On Crunchy Numbers

In the Ike era, we grew up on Wonderbread® :  a sort of Brot ohne Eigenschaften whose edulcorated transmogrification is known as Twinkies.

Since that time, we have learned to abjure, not only such treif, but anything not calling itself wholegrain.   Or, better still, multigrain: some brands boast seven grains, a few claim twelve; disparate mixtures  full of gritty, grainy, crunchy goodness.

 

Now  our local upscale supermarket offers a variety of own-branded bread, that boasts (in large letters) no fewer than

27 GRAINS

That really surprised me.   It’s one of the main points of Jared Diamond’s Guns, Germs, and Steel  that digestible, domesticable, feasibly growable grains  are not to be had for the asking;  there just aren’t that many of them.   The explanation is that, a little lower down and in smaller font, the label reads

AND SEEDS

In other words, this brand of bread is equally at home in the bakery and in the bird-feeder.

Our son, scoffing at this terminological legerdemain, inquired why the market chose “27” of all things.   Unhesitating I replied, “Because it is three to the third power -- the trinitarian pinnacle of the Perfect Cubes”.  -- Said heir and offspring, himself a nascent mathematician, appreciated the point, but doubted that your average shopper was aware of such things.

And indeed, there is a larger point.   For number-theorists like Ramanujan, each integer has its own flavor, its own biography and backstory -- cf. the famous incident of the taxicab numbers (in which G.H. Hardy plays the straight-man or fall-guy).  But for ordinary folks, like rocket scientists (who deal with contingent analogue quantities, rather than integral transcendent entities) or English professors (surrounded by a midge-cloud of  pre- or sub-arithmetical post-modernists), all but a very few -- small -- integers  will be featureless.   Some will be visually familiar:  3 and 4 (the triangle, the square), 5 (the quincunx on dice), 6 (boxcars, ditto),  7 (a “lucky” number for the superstitious) or 23 (ditto, for the more cerebral), 10 (count your fingers), 20 (with its portrait of Jackson).   Some will be familiar for incidental, non-mathematical reasons, like 100 and 1000 (which owe their prominence to the accident of base-10 notation -- a case of decimal fetishism).   --  Those associations are widely shared;  but there may be others  more individual;  as, (mostly for girls), “sweet sixteen” (or in Latin America, la quinceañera).   Indeed,  any integer small enough that you have lived that number of years (particularly those for which you still kept track of your birthdays).    As, the poem(-collection) “Now We Are Six” (by A.A. Milne), an anniversary in memory still green (and which I teach to the neighborhood children when they reach that delightful threshold).  Or…. 27;  which, even before I had attained that age, always seemed numinous (probably, indeed, for its prime-power nature), and which was ratified as such by my marrying at exactly that age, quite close to my birthday.    Whereas, for most folks, a number like 81 (pourtant a perfect square, as well as  3^2^2, to boot) tells no tale, sings no melody.

~

The preceding remarks are of psychological or anthropological interest, but of no moment for mathematics itself:  They present an external, human-centered view of the integers.  But further considerations suggest a subtle distinction between two ways a given integer may be “interesting” (a more bloodless equivalent of our anthropomorphic term crunchy).   We might dub these internal and external:  both times internal to mathematics as a whole, but in one case only, internal to number theory in its most elementary sense.

Thus, consider 5.  Number-theoretically, it’s a prime and that is pretty much that; but in the geometry of three-space, it is the number of Platonic solids.  Or 17:  Number-theoretically it is both a prime and a Fermat number; but in the geometry of two-space, it is the number of crystallographic planar symmetries.  

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

Or: 230, the number of space groups.

Here, the integer in question is not a creature or crystal considered distinct and in itself, but a stopping-point one arrives at by calculating and counting.  There might have turned out to be, say, 18 plane symmetry groups, without upending the world; but the internal structures of 17 and 18 are unrelated.

~

This dichotomy of internal versus external interest  chez the integers, represents ideal poles, which are not exhaustive, but bookend a spectrum.  As, probably intermediate: the “crunchiness” of natural numbers as viewed by students of finite groups.    (Here the masticatory metaphor  returns unbidden, for I always imagine finite groups along the lines of a wrinkly-surfaced walnut.  Finite simple groups -- those for which no homomorphism can split out any subchunk as its “kernel” -- the hardest nuts, the ones you can’t crack.)  Simon Norton seems to have had such an intimate gustatory appreciation of individual finite groups, before he Threw It All Away and went off to ride the bus.

~

In light of our training, we know at least one thing to do, should we ever be given an unfamiliar integer and locked in a room, with no other toys to play with.  Namely, we can probe for primality.  (An activity that becomes, indeed, crucial, for cryptographers.)

But now imagine that instead we had been handed some fraction like

      (2 +  √137) /4081

Untrained, we react with dismay.

Yet later, learning of the Golden Section, (1 + √5)/2, and its many remarkable properties -- not the least of these being that it can be represented as the infinite continued fraction consisting of nothing but ones -- we conclude that the critter is crunchy indeed;  and that there are more things in heaven and earth, than are dreamt of here below, and that we must anticipate the afterlife, before we could begin to embrace them.

~

Our treatment focused on what the innumerate are missing, much like what the Daltonist, unbeknownst, lacks of the hues.   But there is such a thing as unearned crunchiness --  a bogus significance assigned to certain numinous numbers, like 19 among the Baha’i’s; the “23 enigma”; 666; 1000 AD as the Millennium; the mumbo-jumbo numbers in “Lost” and "Touch"; for all which, cf. Wikipedia on apophenia.   In the face of such things (which have snared some otherwise rational people -- a close friend of mine, critically brilliant but unmathematical, fell for the “23” business), we are inclined to say, along with Nulla  extra ecclesiam  salus,  that  Nulla  extra mathematicam  ratio.

~

For a rather recondite example of crunchiness, consider the ‘amicable numbers’  (  الآعداد المتحابة) reported by the medieval historian Ibn-Khaldun in his Muqaddima .   Apparently only two of these were known to the Arabic medievals (or: they are the only two numbers characterized by a theorem of Thabit ibn Qurra), and they are not much to look at:  220 and 284;  but they meant something to contemporary practitionars of the talismanic art.

The modern view is summarized here:

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

A mathematical scratchpad (even further scratched)

[There simply isn’t time, at least before retirement, to integrate each thought-balloon as it bubbles up -- a proto-insight or pre-idea -- into the appropriate essayistic context in finished form.   Yet to leave these on the desktop equivalent of a desk drawer is to tempt the Reaper.   Therefore I shall place some of them here -- philosophical post-it notes;  mathematical Zettel.]

[Cf. De stultitia]

Our focus  in the essay of that name, is on the plight of those sorry souls (99.9999999 % of us) who fail to grasp what Grothendieck, or Witten, or whom-have-you, saw easily enough.

Distinct from, though related to, this, are questions of which (at the forefront of science) we are permitted a glimpse,  but which nobody understands.  As:

On cosmogenesis:

The whole vast imposing structure  organizes iteself  from absolutely  nothing.

This is not simply  difficult to grasp.   It  is    incomprehensible.

-- David Berlinski,  “Was There a Big Bang?” (1998), collected in :  The Deniable Darwin (2009), p. 229

And:

All this leaves us  where we so often find ourselves.  We are confronted with certain open questions.  We do not know the answers, but what is worse, we have no clear idea -- no idea whatsoever -- of how they might be answered. 

But perhaps that is where we should be left:  in the dark, tortured by confusing hints, … and a sense that, dear God, we really do not yet understand.

-- David Berlinski,  “God, Man, and Physics” collected in :  The Deniable Darwin (2009), p. 270

~

The hardest part of a subject is the beginning.  Once a certain stage is passed, we gain confidence  and feel that, if need be, we could carry on by ourselves.

-- John Synge & Byron Griffith,  Principles of Mechanics (1942, 1959), p. 506

Alas, that has not been my experience at all.

Any technical subject is like a whirligig, which rotates faster and faster until the centrifugal force throws you off.   It’s like the Peter Principle, everyone eventually reaching his own personal level of incompetence;  only, in math and in physics, these levels stack indefinitely towards heaven, so that a few of us can ascend quite a ways, before we are finally out of our element.

[Cf.  Any Ideas? ] 

Recent years have seen striking developments in the conceptual organization of mathematics.  There developments use certain new concepts  such as “module”, “category”, and “morphism”  which are algebraic in character.

-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. vii

The reason they speak here of new “concepts” rather than additional structures  is that the notions referred to do not exist merely within algebra, but serve to organize other mathematical fields as well.

~

In an exterior view of the finished product, we see structure mathematics as largely logical or deductive:  P entails Q.

But from the interior standpoint of the practicing mathematician (and here, though we refer to the ‘actio’ sense of mathematicizing as opposed to the actum or product, the interest is not psychological but ideational), a key verb is rather motivate:  P motivates Q.   An illustration of this special vocabulary:  “The desire to extend Fourier L² to L^p spaces  motivates the Riesz interpolation theorem.”

~

More vocabulary from the conceptual domain:  thrust, as in the following passage

The Heisenberg uncertainty principle:  The mathematical thrust of the principle can be formulated in terms of a relation between a function and its Fourier transform.  The basic underlying law, formulated in its vaguest and most general form [i.e., its most intuitive formulation], states that a function and its Fourier transform cannot both be essentially localized.

-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p. 158

~   ~   ~

[Cf.  On Depth]

When I made my original discovery of radiation from black holes, it seemed a miracle that a rather messy calculation should lead to emission that was exactly thermal.  However, joint work with Jim Hartle and Gary Gibbons  uncovered the deep reason.

-- Stephen Hawking, in: Stephen Hawking & Roger Penrose, The Nature of Space and Time (1996), p. 44

For the mathematician, contrasting with messy  are simple and elegant -- yet in the following, even these don’t get you to the yonder side, where Depth dwells:

Having derived the equation for the vibrating string, we now explain two methods to solve it:

(1) using traveling waves;

(2) using the superposition of standing waves.

While the first approach is very simple and elegant, it does not give full insight into the problem.

-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p.  8

String theory is sometimes described as a theory that was invented backwards … People had pieces of it quite well worked out  without understanding the deep meaning of their results. … Math is funny that way.  Formulas can sometimes be manipulated, checked, and extended  witnout being deeply understood.

--Steven Gubser, The Little Book of String Theory (2010), p. 2

[Cf. Consilence in Mathematics]

Horizontal consilience:

… the structure theorem for finitely generated groups -- a fine illustration of conceptual unification.

-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3^rd edn. 1999), p. vi

Mathematics is a coherent, interlocking whole, and advances in one area  often lead to advances elsewhere.

-- Ian Stewart,  How to Cut a Cake (2006), p. 89

*

Commercial Break

A private detective  confronts the uncanny;

an ecclesiastical mystery:

Murphy Calls In a Specialist

*

[Cf. Language and Math]

Expressing himself in the language of fluxions and fluents, Newton managed to conceal his insights in a notation that was miraculously maladroit.  Not so Leibniz.  The language of mathematics and mathematics itself  are mutually sustaining.

-- David Berlinski, Newton’s Gift (2000), p. 57

Note:  The first clause of that observation does not actually relate to the point about notation (as opposed to vocabulary), and is silly in itself.  The concepts were new, so obviously any term for these would either be an out-and-out neologism, or a semantic hijacking of an extant word.   There is nothing lexically more rebarbative about fluent and fluxion than about derivative, differential, infinitessimal. 

*

[Cf. What is Mathematics.]

Betrand Russell, in The Principles of Mathematics (1903):

Mathematics is the class of all propositions of the form ‘p implies q’ …

The appended dribble of dots replace additional uninteresting clauses, which rob the sally of its epigrammatic pithiness, while yet failing to throw any light upon the subject.   It is a definition for people with no interest in the dark loamy richness of actual math as such;  and worthy of the author whose massive Principia Mathematica could as well have been titled Why Math Isn’t Interesting After All.

The characterization becomes even less interesting when you reflect that the expression “p implies q”, in logician’s lingo, is mere ‘material implication’ -- what would be better dubbed immaterial implication, since it involves no notion of causation or even logical entailment (and is thus immaterial to any actual problem), but is neither more nor less than another way of saying “either not-p, or q”.

[cf. Axiomatics in its Element]

Two citations illustrating the insight that axiomatizations, though perhaps logically prior, are pragmatically post-hoc:

The order of nature, and the order of logical dependence, are not the same as the order of our discoveries.

-- Morris Cohen & Ernest Nagel,  An Introduction to Logic and Scientific Method (1934)

Not all axiom systems are formal systems, and formalization need not lead to axiomatization.

The axiomatic method is an orderly way of summarizing experience.

-- Hao Wang, Popular Lectures in Mathematical Logic  (1981), p. 11

[Cf. Oligophrenia mathematica]

I am not a mathematician, but a math groupie;  not even a math wannabe (as I once was, back in Math 55), but a math wannedabe.

And in fact, considered coldly, I did not then  even rise to the level of a wannabe, but only a meta-wannabe, a wannawannabe:  someone who wished that his dearest wish was for mathematics, but who, truth to tell, was more interested in history and literature.

[Cf. Minimalism in Mathematics]

On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.

-- Robert Kanigel, The Man who Knew Infinity, p. 204

The reasons for this were twofold.  Ramanujan himself was not particularly aphoristic.   But he had never absorbed the modern notion of proof, which would take up so much more space;  and as a poor man in India, he suffered from a shortage of paper.

*

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in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

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*

[Sui generis]

On Ramanujan, who grew up in India, and in mathematics  was largely self-taught:

He was like a species that had branched off from the main evolutionary line  and, like an Australian echidna or a Galapagos tortoise, had come to occupy a biological niche all his own.

-- Robert Kanigel, The Man who Knew Infinity (1991), p. 61

The unusual career of Ramanujan  is one of the most celebrated biographies in the history of mathematics.   His achievements in the face of relative intellectual adversity as a child of modest means in the rural subcontinent,  are indeed inspiring, and warm the hearts of those in quest of Diversity -- whence, for those who can decipher the trobar clus of modern peri-academic patois, the book’s subtitle,  “A Life of the Genius Ramanujan”.   (Genius he indisputably was;  but the word these days is mainly used to celebrate anyone other than straight white males -- a “genius at basketball” or whatever.)

Yet the larger lesson is not how divergent Ramanujan was, but how much in the mainstream of things:  He did not found a new field of mathematics, he worked within number theory.   And this fact in turn reminds us of two characterizations of math as a whole, on which we have often dwelt:

            (a)  It is not something we invent out of whole cloth, it is something we discover.   This must channel our discoveries, just as the facts of the actual universe  discipline physics.

            (b)  Mathematics has already, for at least two hundred years, been uniquely rich conceptually  among human endeavors.   In a landscape embracing Cantorian set theory, algebraic geometry, and topos theory, it is next to impossible to come up with something unprecedentedly deep and strange;  in any case, Ramanujan did not.   To return to the metaphor:  We certainly treasure our quirky friend the echidna;  but only to someone whose zoological experience extended no further than a European barnyard, would he seem all that aberrant.   In a world of social insects, benthic hypothermophiles, and communal slime-molds, the echidna seems like just one more furry friend.

~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(My name is Ramanujan, and I approved this message.)

~         ~

~

.

A Subcontinental Platonist

It is not only Europeans, or Christians, or inheritors of the Greek tradition, or those philosophically schooled, who have arrived (semi-independently) at a position of Realism in mathematics.

This, from a most engaging biography of the Indian number-theorist Ramanujan:

In the West, there was an old debate as to whether mathematical reality was made by mathematicians  or, existing independently, was merely discovered by them.  Ramanujan was squarely in the latter camp:  for him, numbers and their mathematical relationships  fairly threw off clues to how the universe fit together.  Each new theorem was one more piece of the Infinite unfathomed.  He told a friend:  “An  equation for me has no meaning unless it expresses a thought of God.”

-- Robert Kanigel, The Man who Knew Infinity (1991), p. 66

For more on the theme, click here:

http://worldofdrjustice.blogspot.com/search/label/Realism

[Footnote:  To Kanigel's credit, he has a section towards the end which examines even-handedly the possible influences of Ramanujan's spirituality upon his style of math.  The account is neither credulous nor dismissive -- that is all we ever ask.]

The Theometry of Paul Erdős

Paul Erdős was no atheist; rather, he had an adversarial relationship to God.  He pictured the Creator as hoarding all the best math proofs in The Book, not wanting to share.   Life is basically a board game played against this miser;  and though we cannot win, we can strive to keep his winnings to a minimum.

The adversarial relationship recalls the tragic case of Lucifer; yet nothing else about Erdos seems actually diabolical.   For Erdős does not oppose the Lord head-on, but at a bias.  Simply, Erdős sees the Deity, not exactly through a glass darkly, but under some distortion -- perhaps a projective transformation.

Now, the same transformation must apply to the rest of the celestial denizens.  We may surmise the results.  The angels get recast as mathematicians (not much of a stretch).  And the Holy Spirit turns into the Friendly Ghost, the S.F.’s mischievous sidekick.   So soon as the Big Guy isn’t looking, Casper scurries off, say to rural India, where he finds the lad Ramanujan puzzled beside a hayrick.

“Psst! You know how, you take the reciprocals of the squares of all the integers, and sum them all up?” (Whisper whisper.)
Ramanujan, suddenly:  “I see it!  Pi-squared over six!”

[Update]  It turns out Heine already wrote a sort of sotie on much this theme:
Die Götter im Exil

[Morphological appendix]

He had studied theology.  But if theology and theosophy, then why not theography and theometry;  why not theognomy, theotrophy, theotomy, theogamy?  Why not theophysics and theo-chemistry?  Why not that ingenious toy, the theotrope?

-- Aldous Huxley, Antic Hay (1923), first page

 

On What There Is (Whether or Not we can See it)

[This is a continuation of a thread begun here.]

[Update III 2013]  The following is an early essay, and somewhat jejune.  The essence of mathematical Platonism concerns, not objects, but objectivity.   We wholly agree with the philosopher Putnam:

It is possible to be a Realist with respect to mathematical discourse, without committing oneself to the existence of 'mathematical objects'.  The queston of Realism, as Kreisel long ago put it, is the questiopn of the objectivity of mathematics ...

-- Hilary Putnam, “What is Mathematical Truth?”, repr. in Mathematics:  Matter and Method (1975, 1979)

~

As a proof-text for today’s sermon on the visible and the invisible, we  may cite so sober-pated an empiricist as Locke (Essay Concerning Human Understanding, II.xxiii.5; p. 270 of the Penguin edition) :

‘Tis plain, then, that the idea of corporeal substance in matter, is as remote from our conceptions, and apprehension, as that of spiritual substance, or spirit, and therefore  from our not having any notion of the substance of spirit, we can no more conclude its non-existence, than we can, for the same reason, deny the existence of body.

Locke’s reticence concerning the hypostasis of matter was prudent, since anything positive he might have said, would have been severely undermined, first  by the later atomic theory (and its successive proton/neutron and quark extensions, though for philosophical purposes these latter developments are minor); then by the mass-energy equivalence; and finally, most radically, by the quantum theory:  by which point our intuitions of just about anything  have gone by the board.

-->

~

~  Posthumous Endorsement ~

"Were I alive today, and in the mood for a mystery,

this is what I would be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am John Locke, and I approved this message.)

~         ~

~

Locke goes on (p. 276):

It is for want of reflection, that we are apt to think, that our senses show us nothing but material things.  Every act of sensation, when duly considered, gives us an equal view of both parts of nature, the corporeal and the spiritual.

            We may concur with the great controversialist, and go him one better:  For in a way, the visible world has a more tenuous hold on reality than the invisible, in particular the mathematical.

            To say this, is no manner of skepticism as regards  the reality of what’s in front of our noses.  No, it’s there all right.  Not for us to second-guess what the Lord hath made and deemed good.  Doctor Johnson’s refutation of – not really idealism, more like nihilism – by giving a stone  a swift kick in the hindquarters, is final.  There is stuff all right; the problem is, are there things?

            For notice: We did not claim merely, nor did Kronecker merely grant, that there is some sort of number-stuff, some quantological porridge  – “there are numbers” like “there be dragons”, vague and unindividuated.  We posited rather (and also observe) an infinitude of neatly individuated entities, as different from one another as – well really, it is difficult to think of even a decent finite collection of physicals, that glitteringly differ among themselves so much as this.  A basket of apples, fine: some are knobbly this way and some are knobbly that; some are worm-eaten, some aren’t.  But to approximate the striking, almost shocking individuality of numbers – this one prime, this one a perfect cube, that one a taxicab number, and all the rest – you would need rather a basket of all manner of fruit, pineapple and breadfruit and durrian and pomegranate.  The seven brides for seven brothers are less distinct among themselves than the first seven integers (especially if we start with zero).

[Footnote: The tag “Taxicab number” springs from an incident in which G.H. Hardy, skeptical of Ramanujan’s apprently intimate acquaintance with the integers – despite a complete absence of formal schooling in the subject, he was on the same familiar terms with them, as Dr. Dolittle with animals – challenged him to find anything the least bit interesting about, oh, say, that integer there, on the number-plate of that cab, whatzit say – “1729 “.   Ho hum, not even prime.  -- Ah yes, said Ramanujan, with a familiar smile.  The smallest integer that can be written as the sum of cubes  in two different ways (1^3 + 12^3 vice 9^3 + 10^3). .  Wherupon he reached out and touseled its forelock, and fed it a hypercube of sugar.]

[Subfootnote: I was kidding, of course, about the integers differing among themselves more than most visible things.  Actual people differ more – but, note, not in their visible envelope.  The radical differences among people stem, precisely, from the realm of the invisible – from their minds, perhaps even their immortal souls.  Integers can’t compete with that.  They’re immortal, but they don’t have souls.]

            So then, what things are there?  The typical examples are:  This table, or this coffeepot.  But it is significant that these examples are usually things  that we (ourselves  created in the image of God, and thus rather already an irruption of the transcendent into the material universe) have crafted to our own ends:  this table, to hold our proofs of the Riemann hypothesis; this coffeepot, to pot our coffee.  Things get a lot more vague when you consider what we have nót remade: which is to say, most of the visible Creation  – wasteland, swamps (or rather: intermittently swampy territory, no license to individualize and pluralize just yet), and starry regions and intersteller detritus.  These things – or rather, this stuff exists all right, but where are the chiseled surfaces of the number “17”, where the English garden and terraced vistas of a really fine Banach space?  If Hilbert space were just a jumble of odd dimensions, mixed up anyhow, jutting out here and there like the spars of a shipwreck, we wouldn’t give a d*mn about it.

            All right, you concede, the physical universe, being all part of one quantum soup, stirred by the overarching Schroedinger equation, does not naturally individuate into midlevel objects.  Still (you contend), the elementary particles at any rate  are absolutely what they are,  and not another thing.  -- But unfortunately, once you get down to that level, new and worse problems arise.  Quite apart from the process, called “decay”, whereby particles spontaneously surrender their essence (to which one might sigh: “We all die…”), and even apart from the wave-particle duality, there are phenomena yet more puzzling. Granted an electron-neutrino is not vague like a swamp or a fog, but it does spend a certain amount of its time cross-dressing as a muon neutrino, so that we barely are authorized to assert, with the Bishop, that “everything is what it is, and not another thing”.  And as for bosons, suppose that they are staunchly now and forever bosons, still, the Bose-Einstein statistics require that no one boson can be separately and distinctly individuated from any other.  If the macroscopic world behaved like that, we would indeed retreat from our picture of the world as peopled by distinct individuals:  Tweedledum in practice and in principle indistinguishable from Tweedledee,  Hyde and Jeckyl randomly phasing in and out.

            Again, in a sense, the contingently-familiar furniture of the world  may be ontologically in worse case than the mathematical.  For, whatever’s familiar, we take for granted; we don’t look too closely into things.  Whereas every single mathematical discovery has been fought for  tooth and nail.  Gauss hid his discovery of non-Euclidean geometry, lest he be mauled by the Boe0tians; it wasn’t ready for prime time until it had been shored up from every angle, and soon even came in concrete models –Klein’s model and that of Poincaré, different visible photographic reductions of some robust pre-existent entity: which you can describe awry, or fail to describe at all, but which you cannot forever ignore.  It awaits you, like the lion.

            So: What value is the general testimony of the populace at large, swearing on a stack of People magazines, that the Real Things of this world are things like – bikini wax and lottery tickets and (oh, but I can’t go on, this is barely worth satire), -- whereas Gilbert space or whatever the hell it is  is just some cockeyed idea of a mad scientist?

            Well.  When it comes to things with which people are particularly familiar, we tend to credit their testimony  (“Tasty Twinkie, that”) and even their predictions-- “That’ll be Midge” or “He’s gonna go long” – particularly if the answer doesn’t really matter. (How about that, a quarterback sneak.  Well, whatever.)  But when it comes to intuitions about probability, or infinity, or angular momentum, or quantum phenomena, or the problem of induction, or the properties of the Cantor set, or the epistemological well-foundedness of what we hold (though typically loosely) in fact to be true, we are – not to offend any sensibilites, but – not to put too fine a point on it ---     but       ---

      ---     to-tal-ly f*cking retarded….

And by “we” I don’t mean:  with the evident exception of you and me and present company, and all of Rabbit’s friends and relations; I mean:  d*mn near everybody, with the possible exception of Feynman (when sober) and just possibly (though I have my doubts) Gauss.

            So, what’s going on in this visible so-called Reality, thing, here?  Ask an eyewitness; just don’t ask two of them, for they’ll tell you different things.  What’s Mary thinking?  No-one knows but Mary, and probably not even she.  What did Caesar say to Antony as they walked into the bar?  Wasn’t there; hard to recover. Nay, why did worm A, spurning the obvious attractions of worm B, chose rather to share its hermaphroditic slime with worm C?  Only another worm could tell you, if even (s)he(it). – Whereas:  Where lie the zeros of the Riemann zeta function?  This question is open to anyone who cares to investigate, regardless of race, creed, color, flavor, gender, nationality, chirality, sign of the zodiac, sexual orientation, membership vs. nonmembership in the National Association of Realtors, galaxy of residence for tax purposes, bodily composition (matter versus antimatter – a perfectly private question of your own personal space), -- height, weight, density, magnetic moment, Gaussian curvature, Euler characteristic (we absolutely do not discriminate on the basis of Euler characteristic  -- you wild ‘n’ wacky  K = -8 folks are totally welcome), …. human vice android vice klingon vice angelic biological status (archangels may participate, but don’t try to pull rank on the cherubim when it comes to the Riemann Hypothesis) … corporeal-status versus disembodied-cloud-of-intellect … existence within time or outside time or astride time, not a problem, yo, come one come all, we ride ‘em six to sixty, step right up, prove the R.H. and win a kewpie doll.

Riemann, Chomsky, Fido

‘Tis of great use to the sailor  to know the length of his line, though he cannot with it fathom all the depths of the ocean.  (Locke, Essay, I.i.6)

It was with the warmest empathy that we read the other day  of the collie with a 200-item Wort- und Spielzeug-schatz -- treasury of words and playthings.
[Note:  This essay was originally written in response to the report of a German dog-prodigy, in 2007.  For the latest such story, see http://www.nytimes.com/2011/01/18/science/18dog.html?hpw ]

As a semanticist, he is the perfect embodiment of what is traditionally known as the Fido-‘Fido’ theory:  this word refers to this object, and the way it gets its reference is – Fido grabs it in his mouth!  A saving detail, somewhat distinguishing him from the complexity of a cash register – or rather (since a cash register, in addition to popping up a digit when you press a key, can also calculate) from a mere inert pegboard with 200 labeled pictures – was his action when presented with an unfamiliar word: he fetched an unfamiliar toy!  Thus, to describe him, we need 201 lines of (utterly non-recursive) code:  if, then, else. (Game, set, match.)

 [For an alternate view of Fido and the Essence of Language, Cf.  Dr. Max Müller’s Bau-Wau Theorie, by Dr. Christoph Gottlieb Voigtmann (Leipzig, 1865).]

            Fido can fetch, but he doesn’t exfoliate.  The structure of his language is the structure of his toybox – a heap of unrelated items. If he could also speak – play with the words as he plays with the toys – magic might happen.  We gaze back with saddened understanding into the empty depths of his eager brown eyes.

(Per Wittgenstein, though, if he could, we’d be disappointed:  “Wenn der Löwe sprechen könnte, wir könnten ihn nicht verstehen.”)

            Empathetic, since we are ourselves in much the same predicament, faced with any subject for which we lack an inborn knack. In language we are all born-geniuses; in mathematics (most of us), born-morons. How poignant to hear the French, who tout their language as logic itself, stumble through the simple act of counting:

“….fifty, sixty…uhh..sixty-ten (soixante-dix)…mmm….four-twenties (quatre-vingts)….four-twenty-ten (quatre-vingt-dix -- I kid you not; and as to the spelling, sic)…”

or say they’ll meet you “today in eight” (aujourd’hui en huit; meaning: seven days from today).

            When  young, the mind at its most resilient, I put my shoulder to the boulder, majoring in math.  Over time  I have found it a labor of Sisyphus.  When I’m not actively pushing it, it rolls back downhill.  And even when giving it my all, I can only fetch things as they are pointed out.  I have never learned to chatter in math.

            By contrast, in learning new words, new languages, new styles, I really am standing on the shoulders of giants, feet firmly rooted in the innate.  No need to constantly “keep up” one’s Spanish.  It’s there.  When an author stretches your syntax – Nabokov or Proust – it gets digested, metabolized, incorporated into the mental flesh.

            At one level, mathematics is a language; and for a time, may give us the feeling of a similar mastery.  The notation is so powerful, it lets us deal with complex sentences with deceptive ease.  But I have found that the various gimmicks and shortcuts become a substitute for thought: I might scramble up to the next terrace, but then I had to pull the ladder up behind me, because the strength of understanding is only as long as that ladder.

            Thus: You learn, with understanding, why a certain integrand can be transformed into another, that lends itself more readily to integration by known rules.  But this understanding then becomes encapsulated, a black box.   Like the law of cosines or any other once-derived, once-felt formula, it becomes formulaic.  Whereas:  from a structure like “Bobby was scolded by his teacher” we get to “the man who is widely believed to have been credited with this discovery” intuitively, without cutting the ties.

            There are other things we’re really really good at, like real-time visual analysis.  The more you learn about what is involved, the more miraculous it seems.  When you realize the fragmented, ambiguous nature of the input, and the coordination required on our part, it is amazing that we can thread our way across a room without tripping over some tensor.

 ***

            One night, math studies years behind me, in the dark without the crutch of a textbook or chalkboard in front of my nose, I tried to recall what I once knew, and for the most part could not.  Floating facts and stray derivations, like isolated lines of remembered verse.  Then with the resolute despair of Descartes at his stove, I tried to discover: all right, what do I know, that isn’t mere memorization?

            It wasn’t much.  I could visualize, intuit, that 2 x 3 = 6, by picturing the boxcar pattern on a die.  This even yielded the commutative truth: 6 = 3 x 2 (just tilt your head to the side).  Fifteen was harder, but was accessible via a triplet of quincunxes, each quincunx mentally held in place with the fingers of one hand.  And already the commutation required a different pattern, one big quincunx, each spot a little triangle.

            Invention gave out by twenty-one.  One can picture a triad of “dotted boxcars”, but what is 7 + 7 + 7 (let alone 3 + 3 + 3 + 3 + 3 + 3 + 3)?  One can dully, dutifully count (which is merely mechanical, bearing no relation to the gut grokking of a quincunx as five), or recall “3 x 7 = 21” from the times-table – a mere boilerplate formula like “a stitch in time saves nine”.

            So, I got as far as boxcars, but shall never catch up to the taxicab, where Ramanujan spotted “1729” as the face of an old friend.

            Shapes in space are also hard.   (Again Locke, Essay II xix.13:  "In a man who speaks of a chiliadron, or a body of a thousand sides, the idea of the figure may be very confused, though that of the number may be very distinct.")
Once, dipping into a bit of knot theory, and finding that I soon had to haul up the ladder to proceed, I went back to square one, and tried to understand a simple knot in the same direct way that we understand a softball or a football.  I made a wire model of a trefoil, turned it every whichway, closed my eyes and followed it with my fingers.  Falling asleep, I would imagine myself (like Mr. Tompkins) on a roller-coaster ride  shaped just like that, sensing when another part of the track would pass above or below, feeling the centrifugal tug.  I can barely do it, strain though I may  -- I throw on the light in a panic and stare at the wire.

            Again Locke (Essay II.x.4):

The memory is very weak:  ideas in the mind quickly fade, and often vanish quite out of the understanding, leaving no more footsteps, or remaining characters of themselves, than shadows do  flying over fields of corn…

            Years of effort have sadly ratified the epigram of Novalis: "Zur Mathematik gelangt Man nur durch eine Theophanie."

            De profundis, ideo, clamo: Riemann – eleison!  Poincaré – eleison!

            Solâ gratiâ.