Is God a Mathematician_ - Mario Livio [116]
Hamming was less convinced, even by the strength of his own argument. He pointed out that
if you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics.
Raskin insisted that “the groundwork for mathematics had been laid down long before in our ancestors, probably over millions of generations.” I must say, however, that I do not find this argument particularly convincing. Even if logic had been deeply embedded in our ancestors’ brains, it is difficult to see how this ability could have led to abstract mathematical theories of the subatomic world, such as quantum mechanics, that display stupendous accuracy.
Remarkably, Hamming concluded his article with an admission that “all of the explanations I have given when added together simply are not enough to explain what I set out to account for” (namely, the unreasonable effectiveness of mathematics).
So, should we close by conceding that the effectiveness of mathematics remains as mysterious as it was when we started?
Before giving up, let us try to distill the essence of Wigner’s puzzle by examining what is known as the scientific method. Scientists first learn facts about nature through a series of experiments and observations. Those facts are initially used to develop some sort of qualitative models of the phenomena (e.g., the Earth attracts apples; colliding subatomic particles can produce other particles; the universe is expanding; and so on). In many branches of science even the emerging theories may remain nonmathematical. One of the best examples of a powerfully explanatory theory of this type is Darwin’s theory of evolution. Even though natural selection is not based on a mathematical formalism, its success in clarifying the origin of species has been remarkable. In fundamental physics, on the other hand, usually the next step involves attempts to construct mathematical, quantitative theories (e.g., general relativity; quantum electrodynamics; string theory; and so on). Finally, the researchers use those mathematical models to predict new phenomena, new particles, and results of never-before-performed experiments and observations. What puzzled Wigner and Einstein was the incredible success of the last two processes. How is it possible that time after time physicists are able to find mathematical tools that not only explain the existing experimental and observational results, but which also lead to entirely new discernments and new predictions?
I attempt to answer this version of the question by borrowing a beautiful example from mathematician Reuben Hersh. Hersh proposed that in the spirit of the analysis of many such problems in mathematics (and indeed in theoretical physics) one should examine the simplest possible case. Consider the seemingly trivial experiment of putting pebbles into an opaque vase. Suppose you first put in four white pebbles, and later you put in seven black pebbles. At some point in their history, humans learned that for some purposes they could represent a collection of pebbles of any color by an abstract concept that they had invented—a natural number. That is, the collection of white pebbles could be associated with the number 4 (or IIII or IV or whichever symbol was used at the time) and the black pebbles with the number 7. Via experimentation of the type I have described above, humans also discovered that another invented concept—arithmetic addition—represents correctly the physical act of aggregation. In other words, the result of the abstract process denoted symbolically by 4 7 can predict unambiguously the final number of pebbles in the vase. What