David Hilbert

A mathematical problem is a touchstone for the sharpness of our thinking.

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

The solvability of every mathematical problem is a definite demand of reason; there is no ignorabimus.

The development of mathematics is a continuous process; it does not proceed by sudden leaps.

Geometry is the science which restores the situation that physics disturbs.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.

A proof is not a proof until it has been understood.

The concept of a group is, as it were, the leading idea of modern mathematics.

The method of 'ideal elements' is nothing other than the logical process of completion.

The theory of algebraic number fields is one of the most beautiful and deepest parts of mathematics.

The solution of a problem lies in the understanding of its deeper contradictions.

Mathematics is the foundation of all exact knowledge of natural phenomena.

The true method of foreseeing the future of mathematics is to study its history and its actual state.

A scientist's work is never finished; it is always in progress.

The axiomatic method is and remains the indispensable tool for all exact research.

The deepest theorems are often the simplest to state.

The progress of mathematics is due to those who dared to look at things from a new angle.

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

To think is to ignore differences and generalize.