Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. Alan Turing
Mathematical Reasoning is not only exact; it has its own criteria of reality. Paul Karl Feyerabend
All acts of mathematical reasoning may... be considered but as applications of a corresponding axiom of quantity. William Stanley Jevons
Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? ...If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism. Henri Poincaré
It soon became clear that the only real long-term solution to the problems that I encountered is to start using computers in the verification of mathematical reasoning. Vladimir Voevodsky