Scientific vs Mathematical Theory

A THEORY FOR ARGUING

The internet is a rich and endless source of flawed arguments. There has been an alarming gradual increase in non-experts dismissing expert consensus as elite conspiracy, as with climate science and vaccinations. Just because a lot of people agree about something doesn’t mean there is a conspiracy. Many people agree that Roger Federer won Wimbledon in 2017. In fact, probably everyone who is aware of it agrees. This doesn’t mean it’s a conspiracy: it means there are very clear rules for how to win Wimbledon, and many, many people could all watch him do it and verify that he did in fact win, according to the rules.

The trouble with science and mathematics in this regard is that the rules are harder to understand, so it is more difficult for non-experts to verify that the rules have been followed. But this lack of understanding goes back to a much more basic level: different uses of the word “theory”. In some uses, a “theory” is just a proposed explanation for something. In science, a “theory” is an explanation that is rigorously tested according to a clear framework, and deemed to be statistically highly likely to be correct. (More accurately, it is deemed statistically unlikely that the outcome would occur without the explanation being correct.)

In mathematics, though, a “theory” is a set of results that has been proved to be true according to logic. There is no probability involved, no evidence required, and no doubt. The doubt and questions come in when we ask how this theory models the world around us, but the results that are true inside this theory must logically be true, and mathematicians can all agree on it. If they doubt it, they have to find an error in the proof; it is not acceptable just to shout about it.

It is a noticeable feature of mathematics that mathematicians are surprisingly good at agreeing about what is and isn’t true. We have open questions, where we don’t know the answer yet, but mathematics from 2000 years ago is still considered true and indeed is still taught. This is different from science, which is continually being refined and updated. I’m not sure that much science from 2000 years ago is still taught, except in a history of science class. The basic reason is that the framework for showing that something is true in mathematics is logical proof, and the framework is clear enough for mathematicians to agree on it. It doesn’t mean a conspiracy is afoot.

Mathematics is, of course, not life, and logical proofs don’t quite work in real life. This is because real life has much more nuance and uncertainty than the mathematical world. The mathematical world has been set up specifically to eliminate that uncertainty, but we can’t just ignore that aspect of real life. Or rather, it’s there whether we ignore it or not.

Thus arguments to back something up in real life aren’t as clean as mathematical proofs, and that is one obvious source of disagreements. However, logical arguments should have a lot in common with proofs, even if they’re not quite as clear cut. Some of the disagreement around arguments in real life is unavoidable, as it stems from genuine uncertainty about the world. But some of the disagreement is avoidable, and we can avoid it by using logic. That is the part we are going to focus on.

Mathematical proofs are usually much longer and more complex than typical arguments in normal life. One of the problems with arguments in normal life is that they often happen rather quickly and there is no time to build up a complex argument. Even if there were time, attention spans have become notoriously short. If you don’t get to the point in one momentous revelation, it is likely that many people won’t follow.

By contrast a single proof in maths might take ten pages to write out, and a year to construct. In fact, the one I’m working on while writing this book has been eleven years in the planning, and has surpassed 200 pages in my notes. As a mathematician I am very well practised at planning long and complex proofs.

A 200-page argument is almost certainly too long for arguments in daily life (although it’s probably not that unusual for legal rulings). However, 280 characters is rather too short. Solving problems in daily life is not simple, and we shouldn’t expect to be able to do so in arguments of one or two sentences, or by straightforward use of intuition. I will argue that the ability to build up, communicate and follow complex logical arguments is an important skill of an intelligently rational human. Doing mathematical proofs is like when athletes train at very high altitude, so that when they come back to normal air pressure things feel much easier. But instead of training our bodies physically, we are training our minds logically, and that happens in the abstract world.