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Pseudomathematics

Pseudomathematics is a form of mathematics-like activity undertaken by many non-mathematicians - and occasionally by mathematicians themselves. The efforts of pseudomathematicans divide into three categories:

• attempting apparently simple classical problems long proved impossible by mainstream mathematics
• generating whole new theories of mathematics or logic from scratch
• attempting hard problems in mathematics using only high-school mathematical knowledge

The first category is doomed to failure; the second is generally futile, as they tend to re-invent existing knowledge at best, and to create nonsense at worst; the third is unlikely to succeed -- but just might, if some Ramanujan-like genius emerges.

Pseudomathematics has equivalents in other scientific fields, particularly physics, where amateurs try to do such things as disprove Einstein using classical mechanics.

Excessive pursuit of pseudomathematics can create mathematical cranks, who regard mainstream mathematicians with suspicion bordering on paranoia because their ideas are continuously rejected. The topic has been studied by Indiana mathematician Underwood Dudley.

Impossible problems

Examples of impossible problems include ruler and compass constructions of:

• Squaring the circle: Drawing a square the same area as a given circle.
• Doubling the cube: Drawing a cube with twice the volume as a given cube.
• Trisecting the angle: Dividing a given angle into three smaller angles all of the same size.

For 2000 years people tried to find constructions within the limits set above, and failed. The reasons were discovered in the 19th century, when it was mathematically proven that they are all impossible. Rather than discouraging pseudomathematicians, statements of impossibility by orthodox mathematicians often spur them on.

Current trends in pseudomathematics

In more recent years, pseudomathematicians have devoted their energies to proving Fermat's last theorem using trivial mathematical techniques (note that there is a lengthy and technical orthodox proof of this theorem, so it belongs to the third category), and to disproving Gödel's second incompleteness theorem (first category: impossible).

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.

The aspiring pseduomathematician very often begins their labors by writing semi-incoherent analyses on either the "true value" of the indeterminate expression 0/0, the "actual meaning" of infinity, or the nature of complex numbers.

See also

• Eccentricity