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Notes for: The World of Simple Programs | How the Discoveries in This Chapter Were Made

*History of experimental mathematics

The general idea of finding mathematical results by doing computational experiments has a distinguished, if not widely discussed, history. The method was extensively used, for example, by Carl Friedrich Gauss in the 1800s in his studies of number theory, and presumably by Srinivasa Ramanujan in the early 1900s in coming up with many algebraic identities. The Gibbs phenomenon in Fourier analysis was noticed in 1898 on a mechanical computer constructed by Albert Michelson. Solitons were rediscovered in experiments done around 1954 on an early electronic computer by Enrico Fermi and collaborators. (They had been seen in physical systems by John Scott Russell in 1834, but had not been widely investigated.) The chaos phenomenon was noted in a computer experiment by Edward Lorenz in 1962 (see page 971). Universal behavior in iterated maps (see page 921) was discovered by Mitchell Feigenbaum in 1975 by looking at examples from an electronic calculator. Many aspects of fractals were found by Benoit Mandelbrot in the 1970s using computer graphics. In the 1960s and 1970s a variety of algebraic identities were found using computer algebra, notably by William Gosper. (Starting in the mid-1970s I routinely did computer algebra experiments to find formulas in theoretical physics--though I did not mention this when presenting the formulas.) The idea that as a matter of principle there should be truths in mathematics that can only be reached by some form of inductive reasoning--like in natural science--was discussed by Kurt Gödel in the 1940s and by Gregory Chaitin in the 1970s. But it received little attention. With the release of Mathematica in 1988, mathematical experiments began to emerge as a standard element of practical mathematical pedagogy, and gradually also as an approach to be tried in at least some types of mathematical research, especially ones close to number theory. But even now, unlike essentially all other branches of science, mainstream mathematics continues to be entirely dominated by theoretical rather than experimental methods. And even when experiments are done, their purpose is essentially always just to provide another way to look at traditional questions in traditional mathematical systems. What I do in this book--and started in the early 1980s--is, however, rather different: I use computer experiments to look at questions and systems that can be viewed as having a mathematical character, yet have never in the past been considered in any way by traditional mathematics.


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