The Entirely Reasonable Efficacy of Mathematics

by F.

John Allen Paulos on the natural, not supernatural, nature of mathematics:

there have been first-rate scientists who have taken mathematics to be some sort of divine manifestation. One of the most well-known such arguments is due to physicist Eugene Wigner. In his famous 1960 paper, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, he maintained that ability of mathematics to describe and predict the physical world is no accident, but rather is evidence of a deep and mysterious harmony.

But is the usefulness of mathematics really so mysterious? There is a quite compelling alternative explanation why mathematics is so useful. We count, we measure, we employ basic logic, and these activities are stimulated by ubiquitous aspects of the physical world. The size of a collection (of stones, grapes, animals), for example, is associated with the size of a number and keeping track of it leads to counting. Putting collections together is associated with adding numbers, and so on.

Another metaphor associates the familiar realm of measuring sticks (small branches, say, or pieces of string) with the more abstract one of geometry, The length of a stick is associated with the size of a number (once some segment is associated with the number one), and relations between the numbers associated with a triangle, say, are noted. (Scores of such metaphors underlying more advanced mathematical disciplines have been developed by linguist George Lakoff and psychologist Rafael Nunez in their book, Where Mathematics Comes From.)

Once part of human practice, these various notions are abstracted, idealized and formalized to create basic mathematics, and the deductive nature of mathematics then makes this formalization useful in realms to which it is only indirectly related.

The universe acts on us, we adapt to it, and the notions that we develop as a result, including the mathematical ones, are in a sense taught us by the universe. That great bugbear of creationists, evolution has selected those of our ancestors (both human and not) whose behavior and thought are consistent with the workings of the universe. The usefulness of mathematics is thus not so unreasonable.

Platonism is just another Iron Age* relic. I can’t believe people take this idea seriously anymore.

*Loosely speaking. If the Iron Age ended in 550 B.C.E, then Platonism couldn’t have developed during it, since Plato was born about 420 B.C.E. But Pythagorus was born in 580 B.C.E., and I think Plato got much of his mathematics woo from the mathematical mysticism of the Iron Age, which the early Pythagoreans were all about.

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