Pseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models. While any given pseudomathematical approach may work within some of these boundaries, for instance, by accepting or invoking most known mathematical definitions that apply, pseudomathematics inevitably disregards or explicitly discards a well-established or proven mechanism, falling back upon any number of demonstrably non-mathematical principles.
Some taxonomy of pseudomathematics
The following categories are rough characterisations of some particularly common pseudomathematical activities:
- Attempting to solve classical problems in terms that have been proven mathematically impossible;
- Misapprehending standard mathematical methods, and insisting 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.
- 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 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.
Pseudomathematics has equivalents in other scientific fields, such as physics. Examples include efforts to invent perpetual motion devices, efforts to disprove Einstein using Newtonian mechanics, and many other feats that are currently accepted as impossible. French psychoanalyst Jacques Lacan, and Bulgarian-French philosopher Julia Kristeva have been accused of misusing mathematics in their work; see Fashionable Nonsense (1998) by Alan Sokal and Jean Bricmont.
Excessive pursuit of pseudomathematics can result in the practitioner being labelled a crank. The topic of mathematical "crankiness" has been extensively studied by Indiana mathematician Underwood Dudley, who has written several popular works about mathematical cranks and their ideas.
Not all mathematical research undertaken by amateur mathematicians is pseudomathematics. Many amateur mathematicians have produced genuinely solid new mathematical results. Indeed, there is no distinction between an amateur mathematically correct result and a professional mathematically correct result: results are either correct or incorrect, and pseudomathematical results, by relying on non-mathematical principles, are not about professionalism but about incorrectness arrived at by improper methodology.
- Sokal, Alan and Jean Bricmont (1998). Fashionable Nonsense: Postmodern Intellectuals Abuse of Science. Editions Odile Jacob, ISBN 0-312-20407-8
- Augustus De Morgan (1872), A Budget of Paradoxes, Volume I a Project Gutenberg
- Underwood Dudley (1992), Mathematical Cranks, Mathematical Association of America. ISBN 0-88385-507-0.
- Underwood Dudley (1996), The Trisectors, Mathematical Association of America. ISBN 0-88385-514-3.
- Underwood Dudley (1997), Numerology: Or, What Pythagoras Wrought, Mathematical Association of America. ISBN 0-88385-524-0.
- Clifford Pickover (1999), Strange Brains and Genius, Quill. ISBN 0-688-16894-9.