What are the characteristics of mathematics, especially contemporary mathematics?
I’ll consider five groups of characteristics:
- Applicability and Effectiveness,
- Abstraction and Generality,
- Logical Derivation, Axiomatic Arrangement,
- Precision, Correctness, Evolution through Dialectic…
Characteristics of Modern Mathematics
In the article What is Mathematics?, I have posited that Mathematics arises from Man’s attempt to summarize the variety of empirical phenomena that he experiences, and that Mathematics advances through the expansion and generalization of these concepts, and the improvement of these models.
But what are the characteristics of mathematics, especially contemporary mathematics?
I’ll consider five groups of characteristics:
- Applicability and Effectiveness,
- Abstraction and Generality,
- Logical Derivation, Axiomatic Arrangement,
- Precision, Correctness, Evolution through Dialectic.
Though each of these characteristics presents unique pedagogical challenges and opportunities, here I’ll focus on the characterisics themselves and leave the pedagogical discussion to (Ebr05d). (Pedagogical matters are discussed in the article Teaching Mathematics “in Tunic”.)
General applicability is a recurring characteristic of mathematics: mathematical truth turns out to be applicable in very distinct areas of application in phenomena from across the universe to across the street. Why is this? What is it about mathematics and the concepts that it captures that causes this?
Mathematics is widely useful because the five phenomena that it studies are ubiquitous in nature and in the natural instincts of man to seek explanation, to generalize, and to attempt to improve the organization of his knowledge. As Mathematics has progressively advanced and abstracted its natural concepts, it has increased the host of subjects to which these concepts can be fruitfully applied.
Abstraction is the generalization of myriad particularities. It is the identification of the essence of the subject, together with a systematic organization around this essence. By appropriate generalizations, the many and varied details are organized into a more manageable framework. Work within particular areas of detail then becomes the area of specialists.
Put another way, the drive to abstraction is the desire to unify diverse instances under a single conceptual framework. Beginning with the abstraction of the number concept from the specific things being counted, mathematical advancement has repeatedly been achieved through insightful abstraction. These abstractions have simplified its topics, made the otherwise often overwhelming number of details more easily accessible, established foundations for orderly organization, allowed easier penetration of the subject and the development of more powerful methods.
For the outsider looking in, it is hard to believe that simplicity is a characteristic of mathematics. Yet, for the practitioner of mathematics, simplicity is a strong part of the culture. Simplicity in what respect? The mathematician desires the simplest possible single exposition. Through greater abstraction, a single exposition is possible at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the higher level.
This is significant: although the mathematician may indeed have found his desired single exposition (for which reason he claims also that simplicity has been achieved), the reader often bears the burden of correctly and conscientiously exploring the quite significant terrain that lies beneath the abstract language of the higher-level exposition.
Thus, I believe it is the mathematician’s desire for a single exposition that leads to the attendant complexity of mathematics, especially in contemporary mathematics.
The modern characteristics of logical derivability and axiomatic arrangement are inherited from the ancient Greek tradition of Thales and Pythagoras and are epitomized in the presentation of Geometry by Euclid (The Elements).
It has not always been this way. The earliest mathematics was firmly empirical, rooted in man’s perception of number (quantity), space (configuration), time, and change (transformation). But by a gradual process of experience, abstraction, and generalization, concepts developed that finally separated mathematics from an empirical science to an abstract science, culminating in the axiomatic science that it is today.
It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics.
This does not mean that there is no connection with empirical reality. Quite the contrary. But it does mean that mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. These abstractions mean that mathematical fact is now established without reference to empirical reality. It may certainly be influenced by this reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics.
Why the contemporary bias for axiomatic Mathematics?
Why is axiomatic mathematics so heavily favored by modern mathematics? For the same reason it was favored in the time of Euclid: in the presence of empirical difficulties, linguistic paradox, or conceptual subtlety, it is an anchor that clarifies more precisely the foundations and the manner of reasoning that underlie a mathematical subject area. Once the difficulties of establishing an axiomatic framework have been met, such a framework is favored because it helps ease the burden of many, complicated, inter-related results, justified in various ways, and inter-mixed with paradoxes, pitfalls, and impossible problems. It is favored when new results cannot be relied upon without complicated inquiries into the chains of reasoning that justify each one.
The value of axiomatic mathematics
What the axiomatic approach offers is a way to bring order to a subject area, but one which requires deciding what is fundamental and what is not, what will be set up higher as a “first principle” and what will be derived from it. When it is done, however, it sets a body of knowledge into a form that can readily be presented and expanded. Appealing and effective axiom systems are then developed and refined. Their existence is a mark of the maturity of a mathematical subfield. Proof within the axiomatic framework becomes the hygiene that the community of working mathematicians adopts in order to make it easier to jointly share in the work of advancing the field.1
Axiomatic Mathematics as Boundaries in the Wilderness
In all cases of real mathematical significance, the selection of axioms is a culminating result of intensive investigations into an entire mathematical area teeming with phenomena, and the gaining of a deep understanding that results finally in identifying a good way to separate the various phenomena that have been discovered. So, though the axioms may sound trivial, in reality, the key axioms delineate substantially different structures. In this sense, axioms are boundaries that separate structurally distinct areas from each other, and, together, from the rest of “wild” mathematics.
For example: the triangle inequality is a theorem of Euclidean geometry. But it is taken as an axiom for the study of metric spaces. By doing so, this one axiom forces much of the Euclidean isometric structure. As such, it becomes a code or litmus test for the “Euclidean-ness” of a space.
Thus, from this point of view, non-axiomatic mathematics is the mathematics of discovery. Axiomatic mathematics has been tamed and made easy to learn, present, and work within. One might regard it is a fenced off area within the otherwise unmarked wilderness of other mathematical and non-mathematical phenomena. One might think of the progress of axiomatic mathematics as paralleling the way in which mankind slowly but inexorably tamed the wilderness, chopping down the trees and pushing the truly wild animals further away, while domesticating and harnessing the desirable easier ones, and setting up buildings, and walkways, farmlands, granaries, and a functioning and productive economy.
The same holds for mathematical definitions: they are attempts to tame certain phenomena and identify them as the subjects for further domestication and as able to be safely put to use, shutting out the untamed disorder of the rest of phenomena, mathematical and non-mathematical.
The down-side of axiomatics
One down-side of the axiomatic presentation of mathematics is that although deep understanding is typically hidden within the axioms, the definitions of the mathematical systems have been designed precisely to make the axioms seem trivial. Which means that it is all too easy to simply state them and move on to the “meat” of the matter.
But this would be a mistake. Time spent understanding why the axioms are there, seeing them as theorems in historically prior investigations, and understanding in what phenomena they arise and where they don’t – time spent this way leads to a much deeper understanding of the significance of taking the axioms on in the first place and understanding the boundaries of the subject that the axioms establish.
Axiomatic mathematics and density of presentation
For those who are interested in learning mathematics efficiently, the axiomatic presentation is most definitely the most efficient both in presentation and in “coverage density” (you get the most amount of reach and the greatest applicability in the fewest steps).
But along with “coverage density” comes conceptual density. The abstract language of axiomatic mathematics can subsume vastly different specific examples within a single abstract statement, examples which may spread across a host of historic sub-disciplines and mathematical objects of interest. You may follow the proof, and be able to turn out your own (within the framework of the axiomatics), but do you really understand the results deeply? Have you actually rubbed shoulders with the individual mathematical animals? Would you be able to recognize the right way to subdue a specific animal if it came across your path in unfamiliar circumstances (i.e. not presented in the efficient abstract language)?
The Language of Mathematics.
Over the course of the past three thousand years, mankind has developed sophisticated spoken and written natural languages that are highly effective for expressing a variety of moods, motives, and meanings. The language in which Mathematics is done has developed no less, and, when mastered, provides a highly efficient and powerful tool for mathematical expression, exploration, reconstruction after exploration, and communication. Its power (when used well) comes from simultaneously being precise (unambiguous) and yet concise (no superfluities, nothing unnecessary). But the language of mathematics is no exception to being used poorly. Just as any language, it can be used well or poorly.
Once correctness in mathematics is separated from empirical evidence and moved into a model-based or axiomatic framework, the touchstone for correctness becomes other, carefully selected, statements that capture the essential elements of the underlying reality: definitions, axioms, previously established theorems. The language of mathematics, and logical reasoning using that language, form the everyday working experience of mathematics.
In earlier times, mathematics was in fact, fully verbal. Now, after the dramatic advances in symbolism that occurred in the Mercantile period (1500s), mathematics can be practised in an apparent symbolic shorthand, without really the need for very many words. This, however, is only a shorthand. The symbols themselves require very careful and precise definition and characterization in order for them to be used, computed with, and allow the results to be correct.
The modern language of working mathematics, as opposed to expository or pedagogical mathematics, is symbolic, and is built squarely upon the propositional logic, the first order predicate logic, and the language of sets and functions.2 The symbolical mode is one which should be learned by the student and used by the practitioner of mathematics. It is the clearest, most unambiguous, and so most precise and therefore demanding language. But, one might say it is a “write only” language: you don’t want to read it. So, once one has written out ones ideas carefully this way, then one typically switches to one of the other two styles: direct or expository, these being the usual methods of communicating with others.3
Evolution Through Dialectic
Mathematical definitions, mathematical notions of correctness, the search for First Principles (Foundations) in Mathematics and the elaboration of areas within Mathematics have all proceeded in a dialectic fashion, alternating between periods of philosophical/foundational contentment coupled with active productive work on the one hand, and the discovery of paradoxes coupled with periods of critical review, reform, and revision on the other. This dialectical process through its history has progressively raised the level of rigor of the Mathematics of each era.4
The level of precision in mathematics increased dramatically during the time of Cauchy, as those demanding rigor dominated mathematics. There were simply too many monsters, too many pitfalls and paradoxes from the monsters of functions in the function theory to the paradoxes and strangenesses in the Fourier analysis and infinite series, to the paradoxes of set theory and modern logic. The way out was through subtle concepts, subtle distinctions, requiring careful delineations, all of which required precision.
A Culture of Precision
Mathematical culture is that what you say should be correct. What you say should have a definition. You should know the definition and limits of what you are saying, stating, or claiming. The distinction is between mathematics being developed informally and mathematics being done more formally, with necessary and sufficient conditions stated up front and restricting the discussion to a particular class of objects.
Thus, I would argue that the modern mathematical culture of precision arises because:
- mathematics has developed a precise, highly symbolic language,
- mathematical concepts have developed in a dialectic manner that allows for the adaptation, adjustment and cumulative refinement of concepts based on experiences, and
- mathematical reasoning is expected to be correct.
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 [Ebr06] and [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 mathematical-literary play Proofs and Refutations, [Lak76].
- Lamport and Djikstra inquire into the place and method of proof in mathematics in [Lam95] and [Dij89b].
Other recommended references are:
The complete article, What is Mathematics?, with its extensive Further Reading section, is available as a PDF here.
A general view of mathematics.
pages 1-64, (Chapter 1), 1956.
Elements of Mathematics: Set Theory.
Mathematics, physics, and information.
Information and Control; Vol. 1; No. 1;, pages 1-5, 1957.
Inevitable experimental errors, determinism, and information theory.
Information and Control; Vol. 2; No. 1;, pages 45-63, 1959.
Poincare’s theorem and uncertainty in classical mechanics.
Literacy in the language of mathematics.
American Mathematical Monthly, pages 735-743, October 1994.
The History of Mathematics: An Introduction.
R. Courant and H. Robbins.
What is Mathematics? An Elementary Approach to Ideas and
Philip J. Davis and Reuben Hersh.
The Mathematical Experience.
On the quality criteria for mathematical writing.
Mathematical methodology – preface.
On hygiene, intellectual and otherwise.
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
R.C. (ed.) Dorf.
The Electrical Engineering Handbook.
CRC Press and IEEE.
A course in the philosophy and foundations of mathematics, and the
search for mathematical method and meaning.
What is mathematics.
Thinking about the teaching of mathematics.
The story of number.
Logic: The study of reasoning.
J. Ferreiros and J.F. Dominguez.
Labyrinth of Thought: A History of Set Theory and Its Role in
Springer, second edition, 2007.
The development of programs for the foundations of mathematics in the
first third of the 20th century.
What rests on what? the proof-theoretic analysis of mathematics.
Why a little bit goes a long way: Logical foundations of
scientifically applicable mathematics.
PSA 1992, vol. 2 (1993), pages 442-455 (with corrections),
Mathematical intuition vs. mathematical monsters.
Does mathematics need new axioms?
American Mathematical Monthly, Vol. 106, No. 2, pages 99-111,
Contemporary Abstract Algebra.
D.C. Heath & Company, third edition edition, 1994.
W. Gellert, H. Kustner, M. Hellwich, and H.; (eds.) Kaestner.
The VNR Concise Encyclopedia of Mathematics.
Van Nostrand Reinhold Company, 1975.
Ron Graham, Donald Knuth, and Oren Patashnik.
Concrete Mathematics: A Foundation for Computer Science.
Dover, 1963; 1977 (corrected).
Mathematics from the Birth of Number.
Finite Dimensional Vector Spaces.
Springer-Verlag, undergraduate texts in mathematics edition, 1987.
The evolution of group theory: A brief survey.
Mathematics Magazine; Vol.59, No.4, pages 194-215, October
Mathematical Thought from Ancient to Modern Times.
Oxford University Press.
Proofs and Refutations: The Logic of Mathematical Discovery.
Cambridge University Press, 1976.
How to write a proof.
The American Mathematical Monthly; Vol.102, No.7, pages
History of Astronomy.
Springer-Verlag, graduate texts in mathematics no. 118 edition, 1989.
Principles of Mathematics.
A.B. (ed.) Tucker.
Computer Science Handbook.
Chapman & Hall/CRC, in cooperation with ACM, the Association for
Computing Machinery, 2004.
The history of logic.
from Introduction to Logic, pages 1-27, 2006.
What is an Answer?
- What is axiomatized now? Logic (Standard logics – propositional, predicate; non-standard logics – modal, etc.). Set theory (Standard – Zermelo-Fraenkel; enhanced – Zermelo-Fraenkel-Axiom of Choice; non-standard). Algebraic systems (Groups, rings, fields, and all manner of systems between these, including arithmetic, and the integers). Analysis (Dedekind cuts of the rational or equivalent Cauchy sequences of rationals). Topology (all manner of spaces). Modern Geometry is now an application of all of these axiomatized mathematical fields to the studies of space. ↩
- I discuss the details of mathematical language, set theory, and logic, in a separate article. ↩
- See (Lam95), and (Dij89b) for discussions by mathematicians on the symbolic mode and its advantages. ↩
- See (Lak76) ↩