Can a definition be given that captures the meaning of Mathematics across the millennia of its recorded history? What unites the practice of mathematics throughout its history and into the present time?
In this article, I will try for a short answer by proceeding iteratively — convergence will be reached in two iterations….
What is Mathematics?
Though mathematical understanding is ancient, stretching back into prehistory, its concepts, organization, scope, outlook and practice have seen profound evolution over the millenia.
So what is this thing called mathematics? Can a definition be found that captures the meaning of Mathematics across the millennia of its recorded history? What unites the practice of mathematics throughout its history and into the present time?
Getting at a satisfactory answer is slippery business. What adequately describes mathematics at various earlier periods of its history is typically inadequate for contemporary mathematics.^{1} The same is true in reverse: an abstract, structure based discussion of mathematics that fits contemporary mathematics would exclude large periods in the early history of mathematics. [CC09].
Criteria for a Good Definition of Mathematics
What should a good answer to the question “What is Mathematics?” look like? I believe it should hold up well against the following three criteria:
 it should cover the practice of Mathematics through its history and into the present.
This means it should apply, not only to the broad range of pure and applied mathematics today, but also to primitive, ancient, and classical mathematics – the mathematics of the Babylonians, Egyptians, Greeks, Indians, Chinese, Arabs, Central Asians, and MercantilePeriod Italians – as well as to the mathematics that accompanied the flowering of science and technology in the modern age.
 it should not exclude periods or activities that do not meet current standards of rigor, or those from a particular historical time;
 it should be in harmony with the manner by which mathematical knowledge develops, both in the past and presently, and with the perspectives of the various users of mathematics today: scientists, engineers, applied and pure mathematicians; and
In this article, I will try for a short answer by proceeding iteratively. Convergence will arrive in two iterations. (For answers that are essentially surveys of mathematics and its variations, directions, and forms, see [Ale56], [DH81], [CR41], and [Mac86].)
First Iteration
I’ll start with a oneliner that has basic versions of the first four out of five ingredients:
Mathematics is a subject concerned with number, shape, change, and relation…
Unpacking this:
 Number has to do with quantity, measurement, and scale;
 Shape is about configuration and arrangement;
 Change considers time and variation; and
 Relation has to do with association and comparison (similarity, difference, equality).
These first four areas, taken together, adequately cover protomathematics and ancient mathematics, including the mathematics of ancient Babylonia, Egypt, China, and India: numeration systems, integer arithmetic, nonsymbolic solution of linear equations and select quadratic equations, mensuration, the properties of various two and three dimensional figures, the fundamentals of statics and mechanics, the keeping of time, division of property, taxation, and arithmetic of fractions.
Second Iteration
For the rest of mathematics, we’ll need more. To cover classical mathematics through to contemporary mathematics requires a tweak to the basic ingredients, the addition of a fifth crucial ingredient, and some exposition:
Mathematics is a subject concerned with four natural phenomena: quantity, space, transformation, and relation. Deepening understanding of the four natural phenomena leads to the development of a chain of evolving conceptual abstractions, to greater generalization, and to correspondingly broader areas of investigation.
This has created a fifth, humanistic area of mathematical activity: the development of logically structured (axiomatic) mathematical systems that generalize and extend empirical concepts, introduce fundamentally new theoretical concepts, and examine the laws that govern their structure, properties and relationships holding between them.
Progress in this last area has led to the development of a variety of mathematical systems. The breadth of modern mathematics is organized within these mathematical systems, and it is from within these systems that they find application in areas beyond the historical core.
Unpacking this:
The fifth, humanistic area of mathematics, structure, has to do with the development of mathematical systems to organize empirical or informal mathematical knowledge, axiomatic foundations for the various areas of mathematical activity, and establishing standards of logical inference and proof.
With its addition, we can cover classical mathematics, including the mathematics of Greece, the golden age of geometric science in Hellenistic North Africa, and the golden age of science in Arabia, Central Asia, and India: the development of axiomatic geometry and associated progress in plane geometry, solid geometry, and analysis, the development of triangle geometry (trigonometry), a systematic development of algebra, the algorithmization of arithmetic, and developments foreshadowing the calculus and symbolical mathematics.^{2}
Once we come to premodern and modern mathematics, crosspollination betweeen areas of mathematics and the earnest development of mathematical structures means that most areas are now associated with various combinations of the five areas:
 Quantity continues to hold matters of number, measurement, and scale. But it is also the subject of arithmetic, geometry, analysis, complex variables, combinatorics, probability, statistics, optimization – indeed almost all of mathematics involves quantity or one of its conceptual abstractions.
 Space continues to hold matters to do with shape, configuration, arrangement, symmetry, perspective. But it is also the subject of the continuum in one, finitely many, and infinite dimensions, the subject of geometry, continuous group theory, linear algebra, analysis, differential mathematics, topology, and set theory, among others.
 Change continues to be about about time, but also includes mathematical tools for representing and analyzing transformation: functions, analytic geometry, the calculus, differential mathematics, geometrical physics, matrix algebra, and analysis, among others.
 Relation continues its association with comparison, similarity and equality, but extends these more generally to equivalence, and the subjects of abstract algebra, set theory, and logic, among others.
 Structure permeates all of modern mathematics, organizing, arranging, rigorizing informal mathematical knowledge, providing axiomatic foundations for the various areas of mathematical activity, and acceptable standards of logical inference and proof. The development and refinement of mathematical structures is never done, and work continues in modern logic, proof theory, set theory, as well as in the intersections between mathematics and physics (gauge theories, string theories, quantum field theories), computer science (computability, recursion, mathematical linguistics), biology (chaos theory, selforganizing systems), and in many other areas.
Three Facets of Mathematics
A definitional answer is, unfortunately, just a start to explaining Mathematics.
To understand Mathematics in a way that is consistent with its history, evolution, and its many diverse applications, as well as with its contemporary, abstract, and highly specialized state, it is helpful to identify three coexisting facets of Mathematics:
 Mathematics as an empirical science,
 Mathematics as a modeling art, and
 Mathematics as an axiomatic arrangement of ideas, their relations, and the conceptual structures built around them.
These facets of Mathematics explain both the historical development, maturity, and modern separation between theoretical and applied considerations. I’ll consider them in turn.
Mathematics as an Empirical Science
Mathematics originates out of science, i.e. out of human interest in the surrounding world, its careful observation, and the empirical verification of mathematical fact. In fact, the world and its patterns are consistently present in the inspiration of all mathematical studies. Even the abstract, abstruse, and seemingly detached topics of advanced higher mathematics are generalizations of patterns observed in the layers of less abstract mathematics, that are themselves an attempt to capture patterns observed in the real world itself.
I would venture, therefore, that the essence of a mathematical concept can always be related back to an original protoconcept that has its roots in empirical observations and the patterns arising out of these.
Mathematics as a Modeling Art
Mathematics as a modeling art involves an effort to develop, maintain, and perfect models of perceived or envisioned reality. I would venture, further, that Mathematics has always involved, and continues to involve, the exploration, explanation and modeling of phenomena.
At the root of this facet of mathematics is the intimate relationship between the physical world and the world of mathematical ideas. Most of the major laws of mathematics are modeled on actual physical occurrences, suitably abstracted. Thus, I would argue that the origin of most of mathematics is a model of something real that has been experienced. Indeed, typical applied mathematics proceeds from a physical context to the context of a mathematical model, performs computations and analysis using mathematical reasoning within the domain of this model, and then finally brings the result back to the physical context for interpretation.
The success of Mathematics in keeping everimproving mathematical models of many and various phenomena, and the fact that the methods behind these models are often applicable in widely different areas and contexts, often lead Mathematics to be viewed enthusiastically (though incorrectly) as the key to the knowledge of all things.^{3}
Mathematics as an Axiomatic Arrangement of Knowledge
The exploration of the logical structure of mathematical knowledge is a relatively recent development, beginning with the ancient Greeks circa 800 BCE. Comparatively, this phenomenon has occupied less than 3 millenia, or less than 10% of the documented history of mathematical knowledge of humankind (30,000 years).
Rapid progress in understanding the logical structure of mathematics occurred since the 1800s CE,^{4} and has led to the flowering of a wide variety of modern mathematical systems and theories whose areas of interests and domains of application go far beyond the historical core of mathematics.
Today, the vast scope of modern mathematical knowledge is organized within structured mathematical systems from within which it finds wide application.
Mathematical structures distill informal mathematical knowledge, identify the important concepts out of the body of informal mathematical knowledge, and provide streamlined logical models to underpin these areas.
Continue reading: >> Characteristics of Modern Mathematics…
The complete article with its extensive Further Reading section, is available as a PDF here.
Further Reading
The following books and papers are recommended for additional reading on the topics discussed in this article.
 An expository survey of elementary mathematics provides an excellent example of the richness and creativity of mathematical ideas, as well as a condensed glimpse into the evolution of mathematical ideas. See [Ale56].
 An expository look at the questions of axiomatic foundations of mathematics is contained in the short paper [Fef99].
 Lakatos develops a particularly vivid presentation of dialectic in mathematics in his mathematicalliterary play Proofs and Refutations, [Lak76].
 Lamport and Djikstra inquire into the place and method of proof in mathematics in [Lam95] and [Dij89b].
Other recommended readings are in the References.
References
 [Ale56]

A.D. Aleksandrov.
A general view of mathematics.
from the book “Mathematics: Its Content, Methods and Meaning”,
([AKL63]), pages 164, (Chapter 1), 1956.  [AKL63]

A.D. Aleksandrov, A.N. Kolmogorov, and M.A. Lavrentev.
Mathematics: Its Content, Methods and Meaning.
MIT Press: 2nd edition, 1969; 1st edition, 1963/1964, dover edition,
1999 (three volumes bound as one) edition, 1963.  [Bou]

Nicholas Bourbaki.
Elements of Mathematics: Set Theory.  [Bri57]

L. Brillouin.
Mathematics, physics, and information.
Information and Control; Vol. 1; No. 1;, pages 15, 1957.  [Bri62]

L. Brillouin.
Poincare’s theorem and uncertainty in classical mechanics.
1962.  [Bur]

David Burton.
The History of Mathematics: An Introduction  [CC09]

Mark ChuCarroll.
What is math?
December 2009.
Article available from the author’s website  [CR41]

R. Courant and H. Robbins.
What is Mathematics? An Elementary Approach to Ideas and
Methods.
Oxford, 1941.  [DH81]

Philip J. Davis and Reuben Hersh.
The Mathematical Experience.
Birkhauser, 1981.  [Dij89a]

E.W. Dijkstra.
Mathematical methodology – preface.
EWD1059; Available from the Internet for download, 1989.  [Dij98]

E.W. Dijkstra.
Society’s role in mathematics, or, in my opinion, the story of the
evolution of rigor in mathematics, and the threshold at which mathematics now
standards unaware.
1998.  [FD07]

J. Ferreiros and J.F. Dominguez.
Labyrinth of Thought: A History of Set Theory and Its Role in
Modern Mathematics.
Springer, second edition, 2007.  [Fef]

Solomon Feferman.
The development of programs for the foundations of mathematics in the
first third of the 20th century.
Available from the Internet at the author’s website.  [Fef92a]

Solomon Feferman.
What rests on what? the prooftheoretic analysis of mathematics.
Available from the Internet at the author’s website., 1992.  [Fef92b]

Solomon Feferman.
Why a little bit goes a long way: Logical foundations of
scientifically applicable mathematics.
PSA 1992, vol. 2 (1993), pages 442455 (with corrections),
1992.  [Fef98]

Solomon Feferman.
Mathematical intuition vs. mathematical monsters.
Available from the Internet at the author’s website., 1998.  [Fef99]

Solomon Feferman.
Does mathematics need new axioms?
American Mathematical Monthly, Vol. 106, No. 2, pages 99111,
Feb 1999.  [Kle86]

Israel Kleiner.
The evolution of group theory: A brief survey.
Mathematics Magazine; Vol.59, No.4, pages 194215, October
1986.  [Kli]

Morris Kline.
Mathematical Thought from Ancient to Modern Times.
Oxford University Press, 3 volumes edition.  [Lak76]

Imre Lakatos.
Proofs and Refutations: The Logic of Mathematical Discovery.
Cambridge University Press, 1976.  [Mac86]

Saunders MacLane.
Form and Function.
1986.  [Pan]

Pannenoek.
History of Astronomy.
Dover.  [Rot97]

GianCarlo Rota.
Indiscrete Thoughts.
Birkhauser, 1997.  [Rus]

Bertrand Russell.
Principles of Mathematics.  [Wal06]

M.A. Walicki.
The history of logic.
from Introduction to Logic, pages 127, 2006.  [Wil82]

Herbert Wilf.
What is an Answer?
1982.
Footnotes:
 Even a cursory look at the modern mathematical literature and its applications, or at the subject classification of the AMS makes it abundantly clear that mathematics covers an enormous breadth of material today and that mathematicians are investigating these areas in many different ways. ↩
 The Greeks encountered the paradoxes of the real numbers, though not their resolution, and were already using the method of exhaustion and converging upper and lower bounds, a precursor to the methods of the integral calculus, apart from a rigorous passing to the limit.
The Arab and Central Asians and Indians advanced algebra to an abstract science, had resolved the solution of algebraic equations including most instances of the general cubic, had developed expansion by Taylor series and precursors of the calculus, had developed the trigonometric identities, applied algebra and trigonometry to astronomical problems, and had fully developed systems of computation for interest rates, taxation, and other numerical calculations using decimal digits including zero, and a place decimal system. ↩
 Successful generalization often leads to that heady feeling of lifting the veils from the face of Mysteries. Indeed, the Rhind Papyrus containing mathematical knowledge of ancient Egyptian begins with a description of what it discusses:
“a thorough study of all things, insight into all that exists, knowledge of all obscure secrets.” (Bur, p.38.)
The mathematical knowledge contained in the Rhind Papyrus, the ability to compile, present, and solve a compendium of practical mathematical exercises using abstract techniques of multiplication and division, was certainly prized among the ancient Egyptians, and known to a limited few. But a study of all things? insight into all that exists? knowledge of all obscure secrets? Anyone who has tasted the heady feeling of succesful generalization and a survey of a branch of the science will perhaps understand the author’s excess of enthusiasm for the mathematical knowledge he was about to share. ↩
 The 200 years of mathematics since 1800 CE is less than 1% of the documented history of mathematics. ↩