What is Mathematics?

4th ed. Jan 2024; 3rd ed. May 2023; 2nd ed. Dec 2009; 1st ed. Sep 2004

“It is not philosophy but active experience in mathematics itself that alone can answer the question: `What is Mathematics?'” – Richard Courant & Herbert Robbins, 1941, What is Mathematics?, Oxford University Press)

“An adequate presentation of any science cannot consist of detailed information alone, however extensive. It must also provide a proper view of the essential nature of the science as a whole.” – Aleksandrov, 1956, Mathematics: Its Content, Methods, and Meaning

‘What is mathematics?’ Much ink has been spilled over this question, as can be seen from the selection of ten respected responses provided in the footnote¹, with seven book-length answers, and three written in the current millenium. One might well ask, is there anything new that can be said, that should be said? We’ll start by clarifying what a good answer should look like, and then explore the answer proposed.

The rest of the paper follows the structure below:

   1. Criteria for a Good Definition of Mathematics
   2. Definition 1: covering mathematics up to the end of the 18th century (1790s)
   3. Two Perspectives 
       Mathematics as Dialectic (Lakatos)
       Mathematics shaped by its Anthropology (Hoyrup)
   4. Definition 2: covering all mathematics, including contemporary mathematics
   5. The emergence of contemporary mathematical practice from 1800s onward
   6. Three Facets of Mathematics
       1. Mathematics as an Empirical Science
       2. Mathematics as a Modeling Art
       3. Mathematics as an Axiomatic Arrangement of Knowledge
   7. Mathematics "from the inside": Mathematicians writing about Mathematics
   8. Continue Reading
   9. References

1. Criteria for a ‘Satisfactory’ Definition of Mathematics

A satisfactory answer to the question ‘What is Mathematics?’ should, in my view, hold up well against the following three criteria:

1. it should cover the practice of Mathematics throughout its long history, as well as current practice, i.e. it should be in harmony with the methods by which mathematical knowledge has developed and continues to develop, as exemplified by practitioners both past and present. Since notions of mathematical truth and rigour vary in time [Grabiner, 1974], it must not exclude historical periods or activities that do not conform to current conventions of style or rigor.
2. it should incorporate the vast range of modern mathematical literature and applications ([Gowers, 2008], [Higham, 2015]), while including former highways that are now less travelled byways (examples: Triangle Geometry [for history see Davis, 1995], Elementary Geometry [for history see Yaglom, 1981], Determinants [see Strang, 1976: “It is hard to know what to say about determinants, at end of the 19th century they seemed more interesting and more important than the matrices they came from, and Muir’s History of Determinants filled four volumes. Mathematics keeps changing direction, however, and determinants are now far from the center of linear algebra.”, p.146], for history see [Petrilli, Smolensky, 2017]).
3. it should incorporate the mathematical ideas, tools and techniques used today and in history, that are of value to scientists physicists, astronomers, biologists, atmospheric scientists, geophysicists, oceanographers, chemists, engineers, computer scientists, economists, financial mathematicians, and other users of mathematics. Included in this must be the fact that mathematicians and mathematically competent practioners of other disciplines are investigating today an enormous breadth of mathematical material in many different ways, including through the use of computers, modeling and simulation, machine learning, automated theorem proving software, computer algebra systems, and artificial intelligence techniques.

Accordingly, it is not just ‘What is Mathematics today?’ that we should be answering, but the broader scoped and more fundamental question: ‘Can we find a definition that adequately describes mathematics across the more than 5000 years of its written history?’ In other words ‘Are there common threads that unite the practice of mathematics today with its long history and pre-history of mathematical practice?’

The difficulty with the above criteria is that 1) our understanding of number, shape, and change stretches back into pre-history, 2) the concepts and applications that have emerged out of this understanding, what we may call mathematical practice, have seen profound evolution over the subsequent periods of mathematical history, in scope, in outlook, and in the organization of this mathematical knowledge, and 3) mathematicians today are investigating an enormous breadth of material, and the range of applications of mathematics is vast (to see this one need only glance at the subject classification of the American Mathematical Society, or the topics contained in the Princeton Companion to Pure Mathematics [Gowers, 2008] and Princeton Companion to Applied Mathematics [Higham, 2015]).

On the principle of requiring more than a current definition, we must therefore reject at the outset any definition of mathematics that is extracted from contemporary pure mathematics. Such definitions are often built around the abstract, deductive presentation of contemporary mathematics, organized axiomatically around a “sets with structure” theme. The problem with these definitions is that they would exclude vibrant, highly productive episodes in the history of mathematics up to the 19th century. They would exclude the pre-modern work of the Old Babylonians on pure mathematics within the scribal schools, the Neo-Babylonians on mathematical astronomy including a precursor of the differential calculus to interpolate the position of Jupiter, Omar Khayyam on the classification of the 19 classes of cubic equations and their complete solution using the intersection of at most two conics, and Fibonacci’s work. They would also exclude the work of Leibniz, Euler, the Bernoullis, Pascal on probability and combinatorics, Fermat on number theory, Lagrange, Laplace on the 3-body problem, and the work of other mathematicians, whose path-breaking advances were was NOT obtained axiomatically, coming after the adoption of symbolism in 1590 (Viete) but before the contemporary reformulation of mathematics axiomatically that began in the mid-1800s and accelerated in the mid-1900s. (For other limited answers also rejected, see Appendix 1.)

In what follows, we will attempt to build a definition of mathematics that covers earlier periods in the history of mathematics and also applies across the enormous range of pure and applied mathematics today. For the purpose of this article, we will consciously set aside additional, not inconsiderable, philosophical questions, such as ‘What is Truth?’, ‘What constitutes Proof?’, and ‘On what foundations does Mathematics build, and with what certainty?’ We’ll discuss a definition of Mathematics that is consistent with mathematical practice across its entire recorded history, from the first written account-keeping in Uruk by the Sumerians (c.3200 BCE), over the ensuing five millenia, to the present era with its its broad scope and modern practice in the present time.

We will proceed iteratively, in two passes. The first pass covers mathematics through to the end of the 18th century; the second pass extends the definition to cover mathematics through to the present.

2. First Pass: A definition covering all mathematics up end of the 18th century

“There are two fundamental sources of ‘bare facts’ for the mathematician, that is, there are some real things out there to which we can confront our understanding. These are, on the one hand the physical world which is the source of geometry, and on the other hand the arithmetic of numbers which is the source of number theory. Any theory concerning either of these subjects can be tested by performing experiments either in the physical world or with numbers.” – Alain Connes (mathematician, Fields Medalist 1982) from Non-Commutative Geometry, 2000

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Definition 1: Mathematics is a subject concerned with number, shape, and change.

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• Number has to do with quantity, measurement, and scale;
• Shape is about configuration and arrangement;
• Change considers variation, often in time, but with respect to any other variable (e.g. position, pressure, etc).

These first three elements adequately cover pre-modern mathematics and modern mathematics upto the end of the 18th century including the mathematics of ancient Babylonia, Egypt, Greece², China, India, Arabia and Central Asia³ This includes numeration systems, integer arithmetic, division of property, taxation, and arithmetic of fractions, Euclidean geometry and its offshoots astronomy and trigonometry, solution of algebraic equations lower than degree five, mensuration, the properties of various two and three dimensional figures, the fundamentals of statics and mechanics, dynamics, infinite series, and even the differential and integral calculus, and physics. In particular, this definition covers the works of Euler, Laplace, Taylor, Newton, etc. Most importantly, the mathematics up to the start of the 19th century, was about the real, physical world, and was a language used to investigate its notions. Mathematics for a long time had been considered either part of accounting, a part of geometry, or a branch of natural philosophy. The mathematics up to the end of the 18th century was concerned with the “bare facts” of this physical and arithmetically coherent world:

To cover contemporary mathematics from the 19th century onward requires further elements, the notion of mathematical structure and the study of relations between mathematical objects and mathematical structures. It requires a deeper understanding of how mathematics has and continues to develop.

Let’s start with the word itself. “Matema” is the ancient Greek term for “that which is learnt,” or “what one gets to know.” So we get the first pillar: Mathematics is a body of knowledge, and area of human understanding. But knowledge and understanding of what?

There are two compelling perspectives: the dialetic perspective of the philosopher-mathematican Imre Lakatos and the anthropology of mathematics perspective of mathematical historian Jens Hoyrup.

3. Mathematics as dialectic and an Anthropology of Mathematics

“In the history of the development of mathematics, three different processes of growth now change places, now run side by side independent of one another, now finally mingle. Plan A is based upon a more particularistic conception of science which divides the total field into a series of mutually separated parts and attempts to develop each part for itself, with a minimum of resources and with all possible avoidance of borrowing from neighboring fields. Its idea is to crystallize out each of the partial fields into a logically closed system. Plan B lays the chief stress upon the organic combination of the partial fields, and upon the stimulation which these exert one upon another. Plan B prefers, therefore, the methods which open an understanding of several fields under a uniform point of point of view. Its ideal is the comprehension of the sum total of mathematical science as a great connected whole. There is still a third Plan C, algorithmic, which, along side of and within the processes of development A and B, often plays an important role as a quasi-independent, onward-driving force, inherent in the formulas, operating apart from the intention and insight of the mathematician, at the time , often indeed in opposition to them.” – Felix Klein, 1908, Elementary Mathematics from an Advanced Perspective, pp.77-85

1. Mathematics as dialectic. In this perspective, mathematics is a great conversation, happening through time, and across traditions, a conversation about ideas, refining, reworking, testing and objecting, ultimately uncovering new and deeper understanding. This is the perspective of [Lakatos, 1976, Proofs and Refutations] and of [Aleksandrov, 1956, A General View of Mathematics]. Lakatos illustrates this with a facinating dialogue between a group of students debating the proof of the Euler characteristic of the polyhedron, artfully compressing into a single dialogue understanding that evolved over the course of approx. 200 years. Aleksandrov focuses on the results of the dialectic, on the steady progress of mathematical concepts: “they are brought into being by a series of successive abstractions and generalizations, each resting on a combination of experience with preceding abstract concepts.[Aleksandrov, 1956, p.17]

There is another aspect of dialectic, a dialectic tension of ideas, concepts, and perspectives, which Felix Klein brings out clearly in the quote displayed at the start of this section [Klein, 1908, 77-85].

In it, Klein describes how in the history of mathematics, the interplay between these three Plans for mathematics moved back and forth, unifying, specializing, unifying again. What drove the impulses between them are both impulses within the individual mathematician and anthropological (social and culture) influences, mainly from the requirement to teach comprehensibly, which ties to a second important perspective.

For an equally fascinating didactic exploration of the nature of mathematics, one is recommended to read Renyi’s “Dialogues on Mathematics” [Renyi, 1967], in which a Socrates discusses the nature of mathematics, an Archimedes discusses the applications of mathematics, and Galileo discusses the ability of mathematics to assist in uncovering the workings of nature. (The first of these 3 dialogues is reproduced in [Hersh, 2006, Ch.1]).

2. The anthropology of mathematics takes the perspective that “the character of mathematical thinking and argument is strongly affected — indeed is almost essentially determined — by the dynamics of the specific social, mostly professional environments in which it is carried” [Hoyrup, 2017], i.e. mathematics is shaped by the interplay between the characteristics of a society and, in particular, the institutions for teaching which influence mathematical thought, research directions, and determine the limits of mathematical practice. The term “anthropology of mathematics”⁴ is due to Jens Hoyrup [Hoyrup, 1980, 1994, 2017, 2019] with the catalyzing idea coming from a paper by Judith Grabiner [Grabiner, 1974]. Grabiner showed how the rise of university teaching of mathematics in France and Germany in the 1800s created pressure to make the mathematics of Newton, Leibniz, Euler, and the Bernoullis more easily accessible to students, which in turn accelerated the re-emergence of deductive mathematics. Taking this as a launching point, Jens Hoyrup in ground-breaking researches from 1980 onward, meticulously investigated and detailed the same phenomenon in the major pre-modern mathematical centres (Babylonia, Greece, Islamic Spain to Afghanistan, and mercantile Italy). Hoyrup showed [Hoyrup, 1994] how the temple culture in the early city-states of Sumeria and the rise of scribal schools shaped mathematical development over the entire Sumero-Akkadian-Old Babylonian period (3200-1600 BCE). He shows how in ancient Greece the decentralization of teaching (sophists were the early itinerant teachers) and the philosophical pre-occupation of aristocratic Greek society, shaped the lens through which the Greeks approached the corpus of 2000 years of Babylonian and Egyptian mathematics and created a new, philosophical, deductive science. [Hoyrup, 2019] With the exception of the Greek experience, a sort of practitioner’s, or utilitarian mathematics (what Hoyrup calls subscientific mathematics).

Perhaps the strongest case for the anthropology of mathematics (culture as a carrier and the socially specific institutions of teaching) is the fact that the uniquely Greek creation of a deductive reasoning did not persist beyond Alexandria, neither in the era of mathematics in the Islamic period, nor in the pre-Renaissance Abacus period, both of which were dominated by the subscientific, practitioner approach to mathematics which also under-pinned both Babylonian and Egyptian mathematics. [Hoyrup, 1990, 1994, 2003].

Hoyrup’s researches since 1980 have shown that: “Old Babylonian ‘algebra’ and Euclidean ‘geometric algebra’ were connected. The geometric riddles of Arabic misaha treatises as well as al-Khwarizmi’s geometric proofs for the basic al-jabr (algebra) procedures belonged within the same network. The Old Babylonian ‘algebraic’ school discipline built upon original borrowings from the ‘neck riddles’ of a lay surveyor’s environment, and that this environment and its riddles, not the tradition of scholar-scribes, was responsible for the transmission of the inspiration to later times [both to the Greeks as well as to the Islamic scientists].” [Hoyrup, 2003, 9] He points out that there is a continuity of problem classes all the way through to the 1200s CE in Jacopo’s Algebra, and Fibonacci (Leonardo of Pisa)’s Liber Abaci. “These belong to a cluster of problems that are found in ancient and medieval sources from Ireland to India. This cluster of problems that usually go together was apparently carried by the community of merchants travelling along the Silk Road and adopted as ‘recreational problems’ by the literate in many places; it is thus a good example of a body of sub-scientific knowledge influencing school knowledge in many places and an illustration of the principle that it is impossible to trace the ‘source’ for a particular trick or problem in a situation where ‘the ground was wet everywhere’.

Hoyrup’s anthropology of mathematics perspective also addresses the WHEN aspect of mathematics:

[We may say that] transition[s] to [M]athematics occurred [in history] when pre-existent and previously independent mathematical practices and techniques were wielded by specialist practitioners who were organized professionally and linked in a network of communication. Such professional groups fall into two main types: in one type knowledge is transmitted within an apprenticeship-system of ‘learning by doing under supervision’ [sub-scientific mathematics]. The other type involves some kind of school [in which] teaching is separate from actual work. In the former type, those who transmit are actively involved in the practical activities of their trade; they will tend to train exactly what is needed, and the understanding they will try to communicate will be that of practical procedures. [In the latter type, the] school teaching of mathematical skill is bound to a writing system extensive enough to carry a literate culture. [While] teachers in the school type may well have as their aim to impart knowledge for practice, the mathematical understanding that they teach will concentrate on inner connections of the topic, i.e. on mathematical explanations.” – Jens Hoyrup, 2017, Perspectives on an Anthropology of Mathematics

Subscientific mathematical culture and pre-modern rhetorical mathematical practice does not mean that the mathematics was primitive, nor that there was no pure mathematics. “On the contrary, at various points in its history, in particular during the Old Babylonian period, mathematical activity turned toward the pure and systematic, pursued for supra-utilitarian reasons, what we might call scientific, or perhaps better systematic.” [Hoyrup, 1980]. In the past 40 years, examples have been discovered of creative and advanced approaches, including a pre-cursor of our Calculus already being used by the Neo-Babylonians of c.500 BCE to predict the position of Jupiter in the night-sky.

Creative progress in European mathematics before the Renaissance (pre-1300s) was constrained by Aristotelian philosophy on the one hand, Euclidean geometry on the other, and the monastic preservation of classical knowledge as a received but not an indiginous intellectual activity. Even when this influenced finally lifted with the rise of the Abacus schools in Italy (1300s-1400s), and when mathematical practice shifted away from rhetorical to symbolic around Francois Viete (1591) who began the extensive use of symbols in calculation, even then the form of mathematical creative culture was not deductive, even if the results were presented that way (see e.g. Newton’s Principia). Only from the mid-1800s did creative mathematical advancement occur again in the deductive style of the classical Greece, this time in the axiomatic works of Boole (logic), Cayley, Weber, Dedekind, Noether (abstract algebra), Cauchy, Bolzano, Weierstrass, Dedekind, Cantor (analysis), culminating in the the formalist program of Hilbert and the structuralist organization of all of mathematics from the 1950s.

We see the similar effects of culture on contemporary mathematics (New Math in the 1960s and its pushback and subsequent splintering of approaches in the 2000s).

But while the dialectic perspective and the anthropology of mathematics perspective do not tell us WHAT this great conversation is about, they do tell us a lot about HOW it happens. And that HOW provides the missing fifth element to complete our definition.

We now have what we need to answer directly:

4. Second Pass: A definition covering all mathematics, including contemporary mathematics

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Definition 2: Mathematics is a body of knowledge, built up over time in a dialetic process, on matters whose origin lies in the three natural phenomena: quantity (number, measurement, scale), space (shape, configuration, arrangement, symmetry, perspective), transformation (change, variation), shaped by the prevailing social culture, and in turn influencing it (often profoundly) through its diverse, and often ingenious, applications.

Deepening understanding of the three natural phenomena lead to the development of a chain of evolving conceptual abstractions, to greater generalization, and to correspondingly broader areas of investigation. Observation of relations (association, comparison, similarity, equivalence) between diverse mathematical phenomenona has created a fifth, humanistic area of mathematical activity: the rational structuring of its accumulated body knowledge through the development of axiomatic (deductively structured) mathematical systems that generalize and extend empirical concepts, introduce fundamentally new theoretical concepts, and examine the laws that govern their structure, properties, and the relationships between them.

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Let’s look in turn at the elements.

The first three are elaborated versions of what we had before:

1. Quantity covers matters of number, measurement, and scale. It is the subjects of arithmetic/combinatorics, geometry/analysis, complex variables, probability/statistics – indeed almost all of mathematics involves quantity or one of its conceptual abstractions (rings or fields).
2. Space covers matters to do with shape, configuration, arrangement, symmetry, perspective. It includes considerations about the continuum in one or more dimensions. It is the subjects of geometry/linear algebra, topology/manifolds, and continuous group theory, among others.
3. Change is about variation and transformation and the mathematical tools for representing and analyzing transformations. It is the subject of functions/mappings, analytic geometry, calculus/differential and integral equations, and physics, among others.
4. Relation is about comparison, similarity, equivalence. It is the subject of set theory, abstract algebra, logic, categories, among others.

   The fifth element brings it all together:

5. Rational structuring of its accumulated knowledge has to do with the validation, systematization, and efficient organization of what is already known, and the use of that structure to open up new knowledge. It includes several ideas that have had a profound influence on mathematics and the way it is organized and communicated: a) Deductive Reasoning, b) the Axiomatic System, c) Formalism (or the reduction of mathematics to logic and set theory), and d) Symbolical Reasoning with an associated Calculus or Grammar. At its core, rational structure is about the validity of what is known (certainty vs. doubt, truth vs. falsity), efficiency in demonstrating that validity and under what limitations, and ways to catalogue what is known. (Note efficiency in cataloguing is different from effectiveness in teaching in much the same way that a dictionary is an efficient catalogue of words, but not an effective way to learn or master a language.) It is the subject of mathematical logic, proof theory, set theory, abstract algebraic theories of lattices, groups, rings, fields, hypercomplex numbers, category theory, probability, statistics, and the abstract formulation of all mathematical areas.

Also worth a further comment is the remark on the influence of mathematics on society through applications. I rather hope that this statement is not controversial. The applications of mathematics are everywhere, amplified through science and engineering. To take but one example, the development of operations research to optimize military supply chains and bombing patterns against U-boats in WWII.

5. Testing the expanded definition against contemporary mathematical practice

We have seen how the first three elements were sufficient to cover mathematics up to the start of the 19th century.

The last two elements (relation and structure) are INTROSPECTIVE views. They are about mathematicians looking at the mathematics that they know and asking how/why they know it and how best to capture/catalogue that knowledge. With the addition of “structure”, we cover the axiomatic mathematics that returned again to the 19th century universities in France and Germany. We are able to cover the development of abstraction in the notions of algebra and number and an abstract algebraic mathematical logic in the early part of the 1800s, led by the English mathematicians George Peacock, George Boole, Augustus De Morgan, and then William Rowan Hamilton, Cayley, and then picked up in Germany by Weber, Noether, Klein, Lie, and others developing abstract algebra, and discovering its power to unlock additional questions.

We are also able to cover the transition from the intuitive mathematics of the 18th-century to the formalist mathematics of the 19th- and 20th-centuries, including the development of set theory as the foundation of the real numbers and analysis (Cauchy, Weierstrass, Dedekind), and the higher cardinalities (Cantor), the rise of point-set topology, functional analysis, linear algebra (geometry in n dimensions), measure theory and the axiomatic foundations of probability (Kolmogorov), and much more.

Is the Fifth Element really new?
What is fascinating in this story is that when one takes Hoyrup’s viewpoint of “anthropology of mathematics”, then this fifth element, introspection on its content, has been with mathematics throughout its known/recorded history, influencing how knowledge is organized, arranged, simplified, rigorized. Hoyrup’s research shows that it appears in every culture and context through the pre-modern mathematical period. It appears differently depending on the social and institutional context of the time (this is the anthropological perspective), but it appears nonetheless. So it is not the case that the current modern axiomatic style is somehow a natural endpoint in a linear mathematical development. On the contrary, there have been periods of more or less rigour, more or less scientific vs. subscientific (utilitarian) styles. Indeed the two high points of the rigorous axiomatic approach have been the Greek period 500 BCE-200 BCE and from the 19th-century onward. Both have seen the primacy of developing logically structured (axiomatic) mathematical systems that systematize, generalize and extend empirical concepts.

Why is this important? Because it is introspection, the fifth element, I claim, that gives mathematics its essential restlessness, its continual development, resifting, refinement of mathematical knowledge, and the exploration of other subjects using a mathematical lens. It has always been the source of advancement in every age of mathematics, and this continues in contemporary mathematics. Because it is the element that introduces new theoretical concepts, and simplifies and clarifies existing concepts by examining the laws that govern their structure, finding deep structural analogies between superficially dissimilar context.

The systematization of mathematics as a science has inevitably led to deeper understanding of the three natural phenomena. 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. Where conceptual abstractions have evolved, and effective generalizations have been found, these have in turn led to correspondingly broader areas of investigation. For example, modern mathematical physics (gauge theories, string theories, quantum field theories), has advanced materially through the contributions of abstract algebra, continuous group theory, and the theory of representations. So also have several other fields, for example computer science (computability, recursion, mathematical linguistics), biology (chaos theory, self-organizing systems), economics, finance, linguistics and fuzzy logic, and others.

6. A Deeper Look: Three Facets of Mathematics

To properly 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, we need to go beyond the definition. Here it is helpful to identify three co-existing facets of Mathematics:

1. Mathematics as an empirical science,
2. Mathematics as a modeling art, and
3. 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. Let’s look at each in turn.

6.1. 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.

Thus, the essence of a mathematical concept can always be related back to an original proto-concept that has its roots in empirical observations and the patterns arising out of these.

6.2. Mathematics as a Modeling Art  

Mathematics as a modeling art involves an effort to develop, maintain, and perfect models of perceived or envisioned reality. 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, 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 ever-improving 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.⁵

6.3. 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,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. To put it in context, the past 200 years of mathematics since 1800 CE is less than 5% of the documented history 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.

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7. Mathematics “from the inside”

I will close with a few observations of mathematics ‘from the inside’, i.e. mathematicians writing about mathematics:

“Mathematics is the backbone of modern science and a remarkably efficient source of new concepts and tools to understand the ‘reality’ in which we participate. The new concepts themselves are the result of a long process of ‘distillation’ in the alembic⁶ of human thought.” – Alain Connes, (mathematician, Fields Medalist 1982), from Advice to the Beginner, 2006

The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. Any one or another of these aspects of mathematics can be at the center of a given achievement. In a far reaching development all of them will be involved. Generally speaking, such a development will start from the “concrete” ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual “reality”. In brief, the flight into abstract generality must start from and return again to the concrete and specific. — Richard Courant (mathematician), from Mathematics in the Modern World, Scientific American Vol.211 No.3, pp.41-49, 1964

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them. — Heinrich Hertz (physicist), from Men of Mathematics, Vol 2, p.16, 1937

Very often in mathematics the crucial problem is to recognize and to discover what are the relevant concepts; once this is accomplished the job may be more than half done. — I.N. Herstein (mathematician), Topics in Algebra

It looked absolutely impossible. But it so happens that you go on worrying away at a problem in science and it seems to get tired, and lies down and lets you catch it. — William Lawrence Bragg \footnote{Bragg, at age 24, won the Nobel Prize for the invention of x-ray crystallography. He remains the youngest person ever to receive the Nobel Prize.}

(For more quotes on Mathematics and the process of doing mathematics, see here.)

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8. Continue Reading

• Characteristics of Pre-Modern Mathematics – an excellent introduction is the collection of essays in Jens Hoyrup’s book “In Measure, Number, Weight: Studies in Mathematics and Culture” [Hoy1994]
• Characteristics of Modern Mathematics

Appendix 1: Answers Deemed Unsatisfactory to the Criteria outlined above

Several authors attempt to answer the title question by providing a survey of mathematics but this in my view is as unhelpful as stating that ‘mathematics is the sum of its contents’. An author should not have to take an intelligent reader through 300+ pages of technical material before providing an clear answer. The caveat of course is that that one must have actively experienced enough mathematics (and for our purposes, enough exposure to its history) to find the answer satisfying.

A second frequently given, but also limiting answer, is that mathematics is the exploration through deductive reasoning of mathematical structures whose properties are abstracted into axioms from objects of practical experience. Simplified, this boils down to mathematics as the study of necessary deductions, which is problematic. It covers much of contemporary mathematics which is built around abstraction and deduction and organized into areas defined by the structures they study (groups, rings, fields, lattices, manifolds, functions, sets, logic structural presentation of contemporary mathematics). While it roots mathematics in practical experience, it denies that there can be mathematics that is not deductive, which is problematic given the long periods of history (before classical Greece and between classical Greece and the 1800s) when much mathematical knowledge was discovered without the use of a formal deductive style of arrangement of that knowledge. [See Hoyrup, 1980, 1994, 2019] To use a metaphor, if the deductive structure of mathematics is like varnish on wood, then we cannot accept that it is only the varnished variety of wood that is mathematics, but not the wood itself. The appeal of varnished wood is aesthetic, its longevity perhaps more assured, but the essential element is the wood, much of which is not easy to discover (as anyone who has tried will find). Hoyrup identifies the separate traditions as “scientific” and “sub-scientific”. Of the sub-scientific tradition, it is rooted in problem-solving and historically persisted even when the scientific tradition waned, as it did after Old Babylonian high point and after classical Greece. (Hoyrup, 1980, 1994, 2001), (Grabiner, 1976). In our view, both are entitled to be called mathematics.

9. References

Group 1: primary materials for engaging in the debate on What is Mathematics
1. [Hoyrup, 2017] – Jens Hoyrup, What is Mathematics? Perspectives Inspired by Anthropology, 2017, Online Article – PDF
2. [Renyi, 1967] – Alfred Renyi, Dialogues on Mathematics, 1967, Holden-Day
3. [Aleksandrov, 1956] – A General View of Mathematics, pp.1-64, Ch.1 in Mathematics: Its Content, Methods, and Meaning [Aleksandrov, Kolmogorov, Lavrentiv, 1963, 1969 (2nd ed)] – This essay provides a condensed glimpse into the dialectical evolution of mathematical ideas, their richness and creativity, drawing only upon knowledge of elementary mathematics for its examples.
4. [Lakatos, 1976] – Imre Lakatos. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, 1976. (Online1 Online2, Lakatos develops a particularly vivid presentation of dialectic in mathematics in his mathematical-literary play.
5. [Klein, 1908] – “Concerning the General Structure of Mathematics”, pp.77-85, in Elementary Mathematics from an Advanced Standpoint, Vol.1, Arithmetic, Algebra, Analysis, by Felix Klein, 1908, (Online) – This is where Klein discusses his 3 plan view of mathematics development: Plan A: self-contained,formal,logical; Plan B: holistic, inter-linked through big ideas; Plan C: algorithmic, calculational, laying out how the subject appears from the various perspectives.
6. [Hoyrup, 1990] – Jens Hoyrup, Sub-Scientific Mathematics: Observations of a Pre-Modern Phenomenon. History of Science, Vol. 28, pp.63-86
7. [Hoyrup, 1994] – Jens Hoyrup, In Measure, Number, Weight: Studies in Mathematics and Culture, 1994, SUNY Press, A collection of 8 essays on the anthropology of mathematics, published from 1985 to 1991
8. [Villani, 2015] – Cedric Villani, Birth of a Theorem: A Mathematical Adventure, 2015, (Online)
9. [Feferman, 1999] – Solomon Feferman. Does mathematics need new axioms? American Mathematical Monthly, Vol. 106, No. 2, pages 99-111, Feb 1999. [Download PDF] (Originating server is Author’s webpage) – This paper provides an expository look at the axiomatic foundations of mathematics.
10. [Hersh, 2006] – Reuben Hersh, 18 Unconventional Essays on Mathematics, 2006
11. [Davis, Hersh, 1981] – The Mathematical Experience, Philip J. Davis and Reuben Hersh, 1981, Birkhauser.
12. [CC09] – Mark Chu-Carroll. What is math? December 2009. Online Article
13. [Ebrahim, 2010d] – A Course in the Philosophy and Foundation of Mathematics, Assad Ebrahim, 2010. Online Article

    Group 2: Expository content that supports the views expressed in this article

14. [Ebrahim, 2019a] – Pre-Historic Origins of Mathematics, Assad Ebrahim, 2019. Online Article
15. [Ebrahim, 2019b] – The Mathematics of Uruk & Susa (3500-3000 BCE), Assad Ebrahim, 2019. Online Article
16. [Ebrahim, 2020a] – How Algebra Became Abstract: George Peacock & the Birth of Modern Algebra (England, c.1830), Assad Ebrahim, 2020. Online Article
17. [Ebrahim, 2020b] – The Rise of Mathematical Logic: From Laws of Thought to Foundations of Mathematics, Assad Ebrahim, 2020. Online Article
18. [Ebrahim, 2010a] – The Development of Mathematics, Assad Ebrahim, 2010. Online Article
19. [Ebrahim, 2010b] – Catalysts in the Development of Mathematics, Assad Ebrahim, 2010. Online Article
20. [Ebrahim, 2010c] – Characteristics of Modern Mathematics, Assad Ebrahim, 2010. Online Article
21. [Grabiner, 1974] – “Is Mathematical Truth Time-Dependent?” (PDF), Judith V. Grabiner, 1974, The American Mathematical Monthly Vol. 81, No. 4 (April), pp.354-365.
22. Denise Schmandt-Besserat, 1977, An Archaic Recording System and the Origin of Writing,”; Syro-Mesopotamian Studies I., 1977, pp.31-70; [Besserat/1977]
23. [Hoyrup, 1980] – Jens Hoyrup, Influences of institutionalized mathematics teaching on the development and organization of mathematical thought in the pre-modern period; Investigations into an aspect of the anthropology of mathematics, 1980, Online Article (PDF)
24. [Hoyrup, 2002] – Jens Hoyrup, Lengths, Widths, and Surfaces: A Portrait of Old Babylonian Algebra and Its Kin, 2002, Springer
25. [Hoyrup, 2019] – Jens Hoyrup, Selected Essays on Pre- and Early Modern Mathematical Practice, 2019
26. [Hoyrup, 2003] – Jens Hoyrup, Practitioners, School Teachers, Mathematicians: the division of pre-Modern mathematics and its actors. 2003
27. [Davis, 1995] – The Rise, Fall, and Possible Transfiguration of Triangle Geometry: a Mini-History, Philip J. Davis, 1995, American Mathematical Monthly, Vol. 102, (Online HTML), (Online PDF)
28. [Yaglom, 1981] – Elementary Geometry, Then and Now, I.M. Yaglom, pp.253-269 in The Geometric Vein: The Coxeter Festschrift, Chandler Davis, Branko Grünbaum and F.A. Sherk (editors), 1981, Springer (Online, pp.253-260)
29. [Pan] – Pannenoek. History of Astronomy. Dover
30. [Wil82] – Herbert Wilf. What is an Answer? 1982.
31. [Ferreiros, 2015] – Mathematical Knowledge and the Interplay of Practices, by Jose Ferreiros, 2015, Princeton University Press (Online)
32. [FeDo07] – J. Ferreiros and J.F. Dominguez. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Springer, second edition, 2007.
33. [Kle86] – Israel Kleiner. The evolution of group theory: A brief survey. Mathematics Magazine; Vol.59, No.4, pages 194-215, October 1986.
34. [Bou] – Nicholas Bourbaki. Elements of the History of Mathematics: Set Theory.

    Group 3: Surveys of Mathematics

35. [Aleksandrov, Kolmogorov, Lavrentiv, 1963] – Mathematics: Its Content, Methods and Meaning, MIT Press, 1st edition 1963, 2nd edition 1969; Dover edition 1999 (three volumes bound as one).
36. [Courant, Robbins, 1941] – What is Mathematics? An Elementary Approach to Ideas and Methods, Richard Courant and Herbert Robbins, 1941, Oxford.
37. [Mac86] – Saunders MacLane. Form and Function. 1986.
38. [Gowers, 2008] – Princeton Companion to Mathematics, Timothy Gowers (ed.), 2008, Princeton University Press
39. [Higham, 2015] – Princeton Companion to Applied Mathematics Mathematics, Nicholas Higham (ed.), 2015, Princeton University Press.
    

    Group 4: Additional Recommended Reading

40. [Couturat/18xx] – transl. Rutherford/2012 – The Logic of Leibniz (PDFs), English Translation of Louis Couturat’s Logique de Leibniz.
41. [Dij89a] – E.W. Dijkstra. Mathematical methodology – preface. EWD-1059; Available from the Internet for download, 1989. Djikstra inquires into the place and method of proof in mathematics.
42. [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.
43. [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.
44. [Fef92a] – Solomon Feferman. What rests on what? the proof-theoretic analysis of mathematics. Available from the Internet at the author’s website., 1992.
45. [Fef92b] – Solomon Feferman. Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics. PSA 1992, vol. 2 (1993), pages 442-455 (with corrections), 1992.
46. [Fef98] – Solomon Feferman. Mathematical intuition vs. mathematical monsters. Available from the Internet at the author’s website., 1998.
47. [Rot97] – Gian-Carlo Rota. Indiscrete Thoughts. Birkhauser, 1997.
48. [Wal06] – M.A. Walicki. The history of logic. from Introduction to Logic, pages 1-27, 2006.
49. [Strang, 1976] – Linear Algebra and Its Applications, Gilbert Strang, 1976, 1st ed, 1980 2e, 1988 3e, 2006 4e (Online)
50. [Petrilli, Smolensky, 2017] – A Brief History of Determinants (Online Article)
51. [Bur] – David Burton. The History of Mathematics: An Introduction

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Footnotes