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. People who practice pseudomathematics are sometimes called pseudomathematicians.

Some taxonomy of pseudomathematics

The following categories are rough characterisations of some particularly common pseudomathematical activities:

1. Attempting to solve classical problems (see Impossible problems section below), in terms that have been proved mathematically impossible;
2. Generating new theories of mathematics or logic from scratch;
3. Taking on difficult problems with only a pre-calculus knowledge of mathematics.(People who work in this way often misapprehend standard methods, and insist that use or knowledge of higher mathematics is somehow cheating or misleading.)

Attempts on classic unsolvable problems

Investigations in the first category are doomed to failure. At the very least a solution would indicate a contradiction within mathematics itself, a radical difficulty which would invalidate everyone's efforts to prove anything as trite.

Examples of impossible problems include the following constructions in

• 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 to inspire more attempts.

This category also extends to attempts to disprove accepted (and proven) mathematical theorem: particularly those which are often found counterintuitive, such as Cantor's diagonal argument and Gödel's incompleteness theorem.

Blue-sky innovation

Work in the second category is generally unproductive, as it tends to re-invent existing knowledge at best, and to create complete nonsense at worst. Some forms of numerology might fall under this category. Much recreational mathematics shares characteristics with this area, without being 'pseudo'.

Elementary proofs

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

As a point of logic, anything provable by higher-level methods can be written out in lower-level language. As a point of heuristic, though, a laborious proof by elementary methods may be no easier to find than a more compressed proof using abstract tools. One sign of the 'pseudo' approach is what one could call a dearth of lemmas, a lemma being an intermediate result or stepping stone to a complete proof.

Amateur mathematics

Although pseudomathematics is primarily engaged in by non-mathematicians, not all mathematical research undertaken by amateurs falls into this category. Occasionally, amateur mathematicians have produced results of genuine interest to the mainstream community.

Crankdom

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 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. Clifford Pickover also considers the "link between genius and madness" among scientists and mathematicians in his 1998 book, Strange Brains and Genius.

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.

Other favorite subjects of pseudomathematicians include the indeterminate expression 0/0, the meaning of infinity, and the nature of complex numbers.

See also

• Eccentricity 偽數學