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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ...

An illustrative contrived example

Consider the following flawed attempt at a theorem:

Theorem: All positive integers are prime, except those divisible by 2, except 2, which is prime.

Proof: By mathematical induction.

Let P = {n | n is prime}.

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.

Q.E.D.

While the above "proof" suffers from various flaws (such as the flawed invocation of mathematical induction and also the fact that 1 is not a prime number), all that is required to topple it is to show a counterexample: the positive integer 33. This number is not prime, and if it 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." Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Look up QED in Wiktionary, the free dictionary. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

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. In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... Look up theorem in Wiktionary, the free dictionary. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ...

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:

1. Attempting to solve classical problems in terms that have been proved mathematically impossible;
2. Misapprehending standard mathematical methods, and insisting that use or knowledge of higher mathematics is somehow cheating or misleading.

Look up Problem in Wiktionary, the free dictionary. ... 1+1=3 and 0=1 are false in all possible worlds. ...

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: Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...

• Squaring the circle: Drawing a square having the same area as a given circle.
• Doubling the cube: Drawing a cube with twice the volume of a given cube.
• Trisecting the angle: Dividing a given angle 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. This square and circle have the same area. ... Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. ... A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... Alternative 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 in the sense of the Gregorian calendar. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

Practitioners

Pseudomathematics has equivalents in other scientific fields, particularly physics, where various efforts are made to continually attempt to invent perpetual motion devices, disprove Einstein using Newtonian mechanics, and other impossible feats. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... This article or section should include material from Parallel Path See also Perpetuum mobile as a musical term Perpetual motion machines (the Latin term perpetuum mobile is not uncommon) are a class of hypothetical machines which would produce useful energy in a way science cannot explain (yet). ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... It has been suggested that this article or section be merged with Classical mechanics. ...

Excessive pursuit of pseudomathematics can result in the practitioner being labelled a "mathematical 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. Crank is a pejorative term for a person who holds some belief which the vast majority of his contemporaries would consider false, clings to this belief in the face of all counterarguments or evidence presented to him. ... Underwood Dudley (born January 6, 1937) is a mathematician formerly of DePauw University who has written a number of research works and textbooks, but is best known for his popular writing. ...

Not all mathematical research undertaken by non-mathematicians is pseudomathematics. Non-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 either are, or are not correct, and pseudomathematical results, by relying on non-mathematical principles, are not about professionalism but about incorrectness arrived at by improper methodology. This is a list of people whose primary vocation did not involve mathematics (or any similar discipline) yet made notable, and sometimes important, contributions to the field of mathematics. ...

Current trends in pseudomathematics

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^[1] using elementary mathematical techniques (third category). The former theorem now has a lengthy and extremely technical orthodox proof drawing on many different areas of advanced mathematics. Even so, numerous attempts to either overturn the orthodox proof or provide a more "elementary" proof closer to the implied "simple" proof to which Fermat is conjectured to have alluded, have been made. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ...

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. 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. Lossless data compression is a class of data compression algorithms that allows the exact original data to be reconstructed from the compressed data. ... Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

Other common subjects of pseudomathematicians include the meaning of infinity, the nature of complex numbers, and the indeterminate expression ⁰/₀. As an example of the last, James Anderson's recently publicized work on transreal numbers would generally be considered pseudomathematics. This article or section is not written in the formal tone expected of an encyclopedia article. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions. ... James Anderson James Anderson is an academic staff member in the School of Systems Engineering at the University of Reading. ... James Anderson James Anderson is an academic staff member in the School of Systems Engineering at the University of Reading. ...

Attempts to solve open problems, such as the existence of odd perfect numbers, are common among pseudomathematicians. This article lists some currently unsolved problems in mathematics. ... In mathematics, a perfect number is an integer which is the sum of its proper positive divisors, excluding itself. ...

References

1. ^ Collected pseudomathematical proofs of RH

• 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.

See also

• 0.999... often claimed to be distinct from 1
• Eccentricity (behavior)