You can stop reading now, if you want. Or can you? Are your decisions really your own, or are you and all other human beings merely spectators in the mind-arena, observing but neither influencing nor initiating what goes on there? Are all your apparent choices in your brain, but out of your hands, made by mechanisms beyond, or below, your conscious control?

In short, do you have free will? This is a big topic – one of the biggest. For me, the three most interesting things in the world are the Problem of Consciousness, the Problem of Existence and the Question of Free Will. I call consciousness and existence problems because I think they’re real. They’re actually there to be investigated and explained. I call free will a question because I don’t think it’s real. I don’t believe that human beings can choose freely or that any possible being, natural or supernatural, can do so. And I don’t believe we truly want free will: it’s an excuse for other things and something we gladly reject in certain circumstances.

Brain on the Plain

Free will is one of the Big Three Philosophical and Scientific Topics, but I’ve never seen it looked at mathematically. Nevertheless, it’s easy to see maths there. For one thing, choices are often binary:

To be, or not to be – that is the question:
Whether ’tis nobler in the mind to suffer
The slings and arrows of outrageous fortune
Or to take arms against a sea of troubles
And by opposing end them. To die, to sleep –
No more – and by a sleep to say we end
The heartache, and the thousand natural shocks
That flesh is heir to. ’Tis a consummation
Devoutly to be wished. To die, to sleep –
To sleep – perchance to dream: ay, there’s the rub,
For in that sleep of death what dreams may come
When we have shuffled off this mortal coil,
Must give us pause. (William Shakespeare, Hamlet, 1623)

“To be or not to be”: it’s a binary choice. But binary or not, choices are influenced by knowledge and beliefs: in short, by information. We don’t know what dreams may come, so we must pause for thought: give more time to the electro-chemical patterns in the brain that underlie thought and that encode information. The brain is a mechanism for processing patterns and altering them according to rules. But a mechanism is itself a pattern. So are rules. The brain is a semi-permanent electro-chemical structure through which ephemeral electro-chemical patterns flow, shaping and being shaped by what they flow through. These ephemeral patterns – data supplied by the sense-organs, memories retrieved from storage – are like rain falling on a landscape: the rain follows the contours of the landscape but also alters those contours.

Rain creates channels, in other words, and the result may be a river with tributaries, sub-tributaries and sub-sub-tributaries (and more). This is a particular kind of pattern: a fractal, a complex and often beautiful mathematical structure created by the interaction of mindless matter. But is neurology, the behaviour of cells in the brain, effectively equivalent to hydrology, the behaviour of water on landscapes? That is, are thoughts and actions ultimately dependant on nothing but the interaction of mindless matter obeying inexorable physics? Is the mechanism running on its own? Is it all bottom-up and not at all top-down?

Satan’s Supernature

It certainly doesn’t feel that way: we do seem in conscious control, but feelings aren’t proofs. For a start, we don’t know how the mechanism of a choice works, because we don’t understand the brain. Almost all of what happens there is opaque to consciousness, which is why subliminal signals or post-hypnotic suggestions can shape our behaviour without our being aware of it. It is certain, then, that apparently free choices can be unfree, determined by factors we are unaware of. But are they ever free at all? Does “free will” really mean “opaque will”? That is, does the subjective impression of freedom in a choice really depend on our ignorance of everything that shapes the choice? That is one possibility. It may be that “free will” is an impossible concept, like “square triangle”. Some philosophers and theologians accept that, yes, free will is impossible in an entirely material universe. They claim it must therefore be a supernatural phenomenon. But the supernatural is no way to rescue the concept of free will. In fact, it compounds the problem. Free will depends on a mechanism of choice and that mechanism is always material in some sense: a more or less permanent structure accepting, processing and outputting information, i.e., making a choice.

In Christian legend, Satan’s choice was to rebel against God. If so, his supernatural mechanism-of-choice was eternal and indestructible, composed of permanent spirit, not transient flesh. In a supernatural Satanic brain, free will becames harder to explain, not less. Supernatural explanations are a kind of psychological accounting trick, because they implicitly contrast the solidity of matter with something ghostly and insubstantial. In fact, it’s the other way round: transient earthly matter is ghostly and insubstantial beside eternal heavenly (or hellish) spirit. But the physical or super-physical nature of the mechanism isn’t fundamentally important, other things being equal. What matters is its structure and the way it shapes the information flowing through it. “1 + 1 = 2” whether one is using an abacus, a computer or simply a brain. However, the physical nature of the mechanism can obviously influence the result of a calculation or a choice. Brains can tire and make mistakes in ways computers cannot.

Fleas and Lesser Fleas

And different kinds of brain can make different kinds of choice. For example, men commit much more crime than women and blacks commit much more crime than whites. Whether we explain these patterns by nature or nurture or both, we are ultimately making statements about the physical nature of the brain as it makes a choice. Although hereditarians explain that physical nature by genes, all hereditarians accept that environment is important too. However, many environmentalists disregard genetics and explain all differences between human groups by environment alone. In either case, mechanism shapes mechanism: genes and the environment shape the brain. There is no escape from mechanism, no escape from entities that contain and process other entities:

So, Naturalists observe, a Flea
Has lesser Fleas that on him prey.
And these have lesser Fleas that bite ’em:
And so proceed ad infinitum.

Jonathan Swift, “On Poetry: A Rhapsody” (1733).

Will from Nihil

Another name for “mechanism” is “mathematics”. If we regard choice as mathematical, it may be possible to decide objectively whether free will is a meaningful concept. Is it possible to have a mechanism of any kind whose output is neither deterministic nor random? And if it is, does that rescue the concept of free will? In books like Erwin Schrödinger and the Quantum Revolution (2012), the physicist John Gribbin has claimed that quantum physics supports the concept of free will:

Classical science describes an utterly predictable world. … For centuries after Newton, classical science posed a real problem for anyone who believes in free will. In principle, if you knew the position and speed of every particle in the Universe, including the atoms we are made of, it would be possible not only to predict the entire future of the Universe, but to reconstruct its entire history in exquisite detail. Leaving aside the practical problems of actually doing this, it seemed to imply that everything, including human behaviour, was pre-ordained. But then came quantum physics. … What physicists painstakingly (and painfully) discovered … is that particles can behave like waves and waves like particles; that quantum entities can be in at least two places at once; that they can get from one place to another without passing through any of the space in between; and that there is no certainty in the quantum world, where everything depends on probabilities. … Great for restoring a belief in free will, but scarcely reassuring in any other way. (Op. cit., Introduction, pg. 16-7)

I don’t think it is great for restoring a belief in free will. Why is a probabilistic universe any friendlier to free will than a deterministic one? Is a completely dark room likelier to contain a square triangle than a brightly lit room? The argument from quantum physics reminds me of the argument from supernature: it’s a kind of psychological accounting trick, substituting vague notions for clear ones and assuming that this solves the problem. It doesn’t. The theoretical behaviour of sub-microscopic particles doesn’t explain how a brain can accept information, process it, and make a conscious choice that is neither determined nor random. Brains and behaviour are macroscopic. If the brain can magnify quantum effects, how does quantum indeterminacy rescue free will? It doesn’t: free will depends on control of one’s own mind and actions. In other words, free will has to be deterministic within the individual who employs it. He determines the choice within the limits set by his own physical nature and history.

Quantum indeterminacy would work against this individual control. It would make the brain’s operation unreliable, as though our decisions depended partly on the roll of sub-cellular quantum dice. In any case, is quantum indeterminacy real or apparent? The universe may be structured in such a way that perfect knowledge and perfect prediction are impossible, but it does not follow that the universe is indeterminate at a fundamental level. The problem of true indeterminacy is related to the problem of true free will and may actually be identical to it. Each seems to demand a creatio ex nihilo, a creation from nothing, an uncaused cause. That is, something must appear that does not depend on what precedes it. Yet an act of free will must relate to what precedes it. How can a brain process information and reach a decision that is independent of the brain, the information and the processing? As I’ve pointed out above, the supernatural is no solution. Free will in a deity’s or an angel’s brain is more difficult to explain, not less. It is merely easier to take on faith, because more removed from our everyday experience.

The Witch in Winter

Furthermore, the more perfect the brain and its knowledge, the less room there is for free will. In a moral sense, God cannot, by definition, possess it: as a perfectly good being, He has no choice but to do what is good, based on the perfect knowledge available to Him. If free will is possible at all, it is possible only to imperfect beings with imperfect knowledge: only to angels and human beings, not to God. But the assumption that human beings possess free will raises all manner of philosophical difficulties. The Christian writer C.S. Lewis (1898-1963) made free will central to his series of books about the land of Narnia, but he doesn’t escape the philosophical difficulties. In The Lion, the Witch and the Wardrobe (1950), for example, the four Pevensie children, Peter, Susan, Edmund and Lucy, discover a way to enter Narnia through a magic wardrobe. It is “always winter and never Christmas” there, because it is under a spell cast by the evil White Witch. Edmund meets her by chance and she seduces him with Turkish delight. He allies himself with her and becomes deceitful and spiteful:

“I say,” began Edmund presently, “oughtn’t we to be bearing a bit more to the left, that is, if we are aiming for the lamp-post?” He had forgotten for the moment that he must pretend never to have been in the wood before. The moment the words were out of his
mouth he realized that he had given himself away. Everyone stopped; everyone stared at him. Peter whistled.

“So you really were here,” he said, “that time Lu said she’d met you in here — and you made out she was telling lies.”

There was a dead silence. “Well, of all the poisonous little beasts—” said Peter, and shrugged his shoulders and said no more. There seemed, indeed, no more to say, and presently the four resumed their journey; but Edmund was saying to himself, “I’ll pay you all out for this, you pack of stuck-up, self-satisfied prigs.” (ch. 6, “Into the Forest”)

So Edmund is blamed for his behaviour. Why? If all four children possess free will, any one of them might have behaved in a similar way if they had encountered the Witch as he did. But the other three treat Edmund’s behaviour as evidence of an imperfection that he should feel ashamed of and try to overcome. Furthermore, the Witch has tricked Edmund: he thought that by allying himself with her that he would get more Turkish delight. In fact, he gets dry bread when he next meets her. So his knowledge was imperfect. If he had known the Witch was lying, he wouldn’t have allied himself with her. The problem doesn’t disappear when Lewis claims that he should have known:

You mustn’t think that even now Edmund was quite so bad that he actually wanted his brother and sisters to be turned into stone. He did want Turkish Delight and to be a Prince (and later a King) and to pay Peter out for calling him a beast. As for what the Witch would do with the others, he didn’t want her to be particularly nice to them — certainly not to put them on the same level as himself; but he managed to believe, or to pretend he believed, that she wouldn’t do anything very bad to them. “Because,” he said to himself, “all these people who say nasty things about her are her enemies and probably half of it isn’t true. She was jolly nice to me, anyway, much nicer than they are. I expect she is the rightful Queen really. Anyway, she’ll be better than that awful Aslan!” At least, that was the excuse he made in his own mind for what he was doing. It wasn’t a very good excuse, however, for deep down inside him he really knew that the White Witch was bad and cruel. (ch. 9, “In the Witch’s House”)

Pain and Poison

What does “deep down inside him” mean? It means that the knowledge wasn’t easily accessible to his consciousness: it was deep, not at the surface. If Edmund fully knew that the Witch was bad and cruel, he would also know that she would not keep her promise of more Turkish delight. So he was acting against his own interests by helping her. In other words, he was being irrational, if we accept that he really knew that the Witch was “bad and cruel”. A rational Edmund, reasoning correctly from the Witch’s true nature, would have chosen differently. However, his actual choice is rationally explicable to outsiders: to the characters in the story and readers of the story. All of his behaviour can be explained: he isn’t acting psychotically or at random. If he were, his siblings couldn’t condemn him as they do. But again, if his actions are products of free will, why do they not accept that they could just as easily have done the same? As it is, they treat Edmund as imperfect: Peter calls him a “poisonous little beast”. But has Edmund chosen to be imperfect? Has he chosen to be younger than Peter and with a lower status that he resents? If he could have foreseen the trouble and misery his choices would bring him, he would not have made them. Not unless he was being perverse and irrational. So his morally bad choices are possible only because of his imperfection: his imperfect knowledge, imperfect foresight, imperfect reason and imperfect psychology.

Edmund is also condemned by Aslan, the giant golden lion who is an equivalent of Jesus Christ in the Narnia series. But if Aslan is incapable of acting imperfectly like Edmund, what basis does Aslan have for condemning him? Is Aslan not rather like a bird condemning a fish for being unable to fly, or a fish condemning a bird for being unable to breathe underwater? In fact, Aslan condemns Edmund for the same reason as Edmund’s siblings have condemned him: in order to punish him, that is, to discourage him from behaving immorally in future. And he won’t: he learns his lesson, because he wants to avoid future pain. Punishment is painful, as the etymology of “pain” suggests: it comes from the Latin poena, meaning “penalty”. In the rest of the book and the rest of the Narnia series, Edmund chooses good, not bad. This is rational, self-interested behaviour, but it can’t be called moral. Edmund knows not to trust evil and he wants to avoid the bad kind of pain, that is, the kind you don’t benefit by.

Passport of Pain

The good kind of pain, the kind you do benefit by, is another constant theme of Lewis’s writing, but this is because it’s part of his presentation of free will. His book The Great Divorce (1945) was written for adults and is about souls from Hell who visit the threshold of Heaven, where they are offered the chance to stay and accept God or return to Hell and reject God. Here is Lewis’s description of a fiery angel offering to help a soul make the right choice. The soul is tormented by a lizard symbolizing impure desire and the angel would like to kill the lizard:

“Get back! You’re burning me. How can I tell you to kill it? You’d kill me if you did.”

“It is not so.”

“Why, you’re hurting me now.”

“I never said it wouldn’t hurt you. I said it wouldn’t kill you.”

“Oh, I know. You think I’m a coward. But it isn’t that. Really it isn’t. I say! Let me run back by tonight’s bus and get an opinion from my own doctor. I’ll come again the first moment I can.”

“This moment contains all moments.”

“Why are you torturing me? You are jeering at me. How can I let you tear me to pieces? If you wanted to help me, why didn’t you kill the damned thing without asking me — before I knew? It would be all over by now if you had.”

“I cannot kill it against your will. It is impossible. Have I your permission?” (Op. cit., ch. 11)

The soul has a choice between salvation and damnation. If it accepts momentary pain, it will be saved and know eternal joy; if it rejects the pain, it will be damned and imprisoned for ever in misery. But it doesn’t understand the choice properly, because it doesn’t have full knowledge or understanding of the situation. It thinks the pain will be bad and, unlike the angel, doesn’t see the pleasure that lies beyond it. Lewis can only present free will on those terms: all the choices in The Great Divorce are made by imperfect souls with imperfect knowledge and imperfect understanding of their situations. Again and again they refuse to humble themselves, be reconciled to God and know Heavenly bliss rather than the self-centred misery of Hell. The narrator of the book sees only one soul accept the offer of salvation, that is, behave in its own best interests, on an objective reading of its situation.

The Strait Gate

This raises the question of salvatory statistics. According to the New Testament, few souls reach the eternal bliss of Heaven and escape the eternal torment of Hell:

7:13 Enter ye in at the strait gate: for wide is the gate, and broad is the way, that leadeth to destruction, and many there be which go in thereat: 7:14 Because strait is the gate, and narrow is the way, which leadeth unto life, and few there be that find it. (Gospel of Matthew)

That isn’t fair. But it is frightening, so it’s an attempt to manipulate behaviour. “Believe and behave or be damned” is a powerful argument. An additional inducement is the thought of being one of the special few. But if most human beings are damned, where does that leave free will? The choice between Heaven and Hell is weighted in favour of Hell, according to both the New Testament and C.S. Lewis’s The Great Divorce. I can see literary and psychological reasons for that, but I can’t see logic or justice. If most people got to Heaven, Hell would become less frightening as doctrine and The Great Divorce less interesting as literature. But what if it’s exactly half of us to Heaven, exactly half of us to Hell? Then free will begins to seem like the toss of a coin: a saved soul could just as easily have been a damned soul, and vice versâ. Furthermore, a saved soul might be damned if it made its choices again, and vice versâ. Again, this seems like coin-tossing. The choices have to be unclear, the situations impefectly understood. And how is that fair? If we assume that it is fair, then true justice would demand that every soul has an equal chance of heaven. But that has not always been Christian doctrine: predestination, or the dooming of some souls to Hell at conception, can be justified from the New Testament. It was taught by St Augustine of Hippo and Protestants like Calvin. And theirs was the Hell of eternal fiery torment.

Lewis rejects that. The Hell he describes in The Great Divorce is neither fiery nor physically frightening, unlike the Hell depicted by a Jesuit preacher in James Joyce’s A Portrait of the Artist as a Young Man (1916):

The preacher’s voice sank. He paused, joined his palms for an instant, parted them. Then he resumed:

… The horror of this strait and dark prison is increased by its awful stench. All the filth of the world, all the offal and scum of the world, we are told, shall run there as to a vast reeking sewer when the terrible conflagration of the last day has purged the world. The brimstone, too, which burns there in such prodigious quantity fills all hell with its intolerable stench; and the bodies of the damned themselves exhale such a pestilential odour that, as saint Bonaventure says, one of them alone would suffice to infect the whole world. The very air of this world, that pure element, becomes foul and unbreathable when it has been long enclosed. Consider then what must be the foulness of the air of hell. Imagine some foul and putrid corpse that has lain rotting and decomposing in the grave, a jelly-like mass of liquid corruption. Imagine such a corpse a prey to flames, devoured by the fire of burning brimstone and giving off dense choking fumes of nauseous loathsome decomposition. And then imagine this sickening stench, multiplied a millionfold and a millionfold again from the millions upon millions of fetid carcasses massed together in the reeking darkness, a huge and rotting human fungus. Imagine all this, and you will have some idea of the horror of the stench of hell.

—But this stench is not, horrible though it is, the greatest physical torment to which the damned are subjected. The torment of fire is the greatest torment to which the tyrant has ever subjected his fellow creatures. Place your finger for a moment in the flame of a candle and you will feel the pain of fire. But our earthly fire was created by God for the benefit of man, to maintain in him the spark of life and to help him in the useful arts, whereas the fire of hell is of another quality and was created by God to torture and punish the unrepentant sinner. Our earthly fire also consumes more or less rapidly according as the object which it attacks is more or less combustible, so that human ingenuity has even succeeded in inventing chemical preparations to check or frustrate its action. But the sulphurous brimstone which burns in hell is a substance which is specially designed to burn for ever and for ever with unspeakable fury. Moreover, our earthly fire destroys at the same time as it burns, so that the more intense it is the shorter is its duration; but the fire of hell has this property, that it preserves that which it burns, and, though it rages with incredible intensity, it rages for ever. (A Portrait of the Artist as a Young Man)

This was Anthony Burgess’s reaction to Joyce’s book:

… when I came to the school retreat and the two sermons on hell I found myself outside literature. I was terrified. Ironically, against the aesthetic canons that were so eloquently expressed by Stephen Daedalus, I found the power of the novel was wholly didactic. I ran to the confessional, poured out my sins of doubt almost sobbing, and received kind absolution and a nugatory penance. I went to mass with my missal, reading the Dies Irae [Day of Wrath] moaning, calling on Christ, fons pietatis [fount of mercy], to drench me in his mercy. (Little Wilson and Big God: Being the First Part of the Confessions of Anthony Burgess, 1987, pp. 140-1).

Hair and Homophobia

A powerful description of Hell frightened Burgess into returning to Catholicism. Like Edmund’s, this was rational, self-interested behaviour, not moral behaviour. If Hell exists, one should minimize one’s risk of going there; if the White Witch is evil and faithless, one should work against her, not for her. There is a paradox here: it is necessary to insist on free will in order to frighten, shame or otherwise manipulate people into not exercising it. It’s also necessary to insist on free will in order to enjoy the pleasures of self-righteousness and moral condemnation. That is the embittered premise of A.E. Housman’s comment on the Oscar Wilde scandal, in which Wilde was tried and imprisoned for buggery:

Oh who is that young sinner with the handcuffs on his wrists?
And what has he been after that they groan and shake their fists?
And wherefore is he wearing such a conscience-stricken air?
Oh they’re taking him to prison for the colour of his hair.

’Tis a shame to human nature, such a head of hair as his;
In the good old time ’twas hanging for the colour that it is;
Though hanging isn’t bad enough and flaying would be fair
For the nameless and abominable colour of his hair.

Oh a deal of pains he’s taken and a pretty price he’s paid
To hide his poll or dye it of a mentionable shade;
But they’ve pulled the beggar’s hat off for the world to see and stare,
And they’re haling him to justice for the colour of his hair.

Now ’tis oakum for his fingers and the treadmill for his feet
And the quarry-gang on Portland in the cold and in the heat,
And between his spells of labour in the time he has to spare
He can curse the God that made him for the colour of his hair.

XVIII in Additional Poems (1940)

Homosexuality, according to Housman, is equivalent to hair-colour: something unsought, unchosen and unworthy of blame or condemnation. But there is possible irony in his irony: if homosexuality is natural, why is homophobia not natural too? Many liberals today regard homophobia with the same self-righteous horror and indignation as Christians regarded homosexuality in Wilde’s day. The liberals are seeking, and winning, the same thrill of feeling superior to sin, but in reverse. In fact, homophobia may be more natural than homosexuality, at least in the sense that it is more heritable and easier to explain on evolutionary grounds. Unlike homosexuality, homophobia is probably not caused by a neuro-pathogen or other insult to the brain. Indeed, does Housman’s poem apply to all crimes and all punishments? Are they all unfair, in the narrow sense that individual criminals lack free will to do otherwise? In a wider sense, of course, laws and punishments may be socially useful as ways of deterring or quarantining criminals, but even the most enlightened justice still contains or provides excuses for moral and literal sadism.

Whip of the Tongue

Housman found excuses for sadism elsewhere: he was famous for his lacerating comments on other classical scholars. But in what sense did Housman choose to be a good scholar and his targets choose to be bad ones? Why was their stupidity or obtuseness worthier of blame than his verbal cruelty? He could have found other and kinder ways to keep the rabble in their place:

Progress there has been, but where? In superior intellects: the rabble do not share it. Such a man as Scaliger, living in our time, would be a better critic than Scaliger was; but we shall not be better critics than Scaliger by the simple act of living in our own time. Textual criticism, like most other sciences, is an aristocratic affair, not communicable to all men, nor to most men. Not to be a textual critic is no reproach to anyone, unless he pretends to be what he is not. To be a textual critic requires aptitude for thinking and willingness to think; and though it also requires other things, those things are supplements and cannot be substitutes. Knowledge is good, method is good, but one thing beyond all others is necessary; and that is to have a head, not a pumpkin, on your shoulders and brains, not pudding, in your head. (The Application of Thought to Textual Criticism)

Housman did not choose to be exceptionally intelligent. And did he really choose to study hard and become highly knowledgeable? After all, study and knowledge depend on traits like conscientiousness and self-discipline, which don’t seem ontologically distinct from intelligence. Why should they depend on free will any more than intelligence does? Do we choose to have them? And why do we need free will in intellectual and empirical affairs anyway? The answer might be: because it’s good to be free. We all want freedom. Except when we don’t – and there are often times when we don’t want to be free. Those who argue in favour of free will sometimes use arguments from emotion. If free will doesn’t exist, then we must be slaves or automata. And who wants to be a slave or automaton? This isn’t a valid argument: not wanting X to be true is no guarantee that X is false, and vice versâ. Besides, there are times when we do want to be slaves or automata, when we don’t want freedom. For example, do you want true freedom to harm or damage someone or something you care for? If you do, true freedom would mean that tomorrow you might choose precisely that. And by choosing like that, you would prove you were not an automaton or a slave of necessity. And who wants to be a slave or automaton? Presumably you do, under those circumstances.

A Day to Slay

Another example: would you like your senses to have free will, so that they could choose freely how to present the world to you? When you’re walking somewhere, for example, your eyes might report solid ground ahead where there is in fact a deep hole equipped with rusty, poison-smeared spikes. Your ears might report bird-song when someone shouts a warning about the hole. Or you might eat decomposing food and think it tastes and smells delicious, thanks to the free will of your nose and mouth. Would you like that? Presumably not. Another example: the next time you take a trip in a car or plane, would you like the designers, engineers and manufacturers of the car or plane to have exercised free will in their work? To have chosen whether or not to rely on previously established rules of mathematics, physics and chemistry during design and manufacture?

If you wouldn’t like those responsible for the car or plane to have exercised their free will, why not? Would you like to have free will yourself if you were an engineer or mechanic? Presumably not. When it comes to truth, physical reality and things we care for, we do want to be slaves and automata. 1 + 1 = 2. I have no choice about accepting that and I am happy that it is so. Here is a more complicated piece of mathematics:

Primes, or numbers like 2, 3, 5 and 7, which are divisible only by themselves and 1, are infinite in number. To prove this, assume the opposite, namely, that they are finite in number. Now, consider the number P, created by multiplying all primes together and adding 1. Is P divisible by any prime in the original list? No, because it will always leave remainder 1 on division. Therefore, P either is itself prime or is divisible by some prime that is not in the original list. Which contradicts the assumption that primes are finite in number. Therefore, primes are infinite in number. Q.E.D.

If you follow the logic, you have no choice about accepting the conclusion. That lack of freedom is both good and awe-inspiring. It is good that we, as far as humanly possible, can know the truth about primes: that they are infinite in number. And it is awe-inspiring that our finite human brains, destined to pass and decay, can know something that is true everywhere and always of the infinite set of integers. If mathematics depended on free will, none of that would be true. If we chose to accept the logic, rather than being forced to accept it, we could not accept the conclusion as certain. Mathematics is the most powerful form of knowledge because it is the least free. And also because it is the most free – that is, the most independent of human preferences and predilections. And I believe it will slay free will one day, proving beyond doubt that there is nothing but absurdity in the idea of a mechanism – material or super-material, fleshly or spiritual – making an undetermined, unconditioned choice.

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Elsewhere other-posted:

• The Brain in Train — determinism and truth
• Scheming Demon – a review of The Screwtape Letters
• At the Mountains of Mathness — general index