On a clear night, step outside to look at the sky, and be awed by a million stars sequined on the blackness of space. On such nights, I feel very small, yet at the same time overwelmed by a sense of wonder at how we, Homo sapiens, an insignificant species on an insignificant planet adrift in a so-so galaxy, can be so creative in our artistic and scientific endeavours. How do the atoms in our puny brains arouse themselves into purposeful energy, the sparks that give us Leonardo’s Last Supper, Michelangelo’s David, the Sistine Chapel, Newton’s Calculus, and the other wonders of modern physics, biology and chemistry? I have no answers to this mystery, but I know it is deep and wonderful.
Nowhere is this paradox more evident than in physics, the queen of the sciences. The physical realm is so dizzyingly complex that one would think it is beyond our comprehension. Yet, the Hungarian-American physicist famously declared the “unreasonable effectiveness of mathematics in the natural sciences”, referring mainly to theoretical physics. Wigner observed that mathematics give scientists an uncanny ability to unlock the secrets of the universe, pointing the way to grand insights as well as down-to-earth applications.
Or consider the words of the legendary American physicist, Richard Feynman (1918-1988) who quipped that mathematics is so effective in the sciences because it is “the language God talks”. He should know. Feynman’s own Nobel prize-winning work was an extension of quantum mechanics called Quantum electrodynamics (QED), and it teems with beautiful power series,integrals and differential equations, and plenty of mental gymnastics involving infinities .
Why is QED a demonstration of the “unreasonable effectiveness” of mathematics, or more generally, our unique creativity?
Without getting into the details, it is because QED is a quantum theory of how light and matter interact, a theory that merges four other elegant theories in one glorious mathematical framework: Maxwell’s theory of electricity and magnetism, the quantum theories of Heisenberg and Schrodinger and Einstein’s special theory of relativity of light. In other words, almost everything (save gravity) that constitutes our universe. That our human minds can conceive of such theoretical unity with mere symbols is sublime beyond words. No wonder Feynman was in love with calculus and his theory’s incredible mathematical beauty. But the story doesn’t end here. QED, like Einstein’s relativity theory, is not only beautiful, it is also accurate – indeed the most accurate theory anyone has ever devised about anything!
The proof is in the experiments. Ever since Feynman’s paper went into print in 1949, experimental physicists have tested its various predictions about the properties of electrons and other particles. And the results of these experiments are in stunning agreement with QED’s predictions – the theory (hold your breath) agrees with reality to eight decimal places, or better than one part in a hundred million! [see note 2].
There is something astonishing about all this, that is not unlike an uplifting spiritual encounter. After all, the differential equations and integrals of QED theory are man-made. They are products of the human mind, the results of electrical impulses zipping across the 100 billion neurons in our little brains. Incredibly, by making certain scribbles on paper and doing certain calculations using our invented methods, we are able to predict nature’s innermost properties and get them right to eight decimal places. What could be more astonishing than that?
 Richard P. Feynman (1949), Space-Time Approach to Quantum Electrodynamics, The Physical Review, 76. Published 15 September 1949.
 See for example, Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Boulder, CO: Westview, 1995. Also, this blog’s musings on which is the most precisely tested physical theory: QED versus Special Relativity: https://scienceblogs.com/principles/2011/05/05/the-most-precisely-tested-theo.