Science is built up of facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.

Geometry is the art of correct reasoning from incorrectly drawn figures.

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

Doubt everything or believe everything: these are two equally convenient strategies. With either we dispense with the need for reflection.

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.

All of mathematics is a tale about groups.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognisable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile and vain.

It is far better to foresee even without certainty than not to foresee at all.

All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living

One geometry cannot be more true than another; it can only be more convenient.

It is through science that we prove, but through intuition that we discover.

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.

In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer.

The mind uses its faculty for creativity only when experience forces it to do so.

Mathematical discoveries, small or great are never born of spontaneous generation.

It is by logic we prove. It is by intuition we discover.

There are no solved problems; there are only problems that are more or less solved.

One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.

...the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely aesthetic feeling, which all mathematicians know

Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.

Experiment is the sole source of truth.

But for harmony beautiful to contemplate, science would not be worth following.

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.

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