The purpose of this article is to illustrate the impact of computer experiments on mathematics and the physical sciences from a historical perspective. We are especially interested in showing the role of visualization and interactivity. Let us start by giving the floor to John Hubbard, who will meet again in the section on complex dynamics :
“I mention computer graphics because faster and cheaper computers alone would not have had the same impact ; without pictures, the information pouring out of mathematical computations would have remained hidden in a flood of numbers, difficult if not impossible to interpret. For people who doubt this, I have a story to relate. Lars Ahlfors, then in his seventies, told me in 1984 that in his youth, his adviser Lindelof had made him read the memoirs of Fatou and Julia, the prize essays from the Académie des Sciences in Paris. Ahlfors told me that they struck him at the time as “the pits of complex analysis” : he only understood what Fatou and Julia had in mind when he saw the pictures Mandelbrot and I were producing. If Ahlfors, the creator of one of the main tools in the subject and the inspirer of Sullivan’s nowandering domains theorem, needed pictures to come to terms with the subject, what can one say of lesser mortals ?”
We do not claim to be exhaustive. Among the noticeable people we could have mentioned there is Derrick H. Lehmer (1905-1991), a number theorist, who worked on the ENIAC. Fortunately, we can refer to the article of Liesbeth De Mol [1] which discusses and contrasts the visions of Lehmer and von Neumann on the use of the computer and its impact on mathematics.
We deliberately avoid technicalities and refer, whenever possible, to some texts understandable by non-specialists. We also point to the original articles which are almost all freely available on Internet.
We did our best to find out who were the programmers behind the computer experiments, for instance Mary Tsingou for the FPUT model, Gary Deem who worked with Kruskal and Zabusky, or Peter Moldave who wrote and run the computer experiments for Mandelbrot. We also did our best to indicate the computers used.
In this article we will not be concerned with exciting topics such as numerical analysis, computer-assisted proofs, proof assistants (formal proof management systems), or machine learning.
Finally, let us mention that this article was originally supposed to be a translation of an article published in French in La Gazette de la Société Mathématique de France, vol. 143 (2015). In fact, the article was almost entirely rewritten and expanded, and many pictures were added.
Acknowledgments. We thank Alix Chazottes for a careful reading.