There is an apparently unnoticed class of truths, which I call

*- statements which may be trivial, may be unexpected, but which are reveal themselves as true after being stated. They may require some thinking over, but they do not require arguments.***“self-revealing”**Or, may be, there is no such thing as a universally self-revealing truth? May be different truths reveal themselves as such to different people? May be even they are true for one person and false for another?

Here are some observations about mathematics, which are self-revealing truths for me, but which, apparently, are not even true for many other mathematicians. I am hardly able to support them by arguments. Or, rather, I am hardly willing to do this before I will face a deep challenge to their validity. But I will try to present them as a coherent whole.

Mathematics is a human activity.

A distinctive feature of mathematics is an attempt to rely only on infallible arguments. This feature is known as the mathematical rigor, as also the concept of a rigorous proof. In fact, a mathematician will tell you that a non-rigorous proof is a non-sense: if something is a proof in mathematics, then it is rigorous.

If mathematics has any value, then only as a human activity.

I expect that here I will be pointed to the applications of mathematics, which should be valued independently of source of the mathematics applied: humans, computer, prophets, aliens from the outer space, etc. I would like to point out that this argument deserves some consideration only because of a historical accident: the fact that some Ancient Greeks interested in geometry were also interested in rigorous arguments. If not them, we would, most likely, have only non-rigorous applied mathematics. The self-revealing truth here is the following.

Applications of mathematics do not require rigorous proofs to be useful.

Let me illustrate this by few observations.

In applications usually one can ignore special cases, or even the cases perceived as anomalous. This is something that mathematicians do not allow to themselves.

It is not widely known that the works of S. Fefereman, H. Friedman, S. Simpson, and G. Takeuti, among others, showed that the part of mathematics which is indispensable for scientific applications can be reduced to the (Peano) arithmetic. The questions about real numbers and sets of real numbers, which fascinated and inspired mathematicians for at least the last couple of centuries, are irrelevant for applications.

The following should be a complete triviality. The mathematical rigor is not needed for applications. We freely use many tools without being convinced or even assured that they will never fail. We use our tools knowing for sure that they will fail, sometimes quite often. Bridges, cars, planes from time to time fail to function as expected. Anybody reading this text on the web is familiar with both software and hardware failures.

One of the most widespread current applications of mathematics is the use prime numbers for encryption. Thanks to the modern cryptography, very big prime numbers even have definite monetary value. But one needs only numbers which are prime with very high probability. They are as good as the numbers which were verified to be prime with full mathematical rigor. If a number sold as a prime number is actually not prime, then a transaction at Amazon.com may fail. But that’s not a problem, occasionally they do fail anyhow by other reasons.

The most distinctive feature of mathematics, the concept of a mathematical proof, is not needed for applications.

Next post: A comment from Timothy Gowers

Обнаглею(я математик только по образованию, т е минимум на 2 порядка слабее Вас, скорее на 3), но не соглашусь. Формально Вы правы - строгость для приложений не нужна. Дело однако в том(кроме прочего), что интуиция работает только в знакомых областях - она всего лишь обобщение и следствие опыта. При вступлении на неведомую территорию необходимо установить полную ясность в основах - т е навести строгость :-) Без этого новых инструментов не создашь :-) Исторически так оно и было - у греков, у Гаусса и Коши, в конце 19-ого века. Противоречащий пример, конечно - создание анализа. Но имхо - не очень противоречащий. Вломились, собрали урожай того что можно было взять нахрапом - и всё, пришлось осматриваться.

ReplyDeleteP. S. Вы знаете, что Ваше identity вычисляется в один клик :-) ? Просто сообщаю, т к Вы вряд ли это хотели :-)