An update re Mathematical (and Natural) Definition (re-updated)

[Continuing this essay:

http://worldofdrjustice.blogspot.com/2013/12/theorems-propositions-dumb-questions.html ]

While we’re on the subject, let us consider further the question of definition in mathematics.

Re Hilbert’s approach to the axiomatization of geometry:

Rather than defining points or lines at the outset  and then postulating axioms that are assumed to be valid for them, a point and a line were not directly defined, except as entities that satisfy the axioms postulated by the system.

-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

This is not quite so radical or ‘post-modernist’ as it might sound, since traditional grammar recognizes many analogous cases in natural language, under the rubrics of synsemantic,  syncategorematic and implicit definition.  [See posts with the Label "incomplete symbol".]  It is a relative notion, with a sliding scale;  but analysis will suggest that a very large set of words and multiword expressions (as, the use of a word in an idiom, especially in an opaque idiom) partake of some degree of syncategorematicity.   However, in the particular perspective of mathematics, this idea harmonizes especially well with a logicist or formalist approach to the subject:

The use of undefined concepts  and the concomitant conception of axioms as implicit definitions  gave enormous impetus to the view of geometry as a purely logical system.

-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

Again, this is much less disorienting and self-bootstrapping than it may seem, since -- outside, indeed, of formal contexts -- virtually all of natural language works exactly like that; and not only expressions like whereas, the moreso as, French ne, German doch, which wear their syncategorematic character on their (empty) sleeves, either:  but plain words like bunny.   You do not learn to use such words on the basis of a definition, formal or informal -- however much it might please linguistic philosophers to invent a terminus technicus such as “ostensive definition”, which labels a phenomenon (or passle of phenomena) without explaining it.   For, as we have seen in our discussions and parables related to matters Quinean, these don’t really work, not logically;  they work pragmatically, to the extent that they work at all, because (since we are all molded from the same clay; or  if you prefer, since our bloodlines have all been subjected to the rigors of Natural Selection) we are all cut to the same cloth.  (To the extent that some individuals fall outside the innate cognitive norms, they fail to acquire the same semantics that the rest of us do:  or else, like some gifted and industrious autists, they acquire this only by dint of an artificial study, like someone learning Sumerian logographics.)   Thus, the following Onomastic Primal Scene does not actually obtain in any real nursery:

“That, Timmy” (pointing -- but at or towards what?) “is a rabbit (noun count, singular).   And by this -- attend now, and please do not misunderstand me -- I do not intend to indicate the entire scene embracing carrots and furballs and playpen and binky (who left that there?) etc., let alone the cosmos as a whole (after all, one has to point somewhere), whether by itself or considered as but one flaky layer in the whole baclava-like complexus known as the multiverse;  but only the, er, furball-related entity.   And by this, I do not mean, so much, (although I do not literally not mean it, either), a pointlike or infinitessimal space-time slice of a leporiform trajectory along the world-sheet, nor a “thickened” (perceptually available) neighborhood of the same;  nor a sort of puddle of rabbit-stuff, undifferentiated from the rest of the puddle; nor a concrete instantiation of the Platonic Form, ‘Rabbit’;  nor a subobject in the Category Leporidae;  nor an agnostically structured pointset consisting of Undetached Rabbit Parts (although I sort of mean that, since, at some point, once you have detached the poor critter to bits  and scattered its disjecta membra over the face of the earth, to be eaten by vermin and recycled as independent atoms, -- at some point, we can no longer confidently say, “That is a rabbit”, in the sense of noun count, singular),  nor -- well, dash it all, I mean just Fluffy, okay?  Fluffy and other creatures that look and hop and act like her.  And by the way it looks like Fluffy wants a cuddle or something, because she is spritzing the wood-shavings in a semantophobic panic.”

~

The scenario above  comports more naturally with a coherence theory of truth, rather than a correspondence theory.

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The quirky, philosophically-minded Intuitionist mathematician Brouwer, harbored similar “mysterian” views on ultimate indefinabilty:

In Brouwer’s opinion, mathematical definitions should not be looked upon in a mathematical way, but should only be used as a support for our memory.  Basic concepts, such as ‘continuous’, ‘once again’, ‘etcetera’, have to be irreducible.

-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p.  45

Nooit nog heeft door de taal iemand zijn ziel aan een ander meegedeeld;  alleen ein verstandhouding, di toch reeds is, kan door de taal worden begleid.

(Caption quotation:  op. cit., p. 32.)

~

Since, outside of the classroom, new words are almost never introduced explicitly, let alone lexicographically or metalinguistically, we must conclude that the language-learner somehow gets the right idea “from context”.   This notion is more problematic than might appear.

For, the history of contexts met-with over the course of a learningful life,  varies considerably from person to person (my own nursery school was wonderfully bunny-rich -- unless the creatures were actually guinea-pigs, come to think of it:  I no longer recall, and after all had nothing to compare them with at the time, they were simply our class mascots and Furry Friends -- but sadly penguin-deficient (of that I am quite sure);  nay, my lifelong platypus-deprivation has been nothing short of absolute), and the fact that we can happily chatter away  among our fellows  about all creatures great and small, without needing to resort to pointing at picture-books (although I do always carry a bunny-book about with me, just in case I should run into Wittgenstein) or red-faced arm-flapping exasperation as we attempt just one more time to make ourselves understood to our perversely thick-witted interlocutors (“Not a ‘triplex of mutually orthogonal rabbit-slices’, dammit!  I mean three  separate  rabbits !!”) strongly suggests that we come from Nature’s Nursery with a lot of shared ontology inborn.  (Chomsky’s school reached similar conclusions many years ago, by a somewhat different path.)

Two-dimensional representation of an imaginary rabbit.  Question:  What is the dimensionality of the *actual* imaginary rabbit?

~

Back to mathematics.

Here definition, in contemporary use, is the intuitive idea, to which axiomatization is the formal counterpart.    You define the term group by simply listing the axioms which any set endowed with an operation must satisfy  if it is to aspire to that dignity.

Yet, having made this move, we see that a vagueness was lurking in our original intuition:  since being ‘axiomatizable’ comes in various flavors:  finitely axiomatizable, axiomatizible in first-order bzw. second-order logic, etc.   And we find surprises, such as when so familiar an item as a torsion group  turns out not to be finitely axiomatizable within first-order logic.   Yet we know what we mean by it, for all that.

Compare: 

mammal:  definable in (cladistic) terms of shared descent

reptile: not so definable

water: definable in terms of molecular composition

blood, wine : not so definable

quartz : definable in terms of mineral composition

granite :  only approximately so definable, or definable at one remove.

~

How you define a mathematical item -- we may even say, how you go about defining it, the tack you take in trying to define it -- depends upon what you are intuitively aiming at.

For example:  How to extend the definition of the multiplication of a finite set of factors, to the infinite case?  (We did so for the case of convergent infinite sums without difficulty.)

Because of the special properties of zero with respect to multiplication, the most obvious definition of a convergent infinite product is not the valuable one.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966)

Or again, from the great Gleason,  ever alert to the lexicographic aspects of mathematics:

The Bolzano-Weierstrass property is often taken as the defining property for compactness, since it is frequently the handiest property  for dealing with compact metric spaces.  However, it is not equivalent to the Heine-Borel property in general topological spaces, and it turns out that the latter is the more valuable in the general case.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 269

Coming up with a useful definition (and here the “coming up with” does seem closer to invention than discovery) becomes an interesting question in its own right, and not a matter of mere fiat. 

Again compare:

Finding a definition (or, really, “characterization”;  yet ultimately the ink-stained lexicographer must needs still define) of:  Romanticism, Minimalism, Idealism;  joke, game; mollusc, microbe, plant;  silver, beige;  etc.

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So for instance, let’s take logicism.

There is evidence that, in 1899, Hilbert endorsed the viewpoint that came to be known as logicism.  Logicism was the thesis that the basic concepts of mathematics are definable by means of logical notions, and that the key principles of mathematics are deducible from logical principles alone.

-- José Ferreirós, “The Crisis in the Foundations of Mathematics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 143

So there you are:  A nice clean definition.  Think what you will of the thesis, and come what may by way of later evidence pro or con, the definition is what it is, right?

Wrong.

Our author goes on:

Over time, this thesis has become unclear, based as it seems to be on a fuzzy and immature conception of the scope of logical theory.  … Historically speaking, logicism was a neat intellectual reaction to the rise of … the set-theoretic approach.

So!  In addition to being confirmed or refuted, apparently a thesis can decay, lose its sharp edges, like an unrefrigerated vegetable.   For:  Any definition of X  itself takes for granted the well-definedness of certain understood entities Y, Z …  Should the latter fall foul of better understanding, X itself can be left high and dry.

The consider the following definitions:

phlogiston:  a material which is the source of light and heat attendant upon combustion

phlogisticated air:  air mixed with phlogiston

monokeratic phlogisticene :  phlogiston mixed with powdered unicorn hoof  (cures scrofula and gout)

These delightful definienda, whose delineation was once so clear, have each met with a sad fate.

Definitions, like dephlogisticated unicorn-hoof, are liable to crumble into dust with the passage of time.

Thus, in mathematics:  Newton’s fluxions, etc.

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Example of a definition  introduced in full awareness that it is merely provisional:

This definition of an affine algebraic variety should be considered only a working preliminary definition.  The problem is that it depends on considerations extrinsic to the objects themselves, namely the embedding of the affine variety in the particular affine space Cⁿ.

-- Karen Smith et al., An Invitation to Algebraic Geometry (1998/2010), p.

This definitio (taking this in the actio rather than the actum sense) is in the spirit of Lakotos’  Proofs and Refutations.

~

Mathematics often sharpens our understanding of any pre-existing conception (“continuity”, “dual”) that comes to swim within its ken.  And so it is for the very notion of definition :  long assumed a matter of free choice, until Russell’s Paradox brought matters up short.   Whereupon he and Poincaré worked out their understanding of impredicative definition or impredicativity. 

Thus, in one formulation of Poincaré’s predicativist  approach:  “All mathematical objects (beyond the natural numbers)” (these being, as even Kronecker concedes, God-given) “must be introduced by explicit definitions.”  And, not just any definition you take a fancy to will do: 

If a definition refers to a presumed totality  of which the object being defined is itself a member, we are involved in a circle:  the object itself is then a constituent of its own definition.

-- José Ferreros, “The Crisis in the Foundations of Mathematics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 146

And this, you understand, is Very, Very Bad.  (We might cheekily dub it Definitional Incest.)

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Mathematicians, like philosophers, and unlike anyone else (including even lexicographers), are given to a certain semantic Akribie --  an extraordinary self-critical attention to their own use of language.   As, consider this:

The conservation “laws” of momentum and angular momentum  are also readily introduced …

-- Robert Hermann, Differential Geometry and the Calculus of Variations (1968), p. 100

I have no idea what subtle mental reserve caused to author to quarantine the word laws in sneer-quotes, nor why he felt it necessary so to caveat -- so to signpost the approach to a possible Occasion of Semantical Sin -- in a work aimed (according to the preface), not at philosophers, nor Jesuit spiritual directors, nor even mathematicians, but to engineers and physicists (those are the grease-stained guys tinkering under the accelerator).   But the fact is, if you move in mathematical circles, your semiotic conscience becomes exquisitely sensitive and attuned.

~

In focusing on definition, I am inadvertently revealing the déformation professionelle of one who used to earn his bread (or rather his hardtack; the profession is ill-paid) as a lexicographer.   For, rather than trying to say what a thing “is” (and here the Korzybskian strictures against the copula  have their full force), we may say, pragmatically rather than ontologically, what a thing is for.   Thus, a hammer “is” a manufactured object of a certain range of shapes and weight, classically with a metal head and wooden handle, (etc. etc. -- “Etc.”, as the Korzybskians have it), if that is helpful to you;  but it is for driving in nails.

Thus -- to take a couple of concepts that always somehow puzzled me definitionally :

Chains and partitions of unity  free our proofs  from the necessity of chopping manifolds into small pieces.

-- Michael Spivak, Calculus on Manifolds

Now that is something a kitchen-maid could understand.

~

[Weiteres zum Thema]

On provisional/dialectical definition:

Menger wrote, in a series of papers on foundational questions  published in 1928:

Dabei möchte ich betonen, daß ich das Wort ‘Konstruktivität’ für ein  wenn überhaupt, so  vermutlich  auf verschiedene Arten und in verschiedenen Abstufungen  präzisierbares (bisher noch nicht präzisiertes)  Wort halte.

-- quoted in Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 199

(For logophiles only:  Let us here salute and savor  that phrase,  “ein  wenn überhaupt, so …”   Impossible to translate this into English  in so compact a compass.)

~

Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 14, quotes Lebesgue:

Bien que je doute fort  qu’on nomme jamais un ensemble qui ne soit  ni fini, ni infini,  l’impossibilité d’un tel ensemble  ne me paraît pas démontré.

Quite aside from the mathematical content to this, as sheer semantic content  that will baffle anyone who

(a) has learned the terms finite and infinite as simple contradictories (infinite iff not-finite);  and who

(b) accepts the tertium non datur

it seems a mere tautology, like the analytical-philosophical lore  of bachelors and married-men.

But this is from Lebesgue, note, as familiar with the intricacies of the various infinities  as anyone on earth.  Clearly something subtler here is meant.  Something I’d never heard of before -- the first worry of the Continuum Hypothesis, so I had understood, concerned the possible existence of wiggle-room between countable infinite and the cardinality of the continuum.

Quite possibly, however, since Lebesgue and Brouwer sometimes shared an intellectual orbit, the explanation may be sought in the following hint (op. cit., p. 66):  “Brouwer distinguishes between species which are abzählbar, zählbar, auszählbar, durchzählbar, and aufzählbar, where some of the distinctions  are related to the question of decidability.”

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Another parallel between mathematics and (e.g.) biology, as regards a certain type of ‘definition’.

Sometimes you are not trying to focus on a new concept in splendid independence, giving necessary and sufficient conditions to ‘be an X’, de-fining (demarcating) its boundaries (Jordan-curve-fashion) between what-all is inside  and what-else is out;  but, rather, starting from some homely, antecedently-familiar item Y, to define this new X as being similar to that Y.    Sometimes you say they’re similar, and leave it at that:

            A hare is like a rabbit.

            A coot is kind of like a duck.

Sometimes you add differentia:

            A zebra is like a horse with stripes.

Or, you may say that the new concept X generalizes Y, without giving necessary or sufficient conditions for membership in the generalization, with or without further examples of members of X:

            Amphibians form a taxon of animals that includes frogs.  (They ‘generalize’ the frog.)

            Amphibians form a taxon of animals that includes frogs and salamanders.

All these strategies are (so to speak) topologically distinct, the one from the other.

Compare, in math (an actual textbook example):

Locally convex spaces are topological vector spaces that generalize normed spaces.

Here the relatively exotic new concept “locally convex spaces” plays the role of amphibians in the example above, with the normed spaces (familiar from the nursery) filling that of our friends the frogs;  with an additional delimiter, topological vector spaces, basically saying:  “generalize, but not too far”.   Thus, if we said

Vertebrates form a taxon of animals that includes frogs.

that would still be a true statement, but the belt would have been let out too many notches to hold up the conceptual trousers.

“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.]

~

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).

~

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)

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~ Commercial break ~

http://www.linguasacrapublishing.com/justice.html

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.”

~

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

~

[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?