Recently, while I was preparing for and delivering a talk on The story of the symmetric group, I came across an interesting maxim of serious combinatorics: think in terms of sets, not in terms of numbers. For instance, the person thinking in terms of numbers says that cardinality of A is cardinality of B plus cardinality of C, while the person viewing things in terms of sets will say that A is a disjoint union of B and C. Similarly, while the number-viewer simply sees one number as a product of two others, the set-viewer will see one set as the Cartesian product of the other two.
The main advantage of dealing with sets, as opposed to dealing with numbers, is that when we have sets, we can consider maps between sets, we can look at elements of sets, and we can do all kinds of funny things which we could notdo with numbers. For instance, sayingthat the number of permutations is more than the number of unordered set partitions, is not as informative as actually describing a surjective map from the set of all permutations to the set of all set partitions. That surjective map will not only show that there are more permutations than set partitions, but will also show that the number of permutations with a given unordered integer partition as cycle type, is more than the number of permutations with that same unordered partition as subset sizes. It can thus also show that unsigned Stirling numbers of the first kind are always bigger than the corresponding Stirling numbers of the second kind, and so on.
I wouldn’t have paid too much attention to this paradigm of thought that I observed myself had it not been that, while discussing Olympiad mathematics with a young aspirant, Ashwath, I happened to repeat these ideas and he really took them to heart. He said that this aprpoach really changed the way he looked at combinatorics, and helped him perceive much more structure in it.
Over the last few days, I have seen a lot of situations wherein I am forced to revise my Think in terms of sets, not numbers paradigm. Rather, I should say think in terms of structures, not numbers where structure usually means a set with additional combinatorial, algebraic, topological or other information encoded into it. Those who are thinking as primitively as numbers, of course need to move to the set-level. But even those who think only in terms of sets need to move upwards.
In fact, it is illustrative of all that I am saying that many laypersons describe mathematics as the study of numbers, while most mathematicians define mathematics as the study of structures and patterns.
I’ll describea few examples.
One of the recent developments in mathematics is the so-called Khovanov invariant theory. Prior to Khovanov invariants, the best way to study a knot (yes, knot in the usual physical sense) was to associate to it a polynomial, the so-called Jones polynomial, whose coefficients basically store some numerical information obtained by trying to perform Reidemeister moves at ea ch crossing of the knot. The Jones polynomial is thus essentially a set of numbers associated with the knot.
Khovanov came up with the brilliant idea of modifying the original construction so that, instead of numbers, we manage to get whole families of vector spaces, such that if we take the dimensions of these vector spaces, and perform suitable summations, we land up with the coefficients of the Jones polynomial. The beauty and power of Khovanov invariants lies in the fact that one can look at these vector spaces (they give a homology theory) and define maps between vector spaces — in fact (for those who care) this is a functor from the category of knots with morphisms as cobordisms, to the category of homology complexes with the usual morphisms of complexes.
Similarly, recently, one of the research students at the ENS, Olivier Dudas, was telling me about his research work on the representation theory of finite groups. It turns out that in order to study certain kinds of representation theory, he again needs to consider other, more complicated, complex-like objects such that, when we take suitable alternating sumations, we get down to the original object. In other words, the representations themselves do not have enough structure — it is necessary to go beyond them to some more structured object to draw inferences and conclusions.
The great (as well as irritating) thing about mathematical structures is that they live on for ever. Once we create a mathematical structure, there is no way of destroying it, except through creating an even more inclusive mathematical structure. Thus, mathematical structures that were created or devised to solve particular problems, have stayed on in history and have soon been studied in their own right.
The recent development of the theory of quantum groups illustrates yet another principle in this structure-oriented thinking. Namely, that if one wants to consider variation of a structure, which itself is fairly rigid, it makes sense to put a lot of other auxilliary structure, and then vary. For instance, onecannot directly vary the structure of a group — it is a fairly rigid object. HOwever, what one can do is look at the group algebra and try to geenralize the notion of group algebra to that of Hopf algebra, and then try to deform the group algebra slightly in such a way that it continues to remain a Hopf algebra. This gives us the notion of quantum group.
It is rather unfortunate that in our school education, even the elementary language of sets is taught only in high school, and is not used to its fullest power even in the most elementary of combinatorics. The strange thing is that set-theoretic language is more concrete than the numerical language, so it should be what the champions of concrete mathematics should favour — ideally every combinatorics problem should be solved with sets and cardinalities should be taken at the very last step, for computations. If students are inculcated with this mindset, it may be much easier for them to grasp the structural thinking of higher mathematics.