I recently came across an article by the highly respected Cal Tech mathematician Terence Tao titled, “Does one have to be a genius to do maths?”
Professor Tao, a man oddly enough with a reputed I.Q. of 230, raises the question of whether or not mathematicians can make contributions to their field regardless of inherit intelligence (within reason of course), and though he does essentially make the case for more or less productive mathematicians, he answers his own question with “an emphatic no.” But, I must say that I feel slightly let down by his answer, although I do agree with him. Its obviously a good thing that one doesn’t have to be a genius in order to be a working mathematician, but surely there must be some qualities that would make someone intellectually well-suited for mathematical research in the first place. Tao lists a few at the beginning of his piece, but these are hardly unique to mathematics – the ability to work hard would be advantageous in almost any number of fields for instance. The interesting question to me then seems to be what qualities would uniquely make someone a good mathematician. Tao doesn’t seem to answer this in any meaningful way, which is a shame since his perspective would be incredibly useful. It’s certainly always possible that Tao doesn’t understand why he himself is a good mathematician, but I’m sure he could at least offer a few interesting insights – if not for the budding mathematician then perhaps for the artificial intelligence and cognitive scientists.
The question isn’t merely philosophical which is a plus, since one could use the information to devise more intelligent machines or better teaching methods. I suppose I could take on the task of setting the limits on what constitutes mathematical ability, but such a task would quickly become too painful to carry on. Setting the sights a little lower and keeping in mind Paul Erdos’s comment that “a mathematician is a machine for turning coffee into theorems,” we could try to understand what makes someone good at deducing theorems. Now, the general procedure in deductive mathematics is to start with some rules (axioms) and then to derive necessary truths (theorems) on the basis of those axioms. However, the axioms themselves are not locked in a vacuum, and plenty of other rules are used in helping someone deduce theorems. Take for instance the axioms defining the real numbers. If one wishes to prove that the square root of 2 is irrational, one must use logical rules that are not contained in the axioms in order to devise a proof. The ultimate question concerning deductive mathematics can then be phrased in the following informal way (perhaps thinking about the problem in this way will yield further insight). Given a set of theorems T and a set of axioms A, how does the set of necessary logical propositions (an admittedly informal notion) L[T:A] look? Well for a starter, it probably contains more theorems than are just contained in T. If T is just a single theorem, then the set L[T:A] is just all logic necessary to prove T from A. So where does this leave us? Well, it leaves us with a good starting point for defining general deductive intelligence and more loosely mathematical intelligence. That is, a general deductive intelligence is able to construct L[T:A]. Someone who is good at mathematics likewise is able to understand at least part of the structure of L[T:A]. Furthermore If one can figure out an efficient method for performing searches that “find” elements of L[T:A], then automatic theorem proving would be trivial because all of the justification for the proof is contained in L[T:A].
In the end, the search for the defining parameters of mathematical intelligence is hard to undertake for a reason. Quite clearly, a mathematician is required to have to certain baseline abilities that make his job possible. However, the current understanding of these abilities is limited due to difficult nature of the problem. One can imagine that there is some structure associated to a given set of axioms and a set of theorems which will represent the logical justification for those theorems. The best mathematicians are then those who best justify.