The axioms of geometry, therefore, are neither synthetic a priori judgments nor experimental facts. They are conventions; our choice among all possib… - Henri Poincaré

In other words, the axioms of geometry (I do not speak of those of arithmetic) are merely disguised definitions.

The axioms of geometry therefore are neither synthetic a priori judgments nor experimental facts.

Deviner avant de démontrer! Ai-je besoin de rappeler que c'est ainsi que se sont faites toutes les découvertes importantes.

Guessing before proving! Need I remind you that it is so that all important discoveries have been made?

It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing the microscope that prodigious smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness.

The mind has the faculty of creating symbols.... Its power is limited only by the necessity of avoiding all contradiction; but the mind only makes use of this faculty if experience furnishes it a stimulus thereto