Is God a Mathematician_ - Mario Livio [114]
Wigner’s Enigma
“Is mathematics created or discovered?” is the wrong question to ask because it implies that the answer has to be one or the other and that the two possibilities are mutually exclusive. Instead, I suggest that mathematics is partly created and partly discovered. Humans commonly invent mathematical concepts and discover the relations among those concepts. Some empirical discoveries surely preceded the formulation of concepts, but the concepts themselves undoubtedly provided an incentive for more theorems to be discovered. I should also note that some philosophers of mathematics, such as the American Hilary Putnam, adopt an intermediate position known as realism—they believe in the objectivity of mathematical discourse (that is, sentences are true or false, and what makes them true or false is external to humans) without committing themselves, like the Platonists, to the existence of “mathematical objects.” Do any of these insights also lead to a satisfactory explanation for Wigner’s “unreasonable effectiveness” puzzle?
Let me first briefly review some of the potential solutions proposed by contemporary thinkers. Physics Nobel laureate David Gross writes:
A point of view that, from my experience, is not uncommon among creative mathematicians—namely that the mathematical structures that they arrive at are not artificial creations of the human mind but rather have a naturalness to them as if they were as real as the structures created by physicists to describe the so-called real world. Mathematicians, in other words, are not inventing new mathematics, they are discovering it. If this is the case then perhaps some of the mysteries that we have been exploring [the “unreasonable effectiveness”] are rendered slightly less mysterious. If mathematics is about structures that are a real part of the natural world, as real as the concepts of theoretical physics, then it is not so surprising that it is an effective tool in analyzing the real world.
In other words, Gross relies here on a version of the “mathematics as a discovery” perspective that is somewhere between the Platonic world and the “universe is mathematics” world, but closer to a Platonic viewpoint. As we have seen, however, it is difficult to philosophically support the “mathematics as a discovery” claim. Furthermore, Platonism cannot truly solve the problem of the phenomenal accuracy that I have described in chapter 8, a point acknowledged by Gross.
Sir Michael Atiyah, whose views on the nature of mathematics I have largely adopted, argues as follows:
If one views the brain in its evolutionary context then the mysterious success of mathematics in the physical sciences is at least partially explained. The brain evolved in order to deal with the physical world, so it should not be too surprising that it has developed a language, mathematics, that is well suited for the purpose.
This line of reasoning is very similar to the solutions proposed by the cognitive scientists. Atiyah also recognizes, however, that this explanation hardly addresses the thornier parts of the problem—how does mathematics explain the more esoteric aspects of the physical world. In particular, this explanation leaves the question of what I called the “passive” effectiveness (mathematical concepts finding applications long after their invention) entirely open. Atiyah notes: “The skeptic can point out that the struggle for survival only requires us to cope with physical phenomena at the human scale, yet mathematical theory appears to deal successfully with all scales from the atomic to the galactic.” To which his only suggestion is: “Perhaps the explanation lies in the abstract hierarchical nature of mathematics which enables us to move up and down the world scale with comparative ease.”
The American mathematician and