The Axiom Delusion

6 min read

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.