Pseudomathematics is a form of mathematics-like activity undertaken primarily by non-mathematicians. The word is adapted from the term pseudoscience, which is applied to ideas that purport to be scientific but are not.
The efforts of pseudomathematicians may be divided into three broad categories:
- Attempting to solve classical problems (see Impossible problems section below) posed in terms that have been proved mathematically impossible;
- Generating new theories of mathematics or logic from scratch;
- Taking on difficult problems with only a pre-calculus knowledge of mathematics - and often insisting that use or knowledge of higher math is somehow cheating or misleading.
Investigations in the first category are doomed to failure. Those in the second are generally unproductive, as they tend to re-invent existing knowledge at best, and to create complete nonsense at worst; some forms of numerology fall under this category.
Efforts in the third category are not necessarily futile since some advanced mathematical results can be proved using more elementary techniques; there is no coherent notion of depth in mathematics. However, unless the investigator possesses a deep intuitive understanding of the subject matter, the probability of achieving a breakthrough is small. (See the article on the Indian mathematical genius Srinivasa Ramanujan for one of the few people who has ever succeeded in this category).
Pseudomathematics has equivalents in other scientific fields, particularly physics, where amateurs continually attempt to invent perpetual-motion devices, disprove Einstein using classical mechanics, and other similarly impossible feats.
Excessive pursuit of pseudomathematics can create mathematical cranks, who regard mainstream mathematicians with suspicion bordering on paranoia because their ideas are continually rejected. The topic has been extensively studied by Indiana mathematician Underwood Dudley, who has written several popular works on the topic. In addition, Clifford Pickover considers the "link between genius and madness" among scientists and mathematicians in his 1998 book, Strange Brains and Genius.
Although pseudomathematics is primarily engaged in by non-mathematicians, it should not be assumed that all mathematical research undertaken by amateurs falls into this category; occasionally amateur mathematicians have produced results of genuine interest to the mainstream community.
- Squaring the circle: Drawing a square having the same area as a given circle.
- Doubling the cube: Drawing a cube with twice the volume of a given cube.
- Trisecting the angle: Dividing a given angle into three smaller angles all of the same size.
For 2,000 years people have tried and failed to find such constructions; the reasons were discovered in the 19th century, when it was proved that they are all impossible. Rather than discouraging pseudomathematicians, however, such statements of impossibility by orthodox mathematicians tend merely to inspire more attempts.
Current trends in pseudomathematics
In recent years, pseudomathematicians have devoted their energies to disproving Gödel's second incompleteness theorem (efforts that fall in the first category mentioned above) and to proving Fermat's last theorem using elementary mathematical techniques (third category). The latter theorem now has a lengthy and extremely technical orthodox proof drawing on many different areas of advanced mathematics. It is particularly tempting for amateur mathematicians, because a note in Fermat's papers claimed that he had developed an elementary proof for it.
Other related activities include attempts to create lossless data compression algorithms which will compress all possible inputs or to disprove the four-color theorem; both of these belong to the first category of problems proven to be impossible. In the former case, there is a trivial proof of impossibility — such an algorithm would need to map a finite large set of input onto a small set of output on a one-to-one basis.