Pseudomathematics, ormathematical crankery, is amathematics-like activity that does not adhere to the framework ofrigor offormal mathematical practice. Common areas of pseudomathematics are solutions of problemsproved to be unsolvable or recognized as extremely hard by experts, as well as attempts to apply mathematics to non-quantifiable areas. A person engaging in pseudomathematics is called apseudomathematician or apseudomath.[1] Pseudomathematics has equivalents in other scientific fields, and may overlap with othertopics characterized as pseudoscience.
Pseudomathematics often containsmathematical fallacies whose executions are tied to elements of deceit rather than genuine, unsuccessful attempts at tackling a problem. Excessive pursuit of pseudomathematics can result in the practitioner being labelled acrank. Because it is based on non-mathematical principles, pseudomathematics is not related to misguided attempts at genuineproofs. Indeed, such mistakes are common in the careers ofamateur mathematicians, some of whom go on to produce celebrated results.[1]
The topic of mathematical crankery has been extensively studied by mathematicianUnderwood Dudley, who has written several popular works about mathematical cranks and their ideas.
One common type of approach is claiming to have solved a classicalproblem that has been proven to be mathematically unsolvable. Common examples of this include the following constructions inEuclidean geometry – using only acompass and straightedge:
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.[5][6]: 47
Another notable case were "Fermatists", who plagued mathematical institutions with requests to check their proofs ofFermat's Last Theorem.[7][8]
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 ofCantor's diagonal argument[9]: 40ff orGödel's incompleteness theorems).[9]: 167ff
The termpseudomath was coined by the logicianAugustus De Morgan, discoverer ofDe Morgan's laws, in hisA Budget of Paradoxes (1872). 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 clean-shaved, and all the rest of the world hairy.[10]
De Morgan named James Smith as an example of a pseudomath who claimed to have proved thatπ is exactly3+1/8.[1] 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."[10] The termpseudomath was adopted later byTobias Dantzig.[11] 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, andperpetuum 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.[11]
The termpseudomathematics has been applied to attempts in mental and social sciences to quantify the effects of what is typically considered to be qualitative.[12] More recently, the same term has been applied tocreationist attempts to refute thetheory of evolution, by way of spurious arguments purportedly based inprobability orcomplexity theory, such asintelligent design proponentWilliam Dembski's concept ofspecified complexity.[13][14]
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.
