A Higher Nonsense

From the UCLA Math Department's Newsletter "Common Denominator" (Fall 2007), a quick bio of a new faculty member:

Chandrashekhar Khare joins the Department as professor in the number theory group. After solving
the Serre modularity conjecture with Wintenberger (and Kisin), Shekhar has become an international
star in Galois representations and their compatible systems. Serre’s conjecture has been outstanding
for almost three decades and essentially asserts that all p-adic Galois representation under appropriate
arithmetic conditions comes from (readily computable) elliptic modular forms (complex analytic
objects). The solution of the conjecture has made headlines in number theory, and summer schools
and conferences are currently being held across the world in the U.S., France, India and Japan on
the proof. Shekhar completed his PhD at CalTech, then joined the Tata Institute of Fundamental Research in Mumbai, India, in 1995 and the University of Utah in 2001. He received the Young Scientist
Award from the Indian National Science Academy in 1999 and was elected to associate membership in
the Indian Academy of Sciences.

Even though I left the UCLA Math Department nearly 20 years ago, I still have a fondness for the place. But my contacts with the Department have been non-existent since they awarded me a Master's Degree (what the super-genius snoots that populated the graduate school called "the booby prize" for not completing the PhD program). Then suddenly, a few months ago, I got on the mailing list for their newsletter, which is obviously a fund raising gimmick. But I love it. It gives me a chance to see what some of my old (now, 20 years later, really old) professors are doing. And what some of the new guys they hire are up to. I've already reported on UCLA's Fields Medal winner Terry Tao. The guy was still watching Fraggle Rock when I started at UCLA (though, as a child prodigy, he already had a few theorems under his belt). Professor Khare above seems to be another accomplished fellow.

What I enjoy about biographies like the one above is that after reading it, although I love math and studied it for three years at one of the top schools for mathematics in the US, I still have almost no idea what the guy does. I mean, I know what number theory is. It's the study of numbers. (Duh!) And I know more specifically how it is distinguished from other branches of mathematics. But looking up Serre's conjecture, which Professor Khare proved, thereby gaining his new job, I see I am completely lost. I see in the Wikipedia article that it is related to Fermat's Last Theorem (the statement of which is easy to understand) and provides a simpler means of proving it than Wiles' original proof. Perhaps the gain in simplicity is like that in going from brain surgery to hip replacement surgery, both beyond the ability of someone who can't tie a knot in fishing line.

Now, number theory is a part of "pure mathematics", and is one of the most arcane areas of that field. I studied applied math. Most mathematicians believe that there is an underlying unity, beyond the basic methods they use, in all of mathematics. And surprising connections between the hitherto seemingly far-flung branches of mathematics are continually being found. But mathematics is a very wide discipline. The connections can sometimes seem very tenuous indeed. But despite my befuddlement in reading biographies like that of Professor Khare, I am secure in the knowledge that what he studies is true.

Mathematics has been called the "Queen of the Sciences", though many would argued that it is not a science at all. Most sciences argue for the truth of their theories by induction on observation of the real world. The truth of a scientific theory is judged by it's correspondence with the world and is thus, in a sense, statistical in nature. Mathematics, even when it talks about probabilities, is purely deductive. (Even "mathematical induction", a method of proof, is in reality deductive.) So, if we believe the underpinning suppositions of a mathematical theory, once the consistency of a mathematical argument has been established, the result is necessarily true. This is another of the reasons why I love mathematics.

But note, I said "if we believe the underpinning suppositions...". This is crucial. The Greeks believed that mathematics, though an abstraction from reality, had a necessary correspondence with reality, and that mathematical axioms were, in a sense, physical ones. Many mathematical physicists carry out their work as if they also still believe this is the case. But by the 19th century, mathematicians had come to believe that their axioms are arbitrary. Of course, there can be a high degree of correspondence between real world and a mathematical description of it. But change an axiom and the mathematical changes completely (though it may still be useful in another context), while the world stays the same. Euclid's geometry corresponds very well with many aspects of our world. But changing (or doing without) his Fifth Postulate results in different geometries, some of which also correspond to our real world, but viewed through a different lens.

So mathematicians are in the business of manufacturing truth. Within a given framework of axioms, mathematics is perfectly true. In fact, the truth of mathematics is, in a sense, tautological. Theorems are just roundabout ways of restating what is already contained in the axioms. But part of the beauty of mathematics lies in this manufacturing of truth.  Even a simple set of axioms, such as those of Euclid, contain all the truths of geometry discovered in the 2500 years since, and many more that have yet to be discovered. There are so many true statements that can be made from a set of axioms that the value of the truth is really contained in the value of the question that it answers. And this value is a subjective, human value. The value can involve how well a branch of mathematics helps understand the physical world, or simply how well the theorem helps advance another field of mathematics which has little connection to reality. (The value of this field may be judged, by those who toil in it, on strictly aesthetic grounds.)

But if mathematics can have an arbitrary basis, people still marvel at it's utility in connection with reality, so much so that some ponder on whether there is some deep, metaphysical connection between mathematics and reality. But that question is well beyond the scope of what I wanted to write about. Suffice it to close by saying that there are other fields where the manufacturing of truth is not nearly so successful as in mathematics. Theorists in the  humanities and social sciences in the last 40 years have been particularly prone to manufacturing truths that are unmoored from reality, or any method of assessing value at all, while also being impervious to criticism. Yet these same manufacturers of truth also have a severe case of "math envy" and "physics envy" because of the successes of those fields. This envy has opened them to ridicule, to which they also seem impervious. The most famous case was the Sokol Hoax, where a physicist played their game and invented a new reality spun from the high nonsense that is axiomatic in some areas of the social sciences.