SimCity: the foundations

Background:   Topology is familiarly, informally characterized as “rubber-sheet geometry”.  That is, unlike the Euclidean geometry that we learned in school, which applies to flat rigid surfaces,  you’re allowed to stretch and bend the space, so long as you don’t tear it or let it intersect itself.

But at some point, we might like -- without returning quite to the simplicities and rigidities of the Euclidean picture -- to make our space… a bit less rubbery.   As the godfather of Calabi-Yau manifolds puts it:

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 77

[The above is an update to this post:

https://worldofdrjustice.blogspot.com/2011/08/urysohn-metrization-theorem-for-real.html  ]

The Trinity (triune unity -- re-renewed)

[In observance of Maundy Thursday,  a re-post  from yesteryear.]

I spend a fair amount of time in the company of Muslims these days;  indeed, at present, by an accident of the seating-chart, I probably spend more time in close propinquity with Muslims, than with Christians, Hindus, Jainists, Jews, and Zoroastrians combined.   (Lotta LDS, though.  Plus all that could change with the next re-org, as my next podmates might be Zoroastrians.)   [Update March 2017:  And now, in fact, our branch chief is a Zoroastrian.] There is an effort of good-will on both sides;  my Sunni neighbor points eagerly to passages in the Koran, where good things are promised to ‘believers’ (mu’miniin) rather than specifically ‘Muslims’ (muslimiin).  A kind-hearted man, he hopes to be with me in Paradise, and not to gaze down on me roasting in Hell.  (`Uqbaalak, ya shaykh.)

Now, we Christians know implicitly, that the doctrine of the Trinity is no polytheism:

that Father, Son, and Holy Ghost, are not at all like Apollo, Zeus, and Hera, say, but more like le père Dupont, at once father, and Frenchman, and fireman.   But to explain this to our Muslim friends, is difficult.   I often have to fall back on saying:  the Trinity is a Christian mystery; it may be true or false (in some transcendental sense of those categories), but it is no descendent, direct or indirect, of the sort of pagan polytheism which stuffed the Kaaba with idols, and peopled the trees with dryads, and Olympus with squabbling gods.  For us as for you, God Himself -- Allah -- Yahweh -- e’en He -- is indeed One.

Yet this unity by no means necessitates or logically entails, that there could be no parts at variance within the Godhead, even to dissension.  (Of course, their absence might be a truth of the Church, and thus beyond dispute;  I am speaking here only of logic, the only subject in which I have been ordained.)  How indeed could we ever know otherwise (save by some enigmatic Revelation)?  After all:  We are made in His image, and we ourselves are a bundle of contradictions;  and He contains us, as a proper part.  (Additionally, that Eloi, eloi passage would seem to point in that direction.)

~

C. S. Lewis saw rightly when he compared the notion of the Trinity (purely as regards its intellectual coherence, rather than anything theological) with the ‘separate’ faces of a cube -- which latter is, however, nothing more nor less than the sum of all of them.  (Indeed, if you were to go with a sort of ‘projective cube’, with antipodal faces identified, our new Cube (topologically a three-torus) would even consist of exactly three parts.)

The historian of physics D’Abro offers a similar parable, along the lines of Abbot’s Flatland (A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 653), his quarry being however, not the Trinity, but the (in some ways similar) “wave-particle duality”:  or, as we might term it, the wave-particle identity.  He imagines our various perceptions of something we believe to be one entity, but which sometimes seems a triangle, and at others, a circle:  eventually we realize that it is a cone, seen now this way, now that.  And, regarding the separateness/unity of electricity and magnetism:

The theory of relativity brought about the fusion of the two aspects, no longer by utilizing the background of 3-dimensional space, but by introducing the more refined background of 4-dimensional space-time.   The underlying entity, the partial aspects of which are electric and magnetic, were found to be the 4-dimensional electromagnetic tensor  situated in space-time.

-- A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 653

The Trinity, we may confide, whatever in its unknowable essence it may be, is at the very least as complex as a  tensor.

~

Pascal (Pensées, 1670 [posthum]), has a very odd passage, asserting the alienating nature of God’s complexity -- or perhaps not -plexity, but monolithicness :

S’il y a un Dieu, il est infiniment incompréhensible, puisque, n’ayant ni partie ni bornes, il n’a nul rapport à nous.  Nous sommes donc incapables de connaître  ni ce qu’il est, ni s’il est.

Ni s'il est ! -- A useful first step towards an antidote  might be to drop that assertion about God's lacking any parts.

.

Refutations of Received Wisdom

Thesis:  “You can’t put a quart in a pint pot.”  (old folk-adage)

Antithesis:  According to Banach-Tarski, you can.

       https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Fixed That 4 U.

Thesis:  “Il faut qu’une porte soit ouverte ou fermée” (Musset).
["A door must be either open or closed."]

Antithesis:  On the contrary:  it might be ajar (a word we first learned in childhood, in the riddle:   Q: When is a door  not a door?”  A: When it’s ‘a-jar’.)  Or more generally, it might be clopen.

      https://en.wikipedia.org/wiki/Clopen_set

Fixed That 4 U.

Featherless biped

Fixed That 4 U.

Problem:  The central problem of computing science is whether P = NP.   How can we go about settling the question?

Answer:  Simple.   We need only demonstrate that N = 1.

Fixed That 4 U.

Internal, External, Universal

[Today’s theologico-mathematical analogy may be stretched, far-fetched;  but ‘tis the Lord’s day, a time meet for meditation  more at large.

For more extensive reflections, focusing on Realism in both domains, consult the essay series that begins here.]

~

Instead of defining the properties of a collection by reference to its members -- its internal  structure -- one can proceed by reference to its external relationships with other collections.

-- R. Goldblatt, Topoi , 2^nd edn. 1984

I am reminded of the Christian critique of narcissistic individualism, so telling for our own day, when it has become a very plague, both sapping the individual character, and corrupting the polity as it forms an algal bloom as identity politics.  This view was made more acute, and very contemporary, by C.S.Lewis in The Four Loves and elsewhere, with its metaphor that health lies neither in religious solipsism (the “inner light”, which he decries) nor in that solipsism-à-deux of “looking into each other’s eyes”, but rather in mutually apprehending some external thing, of which we each see aspects, though along different sight-lines.

There are traditional notions of something large and out-there, above us and beyond us;  but these are vague and unstructured, and have perhaps grown stale through overfamiliarity (though we have never understood them well enough to have leave to dismiss them out of hand).   So let us turn to consider a mathematical notion of something containing -- something larger than what you started with, yet perfectly contained within itself:  not spreading over us like a fog, but rounding us out.  The technical name for this is comforting, downright cozy:  compactification.  (The Water Rat of Wind in the Willows  pictures his snug and tidy den.)

Compactness has turned out to be one of the most central notions of topology, a field which itself is about as central as you can get.  For details, see Wikipedia (that paradisal repository of all that is known, or could ever be known);  but the takeaway is, that it is a quite vaunting generalization of the idea of finiteness.  Such spaces are nice to work with.

Thus for instance:  consider the open interval (0,1).  It is not too intimidating (apart from its harbored continuum), but it is irksomely incomplete, in that a well-regulated sequence of points -- ½,¼, 1/8 … -- can march off towards nullity,  yet nullity they find not, nor unity neither  should they march the other way.  We can complete this space, and simultaneously compactify it, in an obvious way:  just add the points zero and one at either end, to get the closed interval [0,1].  Now all is well.

But there exists a less obvious kind of compactification, involving the addition of but one point (we pause, that you might wonder:  Yet how can this thing be?).  In turns out to be deeper, in that such a one-point compactification (via Alexandroff extension) is available for any locally compact Hausdorff space.  In the simple case of our open interval, conceptually you add a point at one end and bend the segment around to meet it.  The result is a little ring:  like all round things, it is ever so perfect and pleasing.

And our pleasure at this maneuver  is more than aesthetic, for the move applies as well to the entire real line R.  This space is complete in the standard Cauchy-sequence sense, yet it too is “incomplete” in a way, namely, in the sense that an infinite sequence might have no convergent subsequence (R is not 'sequentially compact', as they say in the trade):  the series (such as 1,2,3, …) may march off forever towards infinity, but “infinity isn’t there”.  We can both ‘complete’ and compactify it  by adding a “point at infinity”, replacing the standard metric with a bounded one (the resulting space being homeomorphic to what we started with), and then “round it around” to a ring-shape as before.

You see where we’re going with this.

Ah, but do you.  For mathematics has latterly progressed in ways considerably more intricate than simply sharpening our intuitions of infinity, so that, when we say that “God is infinite”, we can have something much more incisive in mind than simply “way bigger than an elephant”, with which our grandsires had to make do.  For geometry has been algebrized: beginning with Descartes, but zooming off in unexpected new directions with algebraic topology.

We have seen that there are varying ways of compactifying a given space.  In the context of Universal Algebra, a question arises:  For any given space, is there one way that is, in some sense, universal or canonical -- the “Mother of all compactifications” (to speak with Saddam Hussein)?  Indeed there is:  it is known as the Stone–Čech compactification. The result is universal in that any continuous map whatever, from our original space to a compact Hausdorff space, can be factored through the Stone–Čech compactification.  (Thus, the closure of (0,1) into [0,1] does not rate as Stone–Čech, since e.g. sin (1/x), defined on the open interval, does not extend to the closed.) -- Whoever can grasp this, will never consort with Nominalists again.

We have considered this matter in a particular area of point-set topology, but the notion of universality, as made precise by this notion of lifting a given map to procede through the universal, is quite general -- hair-raisingly general, in fact.  In general, “a morphism [is said to be] universal  [iff]  any other morphism into a system with this property  factors uniquely through the universal morphism.” (Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. 129.)

~   ~   ~

So much for the math.  And now for our dominical metaphor, offered in all humility.

We are, according to Scripture, but now also in a sense which might possibly someday be made relatively precise, made in (or better:  from) the image of our Maker.  Only, not visually (that were absurd, and gives rise to all the idolatries), nor yet (abstractly, or spiritually) isomorphically,  but rather: homomorphic images, of various types and sizes.  (Bonus:  homomorphic now becomes a graeco-latin pun.)  Whatever can apply to us, can apply to and through Him, in a manner made familiar by Category Theory.

And by what seems a kind of anticipation of the functorial view, the Historical Church chose precisely universality as its defining epithet:  catholicus.

(Yet who are these, streaming across the blasted landscape in despair, the wretched remnants of their mockeries  strapped to their backs?  Why, ‘tis the very tribe of atheists, quite put to flight!)

Within Set Theory, there is a notion reminiscent of all this:  the Reflection Principle.  It is very counterintuitive -- but then, so is life.

~

Appended Epigram

That God is simply the sum of All that Is, is mere pantheism.  We shall posit rather, that He is its Stone–Čech compactification. 

(Here we tread, not on dangerous, but on spongy ground, the sort that led into the swamp of the ‘God particle’.

Various defenses spring to mind, but I have a feeling that they are self-serving.  Taceamus igitur.)

Similar to our image of the lower thing being the homomorphic image of the higher:

The highest things often have “footprints”, as the medievals put it, among the lower things.

-- James Schall, S.J., The Order of Things (2007), p. 22

~

(All right, now we do something very wrong.  But my character, sapped by whoring after epigrams -- e’en as the bard  was slain by a pun --  cannot resist.

An early post against ultra-Darwinism  mentioned -- purely in passing -- the Urysohn Metrization Theorem;  after which, to my embarrassment, this site received a number of serious enquiries after that worthy result.   Actually  it was kind of cool.  And so, to accommodate surfers who are mathematically advanced but lousy spellers, we add these:

Stone-Cech

Stone-Čeck

Stone-Ček

Stone-Czech

Stone-check

Stone- tchèque

Stone-Tscheck
pStone-pČech  [the p is silent ...])

~ ~ ~

All that is rather by way of somewhat remedying the obvious insufficiences of St Anselm’s Ontological Argument, while yet retaining sympathy with his project.

The images/metaphors  of the Scala Naturae, and the Ladder of Abstraction, both point ever-upwards, as if to some final lodestar or ultimate Utmost, without  of course  proving the existence of any such thing.  There is also something empirically amiss, in that both visions are linear -- and reality is generally not like that.    More to the point would be Partially Ordered Sets -- and that gets us straight to the door of Zorn’s lemma:

Suppose a partially ordered set P has the property that every totally ordered subset has an upper bound in P. Then the set P contains at least one maximal element.

Now, that Maximal Element -- remind you of Anyone?

Stairway to Paradise

This is a more robust analogy than that of the long extension-ladder, but it probably won’t buy us anything of theological import.   Note in particular that the various upper bounds referred to must lie in P:   P is already complete.   Whereas a simile for the Godhead would more likely be along the lines of Inaccessible Cardinals, or Proper Classes,  ever beyond iterative reach.

C.S. Lewis drops a remarkable aside, in the final paragraph of his essay “The Language of Religion”:

I sometimes wonder whether the Ontological Argument did not itself arise as a partially unsuccessful translation of an experience without concepts or words.

-- Christian Reflections (1967), p. 141

(Nota bene:  There are intellectual as well as emotional such experiences, as in mathematical insight -- at least, without words.  Brouwer once characterized mathematics as “an essentially languageless activity of the mind”.

More here.)

Lewis’s essay, incidentally, is  gem, developing at length  an idea he has often sketched, concerning the evolving adequacy of language to non-everyday puzzles like theology and math.  In that spirit, we have offered a couple of vizualizable new analogies to play around with:  Universal Compactification, and Partially Ordered Sets.

Lewis’s linguistic point is continuous with his opposition to intellectual “Whig history”.   Thus, if our ancestors spoke of God as though He had a white beard, and depicted him this way in art, it is not because they were morons;  indeed, such a depiction did not, at the time, constitute an asserted denial of the thesis that God is incorporeal:  for that later thesis simply lies (intellectually and chronologically) beyond the original level of discussion.

(In similar fashion, if I state that “the red vehicle was stationary at the time of the collision", that is not meant to deny the thesis that the earth rotates on its axis, and moreover revolves around the sun.)

Exactly the same point can be made with respect to the praxis of mathematics.  (I mean its ever-evolving practice by actual mathematicians, rather than the arguably  timeless, transcendental truths of Mathematics itself, as it resides in the mind of the Creator.)

Thus, Wikipedia (re Imre Lakatos):

Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysis. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.

Lakatos’ dialectical insights are worked out at length in the multisided dialogue (a ‘polygonal’ conversation, as it were), Proofs and Refutations.

[Update April 2017]  I had rather hoped to have added a “Footnote to CSL” with that shtick about creatures as homomorphic images (of various cuts and complexity) of their Creator, a more flexible metaphor than Lewis’ example of the faces of a cube.  But upon re-reading his essay “Transposition”, I learn that Transposition is his term for much the same thing -- he even uses the term algebraic in that connection.  The whole idea is worked-out exquisitely in that place.

Categories and Sameness (with linguistic appendix)

The free product is a sum in the category of groups;

and the direct sum is a sum in the category of R-modules.

-- old folk saying

We earlier had some fun with (in effect) the notion of hierarchical and cross-cutting patterns of Natural Kinds, in the essay Categories for the Working Mom.  And now that the laundry is fresh and folded and tucked away, we can all relax with a cup of your favorite beverage, and take another look.

“Feynman,” said Wheeler, “I know why all electrons have the same charge and the same mass.”

“Why?” asked Feynman.

“Because," said Wheeler, “they are all the same electron!”

-- Martin Gardner, The New Ambidextrous Universe (1964; third revised edn. 1990), p. 305

Less Zen:

Particles transforming in the same representation of the Poincaré group  and having the same additional quantum numbers  are said to be identical particles.

--  Matthew Schwartz, Quantum Field Theory and the Standard Model (2014), p. 207

~

Linguistic philosophy is familiar with the caveat that to say two things are “the same” must be interpreted under some description.  (The technical name for this:  sortal identity.)

Thus, it is commonly said that the Chevy Whatsis and the Toyota Whomever are the “same” car, because they are built on the same chassis and are mechanically identical, just differently branded.  Now imagine an automotive expert given a pile of paired photographs, each depicting one vehicle; for each pair he must give thumbs up or thumbs down as to whether the vehicles are the “same” in this sense.  He will give thumbs-up to a Whatsis/Whomever pair, though now his assessment is in two steps:  First identifying the particular vehicles depicted as a specific representative  each of a specific brand;  and next, assessing those two models as being “the same” under the applicable metric.

More narrowly -- this, say, for a car-dealer rather than a mechanic -- two vehicles are “the same” only if they are in the same SKU -- a Ford Taurus now being categorized separately from the Mercury Sable, though they are the same for most practical purposes.

More narrowly still, two examples of a Ford Taurus might be accounted equivalent  iff -- we are back in the auto-repair shop -- they are both in the same set of models, these being drawn up according to which parts they require.  The sets will be relative to the part in question:  this group uses drive-train X1, this other requires X2; cross-cutting these, another group uses such&such style of alternator, another another…

Even more narrowly:  the police want to know whether this vehicle sitting here in Arthur J. Mungo’s garage  is the same as the one used as the getaway car in last night’s robbery. 

More narrowly still:  I buy a certain car from you, new, for $20,000.  Five years and many fender-benders later, I demand you buy it back for the same price -- after all, it’s the “same car”.  And indeed, in the sense immediately above, it is, and will still serve to convict you if the robbery happened five years ago rather than last night.  But as seen by the eyeshades of an insurance company, it is not the same at all.

We could keep going, until  ultimately  the only pair of photos that will pass the green-eyeshade test  is ... two identical photos.

I have my mother's eyes ... He has his father's nose

Epigrammatic reflections of the paradoxes of ‘sameness’:

The successive unstable cabinets of the 3^me République, were mostly a reshuffling of the usual suspects.  Thus,

There was some truth in the quip of Clemenceau, when he was criticized by a deputy for having overthrown so many governments.  “I have overthrown only one,” he replied.  “They are all the same.”

-- Wm Shirer, The Collapse of the Third Republic (1969), p. 101

A similar use of same in the sense ‘same-old’, mocking Anglo-American journalists who take the Russia tour and become pundits:

After a year  they went home … sat down at their typewriters, and hastily wrote the same book.  I have been reading that book all spring, under several different titles.

-- Malcolm Cowley, The Flower and the Leaf (misc., collected 1985)

(Here, the determiner "that" actually becomes witty, in context.)

 

These recall Wheeler’s quip about the electron, although there the meaning was more subterranean, having to do with fermion statistics;  cf. clonal colonies of trees:  One aspen, or many?

[Update -- a note to our readers.
It has come to our attention  that a number of you have reached this oft-viewed post   under perhaps a misapprehension:  not out of ontological or taxonomic curiosity, but  by searching for a jpeg of "Homer strangling Bart".   This delicious image  we do indeed offer below;  but it is by no means the meat of the essay.  For those of you whose interest runs to cartoons rather than ontology, you may consult our meagre offerings here:
http://worldofdrjustice.blogspot.com/search/label/cartoons ]

~ Sigmund Freud  und  Sherlock Holmes: ~

~ ~ Seelen-vertausch! ~ ~

~     ~     ~

In our discussion of “analogy” and “sameness” in the essay Consilience in Mathematics, we saw that this informal term gets formalized in various ways -- isomorphism, homeomorphism, diffeomorphism, bijection … -- that, despite the variegated terminology, are really just one central idea, namely:  detailed pairings (“maps”) between objects, which preserve structure of some sort.  Which sort, depends on the mathematical category you are working in.

*

Commercial Break

A private detective  confronts the uncanny;

an ecclesiastical mystery:

Murphy Calls In a Specialist

[Kindle] [Nook]

~     ~     ~

Mathematical structures being immutable and eternal, there is no problem about cross-identifying them across time.  But in the peopled world, things change all the time -- or rather, as we now are careful to notice:  change under certain descriptions, and not under others.

The following is an excerpt from an essay, “Continuity of Identity”, which I hope to post someday.

Here’s a really practical case, of current political relevance.

In the year 19xx, a group of local politicians, and a state or municipal employees union, agreed to keep current wages at a specified modest level, against the promise of lavish pensions in the future:  the latter to be paid by “the taxpayers”.

Now, many of these deals were hatched behind closed doors.  Still, there was little pressure to expose them, for “the taxpayers” ca. 19xx  did indeed benefit from this arrangement:   Their taxes were lower than they would otherwise have been, had the services of police and fire and teachers and what have you had to be bid for on the basis of straight salary and current benefits;  these services were acquired relatively cheaply, the full bill being deferred.

And now, circa half a century later, “the taxpayers” are being asked to pay that bill.  Only… they are not the same people who benefited from the arrangement half a century or so ago-- in many cases, before they were born.  Not the same… as individuals;  but as a corporate body, yes, these are indeed “the taxpayers”.  So: legally, morally:  Can a politician and a union seal an agreement that binds, not themselves, but some third parties in the future, who at the time of the agreement were not even born?

Before you answer too hastily with a resounding outraged No, consider that our entire society is webbed with agreements exactly like that, and would likely fall apart without them.  Every time our nation signs a treaty, or issues a bond, or even launches a project like building highways and bridges (which future citizens will need to keep in repair, or much of the money will have been wasted), we are doing this.

~     ~     ~

An incident earlier this evening reminded me of an old conundrum:  the seemingly uncontroversial matter of what constitutes the same tune.

I happened to overhear on the radio, a snatch of a tune of which I’m very fond, of which I did not know the name.  It is melodic, and simple  apart from some rhythmic oddities.  Emotionally, its effect on me is like that of the brief beautiful interlude in waltz time, in Corelli’s Christmas Concerto.

As the piece segued into the next movement, I realized it had to be Copland, probably quoting someone else -- as he quoted “Simple Gifts” in Appalachian Spring.  Fortunately the announcer eventually came on and said the overall suite of which that piece was a part  was Rodéo. 

Looking at the program for this, I was puzzled.  “Corral Nocturne”?  Surely, for the dreaminess.  But no, it turned out to be -- “Saturday Night Waltz”.

Now, I would never have thought of searching for the piece under that name, since, firstly, it didn’t at all seem to be what some farmer-cowboy couples would be up to on a Saturday night, and futhermore -- okay, this is embarrassing -- I did not realize it was a waltz.  There’s an insinuating syncopation;  certainly if I ever tried to actually waltz to it, I would fall on my punkin haid…

Seeking to learn the mystery of this exotic composition, I got together a safari team -- seasoned explorers and native bearers -- and made my way through malarial jungle and crocodile-infested swamps  over the space of many months, finally scaling an icy peak and putting the question to the loincloth-clad hermit who sits at its summit and --  Well, that’s how it would have been in the old days;  to save time, I simply looked it up in Wikipedia.

And learned, to my astonishment, that the quotation in question is from … “I Ride an Old Paint”.   Astonishing because I absolutely grew up on that song, as sung by Burl Ives.  Our family only owned a half-dozen vocal recordings, so I listened to them over and over; and of these,  “I Ri-ide… an old Paint;  I le-ead…. an old Dan” was one of an even smaller handful at the top.  Never did learn what the lyrics meant, but I heard it and sang it   over and over.  And, being told -- like a puppy by the scruff of the neck -- that the tune right in front of me, in Copland’s ballet, is that; well, yes, I can hear the resemblance.

How could I not have heard the sameness in the first place -- the sameness beneath all the strangeness?  The simplest hypothesis is:  Musical retardation.  To which I partly plead guilty, only -- there are other, even grosser cases of mental compartimentation, which require a different explanation (nobody’s that retarded).

Thus, from childhood:  Consider “Twinkle Twinkle Little Star” and the Alphabet Song.  The tunes are identical, note for note;  failure to perceive similarity might be chalked up to incapacity, but failure to note identity (which was my failure until recent times) -- refusal to assent to "A = A" … something else is going on.

A hint at what this is, is provided by cases in which we children were not mere consumers, but producers of isomelodic products.

Thus:  “Nyaah, nyahh, nya-nyaah nyaah.”

And, on the same ‘tune’, any number of smug worthless despicable taunts:  “TIM-my’s Got a GIRRRL-friend…!!!” or what have you.

[This just in -- probably from the Vatican:  

Homer Simpson, delivering a stern lecture to his wayward boy

The taunts here cited, are sufficient proof that the Fall of Man spares no-one, not even little children, fancied by sentimentalists  to be innocent.   From a secular perspective, such productions suffice to justify strangling the little bastards in the cradle.  That such action is not in fact justified, is a miracle, unexplained by any reasoning within the purely secular realm.  The overriding fact is, that God loves us -- Lord only knows why…]

Clearly, the melodic medium and its message form a gestalt.  To the child, and to some extent later, it would be as artificial to peel off the tune, as to say that two quite distinct people are actually identical (as regards having two eyes, a nose, etc. etc.), their secondary differentia being whatever is left over.

The nec plus ultra of such a perspective  is spoken sentences:  for these each have an intonation which is partly conventional, like a tune:  differing between French and English, or between British and American English.  Hearing a sentence, we do not  on any level  equate it  with all the (thousands of) other sentences we have heard  similarly intoned.

~     ~     ~

To avoid the distraction of same in the sense ‘self-identical’, let us continue the discussion using the term equivalent.  The latter is actually more mathematical, in that there it has a special sense, that of equivalence classes.  (Check Wiki and read all about ‘em.)

So:  Items will be equivalent or not, depending on whether they are viewed sub specie this category or that.  As, in biology:  When working with multiple species, two individuals are equivalent if conspecific.  When working within a species, two individuals might be identified  if, beyond being conspecific, they are of the same sex, and -- for species which show distinctions such as larva vs. adult, sessile vs. vagile, pupa/chrysalis/imago -- in the same stage of life.

*     *     *

~ Commercial break ~

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

We now return you to your regularly scheduled essay.

*     *     *

In mathematics, two objects can be topologically equivalent -- homeomorphic; smoothly equivalent -- diffeomorphic; algebraically equivalent -- isomorphic;  metrically equivalent -- isometric; numerically equivalent -- equinumerous; and so forth.  An example of such usage, in a Hilbert space context:

…the latter direct sum itself is isomorphic (unitarily equivalent) to H …

…It is unitarily equivalent (I might as well say that it is the same as) the bilateral shift.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), pp. 7, 12

[Remarkably, Wikipedia seems not to devote an article to this subject, the closest being this rather abstract one: http://en.wikipedia.org/wiki/Equivalence_of_categories

So I’ll say a little more.]

Further, we speak of two norms on a space as being equivalent if they generate the same topology on that space.  Two group-representations are equivalent if they are interrelatable via conjugation by a nonsingular matrix.  Two signed measures are equivalent if each is absolutely-continuous with respect to the other.  Two vector bundles are equivalent if they are related by a homeomorphism that preserves fibres. Two normal series of a group are equivalent if there’s a bijection between the factor groups with these being isomorphic.

There are even finer distinctions:  Two set-theoretic structures are “equivalent if just the same sentences are true in each, and elementarily equivalent if just the same first-order sentences are true in each.  Isomorphic structures are evidently equivalent, and hence  in particular  elementarily equivalent.” -- Michael Potter, Set Theory and its Philosophy (2004)

As it begins to appear, these different labels are not simply definitional and contentless;  sometimes you arrive at them only after doing some work.  E.g. one reads:

A fundamental fact of differential topology is that the notion of isomorphism in the categories Top[ology], P[iecewise]L[inear] and Diff[erential], is the same in dimensions three and below.  In dimention four, PL and Diff agree, but Top differs.  In dimensions above six, they all differ.

In particular, things go hog-wild in dimension seven.  We thought we knew what roundness was, it turns out we did not.

~ ~ ~

[Update 3 Dec 2011]

This phenomenon of tunes known  but hiding their identities beneath disparate lyrics and thus subjectively unconnected,   deserves a name:  we shall dub it cryptomelodia. 

(©2011 Dr J Worldwide Enterprises Inc.)

[Pronounce this krip-to-meh-LO-dee-a.  The word is modeled upon that other stupendous vocable, cryptomnesia ‘hidden memory’, from crypto- ‘hidden’ and -mnesia ‘memory’, as in amnesia (i.e. a-mnesia ‘no memory’).  Those roots are all Greek;  and I wanted to neologize on a proper Greco-Greek model, rather than conjure up some vulgar Greco-Latin hybrid-hippogriff such as Donald Trump would no doubt coin.  But the Latin-looking aspect of “melody” gave me pause. 

So we consulted with Dr Massey, the official philologer for this site, and he reassured us:

Melodia is of Greek origin.  μελῳδία : from melos, musical phrase, and aoide, song. Cryptomelodia is a great rendering.

This is excellent news.  Dr Massey and I shall split the royalties from this outstanding new word, and retire in opulence.]

A startling example of cryptomelodia occurred at work yesterday.   I work at a pretty patriotic place;  the political spectrum among the employees is much broader than you might imagine, ranging even remarkably far to the left, especially on antiwar issues.   Still, I was startled that morning to suddenly hear the song -- or I should say, the tune -- of “Solidarity Forever” blaring from a computer in the neighboring pod.  I leapt from my seat to see what was going on.  Had the Occupy movement spread to us -- Occupy the Fort ?   -- No, my companions scoffed, that’s the Georgia fight song.  (The woman whose computer harbors this tune on boot-up is from Georgia.)

That only raised another issue:  Why would Georgia -- a pretty conservative neck of the woods -- choose “Solidarity Forever” for their fight song ?  -- Actually, they replied, it’s the fight song of lots of schools.  And then it dawned on me:  The words being different, and the setting (a football stadium) utterly so, probably very few of the fans make the connection -- if indeed they have even heard of the old Wobbly anthem.  (And if they had, they might deem it ironic that the hymn of One Big Union should be hijacked  by gridiron jingoism.)  The tunes inhabit different compartments of the mind.

There was some argument over whether which song actually came first -- some voted for a football origin, which I thought absurd.   So upon reaching home, I burnt some incense before the altar of the All-Knowing -- and Wikipedia promptly informed me that the tune goes back  not only way before football, but way before the IWW:   that I have myself been the victim of a multiple cryptomelodia in this regard.  For, “Solidarity Forever” is simply a re-lyriced “Battle Hymn of the Republic”, which in turn lyrically recycles “John Brown’s Body”.  I am of course quite familiar with both these songs, having sung the first one  in particular  many times in elementary school.  But I never made the connection among these three.

 

(And upon further reflection … the transposition  to the football field  of the tune that’s held in common by all these anthems, is appropriate after all:  since all of them are  in some sense  fight songs.)

As for the tune itself -- its origins are lost in time;  Wiki traces it to “the folk hymn tradition of the American camp meeting movement of the 19^th century”;  before that, all is mist.

Yet hark -- what lyre upon th' aeolian there wafts ?  Nay, ‘tis Aeneas’ bark, plunging westwards from the flames of Troy !  And what air sets the oarsmen bend their strength as one?  Why -- ‘tis … ‘tis that very tune !

~     ~     ~

[Further examples of the notion ‘same’, this time in linguistics.]

If we project backwards to the Pre-Latin form of a third-conjugation infinitive such as dûcere, we note, for Pre-Latin *doukesi,  formal identity with the locative case of a genuine Indo-European s-stem noun.

-- Robert Jeffers & Ilse Lehiste, Principles and Methods for Historical Linguistics (1979), p. 67

Here meaning:  phonetically identical, though with an ultimately different grammatical role.

Another extraordinarily cavalier positing of equivalence:

Chinese and English, for example, may have the same case system as Latin, but the phonetic realization is different.

-- Noam Chomsky, New Horizons in the Study of Language of Mind (2000), p.

(This recalls an anecdote from Charles Fillmore, author of the classic article “The Case for Case”, from back before the idea of abstract/semantic case was as familiar as the traditional notion that hewed to overt morphology.  He began presenting a paper on “Case in Chinese”, using this expanded or deepened sense of the term;  and his elder examiners, leaning forward in alarmed concern, said: “Aren’t you aware, that Chinese doesn’t have case?!?”)

~

… ‘free variation’:  If two forms are equivalent (that is, freely substitutable for each other), their alternating segments may nevertheless be phonemically distinct.

-- D. Hymes & J. Fought, American Structuralism (1975), p. 214

(Compare that elegant phrase, which slips so smoothly from the pen of Quine:  “substitutable salvâ veritate”.   I have here further spiffed up the expression, by giving the â its neat little ablative hat.)

[Update 12 June 2015]
http://worldofdrjustice.blogspot.com/2015/06/the-latest-in-math-porn.html

“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)

*     *     *

~ Commercial break ~

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

We now return you to your regularly scheduled essay.

*     *     *

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

We are by now familiar with the idea that some notions cannot be limited by a definition;  Wittgenstein classically makes this point in his discussion of games.  Mathematics is a sprawling field of activity  and lapidary characterizations are necessarily impressionistic.   Our chances are rather better when it comes to subfields or branches of mathematics.  

Some of these “definitions” are really more by way of elucidating epigrams, and come with appropriate caveats.  Thus:

Roughly speaking, the 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

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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!")

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[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?