DRMacIver's Notebook

I've been trying to come up with a definition of mathematics that I like and think would be useful in the course of teaching people mathematics.

This is of course a big ask, as according to Wikipedia there is a great deal of spirited philosophical debate on the subject, but on the other hand I think most of those definitions are terrible, so I don't feel too bad about trying myself.

The one I dislike the least from that list is Eric Weisstein's:

Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined.

It's a bit long-winded but mostly captures the sense I want. The phrasing I've been thinking of in preference is something more like:

Mathematics is the rigorous study of hypothetical objects.

The idea is that in mathematics we're not really concerned with real life physical objects, we can just say "Suppose there were objects satisfying the following properties, what can we reliably say about them?"

Sometimes those objects are ones that can easily be realised as real physical objects. For example the Mathematics of Chess studies hypothetical chess boards, but those hypothetical chess boards can easily be realised by going out and buying an actual physical chessboard. However, many of them can not be. There is no way to construct a real physical set of natural numbers, but from a mathematical point of view that's OK - we can reason about the properties of the hypothetical one perfectly well.

There are a couple axes of variation on which people differ about the nature of mathematics:

  1. Is informal mathematics legitimate, or should all mathematics be considered a (possibly bad) approximation to an entirely formal set of reasoning rules?
  2. Are some hypothetical objects privileged as the true platonic mathematical objects in a way that others are not?

I think this definition is more or less compatible with any combination of answers to these questions: Formalism is a question of what we count as "rigorous", and even if there are platonic mathematical objects, we still must study them as if they were hypothetical because by its very nature we cannot have access to the platonic realm.

Traditionally the answers to these questions have been correlated more than I think is logically required: The formalist position is that mathematics doesn't real and that everything is formal manipulation of symbols, while the platonist position is that we are seeking to discover truths about the ideal platonic realm and the truths are what matter regardless of how we reason about them.

I think there's room for a third position though, which is that formalism is interesting but not strictly required, but the objects we describe have no inherent reality and really are allowed to be purely hypothetical. I've historically self-described as a formalist, but I think this third position is closer to my true beliefs: I don't think Platonism is philosophically defensible, but I do think there is a lot of interesting mathematical content and activity that cannot be adequately captured by the formalist position.

In many ways this third position is that of Lakatos in his "Proofs and Refutations". Most of the interesting mathematics happens in a fuzzy middle-ground where you are making your definitions precise enough to be defensible. This could go all the way to formalism, but it doesn't have to.

The mathematics of chess is again an interesting test case here: Chess is a purely arbitrary set of rules. I think it would be hard to argue that there is a platonic game of chess that is in some essential way different than it would have been if, say, kings moved like knights or you could win by killing the queen or the king. These are both perfectly valid games that someone could play, and there is a perfectly valid mathematics in studying them, but we study the mathematics of chess in preference to them because that is the actual game people play.

Conversely, there really is a set of true statements about the game of chess (in an informal sense of chess), and while mechanising and formalising the study of them might be useful for determining what they are, I think it's fair to say that what actually matters is whether the statement is true of real games of chess, and the formalisation only matters to the degree that it helps us discover those truths.

I don't think the above definition is enough to fully reconstruct an idea of what mathematics is like, because it leaves open two big questions:

  1. How do we select which hypothetical objects to study?
  2. How do we study them?

The answer to the first is comparatively easy, which is that it's based on what I think of as "The Three Good Reasons To Do Things":

  1. It's useful.
  2. It's interesting.
  3. Some asshole is forcing you to do something useless and boring.

(Most people's encounters with mathematics is of type 3, sadly, which is why I always hear "Oh I hated mathematics at school" when I tell people I did mathematics at university)

Of course, point 2 is slightly subtle, because doing mathematics is much easier if other people have done similar mathematics, so you're constrained not just by what you think is interesting, but by what you can convince other people is interesting.

The second question is the hard part, and I think we currently do a very poor job of explaining it to people. I need to think further about it.