HENRI POINCARE AS AN INVESTIGATOR 213
abruptly; and integro-difference equations would mean that they de- pend 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 dis- continuous. 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, 5 about which Poincare said shortly before his death :
A physical system is capable of only a finite number of distinct states; it abruptly jumps from one state to another without passing through the inter- mediate states.
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 Poincare 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 Poincare 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 geo- metric 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 Poincare was made one of the forty immortals, he said :
You were born, you have lived, you will live, and you will die a mathe- matician; the vital function of your brain is to invent and to resolve more cases in mathematics; everything about you relates to that. Even when you seem to desert mathematics for metaphysics, the former furnishes the examples, the reasoning, the paradoxes. It is in you, possesses you, harries you, dominates you; in repose, your brain automatically pursues its work, without your being aware of it — the fruit forms, grows, ripens, and falls, and you have yourself told us of your wonder at finding it in your hand so perfect. You furnish an admirable example of the mathematical type. Since Archimedes it is classical but legendary. Earely will historian have found an occasion so fit to note in life its external characters, and in place of relating your works, rather is not this
6 See Jour, de Physique, 1912 (5), 2; pp. 5-34; 347-360.