Mathematical folklore

See also folk theorem for other uses of this expression.

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.^[1] 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.^[2]

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.

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Mathematical folklore may also refer to unusual (and possibly apochryphal) stories or jokes involving mathematicians or mathematics that are told verbally in mathematics departments. Compilations include tales collected in G. H. Hardy's A Mathematician's Apology and (Krantz 2002); examples include:

• Galileo dropping weights from the Leaning Tower of Pisa.
• An apple falling on Isaac Newton's head to inspire his theory of gravitation.
• The drinking, duel and early death of Galois.
• Richard Feynman cracking safes in the Manhattan Project.
• Alfréd Rényi's definition of a mathematician - "a mathematician is a device for turning coffee into theorems".
• The "turtles all the way down" story told by Stephen Hawking.
• Fermat's lost simple proof.
• If anyone were to successfully divide by zero, a massive paradox would form, destroying the known universe.
• The unwieldy proof and associated controversies of the Four Color Theorem.