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.
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Types of pseudomathematics
One common type of approach is attempting to solve classical problems in terms that have been proven mathematically impossible. Common examples the following constructions in Euclidean geometry using only compass and straightedge:
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.
Another common approach is to misapprehend standard mathematical methods, and insisting that use or knowledge of higher mathematics is somehow cheating or misleading.
Practitioners
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.
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. Because it is based on non-mathematical principles, pseudomathematics is not related to attempts at genuine proofs that contain mistakes. Indeed, such mistakes are common in the careers of amateur mathematicians who go on to produce celebrated results.