If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.
Intuition is more important to discovery than logic.
We also know how cruel the truth often is, and we wonder whether delusion is not more consoling.
Why is it that showers and even storms seem to come by chance, so that many people think it quite natural to pray for rain or fine weather, though they would consider it ridiculous to ask for an eclipse by prayer.
Often when works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, long or short, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind.
Einstein does not remain attached to the classical principles, and when presented with a problem in physics he quickly envisages all of its possibilities. This leads immediately in his mind to the prediction of new phenomena which may one day be verified by experiment.
Analyse data just so far as to obtain simplicity and no further.
Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means...
A small error in the former will produce an enormous error in the latter.
It is not order only, but unexpected order, that has value.
It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.
The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.
Thought is only a flash between two long nights, but this flash is everything.
Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty.
Guessing before proving! Need I remind you that it is so that all important discoveries have been made?
Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.
For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.
Doubting everything and believing everything are two equally convenient solutions that guard us from having to think.
Geometry is the art of correct reasoning from incorrectly drawn figures.