That kind of title generally sort of annoys
me. It suggests the sort of ersatz profundity you get on
public television, or commercial television when it is (owing to a consent
decree with the FCC or for any other reason) attempting to sound responsible.

Note further that, although the subject-line of this
post is quite widely to be met
with, it is not the norm in scientific subjects. No-one writes a think-piece called “What is physics?” or
“What is chemistry?” or “What is botany?” or “What is meteorology?” You might find “Advice to a young
physicist” (aimed at those who are not in fact physicists, but are contemplating
entering the field) or “So! You
want to be a meteorologist”, addressed to nine-year-olds; but there is really little mystery
about

*what*those fields are; though, once you are in them, there are subtleties, to be sure.
[Note that I am pretty much pulling all this out of my
&ss. I attempted to test these
assertions by searching the title field on Amazon.com, <”what is” *>, but it does not
allow such a search -- although, really, once the user has gone that far, you have
truly spelled it out for them; and
the results you do get are indefinitely depressing.]

O.t.o.h., it is quite common to find that question posed
anent philosophy, or literature (

__Qu’est-ce que la littérature__? -- Sartre) or even “thinking” (__Was heisst Denken__? -- Heiderschnitzel) -- Most such titles, though, relate to abstruse-but-nontechnical subjects: “What is theosophy?” “What is__oahspe__?” "What is the Bill of Rights?"
Anyhow, my purpose here is not to explain, finally, for the
tired business-man, what mathematics is all about. My purpose is essentially lexicographic (here I speak as a
veteran of

[More on the philosophical status of Definition in the natural sciences here.]

__the little red schoolhouse on Federal Street__): If your task is to define**mathematics**in a few words, what do you say? Points are awarded for clarity and concision.[More on the philosophical status of Definition in the natural sciences here.]

~

From works for an educated general audience:

Next, attempts to extract the essence (at a high level of generality and abstraction):

First, a sociological/tautological non-definition:

What is mathematics? One proposal, made in desperation, is ‘what
mathematicians do’.

-- -- Ian Stewart,

__How to Cut a Cake__(2006), p. 27
[That stab in fact fails to offer even the virtue of a
tautology, since it isn’t even

*true*, without the further qualification that it is what mathematicians do…*when*they are doing math. -- If you’d tried similarly, without qualification, to define**linguistics**based simply on the activities of linguists (ex officio: faculty and students in the Linguistics Department) at Berkeley during the years I was there, you would conclude that the field consisted of: fixing your Volkswagens; eating Chinese food; and dabbling in neighboring fields like psychology and philosophy (later all these fields hopped into the hot-tub together and were newly baptised as Cognitive Science) -- all this while studiously ignoring most of the work done in the previous centuries of philology and language-sciences.]
A physicist’s
unruffled take:

Mathematics is just organized
reasoning.

-- Richard Feynman,

__The Character of Physical Law__(1965)
And, in a later lecture to elementary-school science-teachers:

Mathematics is looking for
patterns. … Mathematics is only patterns.

Next, attempts to extract the essence (at a high level of generality and abstraction):

Mathematics is the science of

**quantity and space**.
-- Philip Davis & Reuben Hersh,

__The Mathematical Experience__(1981), p. 6
Within the limits of the word-count, this is a very good
definition; the shade of Noah
Webster nods and smiles.

Thus their essay
at the outset of their excellent book. By the end, they have grown more abstract, more … ineffable:

*The study of mental objects with reproducible properties is called mathematics*.

-- Philip Davis & Reuben Hersh,

__The Mathematical Experience__(1981), p. 399; breathless italics in original.
(Old Noah frowns and cocks an eyebrow for assistance.)

Hao Wang (reprinted in Tymoczko 1998), addressing the
question, lists some “one-sided views” of what math is:

* Mathematics coincides with all
that is the exact in science.

* Mathematics is axiomatic set
theory.

* Mathematics is the study of
abstract structures.

Thus the proverbial blind-men, fondling the ineffable elephant.

The first of that triad is similar in spirit to the
following, which however is offered
less as a definition than
as an epigram:

Mathematics is the part of physics
where experiments are cheap.

-- V.I. Arnold, “On Teaching Mathematics” (lecture, 1997)

-- V.I. Arnold, “On Teaching Mathematics” (lecture, 1997)

And -- off the beaten path, but not awry for all that:

Mathematics is the science by which
a finite intelligence purports to plumb the infinite.

-- Charles Gillispie,

__The Edge of Objectivity__(1960), p. 188
(That sounds overly general, but it’s hard to imagine what
other activity that definition applies to, apart perhaps from theology, though
there the “science” part would meet dispute.)

More like a witticism, from a semi-popular volume by a giant of mathematics:

More like a witticism, from a semi-popular volume by a giant of mathematics:

Mathematics
is the art of giving the same name to different things.

-- H. Poincaré

-- H. Poincaré

Discussion of that sort of thing belongs not in this essay (which focuses on substance), but this one (which deconstructs rhetoric).

~

From works for professionals:

Modern mathematics might be
described as

**the science of abstract objects**, be they real numbers, functions, surfaces, algebraic structures or whatever. Mathematical**logic**adds a new dimension to this science by paying attention to the**language**used in mathematics.
-- Jon Barwise, “The Realm of
First-Order Logic”, in: Jon Barwise, ed.

__Handbook of Mathematical Logic__(1977), p. 6
Mathematics is, as it has always
been, largely

**the science of measurement**. But “measurement” must here be understood as referring to more than the meter stick. The genus of a topological figures measures one of its aspects; objects of genus zero are in a sense simpler than those of higher genus.
There are many dimensions of
measurement ….: characteristic,
transcendence degree, cardinality, fundamental group … Occasionally we are so
successful in the science of measurement
that we can completely characterize an object … by giving, as it were,
its latitude and longitude: its
measurements in the relevant dimensions.

-- Herbert Enderton, “Elements of
Recursion Theory”, in: Jon
Barwise, ed.

__Handbook of Mathematical Logic__(1977), p. 554
How different these are, in their top-level
description! The first makes sheer
abstraction part of the very definition
-- and indeed, abstraction at the level of the very object. (But triangles and circles are not so
abstract as all that...) The second
plunks for measurement -- in prototype, much more concrete, like a housewife
weighing a potato.

Our own stab at a definition, suitable for a children’s
dictionary, is “the scientific study of

**patterns**”. -- Actually, while Editor-in-Chief at Franklin Electronic Publishers, I did write a children’s dictionary, marketed as the “Homework Wiz”. Here is what I came up with back then: “the study of measurements and numbers”. This is obviously a bit of a kludge, but defensible, I think. For a child, “numbers”*must*be mentioned, since initially mathematics is just arithmetic, first of the integers and later of fractions. Slipping “measurements” in there was an anticipation of the calculus. But privileging*just*measurement, as Enderton does, seems dicey. Number theory (e.g. Fermat’s Last Theorem, the Goldbach Conjecture) studies the*patterns*of the integers; you’re not really*measuring*anything.
-- Okay now I’m curious. How does the flagship of the company I used to work for define the thing? Answer:

**mathematics**: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

--

__Merriam-Webster’s Collegiate Dictionary__, eleventh edition (2003)
One senses a sort of “Wait wait I’m not finished!” note in
this: one which is foredestined to
defeat. Clearly the definer
had in mind: arithmetic (and its
generalization into number theory and algebra), and geometry. In fact, supposing the lexicographer
is mulcted for every definition longer than three words, we might define math
as: “arithmetic & geometry”.
These do not, of course, exhaust the subject, but they certainly exhaust
what most students meet through high school. Neither that, nor the wordier Collegiate attempt,
really encompasses function theory, set theory, category theory …

Note: Steven
Wolfram, in a video “Is Mathematics Invented or Discovered?” (https://www.youtube.com/watch?v=RlMMeqO7wOI)
denies the coherence of the definiendum.
He coins the rebarbative relativist plural

**mathematicses**, and even manages to pronounce it online.
~

Let us leave the last word to G.H. Hardy:

A mathematician, like a painter or
a poet, is a maker of

**patterns**. If his patterns are more permanent than theirs, it is because they are made with*.*__ideas__
--

__A Mathematician’s Apology__(1940)
* * *

~ Commercial break ~

We now return you to
your regularly scheduled essay.

* * *

[All right, so sue me;
we have more to say. But
that was indeed the last word on “What is mathematics” proper.]

We are by now familiar with the idea that some notions
cannot be limited by a definition;
Wittgenstein classically makes this point in his discussion of

*games*. Mathematics is a sprawling field of activity and lapidary characterizations are necessarily impressionistic. Our chances are rather better when it comes to*subfields*or*branches*of mathematics.
Some of these “definitions” are really more by way of
elucidating epigrams, and come with appropriate caveats. Thus:

**Roughly speaking**, the

*n*th homology group [on a topological space X] tells you how many interestingly different continuous maps there are from closed n-dimensional manifolds to X.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 185
Distributions

**can be thought of**as asymptotic extremes of behavior of smoother functions, just as real numbers can be thought of as limits of rational numbers.
-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 187
Taken a step further, we arrive at a pure epigram or
witticism -- which, however, still contains a kernel of mathematical truth:

The beginner in differential
geometry will find that the matter of notations is the most annoying obstacle
to grasping the fundamental ideas.
In fact, there is an amusing

**definition of modern differential geometry**as “the study of invariance under change of notation”.
-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. vi
(The allusion is to such matters as tensor formalism vs.
that of differential forms -- no mere ‘notational variant’ in the slighting
sense of the Chomskyans.)

One step further, and we arrive at the classic ludic
definition “

**Time**is Nature’s way of keeping everything from happening all at once” -- which thus also allows certain pageants to play out in**Space**, which we may define (epigram ©2014 WDJ International Enterprises, All Rights Reserved*y compris en URSS*) as “Nature’s way of giving objects some elbow-room.”
~

We proceed, then, to a collection of insightful or intuitive
or epigrammatic characterizations of various subfields of mathematics.

Our purpose is twofold; indeed, the twin goals “can be thought of” as

That
is, as an old Websterian, I am concerned with how to go about giving, not
something that a **dual**to each other. The ostensible aim is (lightly) mathematical: to provide pithy thumbnail sketches of complex fields of research. The more substantial project takes place rather in the lexicographic ‘**conjugate space’**which maps the items so defined into intuitive English.*computer*could understand, but something a

*person*could understand; virtually the reverse of the sort of formal, exhaustive, seemingly unmotivated,

*stipulative*definitions you meet up with (much the way a bicyclist meets up with a stone wall -- I still have scars) at the outset of works by such pitiless authors as Eilenberg & Steenrod, Spanier, Lang, or Bourbaki.

The more general question is, how to

*get at things*with words [compare philosopher John Austin’s celebrated, deceptively-simple title,__How to Do Things with Words__] -- things which, not being verbal confections themselves, would not seem, by their nature, to offer necessarily any purchase whatsoever to our lexical grapping-hooks. Truly describing or defining*anything*is really quite difficult, if the words themselves must do all the work. Try to “define” a carrick bend or a surgeon’s knot in a way that picks them out and differentiates among them. “Definitions” of color (apart from the physicists’ descriptions in terms of wavelength, which is decidedly post-hoc) are really not definitions at all, but ways of*reminding*you of what you already somehow knew by other means, evoking things like apples and fire engines. If you’re a Daltonist, you’re out of luck: no amount of verbiage will sort out*red*and*green*for you, though with practice you can learn to manipulate such terms plausibly in prose, much the way an autist, by means of diligent study, can discourse of things like “empathy” and “embarrassment”.
Here are some examples.

**affine connection**

**Intuitively**, an affine connection is a law of “covariant differentiation”.

-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 261**algebra**

Algebra is the mathematics that
places more emphasis on abstract structure than on intrinsic meaning.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 539**algebraic geometry**

In the very first sentence of the chapter so titled, in
Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 363, János Kollár gets right down to business:
Succinctly put, algebraic geometry
is the study of geometry using polynomials and the investigation of polynomials using geometry.

This no-nonsense formulation, which is not really very
revealing, might have come from Mary Poppins. Yet in the final paragraph, the author goes all gooey:

To me, algebraic geometry is a
belief in the unity of geometry and algebra.

Compare, indeed, the no-fuss/no-mess definition of
math-in-general (quoted above) with which Davis & Hersch began their book,
and the space-launch into the noösphere (likewise quoted) with which they
conclude it.

**analysis**

analysis -- calculus and its more
esoteric descendants

-- Ian Stewart,

__How to Cut a Cake__(2006), p. 132
Stewart is a gifted writer for the general public; and for that purpose, his definition is
excellent.

**Calabi-Yau**

A Calabi-Yau manifold

**can be thought of informally**as a complex manifold with complex orientation.
-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 163
This is no doubt intended to be one of those epigrammatic
gems which one savors over
brandy; but personally I
find it opaque. In particular,
what does that have to do with compactified dimensions, which is the realm in which Calabi-Yau
got launched on its superstar career?

**calculus of variations**

The calculus of variations should be regarded as the “theory” of a
real-valued function on an infinite-dimensional space: namely, the space of curves on the
underlying configuration space.

-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 261
(Not sure why he places the word “theory” in scare-quotes here,
particularly in a work which, according to its preface, is addressed to
engineers and not to philosophers.)

**chaos theory**

Chaos is extreme sensitivity to
ignorance.

-- John Barrow,

__One Hundred Essential Things You Didn’t Know You Didn’t Know__(2008), p. 270**combinatorial group theory**

Combinatorial group theory is the
study of groups defined in terms of

*presentations*: that is, by means of generators and relations.
-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 43**De Rham cohomology**

*De Rham cohomology*,

**roughly speaking**, measures the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 175(Definition in terms of failure! Hm! Cf. that of "ideal class group" below.)

**differential topology**

curved spacetime without metric or geodesics or parallel
transport

-- Ch. Misner, K. Thorne, & J. Wheeler,

__Gravitation__(1973), p. 225**elliptic functions**

Unaccountably, the theory of

**elliptic functions**has virtually disappeared from recent mathematics or physics literature, despite the fact that it is amazingly rich in structure, theorems, and mathematical or physical intuition. … We shall limit ourselves to some properties that follow from the fact that they**can be**.*defined*as the functions describing rigid-body motion
-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 232**functionals**

Functionals can be regarded as
‘functions of infinitely many variables’ [i.e., the values of the function y(x)
[to which it is applied] at separate points], and the calculus of variations
can be regarded as the corresponding analog of differential calculus.

-- I. M. Gelfand & S. V. Fomin,

__Calculus of Variations__(rev. Engl. tr. 1963), p. 4**general relativity**

This one’s a surprise, since most folks think of general
relativity as a branch of physics, not math, but here is a complementary view:

General relativity … can be thought
of as the study of Lorentzian manifolds.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 431**geodesic**

For the most part, this section of the essay is focusing on
intuitive thumbnail characterizations of abstruse ideas. In the case of “geodesic”
(familiar to a broad educated public owing to the popularization of Einsteinian
physics), such a thumbnail is well-known:
“the shortest distance between two points” (with due allowance made for
locality vs. globality). Here we
turn the turtle upon his back, and cite a formal re-visiting of the intuitive
notion (cf.

**point**, below):
A curve is a geodesic iff its tangent field is a parallel field along the curve.

-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 57
(Cf. further

**Riemannian geometry**, below.)**geometry**

Riemann proposed that geometry was
[to be] the study of what he called

-- Timothy Gowers, ed.,

*manifolds*-- “spaces” of points, together with a notion of distance that looked like Euclidean distance on small scales but which could be quite different at larger scales.-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 91**Hamilton-Jacobi theory**

Classically, Hamilton-Jacobi
theory is the study of the formal
properties of the solutions of ordinary differential equations of the Halmilton
type: [….]

We shall interpret Hamilton-Jacobi
theory in the wider sense as the study of the characteristic
curves and maximal integral
submanifolds of a closed 2-differential forms.

-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 122*Hamiltonians*are well-known to physicists, both classical and quantum. This gambit of a “wider sense”, though utterly par-for-the-course among mathematicians, rings oddly in a work supposedly aimed (according to its Preface) at “engineers and physicists” (a phrase which, to a hard-core mathematician, suggests “shoe-shine boys and chambermaids”).

**Hilbert Spaces**

… can be thought of as norms given
by distances that stay the same
not just when you translate, but when you rotate.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 253**ideal class group**

The

*ideal class group*is a way of measuring how badly unique factorization fails.
-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 221**inner-product space**

An inner product space can be
thought of as a vector space with just enough extra structure for the notion of
angle to make sense.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 91**Lie group**

Roughly speaking, a group in which one
can meaningfully define the concept of a smooth curve.

-- Timothy Gowers, ed.,

__The Princeton Companion to Mathematics__(2008), p. 230
Thus far the “What is X?” viewpoint. Compare the “Why was X invented? What is it for?” perspective, discussed
in our companion essay, “

__Why is Mathematics__?”**mathematical logic**

Mathematical logic is the study of
formal languages that are used to describe mathematical structures and what these can tell us about the
structures themselves.

-- Terry Gannon, in Timothy Gowers,
ed.,

__The Princeton Companion to Mathematics__(2008), p. 539
(That is not an especially successful or enticing
description; we cite it merely for
purposes of contrast.)

**point**

Of course, we have an intuitive notion of a ‘point’ in
three-dimensional Euclidean space, but the aim here is to pry you slightly
loose from that intuition: here, largely
for algebraic purposes, though the newer perspective generalizes better to
vector spaces of arbitrary dimension:

A precise definition which realizes
this intuitive picture may be obtained by this device: instead of saying that three numbers

*describe the position*of a point, we define them to*be*a point.
--Barrett O’Neill,

__Elementary Differential Geometry__(1966), p.
Thus, a ‘point’ is now an ordered triple of real numbers.

**representation theory**

Here is a nice variation on the definitional style: Not what a subject “is”, but what it

*aims*at:
The aim of representation theory is
to understand how the

*internal*structure of a group controls the way it acts*externally*as a collection of symmetries.
-- Terry Gannon, in Timothy Gowers,
ed.,

__The Princeton Companion to Mathematics__(2008), p. 539
This definition, I must say, is truly appetite-whetting.

**Riemannian geometry**

The study of the geometric
properties of the extremal curves of this Lagrangian (here called, in this
special case,

*geodesics*) constitutes Riemannian geometry.
-- Robert Hermann,

__Differential Geometry and the Calculus of Variations__(1968), p. 261**topology**

“Topology is rubber-sheet
geometry.”

-- Old wives’ adage, which you
probably learned from your nurse.

Cf. the Zen-flavored updating of this, “Topology is that
style of geometry in which the donut is considered equivalent to the mug that
you dunk it in.” Of course, that
scores only as an epigram, not as imparting information to someone who has no
prior knowledge of what topology is about.

topology, the mathematical study of
the salient properties of geometric shapes

-- Edward Frenkel,

__Love & Math__(2013), p. 252
This one is economical, and cleverer than it looks. “Salient” is a choice word; and here alludes to more than will be
apparent to a layman, though (unlike the donut/mug koan), it will still make
sense to a layman.

The definition given by English Wikipedia (in the apparently
older version that is the only one available to me at work) is surprisingly (over)specific: “Topology is the mathematical
study of surfaces.” The more
current Wiki says rather: “Topology is the mathematical study of shapes and
spaces.” The French version gives “La
topologie est une branche des mathématiques concernant l'étude des déformations
spatiales par des transformations continues (sans arrachages ni recollement des
structures)."

Quite at variance -- ostensibly, at least -- with that “study
of surfaces” thing is this:

Topology [is] an abstract study of the limit-point concept.

-- John Hocking & Gail Young,

__Topology__(1961), p. 1
Here, we feel in the domain of the ‘blind men and the

__elephant’__; but what is really going on is this: There is a geometric, and an analytic, “moment” to the world-historical project of Topology; and this last grasping, accentuates the latter.
Along the lines of Hocking & Young:

Point-set topology is … an

**analogy-based theory**, comprising all that can be said in general about concepts related, though sometimes very loosely, to “closeness”, “vicinity”, and “convergence.”
-- Klaus Jänich,

__Topology__(1980; Eng. trans. 1984), p. 1
A similarly aetiologically tinged characterization:

The concept of topological space grew out of the study of the real line …
and the study of continuous functions…

-- James Munkres,

__Topology: a First Course__(1975).
From a popular article (“Fun with Möbius Bands”):

Topology … deals with shapes and
structures.

-- Martin Gardner,

__Are Universes Thicker than Blackberries__? (2003), p. 57**torsion tensor**

Contemporary algebra is ferociously abstract; really, you should have a physician
check you out before you go anywhere near the subject. And as a relief for those who, ascending the heights, start
feeling light-headed, intuitive relief often comes in the form of a more

*geometrical*interpretation, if any should be available. As:
The above proof provides a geometric
interpretation of the torsion tensor of a connexion as measuring the difference between covariant
differentiation in the given connexion
and covariant differentiation in the torsion-free connexion with the same geodesics.

-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 65
Granted, that is still not exactly “Twinkle Twinkle Little Star”, but
compared with the algebraic thicket of co- and contravariant tensors, it is as
concrete as a pastrami sandwich.

And yet, we fear, the valiant author is here striving against the
tide: compare this other item from
the same work:

The

**torsion tensor**of a connection is a vector-valued function that [blah-de-blah].
(Note: As far as we know,

**In particular, it has nothing to do with the “torsion of a space curve.”)***there is no nice motivation for the word “torsion” to describe the above tensor.*
-- Noel Hicks,

__Notes on Differential Geometry__(1965), p. 59
~

Meanwhile, as a placeholder, here's a hamster:]

Link to this blog or the hamster gets mapped to the empty-set! |

~

[Update 15 January 2014]
Thanks largely to a timely intervention by Edward Frenkel, this post has
found its audience. The hamster is
saved !!

Any of you who would like to add your own terse or chiseled or epigrammatic
intuitive insight into your favorite mathematical subfield or structure, such
as might fit decoratively on a coffee-mug, please Comment or else write me at

Meanwhile, for as much math as can be packed onto the back of a
cocktail napkin, and suitable for framing, try this:

For another “What is …?” essay, there’s this:

[Warning: Funny. Don’t spill your coffee.]

And for a “Why is …?”:

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