Pseudomathematics, or mathematical crankery, is a

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

-like activity that does not adhere to the framework of

rigor Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as ma ...

of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved 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 a pseudomathematician or a pseudomath. Pseudomathematics has equivalents in other scientific fields, and may overlap with other topics characterized as pseudoscience. Pseudomathematics often contains

mathematical fallacies In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a proof ...

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 a crank. Because it is based on non-mathematical principles, pseudomathematics is not related to attempts at genuine

proofs Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

that contain mistakes. Indeed, such mistakes are common in the careers of

amateur mathematicians An amateur () is generally considered a person who pursues an avocation independent from their source of income. Amateurs and their pursuits are also described as popular, informal, self-taught, user-generated, DIY, and hobbyist. History Hi ...

, some of whom go on to produce celebrated results. The topic of mathematical crankery has been extensively studied by mathematician

Underwood Dudley Underwood Dudley (born January 6, 1937) is an American mathematician. His popular works include several books describing crank mathematics by people who think they have squared the circle or done other impossible things. Career Dudley was bo ...

, 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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

—using only a

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

: *

Squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...

: Given any

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

drawing a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

having the same

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

. *

Doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...

: Given any

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

drawing a cube with twice its

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

. *

Trisecting the angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

: Given any

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

dividing it into three smaller angles all of the same size. 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. Yet another notable case are "Fermatists", who plague mathematical institutions with requests to check their proofs of

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...

. 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 In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research i ...

History

The term ''pseudomath'' was coined by the logician Augustus De Morgan, discoverer of

De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...

, 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 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 is exactly . 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 Tobias Dantzig (; February 19, 1884 – August 9, 1956) was an American mathematician, the father of George Dantzig, and the author of '' Number: The Language of Science (A critical survey written for the cultured non-mathematician)'' (1930) and ''A ...

. 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.

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. More recently, the same term has been applied to

creationist Creationism is the religious belief that nature, and aspects such as the universe, Earth, life, and humans, originated with supernatural acts of divine creation. Gunn 2004, p. 9, "The ''Concise Oxford Dictionary'' says that creationism is 'th ...

attempts to refute the

theory of evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation t ...

, by way of spurious arguments purportedly based in

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

or complexity theory.

Examples

History

See also

References

Further reading