Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy

11.12.SHUT UP AND CALCULATE!

Philosophical questions rarely find generally accepted answers. Also, the question of whether numbers were discovered or invented has never found a definite answer accepted by a clear majority of mathematicians. Probably, there will always be a certain plurality of ideas and approaches, as described in the previous sections.

But for most mathematicians the questions mentioned above have little influence on the actual mathematical practice. Some even have a very skeptical position toward the usefulness of philosophy. In his 1994 book Dreams of a Final Theory, American physicist Steven Weinberg (1933–) writes in a chapter called “Against Philosophy” that we should not expect philosophy “to provide today's scientists with any useful guidance about how to go about their work or about what they are likely to find.”²² Indeed, any strict philosophical position could inhibit free and unprejudiced thought and thus make progress more difficult. For example, if you strictly adhered to the ultrafinitist position, you would exclude yourself from most of mathematics—in particular from many branches that are very useful for practical applications.

Many mathematicians therefore consider thoughts about the foundations of their science as a “waste of time.” In 2013, in notes for a paper titled “Does Mathematics Need a Philosophy?” British philosopher Thomas Forster (1948–) writes, “Unfortunately most of what passes for Philosophy of Mathematics does not arise from the praxis of mathematics. In fact I even believe that the entirety of the activity of ‘Philosophy of Mathematics’ as practiced in philosophy departments is—to a first approximation—a waste of time, at least from the point of view of the working mathematician.”²³

The success of mathematics, when applied to the solution of concrete problems, fortunately does not depend on philosophical positions. Even if two mathematicians disagree about the foundations of their science, they would usually agree about the result of a concrete calculation. Whether or not you believe in the independent existence of numbers, a statement like “5 + 3 = 8” remains valid and useful in many concrete situations. All that is important is that the existing framework of mathematics allows us to solve real problems. It is quite a common position that as long as the application of mathematical models is successful, we need no philosophical interpretations. This is called the “shut up and calculate” position. The expression was coined by American physicist David Mermin (1935–), who used it to describe a common attitude of physicists toward philosophical problems with the interpretation of quantum mechanics.

According to Reuben Hersh, most mathematicians seem to oscillate between Platonism and a formalist's point of view. As these two positions are rather incompatible, one can see that philosophy is not a typical mathematician's primary concern. On the other hand, a few sentences could hardly ever be “a full and honest expression of some flesh-and-blood mathematician's view of things,” as stated by Barry Mazur, who describes the meandering between philosophical positions and motivations for doing mathematics as follows:

When I'm working I sometimes have the sense—possibly the illusion—of gazing on the bare platonic beauty of structure or of mathematical objects, and at other times I'm a happy Kantian, marveling at the generative power of the intuitions for setting what an Aristotelian might call the formal conditions of an object. And sometimes I seem to straddle these camps (and this represents no contradiction to me). I feel that the intensity of this experience, the vertiginous imaginings, the leaps of intuition, the breathlessness that results from “seeing” but where the sights are of entities abiding in some realm of ideas, and the passion of it all, is what makes mathematics so supremely important for me. Of course, the realm might be illusion. But the experience?²⁴