From the Notices of the American Mathematical Society
Or I could have titled this post, "Even Mathematicians can be Very Illogical."
In the September 2007 issue of the Notices of the AMS, there is published an interview with Stephen Smale. The interviewer was George Szpiro. There is an editor's note which explains that Stephen Smale won the Fields medal (the mathematics equivalent of the Nobel Prize) in 1966, I suppose to imply that Smale is a good mathematician. I have no doubt he is. However, he can be just as guilty of illogical thinking as the next man. The third question in the interview goes like this.
Szpiro: Why is mathematics so effective in explaining phenomena, as opposed to, say, narratives?
[Note by Adrian: this was an incredibly good question, perhaps even better than Szpiro knew, though I would hardly argue that narratives are a bad way of explaining phenomena. Aside from the Bible, even mathematicians have to use tons of regular language, for me, English, to explain anything, though perhaps you might not think of that as narrative.]
Smale: Mathematics is a kind of formalized way of thinking. One can be much more precise in mathematics than in literature, express relationships in a more precise way, include magnitudes. And even fuzziness can be incorporated in mathematics by using probabilities. I use that a lot because when moving from physics to vision and biology one has to overcome some kind of fuzziness. The way I do that is - in the mathematical tradition - by using probability.
Mathematics is so effective because one can look for universal laws more easily with mathematics than without. It enables us to abstract the main ideas. With formalization and symbols one is able to see what is universal. The abstraction allows us to see universal ideas. I have been very inspired by Newton who could see a falling apple and the motion of planets and recognize them as part of the same phenomenon. I would like to see a language that allows us to translate what we see and then recognize it as part of a broad phenomenon.
When I first read the question, my heart leapt! Here was a chance to wax theological and explain how, because the universe was created by God and is sustained by God, and since God is constant and never goes against His own nature, and since His creation reflects Him, and since there is no contradiction in God, that therefore the universe reflects that in its logic and in the physical laws we see around us.
Instead, we get the incredibly circular attempt to be pragmatic: "Mathematics is so effective because one can look for universal laws more easily with mathematics than without." The entire second paragraph of Smale's response goes to support this thesis. He says mathematics is effective because it makes things easier. But effectiveness and ease are really synonyms here. So we have A because of A. Wow. I'm thoroughly underwhelmed.
I should also point out that Smale is by no means the only mathematician/physicist to miss the mark on this question. At least one great physicist, either Feynman or Penrose, said something, "There is no answer."
Now do not interpret me as claiming that everything Smale said was wrong; modern mathematics really is about abstraction, though I would hasten to add that the re-concretization (if that's a word) is the summit, the final push, that a lot of mathematicians don't want to do, but ought to.
In 1 Corinthians 1, Paul says that God chose the foolish things of the world to shame the wise. My answer to the effectiveness of modern mathematics is that it is effective because of the way God designed the universe. He designed it to exhibit an order closely related to His own. I have no need of any other answer. Interestingly, I also get my marching orders from God, who in Genesis said to multiply, fill the earth, and subdue it. What is subduing the earth if not science and technology?