Counting Intuitions

 This post is an exercise in thinking about vague things.

I think everyone can agree that there are infinity more ways for things to be bad, to not work, as there are for them to work. For things to work, you need a big conjunction of things, you need to be healthy, you need a decent house,  you need good food, you need friends, you need amusements, you need purpose, you need things to be good for your friends, etc...you know the list just keeps growing. While for things to not work, only one of them must be missing. So clearly it's much easier for things to not work, then for them to work, regardless of whichever priority order you put on your personal list.

Even people like me, who do not like probabilities and that have difficulties with them can see that the bad scenarios are much more probable in the big scheme of things than the good ones.

But there is some help to be had. By and large it seems that verifying that something is true is much easier than discovering when something is true. This (possibly very large) gap between the easiness of checking a given answer versus the hardness of coming up with a possible answer is one of the mathematicians most used tools. Note that we do not have a proof for it, it is just obvious for, say, equations of second degree or systems of linear equations. But we extrapolate, we reckon it might happen in all branches of mathematics.

Another helping tool is symmetry: we believe it is everywhere and we love it. It does help to half the work in many situations and the Universe does seem to have a penchant for it. or maybe it's just us, humans, that see it everywhere, when it's only there in some very special places. I don't know of any attempt to measure the amount of symmetry of the Universe, but I know that mathematicians, if they can,  will make things symmetric: symmetric things are prettier.

And a third helping tool is `adversarial thinking', in whichever way you may want to think about it. So this might be games, where there are proponents and opponents and they battle their wits over the truth or falsity of propositions (the mathematician thinking about it might play both sides and hence, perhaps see more clearly the weaknesses of arguments of the other side). Or it might be adversarial training in machine learning, which I don't really enough about to pass judgement on.

In any case, these generic tools are about trying to make the problem easier, about simplifying problems by trying to see what would happen, if they were indeed simpler and easier. 

But of course we know that many times things that are simpler, that look intuitive and clear are just plainly  wrong. The Sun does not move around the Earth; heavier things do not fall faster; things that do seem to stop if not subject to forces non-stop,  actually would keep going non-stop if the attrition of other forces did not stop them. Similarly, lovely graphs in Geometry prove wrong things, because you cannot trust graphs (Escher pictures anyone?)

So one of main points of the apprenticeship in Mathematics is learning to distrust your intuitions. A bit like philosophers who start asking "why" about any and everything, mathematicians have to learn to read the books of they favorite authors doubting every word and trying to prove or disprove every sentence. Being a mathematician is about verifying always; trusting only in special occasions, if at all.