Mathematics is in many ways the most elaborate and sophisticated of the sciences or __________ so it seems to me, as a mathematician. So I find both a special pleasure and constraint in describing the progress of mathematics, because it has been part of so much human speculation; a ladder for mystical as well as rational thought in the intellectual ascent of man. However there are some concepts that any account of mathematics should include: the logical idea of proof, the empirical idea of exact laws of nature (of space particularly), the emergence of the concept of operations, and the movement in mathematics from a static to a dynamic description of nature. They form the theme of this essay.
Even very primitive peoples have a number system; they may not count much beyond four, but they know that two of anything plus two of the same thing makes four, not just sometimes but always. From that fundamental step, many cultures have built their own number systems, usually as a written language with similar conventions. The Babylonians, the Mayans, and the people of India, for example, invented essentially the same way of writing large numbers as a sequence of digits that we use, although they lived far apart in space and in time.
So there is no place and no moment in history where I could stand and say `Arithmetic begins here, now.' People have been counting, as they have been talking, in every culture. Arithmetic, like language, begins in legend. But mathematics in our sense, reasoning with numbers is another matter. And it is to look for the origin of that, at the hinge of legend and history that I went sailing to the island of Samos.
