Mathematics on a distant planet

The purpose of this talk is to get you to think seriously about the extent to which mathematics is arbitrary and the extent to which it is fixed, and about what you think mathematics is. This talk began in my mind long ago during the early SETI (search for extra terrestrial intelligence) when the giant radar dish at Arecibo was being used for the search. Now, many years later, we are searching with far more efficiency and still have no definite signs of life. There are three aspects to mathematics. First, there are the postulates, axioms, or assumptions, however you may wish to call them. The Bourbaki approach has tended to focus on them. Alternate postulates that describe the same mathematics are not to be considered as different mathematics. Second, there are the definitions; these determine a great deal of mathematics. For example, the current definition of a function allows a lot of peculiar things. If I have a continuous saw tooth function with alternating slopes of + 450 and rotate the coordinates 45? then the function is no longer continuous; it is not even a function! Neither Euler nor Fourier would have agreed with this! The idea of continuity arose from the concept of drawing the curve without lifting the (ideal) pen and had no relationship to the coordinates you happen to choose. To have distorted the primitive, intuitive idea of continuity so that it is now coordinate dependent seems to me to be foolish. Thus the idea of a function and the idea of a curve are now distinct concepts. Third, there are the kinds of logical deductions that are permitted. The patterns of reasoning are seldom examined these days and will be looked at briefly, and in only one direction. But before going farther I need to mention a few things in my life that have shaped my opinions. The first occurred at Los Alamos during WWII when we were designing atomic bombs. Shortly before the first field test (you realize that no small scale experiment can be done-either you have a critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, "It is the probability that the test bomb will ignite the whole atmosphere." I decided I would check it myself! The next day when he came for the answers I remarked to him, "The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen-after all, there could be no experiments at the needed energy levels." He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, "What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?" I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, "Never mind, Hamming, no one will ever blame you." Yes, we risked all the life we knew of in the known universe on some mathematics. Mathematics is not merely an idle art form, it is an essential part of our society.