Abstract: MathLang: A language for Mathematics.

Fairouz Kamareddine

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.

In the second half of the twentieth century, programming languages were being developed and this led to the creation of softwares, systems and tools to computerize and check mathematics on the computer, and to give some help and assistance to teachers, students, and users of mathematics. But, a mathematical text is structured differently from a machine-checked text proving the same facts. Making the latter involves extensive knowledge and many choices:

*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.

Furthermore, checking mathematics on the computer has other problems:

*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.

In this talk, I argue that both the logic for mathematics and the computation of mathematics have forgotten the mathematician and the language of mathematics during their development. I start from de Bruijn's mathematical vernacular which he refined almost twenty years after the begin of his Automath project (Automating Mathematics). I compare this vernacular to modern approaches for putting mathematics on the computer (e.g., OMDOC, MathML, versions of XML, etc.) and discuss how we can find a language of mathematics that is open to future developments through logic and computation.