The Structure of a Mathematical Proof

WHAT DOES A PROOF LOOK LIKE?

Before we start a proof in mathematics we have to lay the groundwork very carefully, like setting up the rules for a sport. There are several aspects of this, and I think it’s illuminating to think about them because many arguments in real life go wrong because of problems with the groundwork rather than with the argument per se. Often in arguments in real life we only realise we’re using different definitions or assumptions at a rather late stage of the argument.

1. We should carefully define the concepts we’re talking about.

2. We should carefully state the assumptions we’re making.

3. We should carefully state exactly what we’re going to prove, in an unambiguous way.

Assumptions in mathematics are a bit different from assumptions in life. In maths it is to do with conditions under which we’ve decided to work, or conditions under which we think our result is true. Then when we apply the result, we first check whether those conditions are fulfilled in the situation we’re trying to apply it to.

For example, we might assume we’re living on the surface of a sphere, and then show that something is true under those conditions. This does not pass any judgement about whether or not we actually are living on the surface of a sphere. It just says that if we do turn out to be living on the surface of a sphere, then this thing will be true.

Assumptions in life should work similarly but unfortunately often don’t. For example, in arguments about why women earn less than men, on average, sometimes people assume that women don’t care about earning money as much as men do. Now, under that assumption it might well be reasonable that they end up earning less than men. But that is a hypothetical world. When we apply that result to the real world, we need to see if our assumption is true: is it really the case that women don’t care about earning money as much as men do?

Note that I don’t actually think the implication is true: even if it were true that women didn’t care about earning money as much as men do, I don’t think that would make it reasonable to pay them less than men for the same job. I think that is exploitation. In any case, if we’re clear about what our assumptions are, we can at least be clear about what aspect of the argument we’re agreeing or disagreeing with.

Once we’ve laid the groundwork, the actual proof consists of a series of statements, each of which follows logically from what is already known to be true. This could either be the assumptions, known truths about the world we’re in, or previous statements in the proof. The series of statements creates a chain of logic from the starting point, which is the assumptions we’re making, to the ending point, which is the thing we’re aiming to prove. Of course, sometimes this chain breaks down, especially in normal life, which is why there are more bad arguments in the world than good ones. These failures separate broadly into problems of knowledge and problems of logic.