Pseudomathematics
Pseudomathematics, or mathematical crankery, is a form of mathematicslike activity that aims at advancing a set of questionable beliefs that do not adhere to the framework of rigor of formal mathematical practice.^{ [2]}^{ [3]} Pseudomathematics has equivalents in other scientific fields, such as pseudophysics, and overlaps with these to some extent.
Pseudomathematics often contain a great amount of mathematical fallacies, whose executions are tied to elements of deceit rather than genuine, unsuccessful attempts at tackling a problem.^{ [2]} More often than not, excessive pursuit of pseudomathematics can result in the practitioner being labelled a crank. Because it is based on nonmathematical principles,^{ [3]} pseudomathematics is not related to attempts at genuine proofs that contain mistakes. Indeed, such mistakes are common in the careers of amateur mathematicians, some of which would go on to produce celebrated results.^{ [4]}
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.
Examples
One common type of approach is claiming to have solved a classical problem that has been proved to be mathematically unsolvable. Common examples of this include the following constructions in Euclidean geometry—using only compass and straightedge:
 Squaring the circle: Given any circle drawing a square having the same area.
 Doubling the cube: Given any cube drawing a cube with twice its volume.
 Trisecting the angle: Given any angle dividing it into three smaller angles all of the same size.^{ [5]}^{ [6]}^{ [7]}
For more than 2,000 years, many people had tried and failed to find such constructions; in the 19th century, they were all proven impossible.^{ [8]}^{ [9]}^{:47}
Another common approach is to misapprehend standard mathematical methods, and to insist that the use or knowledge of higher mathematics is somehow cheating or misleading (e.g., the denial of Cantor's diagonal argument^{ [10]}^{:40ff} and Gödel's incompleteness theorems).^{ [10]}^{:167ff}^{ [2]}
History
The term pseudomath was coined by the logician Augustus De Morgan, discoverer of De Morgan's laws, in his A Budget of Paradoxes (1915). De Morgan wrote:
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 cleanshaved, and all the rest of the world hairy.^{ [11]}
De Morgan gave as example of a pseudomath a certain James Smith who claimed persistently to have proved that π is exactly 3+1/8.^{ [4]} 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."^{ [11]} The term pseudomath was adopted later by Tobias Dantzig.^{ [12]} 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 circlesquarers, trisectors, duplicators, and perpetuum mobile designers, loudly clamoring for recognition of their epochmaking 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.^{ [12]}
The term pseudomathematics has been applied to attempts in mental and social sciences to quantify the effects of what is typically considered to be qualitative.^{ [13]} More recently, the same term has been applied to creationist attempts to refute the theory of evolution, by way of spurious arguments purportedly based in probability or complexity theory.^{ [14]}^{ [15]}
See also
 0.999... often claimed to be distinct from 1
 Indiana Pi Bill
 Eccentricity (behavior)
 Invalid proof
 Pseudoscience
References
 ^ See e.g. Squaring the circle#Other modern claims
 ^ ^{a} ^{b} ^{c} "The Definitive Glossary of Higher Mathematical Jargon – Pseudomathematics". Math Vault. 20190801. Retrieved 20191211.
 ^ ^{a} ^{b} "What does Pseudomathematics mean?". www.definitions.net. Retrieved 20191211.
 ^ ^{a} ^{b} Lynch, Peter. "Maths discoveries by amateurs and distractions by cranks". The Irish Times. Retrieved 20191211.
 ^ Dudley, Underwood (1983). "What To Do When the Trisector Comes" (PDF). The Mathematical Intelligencer. 5 (1): 20–25. doi: 10.1007/bf03023502.

^
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 selfstyled mathematicians, particularly circlesquarers, angletrisectors, and cubeduplicators, although it can be extended to include those who deny the validity of nonEuclidean 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.
 ^ Johnson, George (19990209). "Genius or Gibberish? The Strange World of the Math Crank". The New York Times. Retrieved 20191221.
 ^ 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) [1969]. Famous Problems of Geometry and How to Solve Them. Dover Publications.
 ^ ^{a} ^{b} Dudley, Underwood (1992). Mathematical Cranks. Mathematical Association of America. ISBN 0883855070.
 ^ ^{a} ^{b} De Morgan, Augustus (1915). A Budget of Paradoxes (2nd ed.). Chicago: The Open Court Publishing Co.
 ^ ^{a} ^{b} Dantzig, Tobias (1954). "The Pseudomath". The Scientific Monthly. 79 (2): 113–117. Bibcode: 1954SciMo..79..113D. JSTOR 20921.
 ^ Johnson, H. M. (1936). "PseudoMathematics in the Mental and Social Sciences". The American Journal of Psychology. 48 (2): 342–351. doi: 10.2307/1415754. ISSN 00029556. JSTOR 1415754.
 ^ Elsberry, Wesley; Shallit, Jeffrey (2011). "Information theory, evolutionary computation, and Dembski's "complex specified information"". Synthese. 178 (2): 237–270. CiteSeerX 10.1.1.318.2863. doi: 10.1007/s1122900995428.
 ^ Rosenhouse, Jason (2001). "How AntiEvolutionists Abuse Mathematics" (PDF). The Mathematical Intelligencer. 23: 3–8.
Further reading
 Underwood Dudley (1987), A Budget of Trisections, Springer Science+Business Media. ISBN 9781461264309. Revised and reissued in 1996 as The Trisectors, Mathematical Association of America. ISBN 0883855143.
 Underwood Dudley (1997), Numerology: Or, What Pythagoras Wrought, Mathematical Association of America. ISBN 0883855240.
 Clifford Pickover (1999), Strange Brains and Genius, Quill. ISBN 0688168949.
 Bailey, David H.; Borwein, Jonathan M.; de Prado, Marcos López; Zhu, Qiji Jim (2014). "PseudoMathematics and Financial Charlatanism: The Effects of Backtest Overfitting on OutofSample Performance" (PDF). Notices of the AMS. 61 (5): 458–471. doi: 10.1090/noti1105.