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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.
An illustrative contrived example 
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Consider the following flawed attempt at a theorem:
- Theorem: All positive odd numbers are prime.
- Proof: By mathematical induction.
- Let n = 1. Then n ∈ P.
- Since n + 1 ∈ P and (n + 1) + 1 ∈ P, and skipping those divisible by 2, all numbers are prime (except those that are divisible by 2), due to induction.
While the above "proof" suffers from various flaws (such as the flawed invocation of mathematical induction and that there is no agreement that 1 is a prime), all that is required to topple it is to show a counterexample, such as the positive integer 33. This number is not prime, and if it is shown by way of arriving at a contradiction that numbers evenly divisible by any number (other than 1 or themselves) are not prime (by definition) and this thus contradicts the definition of primes, the counterargument might make an appeal such as "then the definition of primes is flawed since the above proof shows that numbers such as 33 (which is not divisible by 2) are prime."
In mathematics, a statement presenting itself as a mathematical truth is provably incorrect (that is, not a mathematical truth statement) if even one counterexample showing it to be false can be found. Indeed, a statement cannot rightly be called a "theorem" if a counterexample disproving it exists. While it is possible to call something a conjecture until a full formal proof is provided, until and unless that proof is provided, it does not become a theorem. Conjectures, too, may be shown to be false if a counterexample exists.
An appeal that a mathematical definition is in itself wrong (i.e., that primes were somehow poorly defined in the first place) is an appeal to an argument that attacks a well-established and well-understood definition: Primes are prime by definition, and such classes of numbers may or may not have properties that make them interesting. Pseudomathematics, however, sometimes appeals to change definitions to suit its claims. At this point, pseudomathematical arguments exit the world of mathematics altogether, and even the appearance of following long-established mathematical models falls apart.
Any statement purporting to be a theorem must hold within the framework of the pre-existing definitions about which it purports to assert a truth. While new definitions may be introduced into a framework to substantiate a theorem, these new definitions must themselves hold within the framework addressed, without introducing any contradiction within that framework. Asserting that 33 is somehow prime because a flawed proof arrives at this eventuality, and then asserting that the definition of primes itself is flawed is pseudomathematical reasoning.
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. This category also extends to attempts to disprove accepted (and proven) mathematical theorems such as Cantor's diagonal argument and Gödel's incompleteness theorem.
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.
Current trends in pseudomathematics 
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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 or the Riemann hypothesis using elementary mathematical techniques (second category). Fermat's theorem now has a lengthy and extremely technical orthodox proof drawing on many different areas of advanced mathematics. Even so, numerous attempts have been made to either overturn the orthodox proof or to provide a more "elementary" proof closer to the implied "simple" proof to which Fermat is conjectured to have alluded. (Such attempts are not pseudomathematical unless accepted mathematical methods are disregarded; perhaps there really is an uncorrectable error in the published proof, and it has not been conclusively demonstrated that Fermat was mistaken in his supposed "proof" or that some other "simpler" proof cannot be found.)
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 (assuming that there is no significant error in the accepted proof of the latter). 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 smaller set of output on a one-to-one basis.
See also 
Further reading 
- 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.