Henri Poincaré

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.

Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?

This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either.

[A]stronomers... define duration in the following way: time... so defined that Newton's law and that of vis viva [or of the ] may be verified. Newton's law is an experimental truth... only approximate... [W]e still have only a definition by approximation.

But these labels can only be finite in number. On that score, psychologic time should be discontinuous.

[W]hat postulate do we implicitly admit? It is that the duration of two identical phenomena is the same; or... that the same causes take the same time to produce the same effects. ...Is it impossible that experiment may some day contradict our postulate?

[T]wo difficulties: (1) Can we transform psychologic time, which is qualitative, into a quantitative time? (2) Can we reduce to one and the same measure facts which transpire in different worlds [of conscious beings]!