The journal Experimental Mathematics, started in 1992, publishes “formal results inspired by experimentation, conjectures suggested by experiments, descriptions of algorithms and software for mathematical exploration, [and] surveys of areas of mathematics from the experimental point of view.” The founder wanted to make clearer and give more credit to an important way that mathematicians come up with new ideas. As the journal’s statement of philosophy puts it, “Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was “through systematic experimentation.”) Yet this tends to be concealed by the tradition of presenting only elegant, well-rounded, and rigorous results.”
When John Tukey wrote Exploratory Data Analysis (1977), he was doing something similar: shedding light on how to come up with new scientific ideas plausible enough to be worth testing. Tukey obviously believed this was a neglected area of statistics research. I was told that the publisher of EDA was uninterested in it; they only published it because it was part of a two-book deal. The other book, with Frederick Mosteller, was more conventional.
My paper titled “Self-experimentation as a source of new ideas” made the same point as Tukey about an earlier step in the scientific process: data collection. How to collect data to generate new ideas worth testing was a neglected area of scientific method. Self-experimentation, derided as a way of testing ideas, might be an excellent way of generating ideas worth testing.
I think of it as crawling back into the water. In the beginning, all math was conjecture and experimentation. In the beginning, all data analysis was exploratory. In the beginning, all science was tiny and devoted to coming up with new ideas. From these came methods of proof, confirmatory data analysis, and methods of carefully testing ideas. Human nature being what it is, users and teachers of the new methods came to greatly disparage the earlier methods. Gary Taubes told me that he spoke to several obesity researchers who thought that the field essentially began with the discovery of leptin. Nothing before that mattered, they believed.
Thanks to Dev Rana.