Computers as a Novel Mathematical Reality: I. A Personal Account

Much ink has been spilled in the last few decades on computer proofs, computer-assissted proofs, computer-verified proofs, etc. It is obvious that the advent and proliferation of computers have drastically changed the applications of mathematics. What is far less discussed, however, is how computers changed mathematics itself, as well as the stance of mathematicians in regard to mathematical reality, both as far as the possibilities to immediately observe it and the apprehension of what we can hope to prove. I am recounting my personal experience of using computers as a mathematical tool and the experience of similar use in the works of colleagues that I observed at close range. This experience has radically changed my perception of many aspects of mathematics; how it functions; and, especially, how it should be taught. This first introductory part consists mostly of reminiscences and some philosophical observations. Later parts describe several specific important advances in algebra and number theory that would had been impossible without computers.