... les traités de mécanique ne distinguent pas bien nettement ce qui est expérience, ce qui est raisonnement mathématique, ce qui est convention, ce… - Henri Poincaré

This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem...
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.

All laws are... deduced from experiment; but to enunciate them, a special language is needful... ordinary language is too poor...
This... is one reason why the physicist can not do without mathematics; it furnishes him the only language he can speak. And a well-made language is no indifferent thing;
...the analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist.
...law springs from experiment, but not immediately. Experiment is individual, the law deduced from it is general; experiment is only approximate, the law is precise...
In a word, to get the law from experiment, it is necessary to generalize... But how generalize? ...in this choice what shall guide us?
It can only be analogy. ...What has taught us to know the true profound analogies, those the eyes do not see but reason divines?
It is the mathematical spirit, which disdains matter to cling only to pure form.

Correlative movement' therefore constitutes the sole connection between two phenomena which otherwise we never should have dreamt of likening.