During this century mathematics has been transformed... - George Frederick James Temple

Logical analysis is indispensable for an examination of the strength of a mathematical structure, but it is useless for its conception and design. The great advances in mathematics have not been made by logic but by creative imagination.

What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy?

There is the definition [of mathematics], boldly proposed by Pierce that 'Mathematics is the science which draws necessary conclusions', and more explicitly formulated by Russel that 'Pure Mathematics is the class of all propositions of the form "p implies q"... it was... the purpose of Russell's treatise to provide a complete, exact and convincing justification of this definition... instead, he and Whitehead collaborated to give a magisterial account of the Principia Mathematica.