Foundations of Physics Anthropology and Einstein's Relativity --Manuscript Draft-Manuscript Number: Full Title:
Anthropology and Einstein's Relativity
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Anthropology, Relativity, Theorem
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Michael Alexeevich Popov Oxford, UNITED KINGDOM
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Michael Alexeevich Popov
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Michael Alexeevich Popov
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May be it is surprisingly, but anthropologists ( V. Bogoraz-Tan (1923), L.Levi-Bruhl (1938-1939), B. Latour (1988) [1-3 ]) have been troubled by an apparent radicalism of Einstein's Relativity . In particular, Lucien Levi-Bruhl in 1938 had found that Einstein's attempt to suggest that merely only physics is the science in which we do not know "primitive" impossibilités et ses absurdités and we always do know rationally what we are talking about is not correct .In his remark on Einstein's article in « Zeitschrifl für freie deutsche Forschung » ( 1938 ) Levi-Bruhl supposed also that Einstein's Relativity itself cannot be free from such sort of ses absurdités as well. In 2012 I made attempt to formulate some anthropological doubts in mathematical language ( it is usual practice of the field anthropologists ) and published my remark on arithmetical counter-example for Einstein's Relativity at Foundational Questions Institute website. It is remarkable, but my arguments were not rejected completely and some theorists had found sufficient foundation for my "nonsense" in Einstein's Relativity, indeed
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Anthropology and Einstein’s Relativity Michael Popov
Abstract May be it is surprisingly, but anthropologists ( V. Bogoraz-Tan (1923), L.Levi-Bruhl (1938-1939), B. Latour (1988) [1-3 ]) have been troubled by an apparent radicalism of Einstein’s Relativity . In particular, Lucien Levi-Bruhl in 1938 had found that Einstein’s attempt to suggest that merely only physics is the science in which we do not know “primitive” impossibilités et ses absurdités and we always do know rationally what we are talking about is not correct .In his remark on Einstein’s article in « Zeitschrifl für freie deutsche Forschung » ( 1938 ) Levi-Bruhl supposed also that Einstein’s Relativity itself cannot be free from such sort of ses absurdités as well. In 2012 I made attempt to formulate some anthropological doubts in mathematical language ( it is usual practice of the field anthropologists ) and published my remark on arithmetical counter-example for Einstein’s Relativity at Foundational Questions Institute website. It is remarkable, but my arguments were not rejected completely and some theorists had found sufficient foundation for my “nonsense” in Einstein’s Relativity, indeed Keywords : anthropology , relativity, theorem. Introduction May be it is surprisingly, but anthropologists ( V. Bogoraz-Tan (1923), L.Levi-Bruhl (Carnets,1938-1939), B. Latour (1988) [1-3]) have been troubled by an apparent radicalism of Einstein’s Relativity in the XX century. In particular, Lucien LeviBruhl in 1938 had found that Einstein’s attempt to suggest that merely only physics is the science in which we do not know “primitive” impossibilités et ses absurdités and we always do know rationally what we are talking about is not correct .In his remark on Einstein’s article in « Zeitschrifl für freie deutsche Forschung » ( 1938 ) Levi-Bruhl supposed also that Einstein’s Relativity itself cannot be free from such sort of ses absurdités as well. [ 1 : 3,18 .7.1938 ] Recently, Bruno Latour published results of his semiotic analysis of theory of Relativity where he expressed some doubts on logical justification of Lorentz transformations in Einstein theory and attempted to describe some sort of absurdity he had found in Relativity . In accordance with Latour , both Special and General Relativity “ are accepted, more frames of reference with less privilege can be accepted, reduced, accumulated and combined, observers can delegated to a few more places in the infinitely large ( the cosmos ) and the infinitely small ( electrons ), and the readings they send will be understandable. [3 : 22 ]. Theoretical physicist Alan Sokal , nevertheless, rejects Latour’s finding as “an example of nonsense “ and had defined 40 pages of Latour’s article as typical “comical misunderstanding”. He suggests Latour “ doesn’t understand what the term “ frame of reference” means in physics – he confuses it with “actor” in semiotics – he claims that relativity cannot deals with transformations laws between two frames of reference, but needs at least three [frames of reference] “ [4:155 ].
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In 2012 I made attempt to formulate some anthropological doubts in mathematical language ( it is usual practice of the field anthropologists ) and published my remark on arithmetical counter-example for Einstein’s Relativity at Foundational Questions Institute website. Interested readers can find my paper and my arguments in http://fqxi.org/community/forum/topic/1426 “Which of Our Basic Mathematical Assumptions in Physics are Wrong ? “ FQXI Essay Contest – Spring 2012 [ 5 ]. It is remarkable, but my arguments were not rejected completely and some theorists had found sufficient foundation for my “nonsense” in Einstein’s Relativity, indeed.
Thesis I am assuming that Physics is not Mathematics. In comparison with pure mathematicians , physicists agree to simplify mathematics because in some cases it “naturally” arises from physical experience. Hence, some basic mathematical assumptions are defined as “ wrong ” (or not sufficiently “ realistic “ ) statements in physical sciences and physicists try to invent own alternative mathematics. However, it may or may not satisfy mathematical “intuitive ontology” of what a pure mathematics should be. For example, Albert Einstein accepted two –dimensional Pythagoras theorem ( in a right –angled triangle , the square on the side opposite the right angle is equal to the sum of the squares on the other two sides ) as fundamental assumption to introduce so puzzling for anthropologists Lorentz transformations in Special Relativity. Following Einstein and modern textbooks, in three dimensions Pythagoras theorem is exactly analogous to conventional Pythagoras theorem in contemporary theoretical physics. In particular, B. Russell in his famous “ABC of Relativity “uses following example: “ Suppose that, instead of wanting merely to fix positions on the plain, you want to fix stations for captive balloons above it, you will then have to add a third quantity, the height at which the balloon is to be. If you call the height z, and if r is the direct distance from O to the balloon, you will have r² = x² + y² + z² , and from this you can calculate r when you know x,y and z. For example, if you can get to the balloon by going 12 miles east, 4 miles north and then 3 miles up, your distance from the balloon in a straight line is thirteen miles, because 12×12 = 144, 4×4 = 16, 3×3 = 9, 144 + 16 + 9 = 169 = 13×13.” [6 : 71] Unfortunately, mathematically speaking, this beautiful physics is not exact! If you can get to the balloon by going x =13,15,17,19,21,23,25,…∞ ( ANY POSITIVE ODD INTEGER ) miles east, and, y= 5,7,9,11,13,15,17,…∞ ( ANY POSITIVE ODD INTEGER ) miles north , Pythagoras theorem in three dimensions CANNOT be TRUE at all, because in accordance with Oliverio’s [7] Pythagoras Quadruple Problem, only for POSITIVE EVEN x and y there exist such z and r , i.e. only for positive even x and y there exists r² = x² + y² + z² in reality ( skeptical reader can test this brilliant result with calculator or even with supercomputer).Thus, there is an infinite number of counter – examples with positive odd x and y for basic Einstein’s assumption and such sort of assumption itself, perhaps, represents pure nonsense, indeed.
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Key Counter- Argument. Because Einstein assumption is the fundamental principle for all known today theoretical constructions based on Einstein’s conception of Space-Time Interval ( deduced as is known from Einstein’s reading of Riemann dissertation “On the Hypotheses which underlie Geometry “(1854) ), my “arithmetical nonsense “ produced some misunderstanding and attempts “to justify” philosophically Einstein’s non-mathematical generalization of Pythagoras theorem among physicists at FQXI . For instance, author of FQXI 2012 award winning essay asked me “But why is that important? The terms in the equation, to which you refer as Einstein's equation (by the way: why?), are namely real numbers, not integers. That means that for any real numbers x, y, and z there is always real number s that satisfies the equation. That real number s doesn't have to be an integer. So where is the problem?”. [5])
Indeed, why physicists cannot re-write Pythagoras theorem in real ( real numbers include all the rational numbers such as the integer – 2, the rational fraction 4/3, all the irrational numbers such as √2 and all transcendental incomputable numbers as π ) numbers if mathematics is only “simply language” in theoretical physics ? Why physicists cannot re-write any mathematical problem in “suitable and a priori solvable “ manner in order to find “solution” always as a kind of physical trick for any unsolvable fundamental mathematical problem formulated by humankind, actually?
My Arguments. [1] As is established, Number theory starts from the positive whole numbers and from the ideas of addition , multiplication, division and subtraction. However, these operations are not always possible unless number theory admits new kinds of numbers. Hence, negative numbers ( 2-5, 3-7 ) as rational fractions are admitted. When mathematicians extended the list of arithmetical operations so as to include root extraction and the solution of equations, they have found these operations are not possible unless we widen our conception of a number, and admit the irrational numbers. Hence, new set of real numbers ( including all the rational numbers, the rational fractions, all the irrational numbers and all transcendental numbers as π ) with necessary are arising. Thus, iff some equations like r² = x² + y² + z² cannot have solutions in the positive integers , we must extend our arithmetic ,to admit the real numbers and to re-write the equation as equation of higher degree, indeed. When x and y are odd , r cannot be found in integers, rational and real numbers at all, because such number r simply cannot exist in general. [2] Following Weierstrass theorem and mathematical association between rational numbers and the points on a line, if we divide interval r (= AA’) into two equal parts, one at least of them must contain infinitely many points at AA’. We select the one which does, or if both do, we can select the left-hand half, and we can denote the selected half by AQ. If AQ is the left-hand half, A is the same point as A’. Similarly,
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if we divide QA’ into two halves , one of them must, correspondingly, contain infinitely many points of this straight line. It is assumed that the points A, …An progress steadily from left to right and A tends to a limiting position, say T. Similarly, Q,…Qn tends to a limiting position T’ . Hence, T’ coincides with T and A and Q both tend to T. Thus T represents so-called a point of accumulation , however if interval r expressed by whole numbers, such kind of a point accumulation cannot exist. Hence, thus, for any real numbers x,y and z there is no always real number r , correspondingly, we cannot use Euclidean methods here ( like in Russell’s example ) to construct irrational and real numbers, actually. [3]. In Relativity theory , Einstein uses notion time as the extraction of the square root of -1. But it is not possible, pure mathematically, unless we go still further and we are admitting the complex numbers. Indeed, Einstein introduced a new kind of complex number of the form x + x’ + x’’ + yi (because the light – time L = ct and L = yi when yi = cti ( i = √ - 1 )) ,however, mathematicians may not satisfy such definition of complex number and Einstein’s formulation of higher degree equation he uses. Indeed, some physicists cannot accept such simplified transition and they developed new twistor algebra (R.Penrose [ 8 ] ) as well as new alternative theory of relativistic gravitation (A. Logunov [ 9 ] ). [4] If we can re-write Pythagoras theorem in real numbers, such theorem could be become undistinguishable from Fermat theorem (1637) where the equation x ⁿ + yⁿ = zⁿ has only the trivial solutions in integers when n >2 ( in other words, when x ⁿ + yⁿ = zⁿ is not Pythagoras theorem ). In light of Wiles’s proof of the Fermat theorem, it is quite natural to make some kind of semiotic experiment and to re-write Fermat theorem in real numbers as it suggested by physicists at FQXI dispute. Indeed, following result by Marvin Jones and Jeremy Rouse [10] we can re-write Fermat cubic equation in quadratic fields in real numbers (i.e. using such irrational real numbers as √2, etc). In particular , we can find an existence of non-trivial solution of the type ( 18 + 17 √ 2)³ + ( 18 - 17√ 2 )³ = 42³ , where √2 is non-constructible by Euclidean methods irrational number . Thus, in famous Russell’s example, if you can get to the balloon by going (18 + 17√2 )³ miles east and ( 18 - 17√2) ³ miles north , your distance from the balloon in a straight line is 42³ miles (!). Thus, Einstein’s Relativity can admit both very different mathematically Pythagoras theorem as well as the Fermat theorem at the same time in order to calculate a fundamental space-time interval. Hence, nonsense is not avoidable.
Acknowledgements 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
I would like to thank both skeptical as well as enthusiastic participants of FQXI 2012 self - organized anthropological experiment. I am indebted in particular Marcoen J.T.F. Cabbolet, Hoang Cao Hai, Vladimir Ragozhin, Christian Corda, and Benjamin F. Dribus help for discussion in this topic at FQXI contest.
References [1]Bogoraz- Tan ,V. Einstein and religion. Petrograd Publisher (1923) [2]Levi-Bruhl,L. Carnets : Part 3 18 July 1938 -http://pages. infinit.net/ sociojmt(1938-1939 ) [3]Latour,B “A Relativistic Account of Einstein’s Relativity “ Social Studies of Science 18 :3-44 (1988 ) [4]Sokal,A, Beyond the Hoax. Science, Philosophy and Culture. Oxford University Press ( 2008 ) [5]Popov,M .A. http://fqxi.org/community/forum/topic/1426 “Which of Our Basic Mathematical Assumptions in Physics are Wrong ? “ FQXI Essay Contest – Spring (2012). [6]Russell, B ABC of Relativity .Routledge ,NY & London (1925-2007) [7]Oliverio,P ‘Self – Generating Pythagorean Quadruples and N-tuples’ Fib.Quart.34, 98-01(1996) [8] Penrose,R "Twistor quantisation and curved space-time", International Journal of Theoretical Physics (Springer Netherlands) 1: 61–99, Bibcode 1968IJTP....1...61P, doi:10.1007/BF00668831 (1968 ). [9]Logunov , A.A. – arXiv gr-qc 0210005v2 ( 21 October 2002 ). [10]Jones, M & Rouse,J , Solutions of the cubic Fermat equation in quadratic fields’ ArXiv : 1211.3610v1 ( November 5 2012 ).
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