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'{{Confusing|date=April 2008}} '''Pseudomathematics''' is a form of [[mathematics]]-like activity that does not work within the framework, definitions, rules, or rigor of [[Formal system|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== {{Original research|section|date=May 2012}} Consider the following flawed attempt at a theorem: ---- :'''Theorem:''' All positive odd numbers are [[Prime number|prime]]. :'''Proof:''' By [[mathematical induction]]. :Let <math>P = \{n \mid n \mbox{ is prime}\}.</math> :Let ''n'' = 1. Then ''n'' &isin; ''P''. :Since ''n''&nbsp;+&nbsp;1&nbsp;&isin;&nbsp;''P'' and (''n''&nbsp;+&nbsp;1)&nbsp;+&nbsp;1&nbsp;&isin;&nbsp;''P'', and skipping those divisible by 2, all numbers are prime (except those that are divisible by 2), due to induction. :''[[Q.E.D.]]'' ---- 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_number#Primality_of_one|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 [[problem]]s in terms that have been proven [[Logical possibility|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. Examples of impossible problems include the following constructions in [[Euclidean geometry]] using only [[compass and straightedge]]: * [[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]]. djasndsjfsafjkasfsajfsf fo4r [42398 21rid 209rmdaksldmdklwfmlscmslsflfmsallf mlsadmf fkaslkasf askf sfjnasdma fmdf fsfisjfsafiusfsjufsuffudfififififufufuffcickickciufuffkc ckck ck ckccpckdckscd ==Practitioners== Pseudomathematics has equivalents in other scientific fields, such as [[physics]]. Examples include efforts to invent [[perpetual motion]] devices, efforts to disprove [[Albert Einstein|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]].<ref>Sokal, Alan and Jean Bricmont (1998). ''Fashionable Nonsense: Postmodern Intellectuals Abuse of Science''. Editions Odile Jacob, ISBN 0-312-20407-8</ref> Excessive pursuit of pseudomathematics can result in the practitioner being labelled a [[crank (person)|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. [[List of amateur mathematicians|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. ==See also== * [[0.999...]] often claimed to be distinct from 1 * [[Eccentricity (behavior)]] * [[Invalid proof]] ==References== {{reflist}} ==Further reading== *Augustus De Morgan (1872), [[A Budget of Paradoxes]], [http://www.gutenberg.org/ebooks/23100 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. {{Mathematics-footer}} {{Pseudoscience}} [[Category:Philosophy of mathematics]] [[Category:Pseudoscience]]'