List of Mathematical Jargon - Philosophy of Mathematics

Philosophy of Mathematics

These terms discuss mathematics as mathematicians think of it; they connote common intellectual strategies or notions the investigation of which somehow underlies much of mathematics.

abstract nonsense

Also general abstract nonsense or generalized abstract nonsense, a tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem.

introduced the very abstract idea of a 'category' — a subject then called 'general abstract nonsense'! —Saunders Mac Lane (1997) raised algebraic geometry to a new level of abstraction...if certain mathematicians could console themselves for a time with the hope that all these complicated structures were 'abstract nonsense'...the later papers of Grothendieck and others showed that classical problems...which had resisted efforts of several generations of talented mathematicians, could be solved in terms of...complicated concepts. —Michael Monastyrsky (2001)

canonical

A reference to a standard or choice-free presentation of some mathematical object. The term canonical is also used more informally, meaning roughly "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes.

There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like:

—The proof that there are infinitely many prime numbers.

—The proof of the irrationality of the square root of two.

—Freek Wiedijk (2006, p.2)

deep

A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. The prime number theorem, proved with techniques from complex analysis, was thought to be a deep result until elementary proofs were found. The fact that π is irrational is a deep result because it requires considerable development of real analysis to prove, even though it can be stated in terms of simple number theory and geometry.

elegant

Also beautiful; an aesthetic term referring to the ability of an idea to provide insight into mathematics, whether by unifying disparate fields, introducing a new perspective on a single field, or providing a technique of proof which is either particularly simple, or captures the intuition or imagination as to why the result it proves is true. Gian-Carlo Rota distinguished between elegance of presentation and beauty of concept, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and some theorems or proofs are beautiful but may be written about inelegantly.

The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory's rigorous expositions. Some beautiful theories may never be given a presentation which matches their beauty....Instances can also be found of mediocre theories of questionable beauty which are given brilliant, exciting expositions.... is rich in beautiful and insightful definitions and poor in elegant proofs.... remain clumsy and dull.... vied for one another in elegance of presentation and in cleverness of proof....In retrospect, one wonders what all the fuss was about.

Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening. We acknowledge a theorem's beauty when we see how the theorem 'fits' in its place....We say that a proof is beautiful when such a proof finally gives away the secret of the theorem.... —Gian-Carlo Rota (1977, pp.173–174, pp.181–182)

elementary

A proof or result is called "elementary" if it requires only basic concepts and methods, in contrast to so-called deep results. The concept of "elementary proof" is used specifically in number theory, where it usually refers to a proof that does not use methods from complex analysis.

folklore

A result is called "folklore" if it is non-obvious, has not been published, and yet is generally known among the specialists in a field. Usually, it is unknown who first obtained the result. If the result is important, it may eventually find its way into the textbooks, whereupon it ceases to be folklore.

Many of the results mentioned in this paper should be considered "folklore" in that they merely formally state ideas that are well-known to researchers in the area, but may not be obvious to beginners and to the best of my knowledge do not appear elsewhere in print. —Russell Impagliazzo (1995)

natural

Similar to "canonical" but more specific, this term makes reference to a description (almost exclusively in the context of transformations) which holds independently of any choices. Though long used informally, this term has found a formal definition in category theory.

pathological

An object behaves pathologically if it fails to conform to the generic behavior of such objects, fails to satisfy certain regularity properties (depending on context), or simply disobeys mathematical intuition. These can be and often are contradictory requirements. Sometimes the term is more pointed, referring to an object which is specifically and artificially exhibited as a counterexample to these properties.

Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose....Nay more, from the logical point of view, it is these strange functions which are the most general....to-day they are invented expressly to put at fault the reasonings of our fathers.... —Henri Poincaré (1913) took on an enormous importance...as giving an incentive for the creation of new types of function whose properties departed completely from what intuitively seemed admissible. A celebrated example of such a so-called 'pathological' function...is the one provided by Weierstrass....This function is continuous but not differentiable. —J. Sousa Pinto (2004)

rigor (rigour)

Mathematics strives to establish its results using indisputable logic rather than informal descriptive argument. Rigor is the use of such logic in a proof.

well-behaved

An object is well-behaved (in contrast with being pathological) if it does satisfy the prevailing regularity properties, or sometimes if it conforms to intuition (but intuition often suggests the opposite behavior as well).