The source
The exact words
Original French
“La mathématique est l'art de donner le même nom à des choses différentes.”
In English
“Mathematics is the art of giving the same name to different things. It is enough that these things, though differing in matter, should be similar in form, to permit of their being, so to speak, run in the same mould.”
Science et méthode (Science and Method). Book I, Chapter II, “L'avenir des mathématiques” (The Future of Mathematics), Paris, 1908. The chapter was first given as an address to the International Congress of Mathematicians in Rome, 1908.
The story
Why Poincaré said it
Poincaré was asked to predict the future of mathematics, and instead of naming problems he talked about method. Facts, he said, are endless; what makes mathematics progress is finding the few ideas that let you see many facts as one. A good definition is a mould into which very different materials can be poured.
The examples are everywhere. A rotation of a cube, a shuffle of cards and the symmetries of an equation are different things that all turn out to be “groups”. The word “uniform” lets one theorem cover infinitely many cases. Once the right name exists, a hundred separate results collapse into one, and the mathematician who coined it has done more than the one who proved the hundred results.
That is why the line is often quoted as a definition of abstraction. It is also a little joke at the expense of language: ordinary speech gives different names to the same thing, while mathematics deliberately does the opposite.
The same mind could also be wrong about what is simple. Poincaré's 1904 question about spheres, the Poincaré conjecture, took a century and Grigori Perelman to settle. For another view of what mathematics is for, read the line by Bertrand Russell on its supreme beauty.
Mathematician
Who was Henri Poincaré?
- Born
- 29 April 1854, Nancy
- Died
- 17 July 1912, Paris
- Known for
- Topology, chaos, the Poincaré conjecture
- Said it
- Science and Method, 1908
Henri Poincaré was born in Nancy and trained as a mining engineer before becoming the leading French mathematician of his age. His prize-winning work on the three-body problem in 1889 led him to the first glimpse of chaos: tiny changes in starting conditions could change everything.
He founded algebraic topology with Analysis Situs (1895), worked out much of the mathematics of special relativity alongside Lorentz and Einstein, and wrote popular books on the philosophy of science that are still read. He was famous for working in short bursts and for ideas that arrived, he said, while stepping onto a bus.
