Axioms

BEYOND LOGIC

11 AXIOMS

LOGIC BY ITSELF HAS no starting point. It consists of a way of making deductions from things we already know. So we have to start somewhere in order to deduce anything logically. Limits are often thought of as being at the end, but there are also limits at the beginning.

We have already touched on this limit to logic that comes from wondering what the root of all truth is. Just like thinking up words for a new language, we have to decide to start with some truths before we can apply logic to find more truths. In mathematics, the things we decide to start with are called axioms, and in life these are our core beliefs.

Axioms are the basic rules in the system. We do not try to prove axioms, we just accept or choose them as basic truths that generate other truths. There are broadly two different ways to approach the use of axioms in maths, which I think of as being externally motivated and internally motivated.

The internal approach to thinking about axioms is where we pick some axioms and see what system this generates, logically. In this case, any axioms are valid because we are assuming them to be true in the system, to see what will result. The only problem is that if the axioms cause a contradiction then the whole system will collapse and become the null system, in which everything is both true and false. This isn’t mathematically incorrect, it just means that there is no sensible notion of truth in that system, so it’s not a very illuminating place in which to understand or model anything.

In the real world this gives us a way of conducting thought experiments. We might, for example, imagine a fantasy world in which there is no gender pay gap. Or a fantasy world in which perpetrators of sexual harassment are not tolerated, particularly not in positions of power and influence. It is informative to imagine a world in which everyone reporting sexual harassment were automatically believed. What would happen? First it would mean that huge quantities of women would report the sexual harassment that men inflict on them every day. (Yes, some men are victims too and some women are harassers.) Huge numbers of men would be removed from positions of power. They would either be replaced by women, or by men who would in turn be at great risk of being accused. Perhaps men would start to become very frightened of false accusations. Perhaps employers would start worrying about hiring men in case they were then accused of sexual harassment. If you think that this is an indefensible state to arrive at, it is worth turning it round and remembering that most if not all women fear sexual harassment all the time; we would be replacing this with men fearing accusation of harassment. It is also worth remembering that some people are loath to employ women “in case they get pregnant” (even though such discrimination is against the law in many countries); we would be replacing this with companies being loath to employ men in case they commit sexual harassment. We will later discuss using analogies to pivot between different points of view in this way. Imagining a world with these new basic axioms doesn’t mean we think it should happen, but it helps us understand the complex web of issues involved. We can understand some things about how far we are from that world and what might need to change in order to get there, together with what some unintended consequences might be.

The external approach is to start with a world that you are trying to understand, say numbers or shapes or relationships or surfaces, or the world we actually do live in. Axiomatising it is the process of looking for basic truths that logically generate everything else that is true. One famous axiomatisation is Euclid’s axiomatisation of geometry, where he came up with five rules from which all other rules of geometry should be deducible.

Axioms in mathematics are analogous to our personal core beliefs.