thumb|[[Squaring the circle: the areas of this square and circle are both equal to pi|. It was proved in 1882 that this figure cannot be constructed in a finite number of steps with an idealized straightedge and compass. Nevertheless, "proofs" of such constructions were still published even 50 years later.]]

Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor 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.

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 proven to be mathematically unsolvable. Common examples of this include the following constructions in Euclidean geometry using only a 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.

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.

Another notable case were "Fermatists", who bombarded mathematical institutions with requests to check their proofs of Fermat's Last Theorem.

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 or Gödel's incompleteness theorems).</blockquote>

De Morgan named James Smith as an example of a pseudomath who claimed to have proved that Pi| is exactly . Dantzig observed:

<blockquote>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. 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, such as intelligent design proponent William Dembski's concept of specified complexity.

See also

  • 0.999..., often fallaciously claimed to be distinct from 1
  • Indiana Pi Bill
  • Mathematical fallacy
  • Pseudoscience

References

Further reading

  • Underwood Dudley (1987), A Budget of Trisections, Springer Science+Business Media. . Revised and reissued in 1996 as The Trisectors, Mathematical Association of America. .
  • Underwood Dudley (1997), Numerology: Or, What Pythagoras Wrought, Mathematical Association of America. .
  • Clifford Pickover (1999), Strange Brains and Genius, Quill. .