The Unreasonable Effectiveness of Mathematics: An RC Inspired by IIM Indore

On 23 September 1846 the astronomer Johann Galle pointed the refractor of the Berlin Observatory at a patch of sky and found a new planet within about one degree of the spot where Urbain Le Verrier had said it would be. Le Verrier had never seen it. He had reached it with pencil and paper, working backwards from small irregularities in the orbit of Uranus. The curious thing is not that mathematics describes nature but that it does so with such uncanny precision, and that it sometimes does so before anyone has looked.

The puzzle was given its famous name by the physicist Eugene Wigner, who in 1960 wrote of the unreasonable effectiveness of mathematics in the natural sciences. Mathematical ideas often arise in pursuits that seem detached from the physical world, and they repeatedly find uses that nobody planned. Non Euclidean geometry, once a speculation about parallel lines, became part of the language of Einstein's general theory of relativity. Complex numbers, long treated as curiosities, now run through electrical engineering and quantum mechanics.

One explanation is that mathematics works because people built it from nature in the first place. Counting grew from the need to tally objects, and geometry from the need to measure land and raise buildings, and many early mathematical practices are tied to trade, farming and construction. On this view the effectiveness is hardly mysterious, since human beings have refined a habit of spotting regularities and writing them down in symbols.

That account weakens when mathematics runs ahead of observation. Paul Dirac's equation for the electron, published in 1928, allowed solutions that he came to read as a particle with the electron's mass and the opposite charge, and the positron was found by Carl Anderson in 1932. Here the symbols did more than tidy known facts. They pointed at something physicists had not yet met. Perhaps the universe has a mathematical structure that human reasoning can reach even where its physical signs are hidden.

Perhaps the universe has a mathematical structure that human reasoning can reach even where its physical signs are hidden.

A rival view warns against reading harmony into this record. Scientists keep the mathematics that explains observations and discard the rest, so the celebrated successes may reflect a filter and not a deep match between number and nature. The record of discarded models is long. The objection has force, yet it leaves a residue. It explains why useful mathematics survives, but not why constructions made for one purpose, or for none at all, turn out to be useful in domains their makers never imagined.

The relation between mathematics and the world may also say something about the limits of understanding. Knowledge is usually pictured as observation followed by explanation. Mathematics scrambles that order, because thought sometimes travels ahead and the world catches up later, as it did in Berlin in 1846 and in the laboratories of the early 1930s. The equations on a physicist's desk are then more than descriptions. They are instruments for finding out what the universe might contain.

The deepest mystery may therefore lie not in the fact that mathematics works but in the mind that can invent something abstract enough to leave experience behind and precise enough to return to it, and no one has yet explained how an invention can turn out to be a discovery.