# Notes

## Section 1: The Notion of Numbers

History of numbers

Numbers were probably first used many thousands of years ago in commerce, and initially only whole numbers and perhaps rational numbers were needed. But already in Babylonian times, practical problems of geometry began to require square roots. Nevertheless, for a very long time, and despite some development of algebra, only numbers that could somehow in principle be constructed mechanically were ever considered. The invention of fluxions by Isaac Newton in the late 1600s, however, introduced the idea of continuous variables—numbers with a continuous range of possible sizes. But while this was a convenient and powerful notion, it also involved a new level of abstraction, and it brought with it considerable confusion about fundamental issues. In fact, it was really only through the development of rigorous mathematical analysis in the late 1800s that this confusion finally began to clear up. And already by the 1880s Georg Cantor and others had constructed completely discontinuous functions, in which the idea of treating numbers as continuous variables where only the size matters was called into question. But until almost the 1970s, and the emergence of fractal geometry and chaos theory, these functions were largely considered as mathematical curiosities, of no practical relevance. (See also page 1168.)

Independent of pure mathematics, however, practical applications of numbers have always had to go beyond the abstract idealization of continuous variables. For whether one does calculations by hand, by mechanical calculator or by electronic computer, one always needs an explicit representation for numbers, typically in terms of a sequence of digits of a certain length. (From the 1930s to 1960s, some work was done on so-called analog computers which used electrical voltages to represent continuous variables, but such machines turned out not to be reliable enough for most practical purposes.) From the earliest days of electronic computing, however, great efforts were made to try to approximate a continuum of numbers as closely as possible. And indeed for studying systems with fairly simple behavior, such approximations can typically be made to work. But as we shall see later in this chapter, with more complex behavior, it is almost inevitable that the approximation breaks down, and there is no choice but to look at the explicit representations of numbers. (See also page 1128.)

From Stephen Wolfram: A New Kind of Science [citation]