The language of mathematics has a vast vocabulary of specialist and technical terms. The central question involved in discussing mathematics as a language can be stated as follows What do we mean when we talk about the language of mathematics The vocabulary of a person is defined either as the set of all Words that are understood by that person or the set of all words likely to be used by that person when constructing It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. For Wikipedia jargon see WikipediaGlossary. For hacker slang see Jargon File. Jargon often appears in lectures, and sometimes in print, as informal shorthand for more rigorous arguments or more precise ideas. Much of this is common English, used in a mathematical or quasi-mathematical sense.

Note that some phrases, like "in general", appear in more than one section.

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. Abstract nonsense, or general abstract nonsense, alternatively general nonsense, is a popular term used by Mathematicians to describe certain kinds of arguments In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

[The paper of Eilenberg and Mac Lane (1942)] introduced the very abstract idea of a 'category' — a subject then called 'general abstract nonsense'!

—Saunders Mac Lane (1997)

[Grothendieck] 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. Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from 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; indeed, this is a canonical example of a canonical proof. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry

There are two canonical proofs that are always used to show non-mathematicians what 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)

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. Many Mathematicians derive aesthetic pleasure from their work and from Mathematics in general 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. Gian-Carlo Rota ( April 27, 1932 &ndash April 18, 1999, known as Juan Carlos Rota

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. . . . [Category theory] is rich in beautiful and insightful definitions and poor in elegant proofs. . . . [The theorems] remain clumsy and dull. . . . [Expositions of projective geometry] vied for one another in elegance of presentation and in cleverness of proof. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. . . . 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)

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. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics a transform is an Operator applied to a function so that under the transform certain operations are simplified 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. In Mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive 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)

[The Dirichlet function] took on an enormous importance. In Mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any . . 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. In Mathematics, the Weierstrass function is a pathological example of a real -valued function on the Real line. . . . 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. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse Rigor is the use of such logic in a proof.

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems' based on fallible intuitions. . . .

Axioms in traditional thought were 'self-evident truths', but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system.

The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. . . . Today, mathematicians continue to argue among themselves about computer-assisted proofs.

—Fundamentals of Mathematics

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). Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space

Descriptive informalities

Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.

almost all

A shorthand term for "all except for a set of measure zero", when there is a measure to speak of. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses In Mathematics, a null set is a set that is negligible in some sense. For example, "almost all real numbers are transcendental" because the algebraic numbers form a countable set of measure zero. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or One can also speak of "almost all" integers having a property to mean "all but finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French For example, "almost all prime numbers are odd". In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic, below.

arbitrarily large, arbitrarily small, arbitrarily close

Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenon as the limit is approached. In Mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in statements such as "&fnof( x A statement such as that predicate P is satisfied by arbitrarily large values, can be expressed in more formal notation by ∀x : ∃y ≥ x : P(y). See also frequently.

arbitrary

A shorthand for the universal quantifier. For the concept of arbitrariness in trademark law see Trademark distinctiveness. An arbitrary choice is one which is made unrestrictedly, or alternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set.

eventually

In the context of limits, this is shorthand for for sufficiently large arguments; the relevant argument(s) are implicit in the context. As an example, one could say that "The function log(log(x)) eventually becomes larger than 100"; in this context, "eventually" means "for sufficiently large x".

factor through

A term in category theory referring to composition of functions. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets If we have three objects A, B, C, a map which is written as a composition with , is said to factor through any (and all) of B, g, and h.

finite

Next to the usual meaning of "not infinite", in another more restrictive meaning that one may encounter, a value being said to be "finite" also excludes infinitesimal values and the value 0. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have For example, if the variance of a random variable is said to be finite, this implies it is a positive real number. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of

frequently

In the context of limits, this is shorthand for arbitrarily large and its relatives; as with eventually, the intended variant is implicit. As an example, one could say that "The function sin(x) is frequently zero", where "frequently" means "for arbitrarily large x".

generic

This term has similar connotations as almost all but is used particularly for concepts outside the purview of measure theory. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its complement satisfies some (context-dependent) notion of smallness. For example, a property which holds on a dense G_δ (intersection of countably many open sets) is said to hold generically. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if In algebraic geometry, one says that a property of points on an algebraic variety that holds on a dense Zariski open set is true generically; however, it is usually not said that a property which holds merely on a dense set (which is not Zariski open) is generic in this situation. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic

in general

In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, with an eye towards identifying a unifying principle. Concisely, this term introduces an "elegant" description which holds for "arbitrary" objects "modulo" "pathology".

Norbert A’Campo of the University of Basel once asked Grothendieck about something related to the Platonic solids. Grothendieck advised caution. The Platonic solids are so beautiful and so exceptional, he said, that one cannot assume such exceptional beauty will hold in more general situations.

—Allyn Jackson (2004, p. 1197)

left-hand side, right-hand side (LHS, RHS)

Most often, these refer simply to the left-hand or the right-hand side of an equation; for example, x = y + 1 has x on the LHS and y +1 on the RHS. In Mathematics, LHS is informal shorthand for the left-hand side of an Equation. Occasionally, these are used in the sense of lvalue and rvalue: an RHS is primitive, and an LHS is derivative. In Computer science, a value is a sequence of Bits that is interpreted according to some Data type.

proper

If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is reflexive), then the qualification proper requires the objects to be different. For example, a proper subset of a set S is a subset of S that is different from S, and a proper divisor of a number n is a divisor of n that is different from n. This overloaded word is also non-jargon for a proper morphism. In Algebraic geometry, a proper morphism between schemes is an analogue of a Proper map between Topological spaces Definition A

resp.

(Respectively) A convention to shorten parallel expositions. "A (resp. B) [has some relationship to] X (resp. Y)" means that A [has some relationship to] X and also that B [has (the same) relationship to] Y.

sharp

Often, a mathematical theorem will establish constraints on the behavior of some object; for example, a function will be shown to have an upper or lower bound. The constraint is sharp if it cannot be made more restrictive without failing in some cases.

smooth

Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to analyticity, and still others which are more complicated. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability Each such usage attempts to invoke the physically intuitive notion of smoothness.

strong, stronger

A theorem is said to be strong if it deduces restrictive results from general hypotheses. One celebrated example is Donaldson's theorem, which puts tight restraints on what would otherwise appear to be a large class of manifolds. In Mathematics, Donaldson's theorem states that a Positive definite Intersection form of a Simply connected Smooth manifold of This (informal) usage reflects the opinion of the mathematical community: not only should such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. A theorem, result, or condition is further called stronger than another one if a proof of the second can be easily obtained from the first. An example is the sequence of theorems: Fermat's little theorem, Euler's theorem, Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see above) is a stronger result than a non-sharp one. Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a Prime number, then for any Integer a In Number theory, Euler's theorem (also known as the Fermat-Euler theorem or Euler's totient theorem) states that if n is a positive Integer Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of Finally, the adjective strong or the adverb strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain satisfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting stronger conditions. In Order theory, a Subset A of a Partially ordered set X is said to be a strong downwards antichain if no two elements have a common In Mathematics, in the area of Order theory, an antichain is a subset of a Partially ordered set such that any two elements in the subset are incomparable Let G = (VE be a Regular graph with v vertices and degree k. G is said to be strongly regular if there are also Integers In Graph theory, a regular graph is a graph where each vertex has the same number of neighbors i When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as "antichain").

sufficiently large, suitably small, sufficiently close

In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. In Mathematics, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand A statement such as that predicate P holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥ x : P(y). See also eventually.

upstairs, downstairs

A descriptive term referring to notation in which two objects are written one above the other; the upper one is upstairs and the lower, downstairs. For example, in a fiber bundle, the total space is often said to be upstairs, with the base space downstairs. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. In a fraction, the numerator is occasionally referred to as upstairs and the denominator downstairs, as in "bringing a term upstairs". In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is the numerator

up to, modulo, mod out by

An extension to mathematical discourse of the notions of modular arithmetic. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers A statement is true up to a condition if the establishment of that condition is the only impediment to the truth of the statement.

vanish

To assume the value 0. For example, "The function sin(x) vanishes for those values of x that are integral multiples of π. " This can also apply to limits: see Vanish at infinity. In Mathematics, a function on a Normed vector space is said to vanish at infinity if f(x\to 0 as \|x\|\to \infty

weak, weaker

The converse of strong. The Language of mathematics has a vast Vocabulary of specialist and technical terms

Proofs and rigorous proof techniques

The formal language of proof draws repeatedly from a small pool of ideas, many of which are invoked through various lexical shorthands in practice. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true

aliter

An obsolescent term which refers to an alternative method of proof. Aliter, which in the Latin language means 'otherwise' or 'by another way' is a traditional part of Mathematical jargon.

diagram chase^[1]

Given a commutative diagram of objects and morphisms between them, if one wishes to prove some property of the morphisms (such as injectivity) which can be stated in terms of elements, then the proof can proceed by tracing the path of elements of various objects around the diagram as successive morphisms are applied to it. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In Category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object That is, one chases elements around the diagram, or does a diagram chase.

for all sufficiently nice X

For all X which satisfy a set of conditions to be specified later. When working out a theorem, the use of this expression in the statement of the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intent is to collect the conditions that will be found to be needed in order for the proof of the theorem to go through. heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method

if and only if (iff)

An abbreviation for logical equivalence of statements. ↔

in general

In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the "induction step", and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that In Mathematics, a sequence is an ordered list of objects (or events

necessary and sufficient

A minor variant on "if and only if"; necessary means "only if" and sufficient means '"if". For example, "For a field K to be algebraically closed it is necessary and sufficient that it have no finite field extensions" means "K is algebraically closed if and only if it has no finite extensions". In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. Often used in lists, as in "The following conditions are necessary and sufficient for a field to be algebraically closed. . . ".

need to show (NTS), required to prove (RTP), wish to show, want to show (WTS)

Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desired theorem; thus, one needs to show just these statements.

one and only one

An especially precise existence statement; the object exists, and furthermore, no other such object exists. In Mathematics and Logic, the phrase "there is one and only one " is used to indicate that exactly one object with a certain property exists

by way of contradiction (BWOC), or "for, if not, . Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile . . "

The rhetorical prelude to a proof by contradiction, preceding the negation of the statement to be proved.

Q.E.D.

(Quod erat demonstrandum): A Latin abbreviation historically placed at the end of proofs, but less common currently. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated"

the following are equivalent (TFAE)

A particular definition is not always the most convenient for certain applications; often one proves theorems stating equivalent rephrasings of the definition.

transport of structure

It is often the case that two objects are shown to be equivalent in some way, and that one of them is endowed with additional structure. In Mathematics, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists Using the equivalence, we may define such a structure on the second object as well, via transport of structure. For example, any two vector spaces of the same dimension are isomorphic; if one of them is given an inner product and if we fix a particular isomorphism, then we may define an inner product on the other space by factoring through the isomorphism. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

Let V be a finite-dimensional vector space over k. . . . Let (e_i)_{1 ≤ i ≤ n} be a basis for V. . . . There is an isomorphism of the polynomial algebra k[T_ij]_{1 ≤ i,j ≤ n} onto the algebra Sym_k(V ⊗ V^*). . . . It extends to an isomorphism of k[GL_n] to the localized algebra Sym_k(V ⊗ V^*)_D, where D = det(e_i ⊗ e_j^*). . . . We write k[GL(V)] for this last algebra. By transport of structure, we obtain a linear algebraic group GL(V) isomorphic to GL_n.

—Igor Shafarevich (1991, p. 12)

without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA), it may be assumed that (WOLOGIMBAT)

Sometimes a proposition can be more easily proved with additional assumptions on the objects it concerns. Without loss of generality (abbreviated to WLOG or WOLOG and less commonly stated as without any loss of generality) is a frequently used expression If the proposition as stated follows from this modified one with a simple and minimal explanation (for example, if the remaining special cases are identical but for notation), then the modified assumptions are introduced with this phrase and the altered proposition is proved.

Informal proof techniques

Some terms are techniques for the avoidance of rigorous proof, though are not logical fallacies. A fallacy is a component of an Argument which being demonstrably flawed in its Logic or form renders the argument invalid in whole They suggest the content of a correct proof without supplying it.

back-of-the-envelope calculation

An informal computation omitting much rigor without sacrificing correctness. The phrase back-of-the-envelope calculations refers to rough calculations that while not rigorous test or support a point Often this computation is "proof of concept" and treats only an accessible special case.

by inspection

A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction. By inspection is a Mathematical term pertaining to the difficulty of a computation

clearly, can be easily shown

A term which shortcuts around calculation the mathematician perceives to be tedious or routine, accessible to any member of the audience with the necessary expertise in the field; Laplace used obvious.

handwaving

A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary. The term handwaving is an informal term that describes either the Debate technique of failing to Rigorously address an Argument in an attempt to bypass the It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.

in general

In a context not requiring rigor, this phrase often appears as a labor-saving device when the technical details of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simple enough case that the computations are reasonable, and then indicates that "in general" the proof is similar.

morally true

Used to indicate that the speaker believes a statement should be true, given their mathematical experience, even though a proof has not yet been put forward. As a variation, the statement may in fact be false, but instead provide a slogan for or illustration of a correct principle. Hasse's local-global principle is a particularly influential example of this. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic In Mathematics, Helmut Hasse 's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation

trivial

Similar to clearly. In Mathematics, the term trivial is frequently used for objects (for examples groups or Topological spaces that have a very simple A concept is trivial if it holds by definition, is immediately corollary to a known statement, or is a simple special case of a more general concept.

Footnotes

1. ^ Numerous examples can be found in (Mac Lane 1998), for example on p. 100.

References

• Monastyrsky, Michael (2001), “Some Trends in Modern Mathematics and the Fields Medal”, Can. Math. Soc. Notes 33 (2 and 3), <http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf> 
• Mac Lane, Saunders (1997), “The PNAS way back then”, Proc. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Natl. Acad. Sci. USA 94: 5983–5985, <http://www.pnas.org/cgi/reprint/94/12/5983.pdf> 
• Eilenberg, Samuel & Mac Lane, Saunders (1942), “Natural Isomorphisms in Group Theory”, Proc. Samuel Eilenberg ( September 30, 1913 — January 30, 1998) was a Polish and American Mathematician of Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Natl. Acad. Sci. USA 28: 537–543 
• Poincare, Henri (1913), Halsted, Bruce, ed. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician , The Foundations of Science, The Science Press, p. 435, <http://books.google.com/books?id=WxLJzL4e4wsC&printsec=titlepage&source=gbs_summary_r&cad=0#PPA435,M1> 
• Pinto, J. Sousa (2004), Hoskins, R. F. , ed. , Infinitesimal methods for mathematical analysis, Horwood Publishing, p. 246, <http://books.google.com/books?id=bLbfhYrhyJUC&pg=PA246&sig=S2vS6a489o-Ez8ORgn0ddDlRhz4#PPA246,M1> 
• Mac Lane, Saunders (1998), Categories for the Working Mathematician, Springer 
• Shafarevich, Igor (1991), Kandall, G. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Igor Rostislavovich Shafarevich ( Russian: Игорь Ростиславович Шафаревич born June 3, 1923 in Zhytomyr) is a A. , ed. , Algebraic Geometry, vol. IV, Springer 
• Rota, Gian-Carlo (1977), “The phenomenology of mathematical beauty”, Synthese 111 (2): 171–182, ISSN 0039-7857 
• Jackson, Allyn (2004), “Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck”, AMS Notices 51 (9,10)  (Parts I and II)
• Fundamentals of Mathematics, Global Media, ISBN 8189940570, <http://books.google.com/books?id=bs0BfTsv5IgC&pg=PT4&sig=XdEBWVDSqf5l37VTfD1JRt2xhjo> 
• Wiedijk, Freek, ed. Gian-Carlo Rota ( April 27, 1932 &ndash April 18, 1999, known as Juan Carlos Rota An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. (2006), The Seventeen Provers of the World, Birkhäuser, ISBN 3540307044