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The efforts of pseudomathematicians may be divided into three broad categories:
Investigations in the first category are doomed to failure. Those in the second are generally unproductive, as they tend to re-invent existing knowledge at best, and to create complete nonsense at worst; some forms of numerology fall under this category. Efforts in the third category are not necessarily futile since some advanced mathematical results can be proved using more elementary techniques; there is no coherent notion of depth in mathematics. However, unless the investigator possesses a deep intuitive understanding of the subject matter, the probability of achieving a breakthrough is small. (See S. Ramanujan for a remarkable example of sophisticated mathematical results being obtained without the benefit of formal mathematical training.)
Pseudomathematics has equivalents in other scientific fields, particularly physics, where amateurs continually attempt to invent perpetual-motion devices, disprove Einstein using classical mechanics, and other similarly impossible feats.
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 extensively studied by Indiana mathematician Underwood Dudley, who has written several popular works on the topic. In addition, Clifford Pickover considers the "link between genius and madness" among scientists and mathematicians in his 1998 book, Strange Brains and Genius.
Examples of impossible problems include the following constructions using only a ruler and compass:
For 2,000 years people have tried and failed to find such constructions; the reasons were discovered in the 19th centuryAlternative meaning: Nineteenth Century (periodical ( 18th century 19th century 20th century more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801- 1900. Events The Little Ice Age ended, when it was proved that they are all impossible. Rather than discouraging pseudomathematicians, however, such statements of impossibility by orthodox mathematicians tend merely to inspire more attempts.
In recent years, pseudomathematicians have devoted their energies to disproving G๖del's second incompleteness theorem (efforts that fall in the first category mentioned above) and to proving Fermat's last theoremFermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem is one of the most famous theorems in the history of mathematics. It states that: :There are no positive natural numbers a b and c such that where n is a natural nu using elementary mathematical techniques (third category). The latter theorem now has a lengthy and extremely technical orthodox proofFermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem is one of the most famous theorems in the history of mathematics. It states that: :There are no positive natural numbers a b and c such that where n is a natural nu drawing on many different areas of advanced mathematics.
Other related activities include attempts to create lossless data compressionLossless data compression is a class of data compression algorithms that allow the original data to be reconstructed exactly from the compressed data. Contrast with lossy data compression. Lossless data compression is used in software compression tools su 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.
Other favorite subjects of pseudomathematicians include the indeterminateIn mathematics, a number of the expressions that may be encountered in calculus and occasionally elsewhere are considered to be indeterminate forms and must be treated as symbolic only, until more careful discussion has taken place. The most common one is expression 0/0, the meaning of infinity, and the nature of complex numbers.
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