''Mathematical Cranks'' is a book on

pseudomathematics Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved to be unsolvable or re ...

and the cranks who create it, written by

Underwood Dudley Underwood Dudley (born January 6, 1937) is an American mathematician and writer. His popular works include several books describing crank mathematics by pseudomathematicians who incorrectly believe they have squared the circle or done other im ...

. It was published by the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

in their MAA Spectrum book series in 1992 ().

Topics

Previously,

Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...

wrote in ''A Budget of Paradoxes'' about cranks in multiple subjects, and Dudley wrote a book about

angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

. However, this book is the first to focus on mathematical crankery as a whole. The book consists of 57 essays, loosely organized by the most common topics in mathematics for cranks to focus their attention on. The "top ten" of these topics, as listed by reviewer Ian Stewart, are, in order: #

squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...

, #

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

, #

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...

and the

parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...

, # the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

, #

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...

s, # the

four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...

, # advocacy for

duodecimal The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is i ...

and other non-standard number systems, #

Cantor's diagonal argument Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infin ...

for the uncountability of the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, and #

doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...

. Other common topics for crankery, collected by Dudley, include calculations for the

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, roots of quintic equations,

Fermat's little theorem In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , t ...

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...

Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...

magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...

divisibility rule A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed Divisor (number theory), divisor without performing the division, usually by examining its digits. Although there are divisibility test ...

constructible polygon In mathematics, a constructible polygon is a regular polygon that can be Compass and straightedge constructions, constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regu ...

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, and the Van der Pol oscillator. As

David Singmaster David Breyer Singmaster (14 December 1938 – 13 February 2023) was an American-British mathematician who was emeritus professor of mathematics at London South Bank University, England. He had a huge personal collection of mechanical puzzles and ...

writes, many of these topics are the subject of mainstream mathematics "and only become crankery in extreme cases". The book omits or passes lightly over other topics that apply mathematics to crankery in other areas, such as

numerology Numerology (known before the 20th century as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, ...

and

pyramidology Pyramidology (or pyramidism) refers to various religion, religious or pseudoscience, pseudoscientific speculations regarding pyramids, most often the Giza pyramid complex and the Great Pyramid of Giza in Egypt.Martin Gardner, ''Fads and Fallaci ...

. Its attitude towards the cranks it covers is one of "sympathy and understanding", and in order to keep the focus on their crankery it names them only by initials. The book also attempts to analyze the motivation and psychology behind crankery, and to provide advice to professional mathematicians on how to respond to cranks. Despite his work on the subject, which has "become enshrined in academic folklore", Dudley has stated "I've been at this for a decade and still can't pin down exactly what it is that makes a crank a crank", adding that "It's like obscenity – you can tell a crank when you see one."

Lawsuit

After the book was published, one of the cranks whose work was featured in the book, William Dilworth, sued Dudley for

defamation Defamation is a communication that injures a third party's reputation and causes a legally redressable injury. The precise legal definition of defamation varies from country to country. It is not necessarily restricted to making assertions ...

in a federal court in

Wisconsin Wisconsin ( ) is a U.S. state, state in the Great Lakes region, Great Lakes region of the Upper Midwest of the United States. It borders Minnesota to the west, Iowa to the southwest, Illinois to the south, Lake Michigan to the east, Michig ...

. The court dismissed the ''Dilworth vs Dudley'' case on two grounds. First, it found that by publishing his work on

, Dilworth had made himself a public figure, creating a higher burden of proof for a defamation case. Second, it found that the word "crank" was "rhetorical hyperbole" rather than an actionably inaccurate description. The

United States Court of Appeals for the Seventh Circuit The United States Court of Appeals for the Seventh Circuit (in case citations, 7th Cir.) is the U.S. United States federal court, federal court with appellate jurisdiction over the United States district court, courts in the following United Stat ...

concurred. After Dilworth repeated the lawsuit in a state court, he lost again and was forced to pay Dudley's legal expenses.

Reception and audience

Reviewer John N. Fujii calls the book "humorous and charming" and "difficult to put down", and advocates it to "all readers interested in the human side of mathematics". Although complaining that famous mathematicians

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

and

Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...

might have been dismissed as cranks by the standards of the book, reviewer Robert Matthews finds it an accurate reflection of most crankery. And

adds that it should be read by "anyone likely to deal with a crank", including professional mathematicians, journalists, and legislators.

References

{{reflist, refs= {{citation , last = Singmaster , first = David , author-link = David Singmaster , journal =

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr = 1189134 , title = Review of ''Mathematical Cranks'' , year = 1993 {{citation , last = Stewart , first = Ian , author-link = Ian Stewart (mathematician) , date = January 1994 , doi = 10.2307/2325140 , issue = 1 , journal =

American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

, jstor = 2325140 , pages = 87–91 , title = Review of ''Mathematical Cranks'' , volume = 101 {{citation , last = Fujii , first = John N. , date = May 1993 , issue = 5 , journal =

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

, jstor = 27968419 , pages = 429–430 , title = Review of ''Mathematical Cranks'' , volume = 86 {{citation , last = Richeson , first = David S. , date = October 8, 2019 , magazine = Lapham's Quarterly , title = Beware of Cranks: Misguided attempts to solve impossible mathematical problems , url = https://www.laphamsquarterly.org/roundtable/beware-cranks {{citation , last = Webster , first = Roger , date = November 1994 , doi = 10.2307/3620224 , issue = 483 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, jstor = 3620224 , pages = 355–356 , title = Review of ''Mathematical Cranks'' , volume = 78 {{citation , last = Johnson , first = George , date = February 9, 1999 , newspaper =

The New York Times ''The New York Times'' (''NYT'') is an American daily newspaper based in New York City. ''The New York Times'' covers domestic, national, and international news, and publishes opinion pieces, investigative reports, and reviews. As one of ...

, title = Genius or Gibberish? The Strange World of the Math Crank , url = https://www.nytimes.com/1999/02/09/science/genius-or-gibberish-the-strange-world-of-the-math-crank.html {{citation , last = Matthews , first = Robert , date = November 2, 1996 , magazine =

New Scientist ''New Scientist'' is a popular science magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organ ...

, title = Review : Going nuts over numbers , url = https://www.newscientist.com/article/mg15220545-300-review-going-nuts-over-numbers/ {{citation , last = Gajda , first = Amy , isbn = 9780674053861 , pages = 163–164 , publisher = Harvard University Press , title = The Trials of Academe: the new era of campus litigation , url = https://books.google.com/books?id=sLPOBoL3PtoC&pg=PA163 , year = 2010 Pseudomathematics Mathematics books 1992 non-fiction books