Frege was frustrated by the use of natural language to describe mathematics. In the preface to his Begriffsschrift he says:

"I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required."

Frege therefore presented in his Begriffsschrift, the first extensive formalisation of logic giving logical concepts via symbols rather than natural language. Frege developed things further in his Grundlagen and Grundgesetze der Arithmetik which could handle elementary arithmetic, set theory, logic, and quantification. In his Grundlagen der Arithmetik, Frege argued that mathematics can be seen as a branch of logic. In his Grundgesetze der Arithmetik, he described the elementary parts of arithmetic within an extension of the logical framework of Begriffsschrift. Frege's tradition was followed by many others: (Russell, Whitehead, Ackermann, Hilbert, etc.). Russell discovered a paradox in Frege's work and proposed type theory as a remedy. Advances were also made in set theory, category theory, etc., each being advocated as a better foundation for mathematics. But, none of the logics proposed satisfies all the needs of mathematicians. In particular, they do not have linguistic categories and are not a satisfactory communication medium. Moreover:

- *Logics make choices (types/sets/categories, predicative/impredicative, etc.). But different parts of mathematics need different choices. There is no agreed best formalism.
- *A logician's formalization of a {\sc Cml} text loses the original structure and is thus of little use to ordinary mathematicians.
- *Mathematicians can do their work without formal mathematical logic.

- *The Choice of the underlying logic.
- *The Choice of how to implement concepts (equational reasoning, induction, etc.).
- *The Choice of the formal system: a type theory (which?), a set theory (which?), etc.
- *The Choice of the proof checker: Coq, Isabelle, PVS, Mizar, etc.

- *An informal mathematical proof will cause headaches as it is hard to turn in one big step into a (series of) syntactic proof expressions.
- *During the checking of mathematics, the informal proof is replaced by a complete proof where every detail is spelled out. Very long expressions replace the clear mathematical text and this is useless to ordinary mathematicians.
- so, despite the enourmous work on logics for mathematics as in Frege's tradition, and computer tools and systems for implementing and checking mathematics as in the second half of the twentieth century, mathematicians remain sceptical about using logic and using computers.