The function of logic in mathematics is critical rather than constructive. - George Frederick James Temple

The 'language theory' is inadequate as a description of the nature of mathematics.

Mathematical activity has taken the forms of a science, a philosophy and an art.

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.