Computers as novel mathematical reality 

In the last decades there was much ado about computer proofs, computer aided proofs,

computer verified proofs, and the like. It is obvious that the advent and proliferation of

computers have drastically changed applications of mathematics.

What one discusses much less, however, and what I find much more interesting, is how

computers have changed mathematics itself, and mathematicians’ stance in regard of

mathematical reality, both as far as the possibilities to immediately observe it, and the

apprehension of what we can hope to prove.

Computers have already changed mathematics in what concerns very basic ideas much

more fundamental than any individual theories: balance of ideas and computations,

large and small (and, most importantly, intermediate size!), finite and infinite, feasible

and unfeasible, deterministic and random, etc.  

Much of the recent progress in mathematics would had been impossible without

computers. In particular, they allowed to reconnect to the history of mathematics

and solve problems in the absolute sense, as they were posed by the XVII--XVIII

centuries classics, rather than merely in asymptotic forms.

I recount my personal experience of using computers as a mathematical tool, and

the experience of such similar use in the works of my colleagues that I could observe

at close range, especially in group theory, and number theory.

This experience has radically changed my perception of many aspects of mathematics,

what it is, how it functions, and especially, how it should be taught.