Eugene Wigner, the physicist, put an exclamation mark on the many many times mathematics (the mathematics that mathematicians, not physicists, work on because they find it interesting) turn out to be exactly what the physicists (and other natural scientists) need to describe Nature’s framework. In his own words from The Unreasonable Effectiveness of Mathematics in the Natural Sciences, “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.” Here is how he began it:
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”
Apart from the fact that pi is only historically “defined” (i.e. most recognized) as the circumference of the circle to its diameter, this is a pretty interesting story. (Pi can be defined by many other formulas.)
Mathematics is the science of the abstract. Physics is the science of the real or more precisely the fundamental elements of the real. Big difference between the two, yet mathematics is the framework, language and the tool of physics. Doing mathematics, the only restriction is logical consistency and we humans have lots of room in choosing what to study and creating a mathematical proof is more of an art than a science. A good mathematician is one who has the insight to recognize which mathematics is interesting and which is not. On the other hand, doing physics, there is the extra restriction of physical reality and we humans have small room in choosing our set of possible physical theories. (Actually usually a set of few expected symmetries leaves room only for one physical model to be taken seriously, as in the case of “deriving” Special Relativity and the Dirac Equation). The mystery and paradox that Wigner talks about is that this humanly-interesting mathematics are the exact tools needed in physics to describe Nature.
On analyzing Wigner’s article and its commentary, I first wanted to understand what he had to say and to check whether his question was really a good one, as I must admit I was very skeptical at the beginning. I have had my doubts due to many things. mainly because I thought that physics might face only one philosophical question, which is about the explanation of the Cosmic Initial Conditions discussed here. But Wigner’s philosophical question seemed to me to be a much deeper one, as he asks about the nature of Nature’s existence and not any reason for Her existence. But now I have come to view the question as arising from discarding the importance of examining historical connection between physics and mathematics.
The problem with Wigner’s question is this: it is being answered too abstractly. He was trying to solve this question as a mathematician not a physicist. But why is this abstractness a problem? Simply because the “real” mathematics and physics that humans do are restricted very much by history. They are human activities done by humans. How mathematics is said to be done don’t exactly coincide with how mathematicians actually do mathematics and the same divide is even more deep between the relations of these sciences to each other and how the scientists relate them to each other. In a word, these sciences are historical. They have a history to which they are bound. A process similar to deconstruction (in post-structuralist philosophy) must be applied to the discourse on mathematics and physics and their inter-relations (not to mathematical proofs or physical theories) to unveil what the mathematicians and the physicists really miss by putting a discussion on a non-historical basis. I am not saying it should be applied to their scientific statements (not to statements like “hydrogen is made of a proton and an electron”) but to what they say about the scientific process or method itself. In our case, this means that what physicists say about the relation of physics to other disciplines must be examined carefully. In his arguments, Wigner takes his view of the historical relation between mathematics and physics for granted. I am not sure why he does this, but his view absolulety helps bring more mystery into the “unreasonable effectiveness of mathematics in the natural sciences.” It also helps make it harder for the reader to relate mathematics and the natural sciences historically, most probably unintentionally on Wigner’s part as his article wasn’t meant for the non-scientist anyway.
Taking a look at how the history of mathematics and physics were written, with their modern beginnings almost happening in the minds of the same scientists (talking about Newton, Laplace, Lagrange and Bernoulli as examples), one might conclude that the Wigner’s mystery is easily solved. But looking at the more recent history of physics, in the development of Quantum Mechanics and Relativity, mathematicians and physicists were not as closely connected as before. And indeed the mathematics used in these two areas of physics weren’t the most interesting mathematics at the time. Neither Linear Algebra nor Differential Geometry was the most active research area in the mathematical world at the time; they were many many active areas. In fact, Special Relativity could be (and was first understood) using elementary algebraic equations which were so old and so regularly used in physics by the time Einstein and others figured out Special Relativity. In today’s world, mathematics has exploded into many many new branches of which only a fraction is used in physics. This makes me think Wigner’s question merely fails on experimental grounds. Even though times do and will change so that both physics and mathematics might grow to a time in which the forefront of physics describes the Universe using the forefront of mathematics, I would still view this as a historical (not social) coincidence rather than a constant conspiracy between Nature and mathematicians.
Let me end with this witty quote from Sir Arthur Eddingtion and a commentary:
“Some men went fishing in the sea with a net, and upon examining what they caught they concluded that there was a minimum size to the fish in the sea.”
If the net is our physical apparatus, the collection of fish is our physical data, and the concept of a minimum size is our (old or newly-invented) appropriate mathematical idea, then this analogy describing how we do physics with mathematics only lacks the most important aspect of the process: the constant clever modification of both physical theory and mathematical ideas that are used to describe the data. This constant modification is driven by a desire for things like simplicity and beauty (symmetry), since most of our current successful physics do exhibit these things so a good guess is that future successful physics will also exhibit them. (This desire is of course not argument-based but belief-based on past experience. If Nature doesn’t keep her promise of simplicity and beauty in the yet-to-be-discovered physics, it won’t be that much of downer. I would be happy with an ugly and complicated theory that explains all known phenomena and correctly predicts new ones. I guess even though I haven’t studied the Standard Model yet, I have sympathy for its ugliness.)
The prospect of change in physics and mathematics seems to be missing in both Wigner’s and Eddington’s remarks. I hope my own remark that physics needs to deal with the Cosmic Initial Conditions philosophical question (it certainly needs to deal with it now, and it has no answer for it yet) will be proven wrong through historical change as well.