abruptly; and integro-difference equations would mean that they depend on all preceding states discontinuously. Each is able to account for certain relations in the states. In the same sense the word atom is the name for a set of relations, and though it may change and the atom itself become a solar system, yet what we really mean by the atom is permanent and represents an objective reality. We are witnesses too of an evolution in science and mathematics from the continuous to the discontinuous. In mathematics it has produced the function defined over a range rather than a line—a chaos, as it were, of elements—and the calculable numbers of Borel. In physics it has produced the electron, the magneton, and the theory of quanta, about which Poincare said shortly before his death:
In biology we have the corresponding theory of mutations. Yet despite this apparent reduction of old ideas into dust, contradictory to our hopes of its permanence; as Poincaré put it: this is right and the other is not wrong. They are in harmony, only the language varies; both set forth certain true relations.
Just as Maxwell and Kelvin were able to invent mechanical models of the ether, so Poincaré is perhaps the most profound genius the world has ever known in devising the more subtle machinery of thought to represent the relations he found not only between numbers and geometric figures, but between the phenomena of physics. His mind seemed to create new structures of this kind continually, finding expression for the most intricate relations. Nowadays this is the same as saying that he was a mathematician, for this ideal world of relations is the one with whose structure mathematics is concerned, and which mathematics claims sovereignty over, verifying Gauss's assertion: "Mathematics is Queen of all the Sciences."
In the address of Masson when Poincaré was made one of the forty immortals, he said:
- See Jour, de Physique, 1912 (5), 2; pp. 5-34; 347-360.