Mathemantical Theories

The word “theory” is used by mathematicians to refer to a body of knowledge about some particular area of their subject. It does, though, have a precise meaning. We begin with the idea of a theorem, which is a statement that has an accompanying proof of its correctness. A mathematical proof is purely logical: it can be done inside a computer, or in your head. Nothing in the physical world, for instance, gets involved.

In a sense all of mathematics is made of theorems. Conjectures — which look like theorems but don’t have proofs — play a very important part in the practice of maths, but they haven’t yet earned their right to be considered part of mathematical knowledge. The Riemann Hypothesis, to pick a very famous example, stimulates a great deal of research and general curiosity, but we don’t yet really know whether it’s true or false. When we do, it — or its negation — will become a theorem instead.

The theory of some mathematical object is the set of all theorems that can be proved from the definitions of that object (broadly speaking). So group theory is just the set of all theorems we can prove about groups. We can make this as rigorous as we like: the relevant definitions are expressed as axioms in a formal language, and the proofs can be expressed as derivations in the same formal language and checked by machine.

So “theory” in maths means “collection of theorems proved to be true”. Nobody, in this context, would say “that’s just a theory” — “just” would suggest there was something deficient about the status of “theory”, which there isn’t here.

Empirical Theories

In the sciences, of course, the word “theory” means something quite different, and this is where the waters get muddied. A scientific theory may be best thought of as a model. This is a simplified description of reality whose behaviour is governed by mathematical relationships that the scientist defines. If the model is good, the scientist can use it to make predictions: given some conditions, she can see what the model will do, and given equivalent conditions in the real world we might expect an equivalent outcome.

Modelling is good, for instance, if you want to know how to hit an enemy ship with a cannon. You set up a simple description of the cannon as a point the ball is launched from and the ball as another point that moves according to an initial acceleration (due to the charge in the barrel) and another ongoing acceleration due to gravity. If you have a mathematical desription for how position varies as a result of these two forces then you can make predictions about where the point representing the cannonball will end up.

Now you can easily calculate what the size and angle of the initial force should be if you want the cannonball to hit the ground, say, half a mile away. Then you can go out and try it on your real cannon. If the real cannon regularly hits the target that the model predicts then we say the theory is good. A scientific theory is really just a general way of constructing models for certain kinds of problems. In this case, the theory of Newtonian mechanics is gives you a pretty good way of creatinging those crucial mathematical relationships that make the cannonball model work.

The construction of a mathematical model, on this account, is a pragmatic practice, not an enquiry into truth. This is where my view parts company with that of many of the new breed of scientific Naturalists, for whom scientific models are truths. By my lights they are precisely not truths but fictions: useful, even necessary fictions that enable us to get things done.

Cannonballs are not really point masses, and forces are not really vectors and so on. If they were the world we live in would be made of mathematics, which is appears not to be. The mathematical entities of which a scientific model is essentially made are useful exactly because they’re unlike the real world they model: they’re simplified, better-behaved, cheaper and so on.

So on my account a scientific theory is a method for model-building. The method may be good or bad. That evaluation is not based on how true the resulting models are — I don’t even know what that would mean — but on how well they enable us to get something done, and especially how well they make predictions.

This jibes well with the fact that scientific theories are never proved like mathematical theorems. The latter are true, and so it makes sense that they have proofs. The former are useful, and are “proved” in the way that a method for hunting ducks might be “proved” by going out and trying it. These two usages of the word are not in any way similar, except inasmuch as both lead to the acceptance or rejection of something. The kind of thing being accepted, the kind of tests applied and the whole social context in which it’s happening are all utterly different. That we use the same word for both is extremely unfortunate.

There’s a philosophical argument to be had over whether the effectiveness of scientific models suggests that the universe really is structured roughly the way scientists think it is; this is a view that’s certainly held by sophisticated, well-intentioned people who’ve thought hard about the issues. I personally don’t think the prospects for that claim are too good, partly because I don’t see how you would find any evidence to support it. (Notice that “scientific models work really well” isn’t evidence for “they work really well because the universe is structured in the way they say”).

[Attentive geeks may have observed that mathematical modelling of scientific theories on my account has little or nothing to do with what mathematical logicians call "model theory". Alain Badiou has (controversially) argued that a certain sleight of hand can make it look as if these two things are two sides of the same coin, and that that's been part of the political success of positivism, which has survived as a general idea long after it was discredited as an epistemology.]

“Just” A Theory Rather Than A… What?

When we say something is “just a theory” we mean, I think, something different from either of the above kinds of activity. Here we mean by “theory” something more like the mathematician’s “conjecture”: something we have a hunch may be true but we don’t really know. We use the term in everyday life all the time without problems. “I have a theory it’s going to rain later,” and so on. Here we mean “just a theory rather than a fact” or “rather than proper knowledge”, whatever that means. But day-to-day we’re not philosophers, and we know what we mean, and everything works pretty well.

When we say a scientific theory is “just a theory” the question is what you think it should be that a theory isn’t. On my account, a scientific theory isn’t an account of the true nature of the universe, for example. On almost all accounts, a scientific theory isn’t immune to skepticism, doubt and even rejection if we discover an inability to do the things we want it to do. If you want something that doesn’t have those kinds of properties, you don’t want a scientific theory. This is, incidentally, why maths isn’t a science.

That’s OK — science is splendid, but it doesn’t have to do everything, and trying to “scientize” all human questions is a project doomed to failure. It’s also a misconceived one, because people often ask for things that scientific models can’t give them and shouldn’t try to. In the interim, though, pretending something is scientific when it’s not really can lead to serious errors of judgement (mathematical finance is an example that springs to mind).