Maturity

Anyone looking to gain some insight into the mind of a leading mathematician, and quite possibly a future Fields' Medallist, should find plenty to interest them on Terence Tao's What's New? pages. His December 6, 2005 entry gives us access to a book he first wrote at the age of 15 on problem solving. Let me just contrast a couple of extracts from the two prefaces:

The 15 year old Tao writes,

Proclus , an ancient Greek philosopher, said: This therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth . . .

But I just like mathematics because it's fun.

Proclus was a fifth century Neoplatonist, perhaps best known for his commentary on Euclid's Elements. Now, as a 30 year old, Tao writes:

Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience. As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non­trivial answers. As a high­ school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems (such as those in this book) and searching for the right "trick'' that would unlock each one. As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today. As a graduate student, I learnt the pride of having one's own research project, and the unique satisfaction that comes from creating an original argument that resolved a previously open question. Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be [sic] guided by very simple, even common­-sensical, principles. The "Aha!'' experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one. And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.

"Very simple, even common­-sensical, principles" is precisely the topic of pages 206 and 207 of my book. They are essential components of mathematics operating at its highest level, and, as such, to overlook them as a philosopher writing on mathematics is to go astray. What would be very useful would be a generous sample of such principles. I gave this one from Timothy Gowers in my book:

if one is trying to maximize the size of some structure under certain constraints, and if the constraints seem to force the extremal examples to be spread about in a uniform sort of way, then choosing an example randomly is likely to give a good answer.

For a pithier example from higher-dimensional algebra: "All interesting equations are lies.", i.e., should be seen as projections of isomorphisms, or higher equivalences. A more technical one tells us that: "certain algebraic structures can be defined in any category equipped with a categorified version of the same structure." In my experience, there's always what I would call a degree of creative vagueness to these principles. While they can be given a formal dressing, this often leaves a residual capacity for future application.

Let's end with a couple of related principles from Tao himself:

If an object is not (pseudo-)random, then it (or some non-trivial component of it) correlates with a structured object.

If A is an arbitrary object, then A (or some non-trivial component of A) splits as the sum of a structured object, plus a pseudorandom error.

I'm left wondering what Tao's thoughts on Proclus are now.