“What is Mathematics?” (expanded)

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 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.]

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From works for an educated  general audience:

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)

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:

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

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

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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.

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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)

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We now return you to your regularly scheduled essay.

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[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 nth 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.”

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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 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.

That is, as an old Websterian, I am concerned with how to go about giving, not something that a 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

Ergodicity means, roughly, that … very long sample paths … end up resembling each other.

-- Nassim Taleb, Fooled by Randomness (2004), p. 59

Ergodicity, … that time will eliminate the annoying effects of randomness.

-- Nassim Taleb, Fooled by Randomness (2004), p. 144

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 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

Holonomy is a measure of how tangent vectors on a particular surface  get twisted up as you attempt to parallel-transport them on a loop.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 129

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

The Lie Bracket of X and Y … informally … represents the net direction of motion if one first moves an infinitesimal amount in the X direction, then in the Y direction, then back in the XS direction and back in the Y direction, in that order.

-- Mark Ronan, “Lie Theory”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 231

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?”

Logic

… [C. S.] Pierce’s last papers on logic, a subject which he defined rather surprisingly as ‘the stable establishment of beliefs’.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 174

Logic is the infancy of mathematics, or conversely, mathematics is the maturity of logic.

-- Bertrand Russell, 1957, quoted in I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 315

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

Spinors are analogues of tangent vectors.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 131

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

-- Noel Hicks, Notes on Differential Geometry (1965), p. 59

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[Note:  Obviously, this pleasant exercise could be extended, with further epigrams as my reading proceeds.  If you would like more, pass this link to your friends; and if reader interest warrants, as evinced by the pageview stats, more may come.
Meanwhile, as a placeholder, here's a hamster:]

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

(Pleads Fluffy, winsomely:  "O  pleeeeease don't place me in the kernel of some mean homomorphism!")

~

[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

david_justice@post.harvard.edu .

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

Differential-geometric monostich

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

What is Algebraic K-Theory *really*, Grandpa?

 

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

And for a “Why is …?”:

Why is Mathematics?

A Dive to the Depths (expanded)

A phrase you will often meet in higher mathematics, and almost nowhere else, is:

“a deep result”

O loveliest of monostichs, thou !

The very notion of what ‘deep’ means, in such a context, is itself deep;  indeed, too deep for me, at present.   This, owing to a crippling condition of mathematical oligophrenia.  --  which, however, I pray that time and diligence might partly palliate.  (For a glimpse into the terrible sufferings of mathematical oligophreniacs, click here, if you dare.)  Yet I am putting up this skeletal promissory-note of a post, so that there will be a space to scribble insights as they wake me in the night.

First off -- the term does not mean simply ‘difficult’; indeed, though such results lie in the depths, and are not to be had for the asking, once you have somehow managed to fish one up, it may seem clarity itself.  Nor does merely being difficult make anything deep.  Any humongous brute-force calculation falls into that category;  for a more-substantive example, consider the Four-Color Hypothesis, which people suspected should be deep, but the proof that changed "Hypothesis" to "Theorem"  is a combination of clever tricks and elbow-grease.  The response of the mathematical community was disappointment:  "So, it turns out it wasn't an interesting conjecture after all."  (Of course, it may yet prove to be "interesting" in our cognitive human sense; that awaits a proof of an entirely different kind.)

We may go further, and put forward that an overarching purpose of mathematical research is to reveal something previously difficult  as now simple, when seen in the right way.  Again and again this has happened in history, beginning with the replacement of finger-counting by symbols, and of clunky symbols like Roman numerals by decimals.  For a more recent example:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Nor is there any simple similarity or opposition between being “deep”  and using what are now called “elementary” methods in number theory (as opposed to analytic methods), e.g. Alte Selberg’s proof of the Prime Number Theorem by elementary methods, compared with earlier analytic proofs by Hadamard and others.  Typically, proofs that restrict themselves to “elementary” methods are harder than those that permit themselves a more capacious toolkit;  but whether they ever, or generally, gain depth via this austere discipline, I have no idea.

In the meantime, some related posts outside of a mathematical context  are these:

            On Depth and Breadth

            On Scope and Difficulty

As appetizers, try the following hors-d’œuvres platter -- to follow which, however, we have as yet prepared no meal (as with our early essays on the Realist vernacular, this is more by way of linguistic warm-up):

The connection between linear transformations  and bilinear functionals  goes quite a bit deeper …

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 38

The Riesz representation theorem depends on some of the deeper parts of the theory of measure and integration.

--George Simmons, Introduction to Topology and Modern Analysis (1963), p.

The analogy between cyclotomic fields  and fields formed from the points of finite order on elliptic curves   is very deep.

-- Neil Koblitz, 1993

The notion of Kan extensions is the deeper form of the basic constructions of adjoints.  We end with the observation that all concepts of category theory are Kan extensions.

-- Saunders MacLane, Categories for the Working Mathematician (1971; 2^nd ed. 1998), p. vii

The continued-fraction representation of real numbers is deeper than the decimal expansion.

-- Roger Penrose, 2004

There are deep ties between enumerative geometry and Ramanujan’s tau function.

Contrast:

The useful fact about products of projections  lies near the surface.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p.  47

In the same work, while acknowledging the possibility of a lattice-theoretic formulation of the subspaces of Hilbert space, he dismisses this possibility as “trite” (p. 22).

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Much commoner than “trite” is trivial, which is virtually a terminus technicus of mathematical practice.  Let one quote stand for all:

One of the useful conclusions we can draw from Theorem 2 [to the effect that the norm of a Hermitian operator equals the supremum of its eigenvalues] is that the spectrum of a Hermitian operator  is not empty.  This is not a trivial conclusion.  We shall obtain the corresponding fact for normal operators  only after the application of a lot more relatively deep analysis.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 55

Psychosociological note:   Attempt to imagine the effect on our pumped-up math-major  freshman or sophomore brains, hearing dismissed as “trivial” (with a wave of the hand) propositions which, but a year before, we ourselves would not have begun to understand, and which indeed most of our countrymen will go peacefully to their graves without understanding.  (For more on the hubris involved, click here.)

Also note:  What counts as “trivial” is relative to where you stand.  Thus, in the very next sentence, Halmos adds:  “We hereby report that the spectrum of an arbitrary operator is also not empty;  since we shall have no occasion to make use of this fact, we shall not enter into its proof.”  The proof, one gathers, is more difficult still.  But when once you have mounted, and stand upon that summit, the fact that Hermitian operators in particular have eigenvalues, is trivial indeed.

Leave it to mathematics to recruit even the notion of triviality into some highly non-trivial constructions.   E.g.

A topological space over X is called a locally trivial fibration if every x in X has a neighborhood over which Y is trivial.

-- Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 129

And:

Sard’s Theorem … is … a highly non-trivial  theorem  which is elementary in the sense that it uses only the notion of a differentiable map.

-- Shlomo Sternberg, Lectures on Differential Geometry (1964), p.

The terms deep and elementary (here in the everday sense, and not the special number-theoretic meaning mentioned above) are not antonyms, but they do contrast:

The spectral theorem implies that every normal operator has a large supply of invariant subspaces;  this is classical  and can be considered elementary by now.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

This dependence of “depth-perception” upon experience accumulated with the passage of time, does not refute the notion of relative mathematical depth as ill-defined.   What is intuitive though difficult to put into words  is a notion of “deeper than” rather than of absolute depth.  To the giant, neither the pond nor the puddle appears deep;  but the pond is deeper  for all that.

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Re the classification of simple algebras (“simple”, to be sure, in a certain technical sense, meaning roughly: incredibly complex and difficult):

The tools employed  are not deep; they are just, so to speak, linear algebra  raised to the nth power.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 122

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Further attestations.

A pioneer of Hilbert space theory, expounding the then-contemporary state-of-the-art for a nonspecialist mathematical audience, particularly as regards dilations and extensions of operators:

There do not seem to be any conspicuous and challenging yes-or-no questions that serve to indicate the direction in which the search for new results might begin,  but I have faith.  There is depth in the subject;  the trouble is that the surface has not been explored enough  to show where the deepest parts lie.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Writes a premier algebraic-topologist:

There are geometric problems which require the use of the multiplicative structure of the topological invariants.  Such problems are deeper than those which can be solved by considering the additive structure alone.

Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 105

Re Lie algebras and a closed subgroup H of the full linear group:

It is furthermore true (and this is deeper) that these one-parameter subgroups  fill a neighborhood of the identity in H, and consequently generate H if H is connected.  …The converse part of the correspondence  involves a subtlety of the type that makes the study of Lie groups a quite sophisticated topic.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 118

Re the Hodge conjecture:

It arises deep within the subject, at a high level of abstraction;  and the only way to reach it  is by way of those layers of increasing abstraction.

-- Keith Devlin, The Millennium Problems (2002), p. 9

This is a truly adult depiction of depth: as lying deep within the layers of the enigmatic onion.  It is not a case where you can just swallow some peyote and see it all in a flash.  
(For more, compare:  The Ladder of Abstraction.)

Writes a philosopher:

The axioms are not logical truths ... Their truth is established by intuitions which lie too deep for proof, since all proof depends on them.

-- Roger Scruton, Modern Philosophy (1994), p. 392

An example from outside the field of mathematics -- though it is a mathematician who is writing this:

Just how a protein manages to organize itself in space, using only the sequence of its own amino acids, remains a mystery, perhaps the deepest in computational biology.

-- David Berlinski, “What Brings a World into Being” (2001), collected in:  The Deniable Darwin (2009), p. 243

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Related vocabulary:

Related to the concept of depth (which focusses on the root of things) is that of richness (regarding the blossoms that bloom from this root).   Hadamard adopts this metaphor explicitly:

Application’s constant relation to theory  is the same as that of the leaf to the tree:  one supports the other, but the former feeds the latter.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 125

Further examples:

The algebra of composition of maps  resembles the algebra of multiplication of numbers,  but its interpretation  is much richer.

-- Lawvere & Schanuel, Conceptual Mathematics (1997), p. 11

… a recent, and surprising, theoretical advance by Winkler.  He shows that, for many … algebraically closed fields, the “free Skolemization” has a model companion.

-- Angus Macintyre, “Model Completeness”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 164

The concept of thickness [in graph theory] is the deep mathematical idea that underlies the recreational puzzle of Earth/Moon maps.
-- Ian Stewart,  How to Cut a Cake (2006), p. 126

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Having convinced ourselves, by the examples of mathematics, that there is more to this assessment-word deep than an emotional or impressionistic grunt,  we look to some cases outside of mathematics where an idea has been similarly assessed.

Some discoveries provide answers to questions.   Others are so deep  that they cast questions in a new light,  showing that previous mysteries  were misperceived.

-- Brian Greene, Fabric of the Cosmos

We are not at home in the world, and this homelessness is a deep truth about our condition.

-- Roger Scruton, Modern Philosophy (1994), p. 464

T.S. Eliot affirms that what is past and what is present, even what might have been, indicate a present purpose.  This is a metaphysical point of great depth.

-- James Schall, S.J., The Order of Things (2007), p. 69

And, more prosaically, but no less tellingly for all that:

Although running Bain Capital required a lot more brains and savvy than playing roulette does -- a lot more brains and savvy than most of us could even pretend to possess -- the job was not conceptually deep.  Romney did not develop a model of the world from the business of private equity. … “He’s not a very notional leader,” [said] Romney’s campaign spokesman …

-- Louis Menand, “Money Pol”, The New Yorker (19 III 2012)

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This is quite aside  from the path of mathematics, but -- it may be, that such depth is displayed in quite distantly allied regions:  all tracing back to Him, perhaps by some functorial construction.  In that spirit, this:

http://www.youtube.com/watch?v=s5xDEyuXaOY&feature=fvwrel

The final anguish  of the Asian bride  suggests the depth  of the Riemann Hypothesis.

The enigma of a woman’s heart,

finally espied  by a Private Eye,

for less than the price  of a Valentine …

This Rose

[Kindle]  [Nook]

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Somewhat less far off the path …  Deep is indeed the term of art  in mathematics, antonymic to trivial.   Now compare, from another discipline, the word profound, in reference to Newton’s perplexing, little-known  philosophical-speculative opus:

That it is exclusively mystical  I do not believe -- that there is a mystical element  seems certain.  I hope that  one day  some profound student -- no one less will suffice -- will study this mass of papers.

-- E. Andrade, quoted in James Newman, ed. World of Mathematics (1956), p. 273

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Above, we saw the distinction deep vs. difficult.  Here now even the latter concept is bilayered:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

In infinite dimensions, it is tedious but not difficult  to construct spaces  strictly  but not uniformly  convex.

-- Prof. Lewis, University of Alberta, course in Functional Analysis, 1982

The proof that Reidemesiter moves  and planar isotopy  suffice to get us from any one projection of a knot  to any other projection of that knot  is not particularly difficult;  however, it is technically involved.

-- Colin Adams, The Knot Book (1994), p. 15

And here the original distinction is made even sharper:  though they are not interchangeable, depth tracks with generality, which in turn tracks (on the plane of praxis -- of proof) with simplicity:

Re the Denjoy-Young-Saks Theorem on the derived numbers of functions:

As we would expect  in view of the great generality of the final statement of the theorem,  the proof due to Saks is of extreme simplicity.

-- F. Riez & B. Sz.-Nagy, Leçons d’analyse fonctionelle [references to the English translation, Functional Analysis, 1955], p. 17

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Here a leading mathematician laments the shallowness of his understanding of something he himself proved (regarding representations of a Kac-Moody algebra, as it happens):

My proof of this result was technically quite involved.  I was able to explain how the Langlands dual group appeared, but even now, more than twenty years later, I still find mysterious why it appears.  I solved the problem, but it was ultimately unsatisfying to feel that something just appeared out of thin air.

-- Edward Frenkel, Love & Math (2013), p. 181

This is setting oneself high standards indeed.   Shakespeare probably did not lie awake o’ nights fretting how the devil he ever came to write Hamlet;  Mozart did not find the bread of pleasure at having written the Sonata in A  turning to ashes at the thought that it might have been dictated to his unconscious  by an angel.

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A near-synonym of the math-word deep, but shorn of all irrelevant aesthetic echo, is:  “highly nontrivial”.   The term is decidedly commendatory, though to a layman it might sound like faint praise, as were one to dub one’s lady-love “seriously unugly”.  The expression may be extensionally impeccable, but ‘twould never pass in a sonnet.

 Further:

In set theory, a forcing extension in Cohen’s sense  is reminiscent of algebraic extensions of a field, but

… the forcing method is far more complex, both conceptually and technically, involving set-theoretic, combinatorial, topological, logical, and metamathematical aspects.

-- Joan Bagaria “Set Theory”,  in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 625

Although he does not use the term "deep" here, that expanded characterization  "complex, both conceptually and technically", especially the "conceptually" part, points in that direction.

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One motive for Frege’s choice  was again generality:

Does not the ground of arithmetic lie deeper than that of all empirical knowledge, deeper even than that of geometry?

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 184

… [Cantor’s] remarks on functions of several variables (where the provability of theorems  was deepening the level of rigour in analysis)

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 223

 

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