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“The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”
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Eugene Wigner wrote these words in his 1960 article “

How is it possible that all the phenomena observed in classical electricity and magnetism can be explained by means of just four mathematical equations? Moreover, physicist James Clerk Maxwell (after whom those four equations of electromagnetism are named) showed in 1864 that the equations predicted that varying electric or magnetic fields should generate certain propagating waves. These waves—the familiar electromagnetic waves (which include light, radio waves, x-rays, etc.)—were eventually detected by the German physicist Heinrich Hertz in a series of experiments conducted in the late 1880s.

And if that is not enough, the modern mathematical theory which describes how light and matter interact, known as quantum electrodynamics (QED), is even more astonishing. In 2010 a group of physicists at Harvard University determined the magnetic moment of the electron (which measures how strongly the electron interacts with a magnetic field) to a precision of less than one part in a trillion. Calculations of the electron’s magnetic moment based on QED reached about the same precision and the two results agree! What is it that gives mathematics such incredible power?

The puzzle of the power of mathematics is in fact even more complex than the above examples from electromagnetism might suggest. There are actually two facets to the “unreasonable effectiveness,” one that I call
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active
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and another that I dub
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passive
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. The active facet refers to the fact that when scientists attempt to light their way through the labyrinth of natural phenomena, they use mathematics as their torch. In other words, at least some of the laws of nature are formulated in directly applicable mathematical terms. The mathematical entities, relations, and equations used in those laws were developed for a specific application. Newton, for instance, formulated the branch of mathematics known as calculus because he needed this tool for capturing motion and change, breaking them up into tiny frame-by-frame sequences. Similarly, string theorists today often develop the mathematical machinery they need.

Passive effectiveness, on the other hand, refers to cases in which mathematicians developed abstract branches of mathematics with absolutely no applications in mind; yet decades, or sometimes centuries later, physicists discovered that those theories provided necessary mathematical underpinnings for physical phenomena. Examples of passive effectiveness abound. Mathematician Bernhard Riemann, for example, discussed in the 1850s new types of geometries that you would encounter on surfaces curved like a sphere or a saddle (instead of the flat plane geometry that we learn in school). Then, when Einstein formulated his theory of General Relativity (in 1915), Riemann’s geometries turned out to be precisely the tool he needed!

At the core of this math mystery lies another argument that mathematicians, philosophers, and, most recently, cognitive scientists have had for a long time: Is math an invention of the human brain? Or does math exist in some abstract world, with humans merely discovering its truths? The debate about this question continues to rage today.

Personally, I believe that by asking simply whether mathematics is discovered or invented, we forget the possibility that mathematics is an intricate combination of inventions
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and
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discoveries. Indeed, I posit that humans invent the mathematical concepts—numbers, shapes, sets, lines, and so on—by abstracting them from the world around them. They then go on to discover the complex connections among the concepts that they had invented; these are the so-called theorems of mathematics.

I must admit that I do not know the full, compelling answer to the question of what is it that gives mathematics its stupendous powers. That remains a mystery.

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Go Deeper
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Editor’s picks for further reading
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NOVA:
The Great Math Mystery

Is math invented by humans, or is it the language of the universe? NOVA takes on this question in a new film premiering April 15, 2015 at 9pm on most PBS stations.

NOVA:
Describing Nature with Math

How do scientists use mathematics to define reality? And why? Peter Tyson investigates two millennia of mathematical discovery.

The Washington Post:
The Structure of Everything

Learn more about the “unreasonable effectiveness of mathematics” in this review of Mario Livio’s book “Is God a Mathematician?”