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The unreasonable effectiveness of mathematics

February 5, 2016

A recent lecture by Prof. Alan Huckleberry made me revisit this old essay by Eugene Wigner, and prepare an episode for Engines about it:

The 16th century astronomer Galileo Galilei claimed that the language of nature is mathematics. As far as we can see, Galileo was right. But why is math the language of nature? And why is that language understandable to us?

These questions puzzled the physicist Eugene Wigner. Wigner and other scientists who were part of the revolution that brought us quantum mechanics and relativity, used math to express the laws of nature. To their surprise, they found that mathematicians had already invented the language they needed. They’d developed this language without thinking of whether or how their ideas could be applied.

This made Wigner ask why math is so unreasonably effective in describing nature. Much is hidden behind this question: First, why do general laws of nature exist? Even if we assume that without such laws all would be chaos, it is still a wonder that we can discover and understand them. But let us accept that that the nature is humanly comprehensible.

What still puzzled Wigner is that we have a language of our own making ready at hand to describe the world around us. The words, phrases and ideas of mathematics we need to talk about the laws of nature are often available when we need them. In math ideas are developed because they naturally flow from previous theories, and because mathematicians find them beautiful. It is then somewhat of a miracle that some of these ideas can be applied not just in physics, but in most other sciences.

Some have argued that math is not as useful as it seems. Maybe we focus too much on problems where math happened to be of great help. In fields like medicine, economics, and the social sciences general laws have been harder to come by. For example, try to find a short list of rules for the English language. Many have tried and failed.

Some data scientists have therefore argued that we need to embrace the complexity of such systems. We should forget the elegance of math for more pragmatic approaches, and let the data guide us. Yet in practice most of these statistical and machine learning approaches still rely on math.

Mathematics may not be able to unlock all mysteries. But it is still very useful – it lets us describe and understand things infinitesimally small, unimaginably large, and events far in the past and the future. Math gives us a glimpse into realms that we can’t directly experience, and where our intuitions are of no use.

Wigner concludes by saying that mathematics is a wonderful gift which we neither understand, nor deserve. He expresses a hope that it will continue to be useful to our continued surprise. I do believe that we will increasingly rely on computers to help us make sense of the world around us. Yet, I am sure that math will remain the language we will keep using in this conversation.

Notes:

Wigner’s original essay can be found here. Although over 50 years old, it asks questions that we are not much closer to answering. There have been many follow-ups to Wigner’s essay. I have drawn from the ideas of the mathematician R. W. Hamming, and the engineer Derek Abbott.

The precise quote from Galileo “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.” More quotes here

I thank Prof. Alan Huckleberry for emphasizing that there is a natural flow of ideas and developments in mathematics that become apparent in retrospect. Yet, I would like to add that there is also a cultural component to these developments. It may thus be better to view the co-development of physics and mathematics as part of a larger cultural evolution of ideas. Perhaps in this larger context the relation between mathematics and physics is more natural. I leave this discussion to historians of science.

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