The Axiom Delusion

The Axiom Delusion

A one-thousand-mile journey begins with a single step. But where to take it? Axioms take your cue! (like all good philosophy, this essay will be aided by limericks)

Photographer: Elvis Bekmanis | Source: Unsplash

Axioms, schmaxioms

There once was a problem unsolved

our thoughts ran quite wild not controlled

so we created the maxim

that questions too taxing

are best to pretend they’re resolved!

Axioms exist because humans suck at doing Philosophy. When starting at a question, a really general question, we get bamboozled. So instead, we assume a starting point and build upwards. The first step we take in this journey is often arbitrary. We keep the axiom if it works. Do you trust Mathematics because you have gone in detail through the different axiomatic framework and been convinced that they represent some sort of self-evident inner knowledge? Or do you accept it because it works? The scary thing is that mathematicians behave likewise! And so do the Philosophers, although they ought to know better. If you don’t trust me on the Mathematics point, that’s okay. Maths is something of a sacred cow in our minds, the holy grail of Truth with a Capital T, and Knowledge with a Capital K. At the end I’ve given a short and hopefully accessible example to where mathematical foundations fall flat, but this isn’t needed to understand the article.

If Philosophy had a platonic form(!) then an axiom would be a self-evident truth from which all other knowledge could be deduced. But, as we will see, things don’t turn out to easy.

The axiom delusion

The axiom delusion is that we forget that any inference based of an assumption is as valid as the assumption is. Put another way, if I think A implies B and I assume A that does not mean that I know B!

Yet this is exactly what Philosophers do. For instance, in Bertrand Russell’s Problems of Philosophy he basically admits that we have no basis for knowledge of the external world… and then proceeds to talk about ethics, existence (or not) of a soul, and other interesting questions. Okay, I get that Philosophers wouldn’t sell many books if they put down their pens after concluding on page 4 that they knew nothing. Yet the inferences made in all subsequent parts of Philosophy rely on the fundamental assumptions. Thomas Nagel, in his book ‘What does it all mean?’, confesses that he doubts anyone will ever have a good answer to the fundamental problem of knowledge… and then goes on to talk about justice, ethics and the like.

Clearly, if I don’t know it the world around me exists, then arguing over which actions are ethical is nonsense. If I have no clue what my actions are, their effects are, nor anything resembling knowledge, then how are we supposed to judge what we should do?

I remember speaking to a PPE (Philosophy, Politics and Economics) student from Oxford, who told me that his only axioms were the axioms of logic. (this in itself was bizarre, and clearly not true, as even if logic is grounds for tautologies, it is far to restrictive to enable the breadth of knowledge we take for granted in everyday life). Yet what are these magical logical axioms? Should we assume the law of the excluded middle? A proposition is true or its negation is true? What a vacuous statement! As if anyone truly had a solid grasp on the abstraction on truth to know what the heck this means. Apparently Quantum Physics negates this finding! Where is the axiom now? Should we refute the empirical findings of scientists on the basis of our intuition about a ‘law’ of logic? A law which we are persuaded of by simple examples! Am I holding up 4 fingers? Is this colour red? Seems reasonable enough that these cannot be true and false. Yet universal laws make a much stronger claim than merely being widely believed examples! Ah, yes, the strength and conviction of a wide set of examples! The Law of the Excluded Middle (it is even capitalised) may have served its time, but in light of recent evidence, should philosophers quietly retire it? And what confidence does that install in the other rules of logic? … or indeed the elements of mathematical proof which uses the laws of logic and on which deduction in physics relies.

I have this image in my head. Imagine flipping a dice to a small child and persuading them there is an unalienable law that it always lands heads or tails. I’m pretty sure this, after several dozen flips, is persuasive, and the child may not be able to imagine a way that it could land otherwise. If our construction of deductive systems is like this, then we will never know anything, as in our next coin flip it might land on the rim down the middle.

A Manifesto to Confusion

If this was a game of snake, then I’ve eaten my tail.

My argument relied on a simple logical argument about what implication is. On my own basis can I form this inference?

It’s unclear. In my reckoning, I both agree with Wittgenstein that ‘doubt presupposes certainty’ yet also think that certainty creates doubt. It is far too easy, taking the ‘certainty’ thinkers like Wittgenstein want to think they have, to dismantle the whole edifice. In this respect I both simultaneously admit that this scepticism is not a positive position (i.e. it undermines itself and solely exists to undermine the basis for other viewpoints), nor does it provide any basis for ‘doing’ anything. All it leads is an empty confusion.

If you are willing to accept the things you think you know, then you can now understand why Philosophers don’t write 5 page long books for two reasons. One: they wouldn’t sell many copies (not that they sell many already). Two: leaving human knowledge as mere ash-or even more reduced than that?-is hardly satisfying.

If you, like me, are less willing to make such bold leaps and assumptions… you will do so anyway, but prepared to be constantly confused and in a muddle, unable to make heads nor tails of anything.

An example from Mathematics

Mathematics: nowhere else has with such painstaking precision watertight deductions laid out from clearly laid out axioms.

But what are these axioms? Let’s look at one of them from set theory.

In the ZFC axioms of set theory, the Axiom of Regularity states that ‘If A is a non-empty set, then there is at least one element of A which is either not a set, or is disjoint [1]from A’[2].

‘Huh?’ I hear you say (or at least hear myself say), ‘Hardly a self-evident truth.’

Why it was tacked on it fascinating. Below is a quote from the Wikipedia page on the Axiom of Regularity, which I will leave you to ponder.

Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties.

As Alice would say, ‘curioser and curioser.’

Humans thought much of their thinking

applying their maths without blinking,

when Truth came to play

we sent her away

pretending that nothing needs fixing

This article was written by sleep-deprived crazed pseudo-intellectual, who studies Economics at Cambridge University.

[1] [disjoint = shares no elements with]

[2] I take my definition from Terence Tao’s book on analysis. Terence Tao is an astonishing brilliant mathematician. His book is for beginners (which I am currently). The exact phrasing is slightly different elsewhere, and is often phrased using formal mathematical logic.

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Maybe Democritus Wasn’t So Bad

Maybe Democritus Wasn’t So Bad

This is a piece against dogmatism, fundamentalism in religion, scientific “fact” and political correctness. This is a piece against people, who refuse to think and who are too enthralled in their phones and daily habits and themselves to experience and learn about the world.

'Democritus meditating on the seat of the soul'

source: By Léon-Alexandre Delhomme – Jean-Louis Lascoux (13 January 2008), CC BY-SA 3.0,

Who am I?

I am a devout Christian and shall expound my views on the world in a series of these pieces. But I am fundamentally doubtful. My views are contradictory, complex and ill-formed because of neglecting to and straight up refusing to think about them enough up to now. I will have to change at least some of them. That does not scare me. What we do not understand is not necessarily wrong and there is much that I do not understand. We must strive to understand more and to accept the new truths that we discover.

How does this relate to Democritus at all?

Yes, yes. Lucretius (a philosopher and poet in the first century BC) wrote the “De Rerum Natura” (On the Nature of Things) which is an exposition of Epicurean philosophy and more importantly the atomic hypothesis. It considers the phenomenon of Brownian Motion[1] to explain the atomic hypothesis. This phenomenon (and much of the reasoning) was then used by Einstein, two thousand years later, to “prove” (to the satisfaction of the scientific community) the atomic hypothesis. TWO THOUSAND YEARS.

You still haven’t gotten anywhere with the explaining about Democritus thing.

Okay, okay. This atomic hypothesis is generally attributed to Leucippus (5th century BC) and his pupil Democritus (460-370BC). It is unclear whose ideas were whose, but it is known that Democritus also wrote about a vast number of other topics. Unfortunately, neither of their works directly survive. Fortunately, some of their ideas concerning the atomic hypothesis survive through rebuttal in the works of Aristotle. They cited many everyday physical phenomena, such as the gradual wearing down of a wheel, to support the theory.

Can you move?

But there is also a metaphysical grounding to the philosophy. They turned Zeno’s paradox on its head. They said that clearly, we can cross a room, so we must not be able to divide up distances infinitely and so there must be atoms.

They also combat Melissus’ argument that we shouldn’t be able to move because we would have to move into nothing. Melissus says that nothing does not exist, and Q.E.D. movement does not exist. They responded that we can move, so clearly nothing, or the void as they termed it, must exist.

In both cases their opponents had assumed that something was ridiculous because it didn’t immediately mesh with their worldview. In both cases it turned out that their worldview was too small, and they were simply refusing to extend it.

This is an interesting type of Philosophical argument. If a set of premises lead to a conclusion, yet you know that the conclusion is wrong, then one of the premises must be wrong. Rather than stick with the premises and conclude that motion doesn’t exist (which is absurd), we should ditch one of the premises.

But I don’t understand why you ever thought this guy was bad; he seems pretty great.

Upon reading Lucretius’ poem, it was very easy to say that it was just another crack-pot theory thought up by the Pre-Socratic Philosophers. It was thought that because the Pre-Socratics had also proposed ideas such as magnets having souls and there being atoms the size and shape of giraffes that all of their work should be dismissed.

It now seems obvious to us that matter is made up of atoms. But if someone now announced that through some abstract reasoning they had disproved the atomic hypothesis, would you be able to accept or even consider their argument? If someone told you that it is possible for a cat to be both dead and alive at the same time, would your first reaction be to say that that is ridiculous or to ask how and why? This proposition seems preposterous and challenges a fundamental part of our worldview. This is Schrödinger’s Cat thought experiment and it comes from Quantum Mechanics, but examples can be chosen from any field: that your friends and family don’t actually exist or that Schoenberg might actually be beautiful or that there may be more to modern art.

Our worldviews are too narrow

I choose science because it illustrates the point well that our worldview is challenged because our worldview is too small. We do not exist on a quantum scale but that doesn’t mean that the rules governing the quantum scale are ridiculous.

It seems to me that if, just occasionally, we considered that there is a world beyond what our eyes and senses and mind and experiences can currently perceive and comprehend and if we were constantly searching for it, then we may be able to appreciate the beauty of the world around us a lot easier.

Plato's cave

Plato knew this millennia ago. We are trapped in a Plato’s cave! We must be careful not to miss the people who have escaped – they have seen a world which is much grander than our narrow worldview.

Democritus was inspired with an idea and it was the right one. He had a glimpse of a world beyond himself. All he had to do was to look for it. This glimpse was an intuition, a feeling, an idea – it was a correct one. There are some wrong ones. We are not infallible. We will see the mirage in the desert and may mistake it for an oasis, but this does not mean that every gypsy, shaman, priest, fortune-teller is scamming you. Only those who pretend to understand the world fully, who pretend that their hypotheses are facts, are scamming us.

Our Glimpses

We have all experienced things beyond ourselves. We have been on a walk and seen the beauty of nature. We have heard a piece of music or read a book or seen a play that inspired us. We have looked up at the stars and seen something more, something calling us. This may just be curiosity. And I know that, when I identify those experiences with my God and I tell you that I have experienced a small part of his infinity, that is not a proof to you. But it is enough to satisfy me. When people conclude that it is curiosity and they chase after it they have found their own God. When people deny that there is a world beyond what they currently know, when they don’t even convince themselves that they have found meaning, when they just blindly follow their assumptions – that is when they are truly lost.

Remember to look up at the stars and not down at your feet. Try to make sense of what you see and wonder about what makes the universe exist. Be curious. And however difficult life may seem, there is always something you can do and succeed at. – Stephen Hawking.

This article was written by co-author 'Devout Doubtful'

[1] by which particles of dust are moved in macroscopically still air.

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