The task of the educator is to make the child’s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.
To invent is to discern, to choose.
Need we add that mathematicians themselves are not infallible?
Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don’t change. Matter is not important, only form interests them.
Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.
But for harmony beautiful to contemplate, science would not be worth following.
One does not ask whether a scientific theory is true, but only whether it is convenient.
Thought must never submit, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, save to the facts themselves, because, for thought, submission would mean ceasing to be.
Hypotheses are what we lack the least.
A very small cause, which escapes us, determines a considerable effect which we cannot ignore, and we say that this effect is due to chance.
Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.
Point set topology is a disease from which the human race will soon recover.
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?
It is far better to foresee even without certainty than not to foresee at all.
The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things.
It is a misfortune for a science to be born too late when the means of observation have become too perfect. That is what is happening at this moment with respect to physical chemistry; the founders are hampered in their general grasp by third and fourth decimal places.
In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer.
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment.
Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.
If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing.