Cogito Issue 1 - Reading School Student Philosophy Magazine

Issue 1

17/12/2021

COGITO The Reading School Philosophy Magazine

About Cogito

Contents About Cogito

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Plato’s Cave

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The Spirit of Science, the Science of Spirit

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An Introduction to Formal Logic

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Game Theory and Ethics

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Should You Be More Moral?

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The Retreat of Liberty, Part 1

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Embracing Nihilism

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The Continuum Hypothesis

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Cogito is Reading School’s new philosophy magazine. In our school, certain subjects are extremely popular and are studied widely by the student base – think Maths, Economics, Biology, Physics and so on.

Philosophy is not one of them.

It is easy to dismiss philosophy as “soft”, “wishy-washy” or “irrelevant”.

This misses the point of philosophy. Philosophy is the study of fundamental questions about the world around us. With philosophy we scrutinise our understanding of existence, of ethics, of knowledge, of mind itself. It is through philosophy that we question our assumptions and arrive at our true beliefs.

Our aim is to inspire students and parents to think and question in the vein of philosophy. We want to share our passion for the discipline with you all, and we want to show you why philosophy is so valuable to each of us, even in our daily lives. The purpose of Cogito is to create a platform for students to explore their views and share them with the rest of the school community.

The magazine is named after the famous philosophical phrase “cogito, ergo sum” (I think, therefore I am), penned by René Descartes in 1637.

Descartes undertook an effort to radically reshape the philosophical thinking of the time and succeeded in doing so through his method of scrutinising every belief he held and subjecting them to the greatest level of doubt, at last arriving at the fundamental and undoubtable truth – I think, therefore I am.

We hope to encourage students and parents alike to develop intellectual capacity, creativity, and confidence akin to that wielded so skilfully by Descartes through engaging with the philosophy within this magazine, and to urge you to become deep and scrupulous thinkers.

Editors-in-Chief: Ryan Lin Daniyal Vemuri Designer-in-Chief: Mudit Pahwa

Writers: Ryan Lin Elijah Gibbons Daniyal Vemuri Eashan Bhasin Lily Hillman

We hope you enjoy this issue of Cogito Sincerely, the editors and contributors

Four minute read

Plato’s Cave In the book The Republic, Plato, an ancient Greek philosopher, created the Allegory of The Cave – an extended metaphor representing us as a society and the effects of education and philosophy.

Plato imagined a cave, with some people living in it who have always lived there and don’t know anything of the outside world. The inhabitants are chained and cannot see something other than a wall on the back of the cave. The cave is dark, and all the inhabitants can see comes from shadows of things from the light of a fire burning at the opening of the cave. Objects in the outside world such as birds, trees, and rocks project shadows on the back wall of the cave for the inhabitants to see. The inhabitants develop a pastime of admiring and describing the shadows and take pride in how intellectual they are. This is all the inhabitants know and think is real because this is all they have ever experienced. Reality for them is images of shadows on a wall of a cave.

One day, one of the inhabitants manage to break from his chains and turns to look at the fire. The light of the fire burns his eyes and makes him immediately want to turn back around. The man does not want to progress in the way he sees things and his understanding of reality. However, after his eyes adjust to the firelight, the man reluctantly makes his way out of the cave into the sunlight. The process is painful and difficult.

The man, after all his life trapped in the cave, finally sees true reality – he sees the colours, the shadows, and reflections of himself and objects in the water. He explores and sees everything in their true form – the birds, the trees, and sky. All of which he had only seen shadows of. Eventually the man sees the sun itself and realises it as the source of the reflections and shadows.

The man realises the truth of everything and feels sorry for his fellow people still trapped in the cave. He goes back and tries to tell his fellow people about the truth of reality.

Upon arriving back in the cave, the man tries to tell his fellow inhabitants the truth.

The other inhabitants think he is dangerous and deluded. He has upset everyone else’s conformist opinions. They even plot to kill him, regarding him as a madman. The prisoners do not want to be free because they are comfortable in their own ignorance, distrustful of the escaped prisoner, and being hostile to those who try to change their views of the world.

Plato’s story of the cave is an allegory of the life of all enlightened people. The inhabitants of the cave are humans before philosophy. The alienation and violent reaction towards the returned philosopher who knows the truth is what all truth tellers can expect when they take their knowledge back to people who have not devoted themselves to thinking.

Plato believed, like the inhabitants of the cave, we live in darkness, unaware of the true reality. We live our lives thinking all of it has meaning when our ordinary life is just a phantom image of what is truly out there. We study hard, get our dream job, and pursue happiness, unaware that all of it has infinitely less substance than the true reality. Plato believed that we could escape the cave, becoming enlightened and seeing reality for what it truly is, through philosophy and seeking truth through reason. The journey to discovering truth, like for the man, is painful and difficult.

However, we will have finally escaped the cave, leaving behind the murky shadows of our illusionary previous life. ryan lin

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Twelve minute read

The Spirit of Science,

Science has found itself in a strange relationship with God. Atheists insist that science and religion are totally irreconcilable- that the basic laws of the universe are in direct conflict with essentially every holy text. Meanwhile, many theists maintain that empirical observation of the universe is exactly what leads us to God in the first place, contending that the complex scientific order of the universe can never be wholly explained without invoking divinity. Given how much we value science in the modern era, and how much we owe to its discoveries, it is unsurprising that we still find both of these camps fighting such a bitter and determined battle for its custody.

But what do the scientists think?

Aside from a few public intellectuals, like Richard Dawkins and Sam Harris, it’s not verry common for people with scientific backgrounds to get too involved in religious debate. But that doesn’t mean that scientists have nothing to say – for example, Einstein famously believed that “Science without religion is lame, religion without science is blind.” Especially as the development of quantum physics cast doubt upon strict materialism and causality, metaphysics became an increasingly popular topic for those caught in the middle of competing scientific and mystic outlooks. Some of the most famous scientists of the 20th Century, physicists like Schrödinger, Heisenberg, Einstein, Planck, among many others – the people who have perhaps come closest to observing the working mechanisms of the universe – have developed some of the most radical interpretations of how the worlds of matter and spirit relate.

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James Jeans, the scientist who discovered the “Jeans length”, the “Jeans escape”, the “Jeans instability” (among other things he didn’t put his name on), wrote once that “God is a mathematician, and the universe begins to look more like a great thought than a great machine.” This is an effective description of the progress of physics in the 20th Century: as we moved away from Newtonian laws and into quantum ones, which fail to be explained, or even really comprehended, through our materialistic understanding of the world, mathematics proved to be the only effective explanation. “Unreasonably effective”, Nobel physicist Eugene Wigner said, but nonetheless unavoidably the only way we can hope to accurately describe the universe. And because of this, Jeans concludes that mathematics is in fact the true essence of the universe- in some sense, the world is made of maths.

But is it true that maths is the only way to describe the world? It’s certainly effective in explaining quantum mechanics, or how cars are designed, or how we put men on the moon, but surely there is some aspect (maybe most aspects) of the human experience which aren’t all just numbers. We’re not merely (or even mainly) mathematical creatures, nor are we particularly logical- we’re built to experience love, despair, beauty, and everything in between. And while some of us might find maths beautiful (and others of us might definitely find that maths puts us in despair), there must be some part of being human that exists wholly beyond its scope. After all, the beauty of an undisturbed lake is no better or worse described by a haiku than a book on fluid dynamics. And the pain

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of an apple falling on your head is not more brilliantly described by the law of gravity than by the word “ouch”.

Werner Heisenberg believed that there are in fact two forms of beauty, both deeply rooted in Ancient Greek thought. The first draws on Aristotle, stating that beauty is the “proper conformity of parts to one another, and to a whole”. The second comes from Plotinus, who described beauty as “the translucence of the eternal splendour of the “one” through the material phenomenon.” By ‘the “one”’ in, he really means oneness – referencing the philosophical problem of ‘the one and the many’, which questions what unifying essence lies behind the universe in order to integrate its many phenomena into a single thing.

Clearly, the first definition is meant to correspond to the kind of systemic beauty found in mathematics and science, whereas the second describes the more poetic and artistic kind, although Heisenberg viewed them as not so different. In both instances, he observed, beauty derives from oneness- as much in the consistency and conformity of natural sciences as in, for example, the harmony and interlocking rhythms of music. Our experiences of both are really very similar, the only difference is that, where scientists grasp beauty via the discovery and proper organisation of natural phenomena, artists conduit beauty from emotion and spirit, by, to quote another Nobel Laureate, the poet Czesław Miłosz, submitting “fog and chaos to distillation.” Science orients itself outwards and art inwards, but both are still ultimately chasing the same oneness. Science and art, mathematics and emotion, physics and spirt – the ‘left-brain/right-brain” modes of our experience – are inverse and complementary methods of representing the same underlying unity of existence.

But Heisenberg did limit them only to representing- he warned against physicists trying to play philosopher, and investigate beyond the apparent physical world – because, he argued, “the philosophic content of science is preserved only when science is conscious of its limits”. The artistic process is characterised by constant failure, because artists are constantly cut off from the essence of what they aim to depict by the limitations of their media, their skill, and (for good ones, anyway) their capacity to even understand the elusiveness of human experience. Scientist are also limited not just by the imprecision of their methods, but also the total disconnect between their measurements – how objects relate to their own environment in the closed system of what we experience – and the real essence of the things they measure.

Arthur Eddington, an accomplished philosopher and physicist who discovered the Eddington luminosity, expounded this view in “Science and the Unseen World”, by demonstrating how quantum physics is circular and self-contained, and thus may only be considered valid within its limited scope. He wrote that, in Einstein’s law of gravitation, empty space is explained via reference to certain quantities called ‘potentials’, which themselves must be calculated from fundamental quantities called ‘intervals’. Intervals are relations between ‘scales’ and ‘clocks’, and a clock is a ‘graduated strip of matter’. Matter

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is the embodiment of “mass”, “momentum”, and “stress”, which are each rendered, in one of the primary achievements of Einstein’s theory, in terms of potentials – the concept we were originally trying to get at. It is not necessary to show the specific equations by which these things are related to understand that this logic is clearly circular.

And this kind of circular reasoning appears throughout science: a more familiar example to those of us who have done GCSE physics (and those fewer of us who can remember it) is the definition of electric force, which is given as something causing the motion of an electric charge, where an electric charge is defined as something that exerts an electric force.

To a philosopher, this circular reasoning would discount the argument entirely, as it means Einstein’s theory, in a purely pedantic sense, can’t technically be proven. But this totally misses the point, because the value of science isn’t really that its true – it’s that its useful. This is probably not the only reason why Socrates never discovered electricity, but at least it is why science, in the view of Eddington, can never discover philosophy; science is practical, and it isn’t concerned with what matter, or potentials, or electricity actually is, as long they can be usefully modelled.

Here, Heisenberg and Eddington both disagree with Jeans by arguing that we cannot conclude that the world is made of maths just because maths comfortably explains it, because these explanations are only pragmatic ones. The validity of science is only ever measured by the degree to which it allows us to make predictions about our physical experience, and thus scientific theories are only ever closed off idealizations of nature, and allow us no knowledge of the actual substance of the world.

This does not at all devalue science – Eddington (astutely) observes that we don’t pluck our eyes from our heads just because they show us colour and not wavelengths, and neither should we discard wavelengths because they show us also only a picture of the world, and not its essence. Rather, Eddington argues that, paradoxically, it is only by leaving open the question of the ultimate essence of matter, energy, mass, etc, that science can attempt to understand these concepts in action. And given that science must pass over not only most human experience, but also the actual essence of material things, we are given no reason to agree with Jeans in thinking that it can ever provide a full explanation of the world, let alone constitute it.

But what system will explain the human experience then? It is clearly not very helpful for us for scientists to stay silent on matters of spirit if our concept of ‘spirit’ is empty and, since the tools and methods of scientific observation are inapplicable to spiritual observation, would leave us in a vacuum. Eddington acknowledged that, as primarily a scientist, the question was almost beyond his scope.

But, also because he was a scientist, he believed that the methodology of science could be reciprocated with some effectiveness to the spirit. Specifically, he argued that, if science is not particularly concerned with the spectre of reality, to any further extent

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than the faux-reality we directly interact with, there is no reason to allow the question of spirit to be any more haunted by it. To some degree we already recognise this – an artist, in trying to bring in soul to their work, doesn’t care in what sense soul can actually be said to exist, only the degree to which this soulfulness accurately explains the human condition. Eddington contends that this is the true aim of spirituality – to create models of mental experience, just as science models the physical – and it is only religion and philosophy which concern themselves with truth.

And, since spirituality is internally directed, the attributes of spiritual colour, he believed, may be allowed to rest on intuition, writing, “I think we should not deny to certain inner convictions, which seem parallel with the unreasoning trust in reason which is at the basis of mathematics.” It was evident to him that such leaps-of-faith were absent to science – the most obvious of which being our choice to trust our senses. But also mathematical statements like “2 + 2 = 4” – those which we consider the most logically rigourous – are not really proven at all. Nobody has ever really proven that two plus two is four, or that minus one gives three; we only accept such quick maths because it’s immediately evident to us that it’s true, but if someone were to ask why 2 + 2 = 4, we would struggle to find a real answer. Similarly, Eddington argued that, just as we trust some internal monitor to tell us mathematical truths, or to trust our senses (which are the ultimate source of physical knowledge), or to accept that the universe is orientated with the future after the past, there are also “convictions of the spiritual sphere where our nature bids us hold to,” such as, for example, that “the universe is orientated with good on top.”

twenty miles a second round the sun-a fraction of a second too early or too late, the plank would be miles away. I must do this whilst hanging from a round planet head outward into space, and with a wind of aether blowing at no one knows how many miles a second through every interstice of my body. The plank has no solidity of substance.

To step on it is like stepping on a swarm of flies. Shall I not slip through? No, if I make the venture one of the flies hits me and gives a boost up again; I fall again and am knocked upwards by another fly; and so on. I may hope that the net result will be that I remain about steady, but if, unfortunately, I should slip through the floor or be boosted too violently up to the ceiling, the occurrence would be, not a violation of the laws of Nature, but a rare coincidence. These are some of the minor difficulties. I ought really to look at the problem four-dimensionally as concerning the intersection of my worldline with that of the plank. Then again, it is necessary to determine in which direction the entropy of the world is increasing in order to make sure that my passage over the threshold is an entrance, not an exit.

Verily, it is easier for a camel to pass through the eye of a needle than for a scientific man to pass through a door. And whether the door be barn door or church door it might be wiser that he should consent to be an ordinary man and walk in rather than wait till all the difficulties involved in a really scientific ingress are resolved. elijah gibbons

Eddington recognises that such inner convictions are not infallible, and are just as prone to illusion as our experience of the physical – we have to allow some room for the pathological condition of the brain to falsely interpret nonsense into transcendent insight – but this does not mean that advance is impossible. Ultimately though, the strongest argument for seeking a spiritual system to explain our experiences of beauty, despair, connection, etc, as more to us than just electrical impulses through myelin and neurexin tissue, Eddington concludes, is not to be found via comparison to science. Because, while the world of science is important, the world of emotion, of mind, and of spirit is the world we live in. Even if one holds theoretically that the physical is all there is, they must acknowledge that a world of scientific readings, measurements, and symbols is an impossible world to inhabit for humans, since to limit your experience to what can be explained by science would certainly make you less than human. And if purely scientific autopsy is insufficient to describe even our basic experience of the world, then it is even more irrelevant to the exalted relationship of the human soul and the divine spirit.

On this, it is best to allow Eddington to speak for himself, uninterrupted: I am standing on the threshold about to enter a room. It is a complicated business. In the first place, I must shove against an atmosphere pressing with a force of fourteen pounds on every square inch of my body. I must make sure of landing on a plank travelling at

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Five minute read

What is Logic? an introduction to

Formal Logic

Logic, as an area of philosophy, is concerned with evaluating arguments – what separates good arguments from bad ones?

An argument is a chain of reasoning that aims to support a conclusion. Merely disagreeing about a matter is not an argument in the philosophical sense; an argument involves giving reasons in favour of a particular claim.

Logic is therefore about evaluating chains of reasoning. But what does it mean to evaluate an argument?

Take the following argument: 1. All dogs are red.

2. Rex is a dog.

3. Therefore, Rex is red.

(1) and (2) are the argument’s premises – they are givens in an argument, statements that provide the starting points of the argument. (3) is the argument’s conclusion – we infer this from the premises.

There are 2 ways in which we can evaluate this argument. First, we can ask whether the argument’s premises are true. Are all dogs really red? Is Rex a dog? Second, we can consider the quality of the inference step within the argument i.e., the movement from the premises to the conclusion. Do all the inference steps make sense? In the argument above, they do.

Logic is not concerned with the first form of evaluation; rather, it is concerned with the deductive validity of an argument, the watertightness of the inference steps. An argument is deductively valid when if the premises are true, it is impossible for the conclusion to be false. Another way of saying this is as follows: an argument is valid if its premises deductively entails its conclusion. The argument above is valid. Whilst all dogs may not be red, and Rex may not be a dog, if we assume (1) and (2) to be true, it is impossible for (3) to be false.

Deductive and Inductive Arguments

It is to be noted that the above argument is a deductive argument, and that deductive validity applies to deductive arguments.

An argument is deductive if, as above, the truth of the premises entails the truth of the conclusion. On the other hand, an argument is inductive if the truth of the premises increases the likelihood of the truth of the conclusion.

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Take the following argument: 1. I have seen a white swan in Slough.

2. I have seen a white swan in London. …

(999) I have seen a white swan in Edinburgh.

(1000) Therefore all swans are white.

This argument is inductive as the inference step moves from sample cases to further cases. Whilst we may consider the above a good argument, and whilst the premises do give us good reason to believe the conclusion, it is not logically impossible to imagine a swan that is not white; indeed, such swans that are not white do not exist. The truth of the premises does not make it impossible for the conclusion to be false.

All inference steps of this form are deductively valid, which is why each of the arguments above are equally valid. This argument template is known as a schema, and we can replace Z with any object and X and Y with any expression that selects a type of object. Returning to the first example, X refers to dogs, Y refers to the property of being red, and Z refers to WRex. Formal logic aims to study in a general manner the valid forms of inference, enabling us to see where these are used in arguments about different matters and therefore allowing us to analyse these arguments for their deductive validity, acting as another tool in the philosopher’s toolkit to be applied to various different quandaries across the subject. daniyal vemuri

Formal Logic

Consider the following arguments: 1. All bachelors are unmarried men.

2. Matthew is a bachelor.

3. Therefore, Matthew is unmarried.

1. All Reading School boys are smart.

2. Mudit is a Reading School boy.

3. Therefore, Mudit is smart.

1. All musicians are eccentric.

2. John is a musician.

3. Therefore, John is eccentric.

All the above arguments have different premises and arrive at different conclusions, and we can create more arguments like these. All these arguments have equally valid inference steps, and they all clearly take on the same form. All these inferential steps involve the same general inferential principle i.e., that if all of one set of items have a certain property, and another item is part of that set, that item has that certain property.

Formal logic is concerned with abstracting away from content (filtering out unnecessary information about the content of the argument) and looking at the form of the inference step, considering whether this is deductively valid. The inference step above is of the form All X are Y. Z is X. Therefore, Z is Y.

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Eight minute read

Game Theory and Ethics Game theory is the mathematical study of strategy and conflict. It has wide applications in economics, mathematics, political science and in a sense philosophy. Game theory deals with games in which all players have preference orderings over the possible outcomes of the game. In this article I will be talking about game theory’s link to ethics and morality (prisoner’s dilemma) and climate change.

Being a part of applied mathematics, game theory can be mathematically applied to human social behaviour, to strategic interaction and to the logic of conflict. This article will introduce game theory and present its applications in different branches of philosophy. It will give a broad view of game theory in philosophy.

Introduction to Game Theory

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One of the most widespread uses of Game theory in ethics has been to propose multiple functions of morality and identify the important features. An example of this is from Edna Ullmann-Margalit who argues that morality can be understood as a system of norms that enable people to coordinate their actions in order to achieve mutually advantageous outcomes in situations where the pursuit of self-interest would prevent this. The classic example that is used is called the prisoners dilemma. The prisoner’s dilemma starts with two soldiers who must choose whether to stay by their mortars in order to slow down the advance of the enemy, or to flee. If both soldiers stay it is certain

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The Prisoner’s Dilemma

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There are two ways in which Game theory is relevant to ethics. The first way is explanatory. Game theoretic tools can be used to explain things ranging from the function of morality and the general features of our moral practices to the emergence of social norms. The second way in which game theory links to ethics is justificatory.

Here game theoretic tools are used to justify things ranging from particular norms or principles to large scale social institutions or general characteristics of our moral practices.

This is an example of the prisoner’s dilemma (there are many more examples) and can be illustrated using grids and pictures. In this example each soldier acts without knowing the other soldiers’ actions. In this situation each soldier has a reason to flee because fleeing provides the only chance of escaping unharmed, which is the preferred outcome of each soldier. However, it is a dilemma because if each soldier flees, they will both end up in a worse position then if neither had fled. The dilemma is made greater as fleeing is each soldiers best response to any strategy the other soldier could choose.

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John von Neumann and Oskar Morgenstern laid the foundations of classical game theory in their treatise called ‘Theory of Games and Economic Behaviour’. Over time Game theory has transformed social sciences, after being refined multiple times, over the next several decades.

that the enemies advance would be halted, but the soldiers would almost certainly be injured. If both the soldiers flee then the enemy will be able to advance and also capture the fleeing soldiers. And if just one of the soldiers decide to stay while the other flees, he will bravely die trying to stop the enemy while the other will just about make it to safety.

Ullmann-Margalit pointed out that if the soldiers understood the nature of their predicament, they might want to do something to prevent themselves from both fleeing resulting in them being captured. This prisoner’s dilemma shown in this argument is shown in many real-world applications. Upon first look, the most obvious solution to the dilemma would be if the soldiers both remained at their posts. However, a problem arises as in these scenarios like this promises of staying at their posts are not reliable and even if they are credible, they are likely to be exploited. Instead, what the soldiers need is a way of literally or figuratively chaining themselves to their mortars.

Ullmann-Margalit suggests that moral norms do this work. More specifically, it’s not

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the promises to remain at their posts that solves the soldiers’ predicament, but their commitment to keeping their promises and there being sanctions for those who didn’t.

This is an example of how morality and ethics are found in Game theory.

Climate Change

Game theory has been crucial to the analysis of many important problems in environmental ethics, the most famous being the tragedy of the commons by G. Hardin.

Here game theory was used to show how resources can be easily depleted or degraded when they are held in common, and the use of the resources are unregulated. The idea of the tragedy of the commons is that if use of and access to scarce resources is not limited then there is a tendency for use of the resources to exceed the environment’s ability to renew the resource. Examples of this are overgrazing land, running down water levels in aquifers and emitting to much pollution into the environment.

There are two schools of thought with respect to how we should confront the problem described above. One school acknowledges the collective and political nature of the problem but maintains that in addition to state action individuals have responsibilities to do things like reduce their carbon footprint to mitigate the effects of climate change.

The other school of thought tends to be sceptical of how much unilateral action (at an individual or national level) can do. So, they argue that our efforts are best spend trying to find workable collective mechanisms to address climate change.

To conclude, many of the early applications of game theory to ethics were limited to theories. However, over time it has become an increasingly important tool in the study of applied problems and real-world scenarios.

Questions of morality have been central to some of the conclusions that problems involving game theory have come to.

In general, most common pool resource problems arise in situations where firstly, individuals would benefit from using the resource to the greatest extent possible and secondly, the actions of individuals would have a small impact on the resource, but everyone’s action together would have a large negative impact on the resource. In cases like this it is easy for an individual to see that the actions of everyone as a whole are a problem, but because any individual decision to refrain from exploiting a resource is unlikely to make an impact on its own, people infer that they might as well exploit a resource for themselves while they can.

eashan bhasin

Common pool resource problems are generally solved in two ways. 1) by taking the resource out of the commons by creating property rights in it, or 2) finding ways to collectively manage the resource, for example, regulating who has access to the resource at any given time. There are many different factors that are taken into consideration when deciding which approach is most suitable to solving these problems. Game theory provides a framework for analysing the strategic behaviour of individuals in different contexts, which is a useful tool for identifying the solution most likely to work for a given case.

Game theory has played an important role in the moral and political analysis of climate change. In particular it has been crucial to the analysis of why addressing climate change is difficult. Here the consensus is that entities interested in addressing climate change are addressed with a massive problem stemming from the fact that many of the things that can be done to prevent further climate change are either near public goods or things which generate multiple positive externalities. For example, reductions of carbon emissions and increased investment in alternative energy technologies are things which benefit most countries regardless of who invests in them. And because investing in these is costly in the short term, we tend to underinvest in them, a problem that is exacerbated by the fact that the most serious costs of climate change arguably fall on future generations.

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Fourteen minute read The question of whether one should be more moral or not has plagued philosophers for centuries. In the modern Western world, it is common for being moral to be considered a crucial virtue – and for good reason. Without a unifying moral code, society would collapse, and unthinkable suffering would be brought upon society.

Should You Be More Moral?

However, through deeper analysis of the metaphysical nature of morality – a branch of philosophy called metaethics, we may have reason to believe otherwise. I will be sharing an essay I wrote which won high commendations, top 30 globally, in the international John Locke Institute essay competition in which I answer the question:

‘Should you strive to be more moral? Why or why not? Are you more moral than most people you know? How do you know?’

I argue for a unique and controversial position, contending that we should not strive to become more moral for multiple reasons. Readers are left to consider, continue to hold, or rethink their previous beliefs.

Metaethics attempts to explain the nature of what morality is (do moral statements objectively exist? – existing independently/outside of human thought.) If metaethics concludes that moral statements do not exist independently of human thought (objectively) then morality exists subjectively. In other words, there are no moral truths out there in the universe, but only exist for your subjective self. For example, a subjective statement, such as ‘I love chocolate’ has no truth out there in the universe so it has no objective truth. If this is the case, we must accept the fact that there is nothing objective about moral statements. In other words, our moral standard has no truth out there, but is just a subjective construct. It is then the case for whether one wants to carry on following this subjective construct or to give up on it as it has no objective truth.

Moral realism is a metaethical theory which argues that morality is objective (there are moral truths out there) whereas moral anti-realism is a metaethical theory which argues that morality is subjective (there are no moral truths out there). I refer to these two theories in my essay.

In my essay, I explore the metaphysical nature of morality, the idea of linguistic memes, and free will. I hope you enjoy it.

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Should you strive to be more moral? Why or why not? Are you more moral than most people you know? How do you know?

I – Introduction

Initially, one must understand the definition of morality in order to further discuss this question as we need to pinpoint what is meant by acting morally. Identifying the target of moral theorizing enables us to see different moral theories as attempting to capture the very same thing.

Two categories exist of the way morality is commonly used linguistically: descriptively and normatively. Descriptive ethics can be thought of as a guide to behaviour that is put forward by the society and that is accepted by almost all members of the society.

Descriptive morality does not make any factual claims about mind-independent moral properties. Normative morality, on the other hand, is the behavioural code that meets the following condition: all rational persons, under certain specified conditions, would endorse it. Normative morality tries to make statements about how one ought to act and thus most argue that mind-independent moral properties exist in the external world.

This essay will be broken down into two sub-questions: 1. Should one strive to be more moral?

2. Am I more moral than most people that I know?

The first sub-question will be answered by initially analysing the metaethical nature of morality – it will be argued that mind-independent moral properties do not exist. Then, it will be explored whether we may wish to pursue moral actions which have been deemed to be by society even though mind-independent moral properties do not exist.

Subsequently, this essay will explore whether we have free will, which will determine whether there is reason to strive for anything, including becoming more moral. It will be contended that one should not strive to be more moral on the basis that we have no free will, and equally on the basis that moral realism is false.

This essay will answer the second sub-question by referring to the metaethical conclusion. It will be argued that I am not more moral than most people that I know, and neither am I less moral because objective morality doesn’t exist.

then there is no obligation to act morally, thus we should not strive to become more moral.

Moral-realism argues that mind-independent moral properties exist in reality whilst moral anti-realism argues that they do not.

Moral-realism may be broken down into two separate theories: moral-naturalism, which argues moral properties are mind-independent natural properties, and moral nonnaturalism, which argues moral properties are mind-independent non-natural properties.

Utilitarianism and Aristotelian virtue ethics may be considered to be moral-naturalist theories whilst Moore’s intuitionism may be considered to be a moral non-naturalist theory.

Utilitarianism is a reductionist, moral-naturalist theory. As Jeremy Bentham argued, ‘nature has placed us under the governance of two sovereign masters: pleasure and pain. It is for them only to point out what we ought to do, as well as what we shall do’, Bentham argued that it is part of our nature to find pleasure morally good and we cannot help but have this feeling. Bentham’s act utilitarianism is a moral-naturalist theory because it claims morally goodness can be directly reduced to pleasure. Pleasure is a property/description of natural beings (humans); thus, goodness is a natural fact because descriptions of natural beings are natural facts.

This essay will look at a few issues regarding moral realism which fail to be sufficiently responded to. Thus, it will be concluded that moral realism cannot be true and moral anti-realism must be true.

Firstly, all moral naturalist metaethical theories commit the naturalistic fallacy, argued by G.E. Moore. Moore described the attempt to identify goodness with natural properties as the naturalistic fallacy, arguing all attempts to equate goodness with natural properties is an assumption/fallacy. As observed in J.S. Mill’s proof of utilitarianism, Mill argues that something being desirable is evidence that it is good, which is an assumption (something being desirable just means it is desirable and nothing else). Virtue ethics also commits the naturalistic fallacy because Aristotle assumes human flourishing is morally good.

Aristotle’s argument for eudaimonia being the ‘final end’ of human action which is morally good is all based upon descriptive morality, rather than deductive reasoning.

Metaethics is the study about what morality is, including philosophy of language, metaphysics, and epistemology. This essay will primarily focus of the metaphysical aspect of moral properties (the debate between moral-realism and anti-realism) which questions whether morality states objective truths and whether mind-independent moral properties exist in reality. If we prove that mind-independent moral properties do not exist,

Secondly, J.L Mackie formulated the argument from relativity, attacking moral realism as a whole. Mackie argued that moral codes differ from one society to another. He argued that if objective mind-independent moral properties existed, moral codes would not be relative if everyone had equal access to the one true moral reality. Intuitionism may explain this difference as extremely bad distorted perceptions of the moral truth which had been falsely conceived by faulty intuitions, however, Mackie argued a better explanation for this difference is that mind-independent objective moral properties do not exist and differences in moral codes are a result of social conditioning. Mackie

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II – Metaethical Analysis of Moral Properties

argued that morality has simply been invented by societies which are bound to differ due to differences in lifestyles and geography. The same issue of relativity occurs for moralnaturalist theories, which Mackie again explained to be just due to differences in societies which led to the formation of slightly shifted moral views. Moral realists could argue that core moral views are held in all societies (murder being wrong) which is evidence that relativity is a result of a faulty perception of moral properties. However, one can argue that major moral views are equal in all societies because cultures all live in similar external conditions (all live on Earth, where there are scarce resources) thus societies have forged morality to be in such a way – where they can survive the longest considering external conditions – which are all similar.

The most important issue, by far, attacking moral realism is Hume’s is-ought gap. Hume argued that it is impossible to derive ought statements (moral statements) from factual statements about the world. For example, we can examine all the factual statements about murder such as the fact that it is vicious, causes pain, and is malicious. However, we will never be able to link these facts together to infer the moral statement that ‘murder is wrong’. We only think murder is wrong, not because of objective facts, but because of our subjective views on malicious actions which cause suffering. Hume’s fork gives us almost certain reason to believe moral naturalism is false as it is impossible for us to have formulated our moral statements from objective natural properties.

These issues have clearly shown that moral realism as a whole is false. If objective mindindependent moral properties do not exist, then it seems as if there is no reason to strive to follow moral codes as they do not truly reflect reality.

III – Memetics – Morality as Memes

Analogous to genes, memes, conceived to be by Richard Dawkins, are defined as ‘unit(s) of cultural transmission’ which are able to replicate via imitation.

This essay will explore the explanation of morality as memes which survive on the basis of natural selection, further building onto the idea of Mackie; that morality originates in culture and society rather than truly existing in a mind-independent moral reality.

In the last chapter of The Selfish Gene Dawkins clarified that ‘memes are tunes, ideas, catch-phrases (…) Just as genes propagate themselves in the gene pool by leaping from body to body via sperms or eggs, memes propagate themselves in the meme pool by leaping from brain to brain via a process which (…) can be called imitation.’ Like genes, memes are replicators; not for genetic material but for cultural units which exist in a purely mental state.

If we accept the meme-to-gene interpretation, we find that memes must behave equally to genes, giving rise to cultural evolution. With this interpretation there must be cultural equivalents for the main evolutionary genetic concepts. The most important equivalent

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of genetic evolution is that of replication, which carries a rate of mutation; similarly, memes which spread through imitation carry also carry a rate of mutation. This allowing cultural evolution to occur.

One category of memes that can arise out of this cultural evolution is of morality.

Via mutations of memes in ancient societies, it would have been inevitable a cultural unit of morality would have arisen. This meme would have given the society a selective advantage (a society which truly believed murder was wrong would be less likely to die out as the people would act more peacefully). Meanwhile, different societies which didn’t have this meme of morality would have a higher chance of dying out. Over time, all surviving societies must have developed the trait of morality – an example of cultural evolution – and these surviving cultural traits have survived to this modern day on the basis that it is more favoured in terms of evolution.

IV – Following the Societal Construct of Morality

If the theories of memetic morality and moral anti-realism are correct, are there any incentives to follow moral codes set by society?

One may argue that we should strive to follow the societal moral guidelines for other reasons even if mind-independent moral properties do not exist.

Sociologically, one can argue that society and social life cannot function without rules, including moral rules on actions.

Psychologically, taking Mackie’s error theory on metaethics to be true, it means the descriptive morality of society is cognitivist, meaning the majority in society believe in the existence of mind-independent moral properties. This means that reputation and social censure will be affected depending on whether one choose to act morally.

Psychologically, humans care about what others think of them, which could motivate morality.

V – Free Will

In 1697, Newton published his Principia where he enunciated his three laws of motion.

All three laws together can predict the future motion of an object given we know its mass and forces upon it. These laws have significant consequences with regards to free will; they imply that the positions and velocities of all the components making up a system at any point in time, together with the forces between them, entirely determine that system’s future state.

This is the theory of Newton’s deterministic clockwork universe; once mass and forces are accounted for within a system, it will run its path identically when repeated. Given a set of starting parameters in a system, its end point is already determined. Applying

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Nineteen minute read this theory to the universe, which began with a set number of parameters, all future outcomes should be pre-determined and predictable. This means that all entities within the universe, including your choices, follow an inevitable path.

One issue regarding the Newton’s clockwork universe theory is the discovery of quantum mechanics. Wave-particle duality seems to suggest that the universe, on the most fundamental level, runs entirely based on probability. For example, the De-Broglie hypothesis claims all matter has a wave nature given by λ=h/p. This probability wave governs all of matter, seeming to suggest that the universe is not entirely pre-determined, but has many possible outcomes.

The Retreat of Liberty

However, the issue of the quantum nature of matter on determinism only exists depending on which interpretation of quantum mechanics one chooses to accept. Many interpretations exist such as the: Copenhagen interpretation, many worlds interpretation, and De-Broglie-Bohm’s theory. The issue of quantum probability exists for the Copenhagen interpretation; however, the De Broglie-Bohm theory is a hidden variable theory meaning it supports determinism, arguing that particle position remains definite at all times. It theorises there must be an undiscovered law that determines quantum probability.

No interpretation has proven to be true, therefore, with regards to free will, if a deterministic interpretation is true, free will does not exist.

If an interpretation of quantum mechanics which disproves determinism is true, one can argue free will may still not exist. This is because, if quantum behaviour affects the choices we make, which causes unpredictable choices, can we really call quantum probability our free will?

I argue that this does not constitute to free will. Thus, we do not have free will.

VI – Conclusion

This essay has shown that neither free will nor moral properties exist. Striving to be moral based on hypothetical imperative reasons fails to be a motivation after proving that free will does not exist because striving to do anything is purely pre-determined or governed by probability. Therefore, on these bases, I conclude that one should not strive to be more moral.

As previously argued, moral-realism is false which means that mind-independent moral properties do not exist. Therefore, I further conclude that I am not more moral than most people that I know because the truth in the concept of morality does not exist. ryan lin

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From the 1st of February, COVID-19 vaccination will become mandatory in Austria.

Austria will become the first country in the developed world to introduce such a requirement and many more governments may soon follow the example set by Vienna.

2 years ago, the society that we live in today would have seemed improbable, if not impossible. It would have been unthinkable to conceive of the government forcing citizens to stay at home, or for the government to require certain garments (masks) to be worn in certain settings. Even now, it is unthinkable to some that the government may force us to forfeit our bodily sovereignty in return for certain freedoms, certain liberties.

The position of liberty world-wide has been harmed during the pandemic. There have been obvious infringements on personal liberty – think lockdowns, mask mandates and the like. But there has also been a silent and subtle expansion of more clandestine activities that hinder our liberty.

The purpose of this article is to highlight some of the ways in which our liberty has been reduced within modern society.

Do not be mistaken – the author is not a raving anti-vaxxer, nor is he an anarchist or a libertarian. Liberty is not always inherently good; it must be restricted in various cases for the collective benefit of society. That being said, there is a strong moral predisposition towards protecting liberty i.e., intuitively we see freedom as a good thing (the extent to which liberty is good and desirable will be expanded upon later in this article). The aim of this article is to enhance our understanding of the concept of liberty so that we can better analyse the trade-offs made when engaging in modern society to determine which trade-offs are in fact worth making.

The article will be divided into two parts; this part shall arrive at a satisfactory concept of liberty, and the second part in the next edition of the magazine will apply this concept of liberty to issues within modern society.

The Concept of Liberty What is liberty?

The words freedom and liberty (used interchangeably in this article) are often bandied about in philosophical and political discourse without a comprehensive or satisfactory understanding of what these terms actually refer to. If we want to identify where our freedoms have been curtailed, we must first be able to recognise what freedom actually is.

In order to analyse liberty, we must first determine what questions a theory of liberty needs to answer in order to define that term. The first question seems obvious – who has liberty? Is the term best applied to individuals or political collectives? The second question equally so. What do people have liberty from? The third question is slightly less obvious, although by no means obscure. What do people have liberty to do?

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Liberty is therefore a triadic relation – we can describe the concept of freedom in the following general terms: X is free from Y to do Z. All theories of liberty address the questions of 1. Who is X, the haver of liberty?

2. What is Y, the constraint on liberty?

3. What is Z, the thing that X is free to do/become?

The philosopher Isaiah Berlin categorised the various conceptions of liberty by various different philosophers into 2 broad categories: negative liberty and positive liberty.

Negative Liberty

Negative liberty is not so called because it labels all liberty as a bad thing. Rather, negative liberty is the concept of liberty as an absence. Under negative liberty, freedom denotes the range of activities that one can perform without being blocked from doing so. If I am blocked from doing something that I would have been able to do otherwise, I am unfree.

For example, take what may be a familiar experience for many parents. I’ve parked my car outside a shop, or in a parking lot. I go in to do whatever I came to do, and I come back to see that someone’s blocked my car in by parking in front of me. I’ve been blocked from removing my car from its space and going home, my day has been ruined, and my liberty has been diminished.

There are some nuances to the theory of negative liberty, and these can be best seen when it is expressed in the triadic form outlined above: X, an agent, is free from Y, human interference, to do Z, whatever they could otherwise do.

Firstly, note that freedom is only constricted by human interference. If my path is blocked by a tree blown over by a storm, my political and social freedom is not diminished. If someone pushed over that tree and left it in the path, my political and social freedom is in fact diminished. The restriction of constraints on freedom to those that are purely artificial is doubly important to maintain the distinction between freedom and ability. I can’t jump 20 metres in the air (sadly), nor can I teleport (even more sadly). That doesn’t reduce my liberty. Not being able to do something due to natural constraints does not constitute a reduction of liberty.

Secondly, note that X is free to do whatever they could otherwise do, not what they otherwise want to do. This prevents strange situations from arising where we say that someone is free when they are in fact not e.g. a happy slave may have their desires

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satisfied but we would not say that they are free.

A potential sticking point or point of ambiguity for negative liberty is what exactly constitutes human interference – this will be particularly relevant when discussing some potential restrictions of liberty faced in modern society. The greatest issue for negative liberty is in fact to do with its account of Y (the constraint on liberty). Negative liberty, critics argue, only recognises external constraints i.e. the actions of other humans. It fails to understand that liberty may be hindered by internal constraints, such as a lack of awareness, desire to conform to a social identity or false consciousness.

Other criticisms of the theory include that it is crude – it reduces freedom to a quantitative measurement in the vein of science rather than acknowledging freedom as a qualitative feature of the human experience – and that it permits liberal despotism i.e. a conception of negative liberty would consider a society run by a dictator that reduces barriers to individual action to be free; some would argue that a conception of liberty must be opposed to despotism and inherently tied to democracy.

It is best to first analyse the other account of liberty, positive liberty, first before attempting to address these issues – this way, we can compare and contrast the two theories in order to determine which one is more convincing.

Positive Liberty

liberty has been reduced by your desire to play Clash Royale, your English revision is forgotten and you probably end up doing not as well as you’d hoped in the English test.

In the triadic form, positive liberty is as follows: X, an individual or (more commonly) a collective, is free from Y, internal interference (desires or urges), to do Z, lead a prescribed form of life.

All theories of liberty address the questions of 1. Who is X, the haver of liberty?

2. What is Y, the constraint on liberty?

3. What is Z, the thing that X is free to do/become?

The key idea here is that of internal interference. Positive liberty is not concerned on whether you act on your desires or whether you are able to do so, but rather how you act on those desires and how you arrive at the decisions that you make.

Positive liberty recognises constraints based on their role in an agent’s deliberations (their reflections and considerations before choosing an action to perform). Your positive liberty is diminished when you fail to deliberate i.e. when you fail to reflect adequately before performing an action.

Negative liberty holds liberty to be an absence, more specifically an absence of human interference. Positive liberty, on the other hand, holds liberty to be a presence. Positive liberty states that liberty is found in the presence of certain faculties or abilities – what these faculties or abilities are depends on the particular theorist’s specific conception of positive liberty (we shall return to this promptly).

Proponents of positive liberty further argue that once an agent has reflected upon the options available to them and the ends that they can choose to pursue, they must act rationally and choose ends that either maintain or increase their freedom. In other words, you can only have positive liberty when choosing acts that don’t restrict your freedom (this will be of particular importance to some issues in the modern day, such as whether advertising reduces our liberty).

Positive liberty can be a more difficult concept to grasp compared to negative liberty because it is not as intuitive as the latter; negative liberty is more in line with what our everyday understanding of liberty involves.

The Problem with Positive Liberty

It is somewhat easier to understand positive liberty in comparison to negative liberty by looking at their respective accounts of Y, the constraint on liberty. Under negative liberty, Y is human interference i.e. a form of external interference. Under positive liberty, however, Y is internal interference e.g. urges or desires.

Take the following example which may be familiar to many of the students reading this article. You’ve sat down to revise. You’ve got an English Literature test tomorrow and you haven’t learnt any Macbeth quotes. You get your copy of the play out, but suddenly you’re overcome by an urge to play Clash Royale. You know you need to revise for this test or you’ll do badly, but your hands go to your phone anyway. You boot up the game and hear the dulcet tones of the loading screen, and you start playing. Your positive

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There are two aspects of positive liberty which combine to create an extremely pressing issue for the theory, the first being the fact that positive liberty requires action, and the second being that positive liberty is often thought of as being achieved through a collective.

As above, under positive liberty you must choose actions that develop or maintain your liberty. Arbitrary choice (choosing whatever you want when you want to choose it) is dangerous in that you risk undermining your own freedom in the future. Freedom, as JS Mill argues, is a fundamental interest of human beings and so we cannot legitimately undermine it through our actions. Positive liberty requires agents to act to increase or sustain their liberty, and to act in a manner that conforms to Z, a prescribed form of life (often a rational form of life).

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Hegel in the Philosophy of Right argues that freedom is the self-actualisation of the human spirit, and that freedom is a type of action (as above) rather than consisting of possibilities for action.

Hegel argues that freedom involves leading a rational life, and that freedom can only be actual in a rational society that is made up of institutions that are felt as rational by the individuals who achieve self-actualisation in those institutions. To put this simply, we have positive liberty only by participating in the society around us.

Another form of positive liberty is that of Jean-Jacques Rousseau. Rousseau argues that individual freedom is achieved by participating in the processes through which a society determines its own affairs in accordance with the general will. Similar to Hegel, Rousseau carries a conception of individual liberty being achieved through participating in a wider collective.

The fundamental danger with positive liberty is that it can lead to authoritarianism.

Firstly, consider the “tyranny of the majority” that JS Mill considered to be so dangerous to a modern society. Minority groups may take part in the self-determination processes of a society and therefore be said to be positively free, but they may have their liberties and rights curtailed either by a majority legislating against them within democratic institutions or by public opinion of the majority being turned against the practices of the minority. These oppressed minorities should surely be considered unfree, yet under positive liberty they are considered to have liberty.

Additionally, positive liberty does not acknowledge that what self-actualisation actually means to an individual depends on that individual’s circumstances and experiences.

JS Mill argues that freedom in the form of self-fulfilment or self-actualisation cannot be achieved or enhanced through a collective body, and rather must be achieved by individuals acting to meet the ends that they set themselves.

Negative liberty, in being concerned with external interference and the interactions between individuals, provides a far stronger barrier to paternalism and authoritarianism than positive liberty ever could. It is not by coincidence that concepts of positive liberty, in particular that conceived by Hegel, were ‘at the heart of many of the nationalist, communist, authoritarian and totalitarian creeds of [Berlin’s] day’. Negative liberty also better provides for individuals to achieve the forms of life prescribed by positive liberty than positive liberty does itself, as it allows individuals to act with reference to their own chosen ends and grow in their own specific way to achieve self-actualisation.

Finally, negative liberty is simply more intuitive than positive liberty and better aligns with our understanding of what liberty actually is. It is therefore the theory of negative liberty that we shall use in Part 2 of this article to analyse reductions of liberty in the modern society, before considering whether these reductions are justified and when these reductions are justified. daniyal vemuri

Furthermore, it is ambiguous as to what constitutes self-determination for a society.

Although we would most likely say that self-determination is achieved via democracy, it is possible to argue that self-determination within a society can only be achieved via bringing together its constituent parts under a rational plan in order to meet a collective good – this sounds ominously similar to rhetoric used by certain characters in the 1930s.

The largest flaw with positive liberty is that it prescribes a form of life to be lead, a rational life. Positive liberty distinguishes between the rational selves of individuals and the irrational selves of individuals (i.e. between reason and emotion/desire/urges).

Positive liberty values the rational self over the irrational, and argues that you are positively free when the rational self is free from influence from the irrational. Some people are more rational than others, in that they are better able to curtail their urges and act according to their rational self. These people know what is rational better than others, and so these people under positive freedom would be justified in compelling those less rational than themselves to do the rational thing which leads to self-actualisation and freedom from interference by the irrational self. Arguably, these more rational people would be required to exert authority and intervene in others’ lives in order to develop and maintain liberty.

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Five minute read

Embracing Nihilism

Nihilism is the philosophical stance that rejects general or fundamental aspects of human existence, such as objective truth, knowledge, morality, values or meaning. Nihilists do not believe that there is any objective meaning of life. Nihilists do not believe that there are any moral values, and reduces human existence to mere coincidence, claiming there is no big reason why we are all here. Nihilism is commonly regarded as negative and as leading to great pessimism. I argue differently.

Since the dawn of time humans have been questioning what the purpose of their existence is. Every religion, every misguided sacrifice and prayer, and every question for truth within the vast expanse of human history has been for the pursuit for the answer for this question.

Historically, religion has been the most popular method of satisfying this question, providing the ultimate source of meaning and guidance in a harsh and brutal universe.

Belief in a divine power provided comfort to believers, knowing there was a greater purpose for their lives. An omnibenevolent God meant there was a higher power beyond us who would protect us, even after death.

For centuries, the open question of the purpose of our existence was filled, through a supreme, omnipotent, omnibenevolent God. We have long since left this period in humanity’s history. Granted, religious beliefs prevail throughout the world, the world is slowly starting to shy away from religion, in favour of science and atheism. Religious beliefs are beginning to decay, and humanity’s unbroken trust in a higher power is rotting away.

German philosopher, Friedrich Nietzsche, saw nihilism as the outcome of this decaying of religious belief. He diagnosed nihilism as a latent presence within the very foundations of European culture and saw it as a necessary and approaching destiny. Nietzsche believed this process would result in humanity devolving into the ‘last man’ – a society of archetypal passive nihilists, who take no risks, seeking only conform security, and ultimately resulting a widespread depression.

In the space of merely one century, we have realised very quickly just how minuscule we are on the universal scale. Scientific discovery and the rise of atheism is causing a sort of ideological crisis. In the face of inescapable facts, and in the increasing absence of a higher power to trust in, now more than ever are people turning to nihilistic pessimism as their only resort. A bleak, but logical outlook on life that believes in what is superficially obvious and tells them that nothing matters.

Does this mean we should go back to believing the lies of religion? I do not believe so.

What we need to do as a society is to embrace nihilism. Not in the pessimistic way, which many attempt to describe nihilism as, but in an optimistic way.

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Twenty-three minute read

There is no meaning of life. There are no moral truths. There are no objective values. We may not even have free will.

The Continuum Hypothesis

and the limits of knowledge in mathematics

So what?

Every day you live is a slap to the face to the likes of probability and the 2nd law of thermodynamics. Every day you live is a chance to experience all that life has to offer – have fun, make the world a better place, and do what you love. We are so minuscule in the vast expanse of the universe, which is what makes this all crazier. Humanity has overcome so many challenges. Humanity has discovered so many scientific breakthroughs, and technological advancements. All on a small, insignificant, blue planet, we have done so many incredible things. Yes, there is no objective meaning of life, but we are free to create meaning for ourselves, and do what we please and what makes us happy. Even if free will does not exist, we are experiencing all of this, and that is all that matters.

So, progress and live your life with internal purpose. It is an incredible and beautiful opportunity. ryan lin

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When we first learn to count, we might know all the numbers up to about ten or twenty.

Then, we learn up to a hundred, a thousand, a million, and beyond. It really begs the question: can we go on forever? Is there a “biggest number”? Of course, in the counting numbers we can continue for as long as we like, but still, we often use both in mathematics and daily life the concept of that which is greater than any conceivable number or quantity – the concept of infinity.

Infinity has been in the psyche of mathematicians (and humanity as a whole) for as long as we have been counting and measuring things. What if there were infinitely many stars in the sky? What if there was a city bigger than any other could possibly be? What if we could live forever?

Hotel Infinity

To understand the counter-intuitive nature of infinity as it appears in mathematics, we begin with a famous thought experiment introduced by David Hilbert in 1924. When an ordinary hotel is full, it has no more rooms to hold any guests, and any new arrivals must be turned away. No matter whether the hotel has ten or ten million rooms, once it is full, it is full.

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Hilbert’s Hotel Infinity, however, can happily welcome a new guest, even when every single room is occupied. All the hotel staff need to do is ask the guest in room 1 to move into room 2, the guest in room 2 to move into room 3, and in general ask the guest n to move into room n+1. Then, with room 1 empty, a new space is created for the new guest, and the hotel returns to being full.

The total number of guests at the hotel before and after seems to be equal: this suggests that if we were to add one to an infinite quantity, the result would be unchanged – that is, “infinity”. We might tentatively write this as the equation ∞ + 1 = ∞. (Of course, we must keep in mind that it is technically inaccurate to include infinity in an equation in this way, because as we have just seen, infinity does not behave as normal numbers do).

The same procedure can be applied with any finite number of guests. For a finite number k of new guests, each guest in room n is moved into room n+k. More surprisingly, the Hotel Infinity can take in infinitely many guests at once. This time, the staff ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 4, and in general the guest in room n to move into room 2n. This leaves all the odd-numbered rooms open for the new guests. Here we might write that ∞ + ∞ = ∞.

For the last trick the Hotel Infinity has to offer in this section, we will show that infinitely many coaches with infinitely many guests in each can be accommodated: informally, ∞ x ∞ = ∞. This can be done in two ways. Firstly, consider a two-dimensional grid, with each row representing a coach (we can treat the original hotel guests as the first “coach”) and each guest listed in order in each row. Then, we make a line starting with the first guest on the first coach, then the second guest on the first coach, then the first guest on the second coach, and so on, as shown in the diagram. This makes a countable list of all the guests, and each room in the hotel can be assigned in this order.

It is not just a coincidence that the two numbers for each guest resemble the numerator and denominator of a fraction. Indeed, we can think of the hotel as the collection of all natural (i.e. whole positive) numbers, with each room representing its room number, and each guest as a fraction with the numerator and denominator being equal to the number of the guest on the coach and the number of the coach respectively. This can be used to show that the size, or “cardinality”, or the natural numbers and the fractions are the same. (Though not shown here, it is not difficult to adapt this argument to omit fractions which can be simplified to a duplicate of another fraction on the grid; or to include negative fractions and negative integers).

For the second demonstration of this, we use the fact that every natural number can be uniquely factorised into a product of prime numbers. Then for guest number g on coach number c, we can assign them room 2g3c knowing that this room number is not shared by any other guest. However, in this arrangement there are still empty rooms in the hotel, meaning that the number of guests is less than or equal to the number of rooms.

However, it is simple to show that the number of guests is also greater than or equal to the number of rooms, implying that they are indeed equal.

This approach also naturally shows us a way in which we can extend our reasoning to show that the hotel can accommodate an infinite number of car ferries, each with and infinite number of coaches, each with an infinite number of guests (∞×∞×∞ = ∞), by assigning each guest to room 2g3c5f with g, c, and f being the number of the guest, coach and ferry respectively. More generally, we now know that the cardinality of an n-dimensional grid of natural numbers (for natural n) is equal to the cardinality of the natural numbers. Of course, this follows from repeated application of the first argument, but in the face of such a veridical paradox as the Hotel Infinity, it is reassuring that two separate lines of reasoning lead us to the same conclusion.

Achilles and the Tortoise

Before we continue to grapple with set theory in the 19th and 20th centuries, let us go back over two millennia to the ancient world and consider a philosophical paradox about a different kind of infinity.

Zeno (495-430 BCE) was a Greek philosopher who was described as “immeasurably subtle and profound” by Bertrand Russel, himself a pioneer on all manner of pure mathematics. Though none of his works survive, from what we know about him from Plato and Aristotle, he held eccentric beliefs about the nature of the world: for example, that motion, or indeed any form of change, is merely an illusion, and that the world cannot be subdivided into ideas such as “sky” or “earth” but exists as a single unchanging entity. To this end, he compiled many paradoxes about motion and change in an attempt to show their logical absurdity, of which Achilles and the tortoise is the most famous.

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Suppose that the legendary hero Achilles was to have a race against a tortoise. However, to make the race fairer, the tortoise was given a 10-metre lead. The question is, when will Achilles catch up?

Assume for simplicity that the tortoise moves at half the speed of Achilles. Then, in order for Achilles to catch up, he would have to run the first 10 metres to reach the tortoise’s original position. Then, during this time, the tortoise has moved 5 metres, so Achilles must run a further 5 metres in order to catch up. Then, during this time, the tortoise has moved 2.5 metres, so Achilles must run a further 2.5 metres, and so on. However many times Achilles catches up, the tortoise goes just a little bit further, continuing the cycle forever. Achilles would have to catch up an infinite number of times in order to pass the tortoise. Surely you can’t do an infinite number of things in a finite amount of time?

To understand this further, let’s model the race as an equation. Let t equal the total time to catch up. Then for the first 10 metres, say Achilles takes 1 second. Then, for the next 5 metres, he takes 0.5 seconds, and the next 0.25 seconds, and so on. Then t = 1 + 0.5 + 0.25 + 0.125 + …

This is called the sum of an infinite series. This series has an important property. Adding the first four terms, we get 17/8; adding the first ten, 1511/512. As we add more terms, the series gets closer and closer to 2 without limit – it converges to 2.

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But how can this be shown mathematically? We can start by inspecting the series and noticing that each term is equal to the previous term divided by 2. Then we can divide t by 2: = 0.5 + 0.25 + 0.125 + 0.0625 + ... and see that it shares terms with the original. Then, subtracting them from each other, we get t - 1/2t = (1 + 0.5 + 0.25 + …) - (0.5 + 0.25 + …) = 1 t=2 This is known as the method of differences.

What may make Zeno’s argument strangely plausible is that if we were to try and list out each point Achilles must reach, we would indeed be there forever. The key difference is that doing so takes roughly the same amount of time every time, while Achilles’ tasks soon take only milliseconds to do. Series like 1 + 1 + 1 + … are known as divergent, because they grow ever bigger without limit. Series like 1 + 1/2 +1/4 + 1/8 + … are known as convergent, because they grow ever closer to a fixed value. Additionally, there are series that neither converge nor diverge, like 1 - 1 + 1 - 1 + …, which fluctuates forever between 1 and 0.

Well after Zeno’s paradoxes, series became an important part of mathematics. In some cases, it can be much more difficult to know whether a series is diverging, converging, or neither. Take the harmonic series: 1 + 1/2 + 1/3 + 1/4+ … Does this series converge?

Adding the first ten terms, we get just under 3; adding the first hundred, just over 5; the first thousand, around 7.5. Intuitively, it seems so: the rate of growth is certainly slowing down. But around 1350, Nicole d’Oresme realised that it actually diverges. The reasoning is as such: after the first 1, split the series into groups of 1, 2, 4, 8, and so on: (1) + (1/2) +(1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + …

Each bracket is greater than or equal to 1/2: (1/3 + 1/4) is bigger than 1/4 + 1/4= ; (1/5 + 1/6 + 1/7 + 1/8) is bigger than (1/8 + 1/8 + 1/8 + 1/8), and so on. In this way the series has been split into blocks of size ½ or more, which can be summed to reach any value without limit.

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This surprising result only holds because of the exceptional slowness at which the harmonic series diverges. In fact, to reach 100, we need to sum over 15 tredecillion (1.5 x 1043) terms!

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1

2

3

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Zeno was not only thinking about heroes and tortoises here: he argued that using the same argument he could show that not only would Achilles never catch up to the tortoise, but he would never even finish a race on his own, because to reach the finish line, he would first have to reach the half-way point, and to do so he would first have to reach the quarter-way mark, and the eighth-way mark, and so on. It may come as no

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The properties of infinity discussed so far largely match with our expectations: no matter what you add, subtract, multiply or divide by, infinity remains the same. But it was one of Hilbert’s contemporaries, Georg Cantor, who showed that this assumption breaks down in a spectacular way.

The rooms of the Hotel Infinity represent elements of a set – a mathematical name for a collection of objects, like the set of prime numbers, the set of cities in the UK, or the set of human beings. As long as a set is defined consistently, a set can contain any and any number of things.

The key idea behind Cantor’s work was to compare the sizes, or “cardinalities” of sets, by using functions mapping one set to another. If all the elements of set A can be mapped onto individual elements of set B (an injective function), then the cardinality of A is less than or equal to the cardinality of B. If two sets can be mapped onto each other - in other words, there is a one-to-one (bijective) function between the elements of two sets, then the cardinalities of the two sets are equal.

8, 8, 8, 8,

.

.

T

4, 5, 6, 7, 4, 6, 4, 5, 7, 4, 5, 6, 7, 4, 6, 7, 5, 6, 7, 4, 7, 5, 6, 4, 5, 6, 4, 6,

} 9, 10, 11, . . .} 10, . . .} 10, . . .} 9, 11, . . .} 9, 11, . . .} 9, . . .} 11, . . .} 9, 10, . . .} 9, 10, 11, . . .} 9, 11, . . .}

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This applies naturally to finite sets: the sets {1, 2, 3, 4, 5} and {6, 7, 8, 9, 10} both have a cardinality of 5, and the function f(x) = x + 5 is a bijection between them. The Hotel Infinity itself represents the set ℕ of all natural numbers, and we saw previously that its cardinality is unchanged by adding or removing finite numbers of elements, adding the elements of ℕ to itself, or taking the Cartesian product of ℕ with itself (a 2-dimensional grid of natural numbers).

f(1) = {1, f(2) = {1, 2, 3, f(3) = { 2, 3, f(4) = {1, 3, f(5) = {1, 2, f(6) = { 3, f(7) = {1, f(8) = { 3, f(9) = {1, 2, f(10) = {1, 2, f(11) = {1, 2,

...

To Infinity and Beyond

Cantor argued that the cardinality of the power set of a set is strictly larger than the cardinality of the set itself, and that this property extends even to infinite sets. He did this by showing that for any function f(s) from S to P(S), there exists a member T of P(S) that is not in the image of f(s) – that is, there is no s in S such that f(s) = T. Consider the set T = {s ∈ S: s ∉ f(s)}, with ∈ and ∉ meaning “is a member of ” and “is not a member of ” respectively. For all s in S, either s is a member of T or it is not. If it is, then by definition s is not a member of f(s), and so T is not equal to f(s). If on the other hand it is not, then by definition s is in f(s), and so again T is not equal to f(s).

... ... ... ... ... ... ... ... ... ...

influence the mathematics we do today: when Newton and Leibniz later analysed motion through infinitesimally small changes, it would become what we know now as calculus.

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, . . .}

This changes when we introduce the idea of a power set. A power set P(S) of a set S is the set of all subsets of S. For example, the power set of the set {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}.

This is known as the “diagonal argument”, since for any list of f(s) attempting to map S to P(S), we can choose T such that the first element is opposite to f(1), the second opposite to f(2), and so on in a diagonal line.

Here ∅, or {}, denotes the empty set: the unique set with nothing in it. The power set also contains the set itself. The power set P(S) can also be written as 2S: if the cardinality of a finite set S is n, then the cardinality of P(S) is 2n. More formally, define the indicator function of a subset A of a set S for x in S as 1 if x is in A and 0 otherwise; then for every subset A of S there is a corresponding indicator function IA of S. The set {0, 1}‌S, the set of all the functions from S onto {0, 1}, consists of all the indicator functions of all the subsets of S: in other words, it is equivalent or bijective to P(S). Since each element can be 0 or 1 under any function in {0, 1} (i.e. there are 2 possible options) the cardinality of {0, 1}S is 2n. Furthermore, since 2 can be written as {0, 1} in von Neumann ordinals, it is valid to write P(S) itself as 2S. In fact, in general, if XY is the set of all functions from Y to X then the cardinality of XY, |XY| = |X||Y|.

Applying this to the natural numbers, we know that the cardinality of the real numbers |ℝ| = |2ℕ|. We shall write c, for “continuum”, to mean |ℝ|. Here we will also introduce the “aleph numbers” using the first Hebrew letter aleph ℵ to denote an ordered sequence of cardinal numbers (transfinite numbers used to describe the cardinality of sets), with the first being |ℕ| = ℵ0. Then, as a real number can be split into an integer and a decimal expansion (a countable sequence of digits), c ≤ ℵ0 · 10ℵ0 ≤ 2ℵ0 · (24)ℵ0 = 2(ℵ0+ 4 · ℵ0) = 2ℵ0.

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On the other hand, by considering 2 arbitrary digits, say 4 and 9, and seeing that the cardinality of the real numbers with decimal expansions only containing 4 and 9 is equal to 2ℵ0, it is clear that 2ℵ0 ≤ c. Thus:

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On the other hand, by considering 2 arbitrary digits, say 4 and 9, and seeing that the cardinality of the real numbers with decimal expansions only containing 4 and 9 is equal to 2ℵ0, it is clear that 2ℵ0 ≤ c. Thus: |ℝ| = c = 2ℵ0 = 2|ℕ| and furthermore ℵ0 < c.

Any attempt to map the natural numbers to the real numbers – or equivalently, create a countable list of real numbers – fails because we can choose a real number such that the first digit differs from the first number, the second digit differs from the second number, etc. This new number is not on the list, and so the list is incomplete.

P(P(ℕ)) (equivalent to the set of functions from ℝ to ℝ), then P(P(P(ℕ))) (or P3(ℕ) for short), P4(ℕ), and so on. Here we can introduce “beth numbers” using the second Hebrew letter beth ℶ, where ℶ0 = ℵ0 and ℶn+1 = 2ℶn. For example, |P3(ℕ)| can be written as ℶ3.

Similarly to the cardinal numbers seen so far used to describe the size of sets, Cantor developed the system of ordinal numbers used to show the arrangement, or order, of numbers within infinite sets. It may seem unnecessary to have two different transfinite number systems for these purposes, as in the finite numbers we can determine the arrangement of a set of objects just by counting them, but this does not work so easily with infinite sets. The beth numbers can be defined on a limit ordinal, which is an ordinal that cannot be reached by applying a “successor function” (e.g. ω in the diagram below, which cannot be reached from any natural number by adding 1) as follows: for limit ordinal λ, the beth number ℶλ is equal to the supremum (the smallest number greater than or equal to all members of a set) of all previous beth numbers ℶα for α < λ. Then we can skip infinitely many steps forward to ℶω, but even that pales in comparison to the repertoire of the modern-day set theorist. The entities they tackle gain descriptions such as “hyper-inaccessible”, being so large that we cannot even be certain that they exist.

Such is the incomprehensible nature of infinity.

Imagine that the Hotel Infinity once again had to take in new guests, but this time there are so many guests outside that they cannot make a queue, even if it is infinitely long. It might be possible to house them all at Hotel Continuum, but here the standard Hotel Infinity has reached its limit: no matter how cleverly you assign rooms, the vast majority will have to be turned away.

This is a highly surprising result: not only did Cantor seem to shatter the conception of infinity as an absolute, unchanging monolith, but he showed that through the process of the power set, an infinite ladder could be constructed of higher and higher infinities.

Starting with ℕ, we can then take P(ℕ) (equivalent to the set of real numbers ℝ), then

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Deus Ex Infinito

Cantor himself believed in an Absolute Infinite, which he denoted as Ω, bigger than any quantity finite or transfinite. He associated this with the idea of God, and wrote about it as such: “The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extraworldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or order-type. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.”

However, in Cantor’s letters to Richard Dedekind he says that Ω as an ordered sequence of all ordinal numbers cannot be consistent. Take Ω’ to be Ω with 0 added: Ω’ = 0, 1, 2, 3, … ω, ω+1, … Then if Ω’ were consistent, then as a set it would correspond to some number δ, which would be greater than any number in Ω; however, as Ω contains all ordinal numbers, this implies that δ > δ, which is a contradiction.

It seems paradoxical that the set of all ordinal numbers is self-contradictory. We said before that a set can be anything, so long as it is defined consistently. But here that approach leads to difficulty. For now, we will note that in the time since different axiomatical systems restrict the formation of sets in various ways to avoid this problem.

We can instead define a class as a collection of sets sharing a specific property. However, this approach somewhat leaves the philosophical question unanswered. Cantor himself rejected the idea of a set of all sets or an “organic totality”, and asserted that the Absolute Infinite he spoke about could not be adequately described by mathematics.

There is an important distinction to be made between the “infinity” discussed before used in calculus, series, limits, etc. and the infinities of set theory. In the former, we do not treat infinity as a constant entity to be manipulated algebraically, but merely to mean “unbounded” or “continuing forever” or similar. Conversely, the latter denotes infinite entities treated as full, actual objects. We can distinguish the two different meanings by use of the terms “potential infinity” and “actual infinity”. These concepts have been regarded as separate since the time of the Greeks, when they were known as “apeiron” and “aphorismenon” respectively. Aristotle described potential infinity as “the opposite of what people say it is; it is not ‘that which has nothing beyond itself ’, but ‘that which always has something beyond itself ’”. He thought that the existence of an actual infinity was impossible because it would have infinite magnitude and be “bigger than the heavens”.

The mathematical community continued to largely oppose the notion of actual infinity until well after Cantor published his work on set theory.

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Cantor wrote himself that “in the Middle Ages all scholastic philosophers advocated Aristotle’s ‘infinitum actu non datur’ [actual infinity does not exist] as an irrefutable principle”. There were some notable exceptions, such as Galileo Galilei, Gottfried Wilhelm Leibniz, and Bernard Bolzano, who first introduced the notion of a set.

Nevertheless, the well-known quote from Gauss defined the attitude of the time:

“I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.”

In time, though, the opinion of the community came around to Cantor’s new set theory, owing to the fact that transfinite arithmetic operations and expressions can be used in a variety of theorems in a consistent and meaningful way. One may write a symbol ω with the meaning of “actual (countable) infinity”, add it to sets, define operations on it, and make valid claims such as “ω < ω + 1”. It became of less importance what people with an intuitionist or finitist leaning actually thought about the validity of a “complete infinity”. in the words of Hilbert, “No one shall expel us from the paradise which Cantor has created for us”.

The Liar

In the exploration of infinity, we touched a few times on the underlying logical systems that determine the mathematical objects we can construct and the logical inferences we can make. In laying out such a system it is paramount that there are no contradictions: our axioms (initial assumptions) must form a clear and consistent network of truths if they are to be of use to us.

The simplest and oldest recorded “paradox” we know of can be attributed to Eubulides in the 4th century BCE. It consists of the simple statement: “This is a lie”. With it, the notion that any statement could be neatly categorised into “true” and “false” was dismantled.

If we take it to be true, then it implies that it is false; and if it is false, then it implies that it is true. The heart of this paradox lies in its self-referential nature. We can adapt this in various ways, e.g: • The following sentence is true • The previous sentence is false but the premise remains the same. The Liar appears in many forms, and multiple times throughout mathematical history it threatened to render useless the life’s work of countless mathematicians.

In the late 19th century, Georg Cantor’s work on infinity utilised the concept of sets to show the existence of many separate infinite cardinal numbers. In a naïve conception of

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set theory, we can make any set we like, so long as we define it properly. One might even consider the set of all sets. Now, being itself a set, this set must also contain itself as a member. It was Bertrand Russell (1872-1970) who realised that this self-referential property could give rise to the same paradox as The Liar. This new paradox took the form of a set of all sets which do not contain themselves. The question now was: does this set contain itself ?

If it does, then that means it is a set which contains itself, and so it cannot be in the set.

If it does not, then that makes it a set which does not contain itself, and so it must be in the set. Russell himself made an analogy to illustrate this. Suppose that in a village, some men shave themselves, and the rest are shaved by the barber. In other words, the barber shaves every man who does not shave himself. Does the barber shave himself or not?

Russell realised that his paradox was about more than Cantor’s infinities.

At the same time as Cantor’s work on sets, higher mathematics was undergoing the process of scrutinising and redefining the foundations from which all proofs were derived. There was an anxiety that, were one to simply assume the basic mathematical truths, one day a discrepancy would be found, and generations of proofs would have to be re-done or abandoned. To this end, mathematicians defined a few completely abstract, basic axioms, and from there reconstructed our familiar assumptions in excruciating logical detail. There is somewhat a sense of irony when, after a few hundred pages of dense logical symbols, the authors of the Principia Mathematica conclude that 1 + 1 = 2.

From there, mathematicians took a few separate approaches to avoid the paradoxes inherent in unrestricted sets. Russell himself decided to instead use the notion of a type – like a set, but having a property called a level, such that a type of level 0 behaves like a normal set of objects, and a type of a higher level may only contain types of a lower level than itself. Consequently, no type may ever contain itself. In collaboration with mathematician and philosopher Alfred North Whitehead, Russell set to work on a new project, the aforementioned Principia Mathematica, a three-volume epic even more inaccessible than that of Frege.

There are other approaches, with the most well-known of all being ZFC. ZFC stands for Zermelo–Fraenkel set theory, with C standing for the additional axiom of choice.

Today, ZFC is the most common form of axiomatic set theory and as such is used as the basis for the vast majority of mathematics. The primary aim of ZFC is to formalise a single concept: that of a hereditary, well-founded set. A hereditary set is a set where each element is also a hereditary set: that is, it is either the empty set ∅ or some set formed by taking sets of combinations of ∅ such as {{{∅}, ∅}, ∅}. A well-founded set is a set S such that every subset of S has a least element (for some defined definition of a “least” element).

One list of axioms of ZFC is as follows [from Kunen, 1980]: Axiom of extensionality: Two sets are equal (the same) if they have the same elements.

Axiom of regularity: Every non-empty set x contains a member y such that x and y are disjoint sets.

Axiom schema of specification: A set can be formed from the elements of another set satisfying a defined property.

Axiom of pairing: For sets x and y, there exists a set containing x and y as elements.

Axiom of union: For any set of sets x, there is a set y containing the elements of the elements of x.

Axiom schema of replacement: The image of any set under any definable mapping is also a set.

Axiom of infinity: There exists a set with infinitely many members. (More formally, if S(w) denotes the union of a set w with {w}, then there exists a set X such that ∅ is a member of X, and for any set y in X, S(y) is also in X.) Axiom of power set: For any set x, there exists a set y containing every subset of x.

Axiom of choice: Given any collection of nonempty sets, it is possible to select exactly one element from each set, even if the collection is infinite.

One such work in particular was Gottlob Frege’s “Grundgesetze der Arithmetik” (The Basic Laws of Arithmetic). Using only sets of objects, it may seem difficult to define an abstract concept such as the number 2. Frege’s basic idea was to recognise that the primary use of numbers was to count: in other words, to find the size of a set. It is comparatively easy to find examples of two books, two socks, two apples, etc. He defined the number two as the set of all such pairs, and painstakingly showed that using such an approach, all the normal arithmetic laws held. It was a tragic blow when Russell’s paradox tore a hole in Frege’s work.

Note that the axiom schema of specification avoids Russell’s paradox by making the restriction that a set of elements satisfying a property must be formed from the elements of another set. This is in contrast to the axiom schema of unrestricted comprehension saying that there exists a set consisting of any members whatsoever satisfying a certain property, which was used implicitly in naïve set theory and explicitly by Frege before being shown to be logically inconsistent.

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Given the first eight axioms, the axiom of choice is also equivalent to the well-ordering theorem, which states that for any set X, there exists a binary relation that well-orders X: that is, every set is well-founded. Furthermore, it is equivalent to Zorn’s lemma, named after Max Zorn, stating that a partially ordered subset containing upper bounds for every totally ordered subset has at least one maximum element. The equivalence of the three under ZF leads to the well-known joke about the intuitiveness of each result: “The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?”

Many have criticised the axiom of choice for the fact that it merely asserts that it is possible to choose an element from each set, without providing any way to do so. If the number of sets is finite, or there is some selection rule that can be found, then a selection can be made without the use of the axiom of choice. However, there exist models of set theory in which the axiom of choice implies that the real numbers are well-ordered but no such ordering is definable. Bertrand Russell used the following analogy: given infinitely many pairs of shoes, it is easy to choose one from each pair – one could choose the left shoe from each pair, for example. In fact, there are uncountably many ways to choose. However, there is no obvious way to choose one sock from an arbitrary pair of socks without invoking the axiom of choice.

Despite these drawbacks, there are many cases in which the axiom of choice is vital or simply convenient in proving some standard theorems, such as that every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact. As such, it is generally accepted in modern set theory.

Hilbert’s Program

“For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either. ... The true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish Ignorabimus, our credo avers: We must know. We shall know!” - David Hilbert, 1930 The work of Frege, Russell, Peano, and many more certainly showed a new interest in the foundations of arithmetic, but it was Hilbert who named their ultimate aim: a completely logical foundation on which to build mathematics. This would comprise a set of underlying axioms, along with rules governing exactly what logical steps are valid.

This system should be consistent, meaning that it does not lead to any contradictions, and, more ambitiously, it should be complete, meaning that every logical statement that could be formulated in the system can be proved to be true or false. It was a truly great aim, but with the publishing of the Principia Mathematica and the successful avoidance of Russell’s paradox, it seemed attainable. This time, it was Kurt Gödel who would show that The Liar would return in a way that no new axioms could avoid.

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Put simply, Gödel realised that the same self-referential property that leads to The Liar exists in the language of set theory. This can be applied to the problem of completeness: “This statement has no proof.” It was first published as a formal theorem in Gödel’s 1931 paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I”, and later became known as Gödel’s first incompleteness theorem.

It states: “Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.” The statement in question is referred to as “the Gödel sentence”, though there are infinitely many unprovable statements formed by the conjunction of the Gödel sentence with any other valid statement.

To prove his first incompleteness theorem, Gödel developed the notion of a Gödel number, which is a number formed from encoding a string of logical symbols into a number in much the same way as strings ASCII characters are encoded in computers.

He then demonstrated that the notion of provability within a system could be expressed using arithmetical functions on Gödel numbers. The Gödel sentence states that, when a particular sequence of steps is used to construct a sentence, that sentence will not be provable. In fact, the constructed sentence is precisely the Gödel sentence itself. When this is converted to an assertion about the properties of its Gödel number, it is clear that it must be provable if the system is complete.

Here the problem arises: if the system is complete, the sentence is provable. Yet the sentence itself states that it is unprovable. The only possible conclusion is that every system which is sufficiently advanced to express the theorems of mathematics contains theorems which cannot be proven.

One objection to this result might be that the Gödel sentence claims that it is unprovable, and we end with the conclusion that it is indeed unprovable, so it must be true. However, since the Gödel sentence cannot specify how it is intended to be interpreted, its truth may only be deduced by meta-analysing it outside its given logical system. One might also ask why The Liar cannot be stated directly as a paradox within the logical system. The answer follows from Tarski’s undefinability theorem stating informally that arithmetical truth cannot be defined within arithmetic. Thirdly, an objection could be made that the Gödel sentence could simply be included as an axiom in the system, and therefore it would cease to be incomplete. However, note here that Gödel’s theorem applies to any sufficiently advanced system, and so adding another axiom to a system merely creates a new system with no less complexity, where the theorem can be applied once again.

Not to do things half-way, in a later theorem in the same 1931 paper, which later became known as Gödel’s second incompleteness theorem, Gödel showed that in general the consistency of a system’s axioms is unprovable within the system. For a system F, let the statement Cons(F) denote the statement “there does not exist a natural number coding a formal derivation within the system F whose conclusion is a syntactic contradiction.”

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Gödel showed that the truth or falsity of Cons(F) was not provable from the axioms of F itself. He had now proved that any system which hoped to describe the whole of mathematics was neither consistent nor complete. Hilbert’s dream was shattered. At the lecture where his result was announced, the great mathematician John von Neumann was reported to say, “It is all over”. In retrospect, it was only just beginning.

The Continuum Hypothesis

The continuum hypothesis, or CH for short, is a theorem first advanced by Georg Cantor in 1878. It states that there is no set whose cardinality is strictly between that of the natural numbers and the real numbers. In ZFC, this is equivalent to the statement 2ℵ = ℵ1.

It was the first of the 23 famous problems presented by Hilbert in 1900, with the second being to prove that the axioms of arithmetic are consistent, and many other problems at the edge of the field at the time. But as with the second problem, the continuum hypothesis was eventually found to be undecidable in ZFC alone. This was due to the combined work of Gödel and Paul Cohen. First, in 1940, Gödel proved that CH cannot be disproved in Zermelo-Fraenkel set theory, whether the axiom of choice is included or not. This was done by proving that CH and AC (the axiom of choice) both hold in the constructible universe L, an inner model (substructure) of ZF. The existence of this inner model shows that these additional axioms are consistent, if and only if ZF itself is consistent. Of course, due to Gödel’s incompleteness theorems, this cannot be shown directly, but within stronger systems it can be proved and many believe it to be true.

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one sense, in the most important axiomatical systems used today in mathematics, we can definitively say that it is true, false, or undecidable. The mathematician and philosopher Joel David Hamkins advocated for a multiverse approach towards the matter in 2012: “I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for [by the universists].” On the other hand, for all the systems we have explored, this idea of the multiverse suggests a plethora of uncharted ones for which we can continue to discover new properties of the continuum hypothesis, the generalized continuum hypothesis, and more.

The existence of undecidable statements remains a source of discomfort for mathematicians today; what if the problem they spent their career working on turns out to be impossible to solve? It is not unreasonable to have such concerns, considering the problems that have existed in one form or another for thousands of years but remain unsolved to this day; to name but one example, those concerning the distribution of prime numbers. But although our attempts to consolidate mathematics into one unified system turned out to be futile, do not be discouraged: the multiverse that we have discovered since is a rich and open world with much left to explore. lily hillman

The second half of the proof was completed in 1963, by Paul Cohen. He developed the technique of forcing, which essentially constitutes expanding a set theoretic universe (a collection of all the relevant mathematical entities in a given situation) to include new sets which may violate the continuum hypothesis, in much the same way as new guests may be accommodated in the Hotel Infinity. By doing so, he showed that ZFC was consistent with the negation of the continuum hypothesis, and as such that CH was completely logically independent of ZFC. For his work, Cohen was awarded the Fields Medal, the mathematical Nobel Prize, in 1966.

For what it is worth, Gödel himself believed CH to be false, and that the failure of ZFC to be able to prove so was merely a failure of its axioms to fully characterise the universe of sets. Cohen, though less willing to freely assert such ideas, tended to reject it. In general, those who viewed universes of sets as controlled structures preferred to accept CH, while those who viewed them as more organic preferred to reject it. In fact, there are many intuitive arguments for and against CH, and many different axiomatic systems for which it is both true and false. There are even stronger theorems, such as the generalised continuum hypothesis, claiming that 2ℵα = ℵα+1 (or that ℵα = ℶα) for all ordinals α, and systems for which they hold, which remain of interest to set theorists today.

The question of whether the continuum hypothesis has been solved is hard to answer: in

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