Mathematical Folklore

As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that are found by investigation and may circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. Knowledge of folklore is the coin of the realm of academic mathematics, showing relative insight of investigators.

Quite important at times for researchers are folk theorems, which are results known, at least to experts in a field, and considered to have established status, but not published in complete form. Sometimes these are only alluded to in the public literature. For example, in tidying up loose ends of the classification of finite simple groups around 2004 (a result which had been claimed, somewhat prematurely, to be proved around 1980), Michael Aschbacher devoted an entire volume to proving various infrastructural results, some of which had not previously been proved in print. A second example is a book of exercises, described on the back cover:

This book contains almost 350 exercises in the basics of ring theory. The problems form the 'folklore' of ring theory, and the solutions are given in as much detail as possible.

Another distinct category is wellknowable mathematics, a term introduced by John Conway. This consists of matters that are known and factual, but not in active circulation in relation with current research. Both of these concepts are attempts to describe the actual context in which research work is done.

Some people, principally non-mathematicians, use the term folk mathematics to refer to the informal mathematics studied in many ethno-cultural studies of mathematics.