but his emphasis on the human personalities involved in mathematics was refreshingly unusual. BERNOULLI, James (Swiss, 1454-1705): one of a famous family of mathematicians, developed the calculus and used it to solve various outstanding problems. The differential equation referred to in the seminar is of the form y’ = P (x), y + θ (x), yA

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BOURBAKI, Nicholas (French, C20): the pseudonym - after a 19th century French general - of the group of mathematicians who worked collectively on a series of books that organized the various branches of pure mathematics in tens of basic structures. Various members of Bourbaki were involved in postwar reforms in mathematics education. BROUWER, Luitzen E J (Dutch, C20): criticized the formalist and logicist foundations of mathematics and developed an alternative approach known as intuitionist. This admitted the possibility of the so-called excluded middle of traditional logic; ‘reductio ad absurdua’ proofs, such as those involved in infinite set theory, were therefore suspect. CANTOR, Georg (German, 1843-1918): developed the theory of transfinite numbers, showing that infinite sets were not all of the sate ‘size’, The rationales are equinumerous with the integers in the sense that these two infinite sets can be put into one-one correspondence. They are then said to be denumerable. But the irrationals are not - they have to be represented by a ‘higher’ infinity. The example given in the seminar shows how a denumerable set of denumerable sets is denumerable. CASSINI, Domenico (Italian, 1425-1712): eventually worked in Paris where he became Astronomer Royal. He introduced a type of quartic curve called a Cassini oval, defined as the locus of a point such that the product of its distances from two fixed points is constant. CASTELNUOVO, Guido (Italian, 1845-1952).

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CAUCHY, Augustin-Louis (French, 17B9-1857): laid the foundations of modern analysis, in particular the theory of complex functions. He was a prolific author with tore than seven hundred published papers. CAYLEY, Arthur (English, 1821-1895): developed various branches of algebra, including the theory of matrices. A fundamental theorem is now jointly named after hit and his Irish contemporary, Hamilton; this states that a matrix satisfies its own characteristic equation, Cayley verified this in the case of 2 Ă— 2 matrices and seems then to have assumed that it was true in general. CHOQUET, Gustave (French, C20): member of the Bourbaki group. He worked for many years with a group of mathematics educators (the International Commission for the Study and Improvement of Mathematics) which Gattegno and others had started in 1952. COPERNICUS, Nicholas (Polish, 1473-1543): famous astronomer who challenged the orthodox view that the solar system revolved round a fixed earth. He still thought that the planets moved in circles, but asserted that they did so round the fixed sun. DALTRY, Cyril (English, C20): lecturer in mathematics education at the London University Institute of Education, noted for his enthusiastic admiration for the work of one of his predecessors, Percy Nunn. Gattegno was an awkward colleague at the Institute in the 50s.

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DARBOUX, Gaston (French, 1842-1717): author of a number of influential textbooks. DEDEKIND, Richard (German, 1831-1916)! a careful and creative thinker who developed a rigorous approach to number. He was a friend of Cantor and shared in the creation of the theory of infinite sets. DESCARTES, Rene (French, 1596-1650)! famous French philosopher, noted mathematically for his algebraic representation of geometry. His work leant that numbers could be represented by lines, and vice versa. Thus the awkward ‘irrationals’ that had been previously treated as geometrical magnitudes could now be allocated a ‘number’, DIEUDONNE, Jean (French, C20): member of the Bourbaki group. Influential speaker at the 1961 Royaumont conference, where he expressed his view that the mathematics curriculum should be thoroughly modernized - with the slogan ‘a bas les triangles’. DURELL, C. V. (English, C20): well-known schoolmaster and textbook writer. His Arithmetic for schools was almost universally used in schools up to the late fifties. ENRIQUES, Federico (Italian, 1871-1946). ERATOSTHENES (Greek, C3 BC): Alexandrian scholar famous for his calculation of the length of the circumference of the earth. Known also for the process by which primes are ‘sieved’ 212


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out of the tale of integers by successive deletion of multiples of numbers retained in the ‘sieve’. EUCLID (Greek, C3 BC): compiler of the Elements, the thirteen books that presented some known mathematics in a systematic form. The books devoted to the elementary geometrical properties of polygons and circles became standard textbooks in Western European universities - and, eventually, schools. Book 10, specifically referred to in the seminar, was a lengthy, sophisticated treatise on certain irrationals EUDOXUS (Greek, C3 BC): one of the greatest of the classical mathematicians known for his work in astronomy and for the subtle theory of ratios that forms the main topic of book 5 of Euclid’s Elements. He worked mainly in towns in Asia Minor and in Athens (and would not be usually be considered an Alexandrian as was implied in the seminar). EULER, Leonhard (Swiss, 1707-1783): prolific mathematician who held posts at Russian and Prussian universities. He contributed to various branches of mathematics. His work in analysis and number theory - though often unrigorous by modern standards - was particularly notable for its elegant, and often surprising, results. GAUSS, Karl (6erman, 1777-1855): one of the very greatest mathematicians, he made fundamental discoveries in a number of fields. He was cautious about publication, preferring to polish and repolish papers before publishing them. In possibly

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controversial cases like his discovery of non-Euclidean geometry, he preferred not to publish at all. GOURSAT, Edouard (French, 1858-1936): influential author of textbooks. HADAHARD, Jacques (French, 1865-1963): noted for his investigation into the ways mathematicians worked. He collected responses from a number of his contemporaries and analyzed them in an influential book, The psychology of invention in the mathematical field, published in 1945. HARDY, Godfrey (English, C20): brought continental standards of analytic rigor to English mathematics. He collaborated with Littlewood and with the Indian mathematician, Ramanujan. Later he wrote a famous defense of pure mathematics, A mathematician’s apology. LABRANGE, Joseph-Louis (Italian-French, 1734-1313)1 outstanding mathematician who made fundamental discoveries in various branches of pure and applied mathematics. The theorem in group theory that the order of a group is a multiple of the order of any subgroup is now usually named after his, although he would not have put it this way. LEIBNIZ, Gottfried (German, 1646-1736): created an approach to calculus which was independent of that of Newton, and which was to be rapidly adopted and developed by European mathematicians. His notation and basic approach is still used today. 214


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LIOUVILLE, Joseph (French, 1809-1882): contributed to the theory of functions, in particular a general theory of functions that could be defined as solutions of a certain type of secondorder differential equation, sometimes named after hit, or more commonly - in conjunction with his friend and contemporary, Stura. LITTLEWOOD, John (English, 1885-1973): fated for his collaboration with 8 H Hardy; this was achieved in an eccentric and impersonal way that does not contradict the view of hid as a loner that was presented at the seminar. OSTROWSKI, A. (C20): professor at Basle; he supervised Gattegno’s doctorate thesis. NEWTON, Isaac (English, 1642-1727): created a dynamic approach to the calculus. His so-called ‘fluxions’ enabled him to develop a theory of gravitation and to solve some outstanding problems in astronomy. They were taken up by his English contemporaries, but a protracted controversy with Leibniz about priority of discovery meant that there was little interaction between the two approaches for some time. NICOLET, Jean-Louis (Swiss, C20): schoolteacher who made a series of animated geometry films illustrating geometrical properties of various conies including some elementary circle theorems. Gattegno explored ways of using these films in classrooms. He also later made some computer-animated versions that developed some of the themes of the original Nicolet films.

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NUNN, Percy (English, C20): influential mathematics educator in the 20s; he wrote various books on the teaching of mathematics. PAPPUS (Greek, C3 AD): a late Alexandrian geometer whose work developed Euclidean geometry in directions that can be seen, with hindsight, as being part of projective geometry. PEANO, Guiseppe (Italian, 1858-1932): was particularly concerned with a rigorous logical foundation of mathematics, in particular a proper axiomatic basis for the natural numbers, which drew attention to the status of the principle of mathematical induction. His axioms were not complete, in that they did not characterize the natural numbers uniquely. PLATO (6reek, C4 BC): the famous philosopher was also the founder of the Academy; there are many examples in his writings that reveal his mathematical interests and knowledge. PONCELET, Jean-Victor (French, 1788-1867): developed projective geometry, in particular introducing imaginary elements. It is said that he began to prepare his book on the subject while he was a prisoner of the Russians after Napoleon’s unsuccessful campaign. POINCARE, Henri (French, 1854-1912): polymath, said to have been the last mathematician who could be said to be familiar with all branches of the mathematics of his time. He conjectured that - in a certain sense - the typical 3D topological object is a

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sphere. The conjecture has, in fact, not been proved for all dimensions, the 3D case being finally established in 1986. POLYA, 6eorge (Hungarian, C20): a mathematician of wide interests, he worked in Switzerland - where Gattegno first met him - and eventually moved to USA. He became well-known for his accounts of problem-solving in mathematics and for his work in mathematics education. PYTHAGORAS (Greek, C6 BC): the almost legendary mathematician known for his work in the arithmetic of whole numbers, for the application of number ratios in harmony, and for the geometrical theorem named after his. This theorem led to the discovery that the diagonal of a square could not be measured in tens of the side. The nature of the Greek reaction to this discovery over the following 200 years is a matter of some controversy - as was revealed in the seminar. RIEMANN, Bernhardt (6erman, 1826-1912): one of the first to try to give a rigorous account of the calculus; for example, in the so -called ‘Riemann sums’ whose limit defines the integral of a function. RUSSELL, Bertrand (English, 1872-1970): his work on the foundations of mathematics invoked the set-theory that had been developed by Cantor. In particular, he emphasized cardinality; he defined number as a class of classes - thus, 2 was the class of all pairs.

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SILLITO, Geoff (Scottish, C20): mathematics educator, involved in post-war curriculum reform. He had been an active member of the Association of Teachers of Mathematics; Gattegno was one of the contributors to the book (‘Mathematical reflections’) written in his memory. THALES (Greek, C6 BC): legendary Ionian mathematician of whose work little precise is known. His name is associated with the similar triangle property that is invoked in the standard ruler-and-compass construction of the division of a line into an integral number of parts. THEATETUS (Greek, C4 BC): follower of Plato who worked on theory of incommensurables and is usually held to be the author of the extensive account in book 10 of Euclid’s Elements. VIETA (French, 1540-1603): the usual Latinized name of Francois Viete, who initiated our current approach to algebra. The reference to a supposed remark about the infidel is a bit obscure - there is a comment to the effect that the ‘barbarians’ have complicated the Greek inheritance unnecessarily in the preface to his book, The analytic art. WEIERSTRASS, Karl (German, 1815-98): an influential teacher who gave a rigorous account of analysis which did not depend on an intuitive appeal to geometric figures. WEIL, Andre (French, C20): influential and important member of the Bourbaki group. (It may interest readers of Simone Neil’s acerbic comments on algebra to know that she was his sister.) 218


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ZENO (Greek, C5 BC): known for his famous paradoxes, typically the story of Achilles and the tortoise. Achilles moves twice as far as the tortoise in any given time, so halves the distance between them in this time. But by successively halving the distance between them, he will never catch up the tortoise there is always a further distance to be halved. ZERHELO, Ernst (German, 1871-1983): worked on problems of set-theory, in particular on ways of avoiding some of the paradoxes that had appeared in the work of Russell and others.

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