David Hilbert

A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts.

Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.

The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms.

He who seeks for methods without having a definite problem in mind seeks for the most part in vain.

How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.

One hears a lot of talk about the crisis in science. It does not exist.

A theory's correctness is not a precondition for its fertility.

The value of a solution in mathematics is always inversely proportional to the number of assumptions it requires.

The life of a mathematician is a fight against the infinite.

Before beginning [to solve a problem], prepare a plan; then carry it out without deviation.

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas.

For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either.

The significance of a problem in mathematics lies not only in its difficulty but in its consequences.

A proof only becomes a proof after the social act of 'accepting it as a proof'.

The foundation of the theory of invariants is, in my opinion, the discovery of the fact that invariants are nothing else but symmetric functions.

The method of integral equations offers a direct route to many of the most beautiful results of analysis.

It must be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated.

The axioms of geometry are not arbitrary, but sensible, statements which are, in general, induced by space perception and are determined as to their precise content by expediency.

The goal of scientific thinking is to find the simplest possible explanations for the most complicated facts.