Pseudomathematics or mathematical crankery is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical practice. Pseudomathematics has equivalents in other scientific fields, such as pseudophysics, and overlaps with these to some extent.
Excessive pursuit of pseudomathematics can result in the practitioner being labelled a crank. The topic of mathematical crankery has been extensively studied by mathematician Underwood Dudley, who has written several popular works about mathematical cranks and their ideas. Because it is based on non-mathematical principles, pseudomathematics is not related to attempts at genuine proofs that contain mistakes. Indeed, such mistakes are common in the careers of amateur mathematicians who go on to produce celebrated results.
The pseudomath is a person who handles mathematics as the monkey handled the razor. The creature tried to shave himself as he had seen his master do; but, not having any notion of the angle at which the razor was to be held, he cut his own throat. He never tried a second time, poor animal! but the pseudomath keeps on at his work, proclaims himself clean-shaved, and all the rest of the world hairy.
De Morgan gave as example of a pseudomath a certain James Smith who claimed persistently to have proved that . Of Smith, De Morgan wrote, "He is beyond a doubt the ablest head at unreasoning, and the greatest hand at writing it, of all who have tried in our day to attach their names to an error." The term pseudomath was adopted later by Tobias Dantzig. Dantzig observed,
With the advent of modern times, there was an unprecedented increase in pseudomathematical activity. During the 18th century, all scientific academies of Europe saw themselves besieged by circle-squarers, trisectors, duplicators, and perpetuum mobile designers, loudly clamoring for recognition of their epoch-making achievements. In the second half of that century, the nuisance had become so unbearable that, one by one, the academies were forced to discontinue the examination of the proposed solutions.
^Schaaf, William L. (1973). A Bibliography of Recreational Mathematics, Volume 3. National Council of Teachers of Mathematics. p. 161. Pseudomath. A term coined by Augustus De Morgan to identify amateur or self-styled mathematicians, particularly circle-squarers, angle-trisectors, and cube-duplicators, although it can be extended to include those who deny the validity of non-Euclidean geometries. The typical pseudomath has but little mathematical training and insight, is not interested in the results of orthodox mathematics, has complete faith in his own capabilities, and resents the indifference of professional mathematicians.
^Wantzel, P M L (1837). "Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas". Journal de Mathématiques Pures et Appliquées. 1. 2: 366–372.
^Bold, Benjamin (1982) . Famous Problems of Geometry and How to Solve Them. Dover Publications.
Underwood Dudley (1987), A Budget of Trisections, Springer Science+Business Media. ISBN978-1-4612-6430-9. Revised and reissued in 1996 as The Trisectors, Mathematical Association of America. ISBN0-88385-514-3.
Underwood Dudley (1992), Mathematical Cranks, Mathematical Association of America. ISBN0-88385-507-0.
Underwood Dudley (1997), Numerology: Or, What Pythagoras Wrought, Mathematical Association of America. ISBN0-88385-524-0.