Deep Mathematics I: “Typical” Theorems

This is the first of what I hope will be a series of (not chronolgically consecutive) posts discussing what makes “deep” mathematics deep. This is, of course, not a new issue and has been dicussed many times before. I suppose this is more an issue of semantics than philosophy, but this is certainly one place the fact that mathematics is done by human beings rather just being being the “cold and austere” * discipline that it is so often viewed as being becomes apparant.

In this post I would like to argue that one sufficient condition for a piece of mathematics to be described as ‘deep’ is that it indicates the “typical behaviour” of somthing. Now any mathematical theorem is, of course, a precise statment, usually of the form “Every X is a Y” or “No X is a Y”, but the sort of theorems that I am talking about are the ones that can often be summed up as “Xs tend to be Ys”, “your average X on the street is a Y”, “Every X looks a lot like a Y” etc. The ability to paraphrase a theorem in this way indicates that it somhow gives us a better understanding of the ‘true nature’ of a particular kind of object. A few examples will hopefully make what I mean a bit clearer.

• Coxeter Groups: The concept of a reflection (a function from to fixing a hyperplane and mapping everything perpendicular to it to minus itself). Since isometries can always be expressed as products of reflections its natural to ask what groups are generated by them. In the 1930’s Coxeter showed that finite groups of this kind are built up from certain “irreducible” reflection groups (a bit like prime numbers) and successfully classified these. What does the classification state? That reflection groups are trpically built from symmetric groups. In what sense does it state this? Of the three infinite families, one of them is the set of all symmetric groups and the other two consist of groups made by sitting asymmetric group ontop of a set of much smaller ones.
• Kuratowski’s theorem: thinking of a graph as simply a bunch of points joined up by a bunch of lines it is natural to try and draw these minimizing the number of edges that cross to make the structure of the graph clear. If we can do this we say the graph is planer. Kuratowski’s theorem chracterizes the graphs that are non-planaer: there are essentially only two non-planer graphs, usually written and . Every other non-planer graph is consists of copies of graphs made from these glued together. (I’m aware that this generalizes to the deep and jaw-dropping Robertson-Seymore theorem, but that’s a little harder to describe and not quite the sort of theorem I’m talking about.)
• Hurwitz’s theorem: a Riemann surface is beautiful kind of surface that looks a bit like contorted version of the complex plain. Hurwitz’s theorem is a precise statment about the number of automorphism (“symmetries”) of a Riemann surface that preserve angles. What does the theorem state? That Riemann surfaces tend to have few automorphisms.

This is by no means an exhaustive or even particularly representative list. Further examples would be appreciated.

Do people agree/disagree with this view? If you agree can you provide further examples of “typical” theorems? I you disagree can you give counter examples?

Comments, Commendations, Castigations…

(*=”Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere” Bertrand Russell)