studies the properties and structure of abstract objects, such as
the E8 group, in group theory. This may be done without focusing on
concrete applications of the concepts in the physical
is the study of mathematical concepts independently of any
application outside mathematics. These concepts may originate in
real-world concerns, and the results obtained may later turn out to
be useful for practical applications, but pure mathematicians are
not primarily motivated by such applications. Instead, the appeal
is attributed to the intellectual challenge and aesthetic beauty of
working out the logical consequences of basic
mathematics has existed as an activity since at least Ancient
Greece, the concept was elaborated upon around the year 1900,
after the introduction of theories with counter-intuitive
properties (such as non-Euclidean geometries and Cantor's theory of
infinite sets), and the discovery of apparent paradoxes (such as
continuous functions that are nowhere differentiable, and Russell's
paradox). This introduced the need of renewing the concept of
mathematical rigor and rewriting all mathematics accordingly, with
a systematic use of axiomatic methods. This led many mathematicians
to focus on mathematics for its own sake, that is, pure
almost all mathematical theories remained motivated by problems
coming from the real world or from less abstract mathematical
theories. Also, many mathematical theories, which had seemed to be
totally pure mathematics, were eventually used in applied areas,
mainly physics and computer science. A famous early example is
Isaac Newton's demonstration that his law of universal gravitation
implied that planets move in orbits that are conic sections,
geometrical curves that had been studied in antiquity by
Apollonius. Another example is the problem of factoring large
integers, which is the basis of the RSA cryptosystem, widely used
to secure internet communications.
It follows that,
presently, the distinction between pure and applied mathematics is
more a philosophical point of view or a mathematician's preference
than a rigid subdivision of mathematics. In particular, it is not
uncommon that some members of a department of applied mathematics
describe themselves as pure mathematicians.
mathematicians were among the earliest to make a distinction
between pure and applied mathematics. Plato helped to create the
gap between "arithmetic", now called number theory, and "logistic",
now called arithmetic. Plato regarded logistic (arithmetic) as
appropriate for businessmen and men of war who "must learn the art
of numbers or [they] will not know how to array [their] troops" and
arithmetic (number theory) as appropriate for philosophers "because
[they have] to arise out of the sea of change and lay hold of true
being." Euclid of Alexandria, when asked by one of his students
of what use was the study of geometry, asked his slave to give the
student threepence, "since he must make gain of what he learns."
The Greek mathematician Apollonius of Perga was asked about the
usefulness of some of his theorems in Book IV of Conics to which he
They are worthy of
acceptance for the sake of the demonstrations themselves, in the
same way as we accept many other things in mathematics for this and
for no other reason.
And since many of
his results were not applicable to the science or engineering of
his day, Apollonius further argued in the preface of the fifth book
of Conics that the subject is one of those that "...seem worthy of
study for their own sake."
The term itself is
enshrined in the full title of the Sadleirian Chair, Sadleirian
Professor of Pure Mathematics, founded (as a professorship) in the
mid-nineteenth century. The idea of a separate discipline of pure
mathematics may have emerged at that time. The generation of Gauss
made no sweeping distinction of the kind, between pure and applied.
In the following years, specialisation and professionalisation
(particularly in the Weierstrass approach to mathematical analysis)
started to make a rift more apparent.
At the start of
the twentieth century mathematicians took up the axiomatic method,
strongly influenced by David Hilbert's example. The logical
formulation of pure mathematics suggested by Bertrand Russell in
terms of a quantifier structure of propositions seemed more and
more plausible, as large parts of mathematics became axiomatised
and thus subject to the simple criteria of rigorous
according to a view that can be ascribed to the Bourbaki group, is
what is proved. Pure mathematician became a recognized vocation,
achievable through training.
The case was made
that pure mathematics is useful in engineering
There is a
training in habits of thought, points of view, and intellectual
comprehension of ordinary engineering problems, which only the
study of higher mathematics can give.
An illustration of
the Banach–Tarski paradox, a famous result in pure mathematics.
Although it is proven that it is possible to convert one sphere
into two using nothing but cuts and rotations, the transformation
involves objects that cannot exist in the physical
concept in pure mathematics is the idea of generality; pure
mathematics often exhibits a trend towards increased generality.
Uses and advantages of generality include the following:
theorems or mathematical structures can lead to deeper
understanding of the original theorems or structures
simplify the presentation of material, resulting in shorter proofs
or arguments that are easier to follow.
One can use
generality to avoid duplication of effort, proving a general result
instead of having to prove separate cases independently, or using
results from other areas of mathematics.
facilitate connections between different branches of mathematics.
Category theory is one area of mathematics dedicated to exploring
this commonality of structure as it plays out in some areas of
impact on intuition is both dependent on the subject and a matter
of personal preference or learning style. Often generality is seen
as a hindrance to intuition, although it can certainly function as
an aid to it, especially when it provides analogies to material for
which one already has good intuition.
As a prime example
of generality, the Erlangen program involved an expansion of
geometry to accommodate non-Euclidean geometries as well as the
field of topology, and other forms of geometry, by viewing geometry
as the study of a space together with a group of transformations.
The study of numbers, called algebra at the beginning undergraduate
level, extends to abstract algebra at a more advanced level; and
the study of functions, called calculus at the college freshman
level becomes mathematical analysis and functional analysis at a
more advanced level. Each of these branches of more abstract
mathematics have many sub-specialties, and there are in fact many
connections between pure mathematics and applied mathematics
disciplines. A steep rise in abstraction was seen mid 20th
however, these developments led to a sharp divergence from physics,
particularly from 1950 to 1983. Later this was criticised, for
example by Vladimir Arnold, as too much Hilbert, not enough
Poincaré. The point does not yet seem to be settled, in that string
theory pulls one way, while discrete mathematics pulls back towards
proof as central.
have always had differing opinions regarding the distinction
between pure and applied mathematics. One of the most famous (but
perhaps misunderstood) modern examples of this debate can be found
in G.H. Hardy's A Mathematician's Apology.
It is widely
believed that Hardy considered applied mathematics to be ugly and
dull. Although it is true that Hardy preferred pure mathematics,
which he often compared to painting and poetry, Hardy saw the
distinction between pure and applied mathematics to be simply that
applied mathematics sought to express physical truth in a
mathematical framework, whereas pure mathematics expressed truths
that were independent of the physical world. Hardy made a separate
distinction in mathematics between what he called "real"
mathematics, "which has permanent aesthetic value", and "the dull
and elementary parts of mathematics" that have practical
some physicists, such as Einstein, and Dirac, to be among the
"real" mathematicians, but at the time that he was writing the
Apology he also considered general relativity and quantum mechanics
to be "useless", which allowed him to hold the opinion that only
"dull" mathematics was useful. Moreover, Hardy briefly admitted
that—just as the application of matrix theory and group theory to
physics had come unexpectedly—the time may come where some kinds of
beautiful, "real" mathematics may be useful as well.
view is offered by Magid:
thought that a good model here could be drawn from ring theory. In
that subject, one has the subareas of commutative ring theory and
non-commutative ring theory. An uninformed observer might think
that these represent a dichotomy, but in fact the latter subsumes
the former: a non-commutative ring is a not-necessarily-commutative
ring. If we use similar conventions, then we could refer to applied
mathematics and nonapplied mathematics, where by the latter we mean
not-necessarily-applied mathematics... [emphasis
^ Piaggio, H. T.
H., "Sadleirian Professors", in O'Connor, John J.; Robertson,
Edmund F. (eds.), MacTutor History of Mathematics archive,
University of St Andrews.
^ Robinson, Sara
(June 2003). "Still Guarding Secrets after Years of Attacks, RSA
Earns Accolades for its Founders" (PDF). SIAM News. 36
^ Boyer, Carl B.
(1991). "The age of Plato and Aristotle". A History of Mathematics
(Second ed.). John Wiley & Sons, Inc. p. 86. ISBN
0-471-54397-7. Plato is important in the history of mathematics
largely for his role as inspirer and director of others, and
perhaps to him is due the sharp distinction in ancient Greece
between arithmetic (in the sense of the theory of numbers) and
logistic (the technique of computation). Plato regarded logistic as
appropriate for the businessman and for the man of war, who "must
learn the art of numbers or he will not know how to array his
troops." The philosopher, on the other hand, must be an
arithmetician "because he has to arise out of the sea of change and
lay hold of true being."
^ Boyer, Carl B.
(1991). "Euclid of Alexandria". A History of Mathematics (Second
ed.). John Wiley & Sons, Inc. p. 101. ISBN 0-471-54397-7.
Evidently Euclid did not stress the practical aspects of his
subject, for there is a tale told of him that when one of his
students asked of what use was the study of geometry, Euclid asked
his slave to give the student threepence, "since he must make gain
of what he learns."
a b Boyer, Carl B.
(1991). "Apollonius of Perga". A History of Mathematics (Second
ed.). John Wiley & Sons, Inc. p. 152. ISBN 0-471-54397-7. It is
in connection with the theorems in this book that Apollonius makes
a statement implying that in his day, as in ours, there were
narrow-minded opponents of pure mathematics who pejoratively
inquired about the usefulness of such results. The author proudly
asserted: "They are worthy of acceptance for the sake of the
demonstrations themselves, in the same way as we accept many other
things in mathematics for this and for no other reason." (Heath
The preface to
Book V, relating to maximum and minimum straight lines drawn to a
conic, again argues that the subject is one of those that seem
"worthy of study for their own sake." While one must admire the
author for his lofty intellectual attitude, it may be pertinently
pointed out that s day was beautiful theory, with no prospect of
applicability to the science or engineering of his time, has since
become fundamental in such fields as terrestrial dynamics and
^ A. S. Hathaway
(1901) "Pure mathematics for engineering students", Bulletin of the
American Mathematical Society 7(6):266–71.
^ Andy Magid
(November 2005) Letter from the Editor, Notices of the American
Mathematical Society, page 1173
quotations related to: Pure mathematics
What is Pure
Mathematics? – Department of Pure Mathematics, University of
What is Pure
Mathematics? by Professor P. J. Giblin The University of
The Principles of
Mathematics by Bertrand Russell
How to Become a
Pure Mathematician (or Statistician), a list of undergraduate and
basic graduate textbooks and lecture notes, with several comments
and links to solutions, companion sites, data sets, errata pages,