Pseudomathematics, or mathematical crankery, is a form of mathematics-like activity that aims at advancing a set of questionable beliefs that do not adhere to the framework of rigor of formalmathematical practice. Pseudomathematics has equivalents in other scientific fields, such as pseudophysics, and overlaps with these to some extent. Pseudomathematics often contain a great amount of mathematical fallacies, whose executions are tied to elements of deceit rather than genuine, unsuccessful attempts at tackling a problem. More often than not, excessive pursuit of pseudomathematics can result in the practitioner being labelled a crank. Because it is based on non-mathematical principles, pseudomathematics is not related to attempts at genuine proofs that contain mistakes. Indeed, such mistakes are common in the careers of amateur mathematicians, some of which would go on to produce celebrated results. The topic of mathematical crankery has been extensively studied by mathematician Underwood Dudley, who has written several popular works about mathematical cranks and their ideas.
Examples
One common type of approach is claiming to have solved classical problems which have been proved to be mathematically impossible. Common examples of this 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 had tried and failed to find such constructions; in the 19th century, they were all proven impossible. Another common approach is to misapprehend standard mathematical methods, and to insist that the use or knowledge of higher mathematics is somehow cheating or misleading.
History
The term pseudomath was coined by the logician Augustus De Morgan, discoverer of De Morgan's laws, in his A Budget of Paradoxes. De Morgan wrote, De Morgan gave as example of a pseudomath a certain James Smith who claimed persistently to have proved that Pi| is exactly. Of Smith, De Morgan wrote, "He is beyond a doubt the ablest head at unreasoning, and the greatest hand at writing it, of all who have tried in our day to attach their names to an error." The term pseudomath was adopted later by Tobias Dantzig. Dantzig observed, The term pseudomathematics has been applied to attempts in mental and social sciences to quantify the effects of what is typically considered to be qualitative. More recently, the same term has been applied to creationist attempts to refute the theory of evolution, by way of spurious arguments purportedly based in probability or complexity theory.
