In his classic paper “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (1960), Eugene Wigner tells a story of two friends meeting after a long while. One of them has become a statistician working in population trends. He shows a reprint of his recent paper to his friend, who, upon seeing a Gaussian distribution, asks what the symbol π means. The statistician answers by telling that π gives the ratio of circumference of a circle to its diameter. His friend accuses him of joking, for surely natural language cannot reliably pick out things like “circles”, “circumferences” and “ratios”, much less abstract entities like π?
I am of course joking, albeit poorly: Wigner’s joke ends with the statistician’s friend doubting that populations have anything to do with the circumferences of circles. What Wigner wishes to convey is the idea that mathematics is surprisingly, even miraculously, good in scientific theorizing. But why stop (or start) here? Surely, if this is surprising, it is even more surprising that we can communicate with each other with natural language.
Look at all the things we do all the time with natural language. We tell our friends and spouses of our day at work, we share our thoughts and dreams, we describe the world around us – and even describe imaginary worlds, events and persons. Soon after encountering something for the first time, we will mirabile dictu have a name for that thing. We even concoct philosophical arguments with natural language! But there is more: we can use metaphors – literally using words against their literal meanings – to extend our language, both in everyday contexts and in science. How on earth is all this possible? How does our language fit so well with the actual and imaginary worlds? Is this miraculous?
To see our way clear of such questions, it is good to look at the history of language – or of mathematics as the case may be. Signaling systems, ways of communication, are tools. They have developed (and been developed) for a use. The use constrains the evolution of the tool: the bits that find a use are kept and honed, the other bits are left to fend for themselves. Once this is understood, the mystery dissolves. The same will turn out true for the case of the effectiveness of mathematics in the sciences: mathematics has by and large been developed hand-in-hand with physical theory. Those bits of mathematics which are built for application, or find an application, are developed further, and will be effective in their applications, because they are tools selected for just that purpose. The history of mathematics is illustrative on this point (see for instance Grattan-Guinness 2008).
So, a look at history might sometimes dissolve mysteries before they are even born. It is only if one sees things like language, mathematics and physical theory as something separate from human activities that such things start to seem mysterious. But the carpenter finds no mystery in her hammer’s effectiveness at driving nails.
Grattan-Guinnes, Ivor (2008). Solving Wigner’s Mystery: The Reasonable (Though Perhaps Limited) Effectiveness of Mathematics in the Natural Sciences, The Mathematical Intelligencer 30: 3, 7–17.
Wigner, Eugene P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications on Pure and Applied Mathematics, vol XIII, 1–14.