This post collects some brief philosophical thoughts about mathematics that appeared as part of my paper “Combinatorics with a geometric flavor: some examples,” from the proceedings of the conference “Vision in Mathematics, towards 2000.” I added two small items (the first and fifth).
1. Mathematical truths - theorems.
“There are infinitely many primes;” “The three angles of a triangle add up to 180 degrees;” ”A continuous real function defined in a closed interval attains there its maximum;” “A non-constant polynomial over the complex numbers has a solution;” If you substitute a matrix A in its characteristic polynomial you get zero;” ”A simply connected closed 3-dimensional manifold is homeomorphic to a sphere.”
These truths appear very different from truths in other areas of life. This sharp difference is the secret to some of the successes of mathematics and explains also its limitation.
What makes a mathematical theorem important, deep, or central?
2. Proofs, more proofs, “proofs from the book” and computer proofs
Science has a dual role: exploring and explaining. In mathematics, unlike other sciences, mathematical proofs are used as the basic tool for both tasks: to explore mathematical facts and to explain them.
The meaning of a mathematical proof is quite stable. It seems unharmed by the “foundation crisis” and the incompleteness results in the beginning of the 20th century, and unaffected by the recent notions of randomized and interactive proofs in theoretical computer science. Still, long and complicated proofs,
as well as computerized proofs, raise questions about the nature of mathematical explanations.
Proofs are gradually becoming intolerably difficult. This may suggest that soon our days of successfully tackling a large percentage of the problems we pose are over. Also, this may reflect the small incentives to simplify.
Be that as it may, we cannot be satisfied without repeatedly finding new connections and new proofs, and we should not give up hope to find simple and illuminating proofs that can be presented in the classroom. For some “proofs from the book”, see the lovely book by Aigner and Ziegler.
Some believe that computer proofs will take over (Doron Zeilberger is a strong advocate for this view). Appel and Haken’s proof of the four color theorem was a landmark in this respect. Can computers be used not just for “symbol crunching” but also for “idea crunching”? (Perhaps, “idea crunching” will be easier for computers?) The role of computers in exploring mathematical facts is already significant. As for explaining mathematical facts, it raises, for instance, the question: explaining to whom? To humans, or to other computers?
3. Problems and conjectures
The posing of problems and conjectures is part of the process of exploring the factual matters as well as of proposing explanations for them. Is the development of mathematics shaped by problems? And what are good problems? Do they arise naturally like the sphere-packing conjecture, or are they perhaps sporadic and ingenious like Fermat’s last theorem and the four color problem? To what an extent are good mathematical problems suggested by other sciences?
Modern combinatorics was greatly shaped by problems posed by Erdös, who was very cautious concerning our ability to predict the future of a problem.
It is not unusual that a single example or a very few shape an entire mathematical discipline. Examples are the Petersen graph, cyclic polytopes, the Fano plane, the prisoner dilemma, the real n-dimensional projective space and the group of two by two nonsingular matrices. And it seems that overall, we are short of examples. The methods for coming up with useful examples (or counterexamples for commonly believed conjectures) are even less clear than the methods for proving.
5. Concepts, methods, ideas and theories
Is a conceptual understanding of a mathematical truth the goal of mathematical understanding? Or perhaps the conceptual framework can restrict and bias our mathematical understanding?
6. Our community
Like musicians who can enjoy and understand complicated scores even in a world with no sounds, for us mathematics is a source of delight, excitement and even controversy. This is hard to share with nonmathematicians.
In our small world we should seek new ways for communication and interaction and for the right balance between competition and solidarity, criticism and empathy, exclusion and inclusion.
7. Applications and Expectations
Mathematicians today have a strong desire to interact and influence other sciences, as well as technology, industry, and even economic life. The trends towards isolationism have been reversed, and there is a greater understanding of the subtleties of applying mathematics to and interacting with other fields. Applied mathematics introduced new paradigms for practicing mathematics which go beyong the familiar theorems/proofs paradigm.
The general public knows very vaguely what mathematicians do. At the same time people have quite clear expectations from mathematics. More than other sciences, and certainly much more than law, religion, politics and the media, mathematics is expected to be rigorous and precise in telling its uninteresting, irrelevant, and uncomforting truths.
The value of mathematics for society goes far beyond its applications through technology; it is indeed a pillar of human culture. After a century of amazing technological development along with rising influence of pseudosciences and the occult, this value is important.