Mathematics, Computers and Zeilberger
This piece is a ramble through a collection of thoughts linked to and influenced by the opinions of Doron Zeilberger. It starts with the uncertainty of proof and discusses the importance of computers before concluding with the future of mathematics in the large but finite.
I am never sure how many mathematicians feel this, but I often doubt my own proofs. I find myself certain of some fact, with a chain of reasoning to support it, that, at a later date, fails on some small detail. Only when all the details are worked out do I start to feel a sense of comfort, but even then I fear something will leap out at me and the ideas I thought were quite brilliant fall apart completely.
I therefore always take comfort from Zeilberger’s opinions. He discusses this problem, and actually takes it further:
Mathematics is arguably the most certain body of knowledge, but of course, nothing is certain in this world, and it is a distinct possibility that the Pythagorean Theorem, and even 2+2=4, are wrong, and it just so happened that Nature and/or God programmed the human mind so that it will overlook the gaps in their alleged proofs. Complete certainty (even of death and taxes, and certainly of mathematical facts) is an unreachable ideal, but one can at least try to improve the reliability…
However he also provides a potential solution:
…we need computers. Computers abhor ambiguity, and trying to teach computers mathematics is also good for us humans, since it forces us to discover hidden ambiguities and resolve them.
I agree with this entirely. It is so much easier to check if I understand something if I can pull out mathematica or python and actually program it. If my understanding is gibberish I will not see what I expect, but get gibberish back from the computer. It is therefore easy to check if something has failed. Furthemore, I can follow through what happened, and find where it failed (assuming it was not a bug in the code) and thus identify where my intuition was wrong. On the other hand if the output of the code agrees with my expectation, I know that something has gone right. I can now take this further, instead of slowly working through examples for a conjecture by hand I can immediately check them. I even learn mathematics better this way. I probably learnt more about Hyperbolic geometry in constructing the image below than in all the lectures I have attended on the subject (though maybe not in discussions).
To me, therefore, there is an obvious, strong case for using computers for both research and teaching, and encouraging all mathematicians and mathematics students to use SAGE or Mathematica (even though I use it, I do feel that it is a little too expensive). However one can take things further. Maybe computers are not just useful, but essential to the future of mathematics, returning to Zeilberger:
According to this criterion [level of abstraction], most of human mathematics is completely useless. It was developed by humans for human consumption. In order for humans to understand it, it had to proceed in tiny steps, each comprehensible to a human. But if we take the “mesh size” of each step, dA, to be larger, one can do potentially much bigger and better things, and the computer’s dA is much larger, so we can (potentially) reach a mountain-top much faster, and conquer new mountain-tops where no humans will ever tread with their naked brains.
To me this vision of mathematics is particularly exciting. Mathematics started by studying the behaviour of the small; low numbers and simple equations. The next step was a mathematics of infinity, resting on the observation that in many cases infinity could be used to approximate large numbers, and the continuum to approximate large collections of small things. The problem is that this only holds in “many cases”, it is not true in general. In fact in the growing quantitative study of biology and sociology, made possible by DNA sequences and the data in the internet, it regularly does not hold. We therefore require a mathematics of the large but finite. Computers, with mathematician trainers, provide a way of achieving this. It should be an exciting adventure, after all we have much to learn even in the simplest cases, to quote Tim Gowers:
…there is more to say about the whole question of multiplying large number than you might think…