Salutes and Semiotics

We earlier offered semiotic analyses of politically-tinged postures and gestures, in Europe

     La quenelle

and in the Middle East

     The Rabi`ah

And now such a controversy has flared up in the United States, specifically at West Point.   In a highly interesting and well-illustrated article in this morning’s New York Times,

     Raised-Fist Photo by Black Women at West Point Spurs Inquiry

we are invited to consider this graduation photo:

Now, as a non-veteran civilian, I shall have nothing at all to say about this Army-internal controversy, leaving that to the warfighters themselves.   But a word about the ambiguities of the symbolism.

The raised-fist gesture has been around a long time, appearing in a wide variety of contexts, though  in general  with a common theme of militancy.   I first encountered it personally in the late 1960’s, as part of the movement against the Vietnam War.  At the time, there were actually two raised-hand antiwar gestures.  The one more familiar to outsiders was that of a raised hand with two lifted fingers, the index and the medius (“Pointer” and “Tall-man” as we called them in kindergarten -- O happy days!);  and that, to be sure, with the palm towards the onlooker; the same gesture with the onlooker facing the back of the hand, is not used in America, but is used in England, and means something quite different, and very very bad.  This gesture was often called the “Peace sign”;  it was the badge of the peacenik.  Although most of us who flashed the thing didn’t know it at the time, it was an odd choice for an antiwar symbol, since the same sign during the second World War was the “V for Victory” sign, universally known.   Well, autre temps, autre sémiotique.

The raised fist, in its beginnings, was subtley different:  it was the signature gesture, not of peaceniks, but of activists, militants, and had roots in the Communist movement (a fact probably unknown to most who came to use the gesture, simply for its jaunty sexiness).  In their purist form, the gestures meant respectively  “Bring the Boys Home” (a moderate slogan, suitable for patriots) and “Bring the War Home” (in the startling formulation of the Weathermen, of insurrectionary intent.).

Back to the posed photograph.   Its evaluation, and the actions to be taken as a consequence, hinge crucially on the fact that these soldiers are in uniform;  that is the essential context, sine qua non. I know of people who have been stripped of their security clearances and even drummed out of the service, for having done something publically in uniform  which they would be perfectly at liberty to do in civvies. As such, it is a matter for the Army in general and West Point in particular;  for the rest of us, it is none of our affair.

But as semioticians, we may consider a bit.

The pose itself is striking -- and, to this observer, strikingly handsome, considered only as a tableau.   It calls to mind  such sculptural compositions as  this (from the Arc de Triomphe):

But more pertinently, we are dealing here, not with such stray, subjective associations, but with a context within a context, the larger context (as the article illustrates) being specifically a long-standing West Point tradition, exemplified by this, from 1884:

Thus, while the raised-fist motif (of ambiguous interpretation, now that the gesture has been so watered-down as to be exploited by pop stars) might or might not seem defiant or truculent (eye of the beholder),  it occurs in a framing that is in fact ultra-Army and ultra-traditionalist.  And quite possibly,  what was in the minds of these young women at the time  was not really that of insurrection (let alone Communist), but more along the lines of another Army tradition:  esprit de corps.

[Aesthetic footnote]  The distaff composition actually seems quite a bit more "Hoo-ah!" than that of the 19th century men.  Most of them can't stand up straight; the guys  towards the bottom  seem to have melted.

[Psychological footnote]  As a complete outsider to all this, what first struck me (and somewhat shocked me) about the picture, was not the fists, any minatory interpretation of which was at least partly neutralized by the merry grin on the woman stage-left, and the cocked-head dreamy half-smile of the woman at the opposite bannister (who resembles a -- white -- woman I work with).  Rather, it was the grey old-fashioned uniforms.  Had they dressed up as Confederates?  Surely not;  the article identifies the kit as “traditional gray dress uniforms”.   To contextualize adequately  requires expert testimony;  you can’t just go by the sight of your eyes.

West-Easterly Divan (II)

O, daß der Sinnen doch so viele sind!

Verwirrung bringen sie ins Glück herein.

Wenn ich dich sehe, wünsch ich taub zu sein,

wenn ich dich höre, blind.

[Quelle: West-östlicher Divan, Buch Suleika]

Too many senses  overload my self,

to seize thee clearly, thereby undermined.

When I thee see, I would that I were deaf;

When I thee hear, would I were blind!

~

Bruce Springsteen:  “I Wish I Were Blind”

O  wär’ ich lieber blind !

Pop ! goes the Positivist

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

Another example of fetishizing the visible (cf. now also this)  occurs in Karl Popper’s Conjectures and Refutations: the Growth of Scientific Knowledge (1962; page refs. to the Harper paperback reprint), p. 284:

In the last pages of Testability, Carnap discussed the sentence ‘If all minds should disappear from the universe, the stars would still go on in their courses.’  Lewis and Schlick asserted, correctly, that this sentence was not verifiable.

They so asserted incorrectly.  After all, we have a pretty good handle on those stars in their courses, beginning from not long after the Big Bang  through the present, during most of which there were no human minds.

And here’s a simpler experiment:  Let’s all just refrain from observing the moon for the next two weeks.  Now we open our eyes.  Surpri-ise!  Right where we expected to find it, and at a phase two weeks more mature than when we last checked (rather than the phase as last seen).

One occasionally observes a very young child shut her eyes and announce in a sing-song: “Youuu ca-han’t seee mee….!”  I doubt the child actually believes that, it’s just fun to say.  But suppose she did. She is then entertaining an erroneous optical theory, but not an erroneous ontological one.  She imagines that the lights went out for you as well; but she never entertains the thought that you yourself have been extinguished, or been transfixed into some mute deaf motionless limbo until such time as she shall be pleased to gaze again – after all, she has just shown this, by addressing you.

Incidentally, the whole Schlick-shop with its merry/morose band of logical positivists, appear to have created enormous problems for themselves, and come up with completely counter-intuitive results, from the basic motivation of wishing to exclude all metaphysical propositions from the outset as strictly meaningless.   (The story here is in line with our overarching defense of theism.)  In other words, it wasn’t enough to denounce as wholly unconvincing,  all purported metaphysical statements hitherto; nor to brand it a particularly uninteresting language-game; nor to marginalize the enterprise with the derision usually reserved for circle-squarers.  Metaphysics as such, and for all time, had to have a stake driven through its heart, the stake itself consisting in pure reasoning from selbstverständlich  (metaphysically a-priori? God-given? -- oops, NOT) first principles.  For a demonstration of the absurdities that apparently resulted, cf. Popper, passim.  As (p. 281, op.cit.)

My criticism of the verifiability criterion has always been this: against the intention of its defenders, it did not exclude obvious metaphysical statements, but it did exclude the most important … of all scientific statements … the universal laws of nature.

Why even go near such a disaster, for the sake of excluding-in-principle all metaphysical statements – which they were ignoring anyway?  Popper goes so far as to hazard a guess in a footnote to p. 175:  “One not need believe in the ‘scientific’ character of psycho-analysis (which, I think, is in a metaphysical phase) in order to diagnose the anti-metaphysical fervour of positivism as a form of Father-killing.”  (Note, b.t.w., the capital initial of “Father”.)

*

Since we were obliged, a moment ago, to deprecate Popper, let us hasten to another example, in which we find him on the side of the angels.  First let us note:

             * Our conduct is guided by moral laws.

             * Our scientific activity is guided by metaphysical principles.

             * Both categories are philosophically vexed, and any concrete application of these laws and principles may prove problematic; but we cannot do without them.

            Here is Popper (p. 287) characterizing the logical positivists: “They implicitly accept the rule: ‘Always chose the most probable hypothesis!’  He goes on:

Now it can be easily shown that this rule is equivalent to…’Always choose the hypothesis which goes as little beyond the evidence as possible!’  And this, in turn [amounts to] ‘Always choose the hypothesis which has the highest degree of ad hoc character!’

an undesirable result for science.   (Cf. Chomsky, passim, re the extraordinarily abstract and roundabout way you may have to proceed  to get valid results, by no means simply tip-toeing a bit beyond “the evidence”.)

            The principles which we require instead  may be best illustrated by an old joke (presented here in a slightly revised form):

     A zoologist, a mathematician, and a positivist  are touring the hinterlands of Bohemia by train.  Through the window they spot, standing motionless at some distance, a purple cow.

     “Fancy that!” exclaims the zoologist. “There are purple cows in Bohemia.”

     The mathematician, in a low voice, amends: “There is at least one purple cow in Bohemia.”

     The positivist, raising his finger, crisply corrects to:  “In Bohemia there is at least one cow, purple on one side.”

            The joke is that the positivist has the most precise view of the evidence; and of the three statements, only his own is certainly true (interpreting an unmodified “purple cow” in the natural way, as “purple more or less all over”); yet his scruples are absurd.  Science would not progress, were it bound by such fragmented literalism.  (Indeed, we should have to replace “cow” with “an apparently bovine organism”, and always assuming that we can rule out a mirage, or a collective hallucination, or some effect of swamp gas; or for that matter an analogue to one of Hilary Putnam’s pet entities, the Mechanical Cat from Mars.)  And we would have to gloss Quine’s celebrated interjection, gavagai!, as possibly:  “Lo, a rabbit-stage, seen from the east.”

(For more on bovine mathematics, see "How Now, Round Cow".)

A gathering of Positivists

Note:  Occasionally one meets with such unilateral positivist agnosticism  in a non-jocular context:

Franz Ferdinand rode erect, his visible foot deep in the stirrup, a saber at his side.

-- Robert D. Kaplan, Balkan Ghosts (1993), p. xxvi

The above scenario has been much discussed among epistemologists, under the rubric “Henry and the barn façades”.   However, the cow version is much funnier.

Ceci n’est pas une grange 

~

To move from this to an outlook useful for science, we need these metaphysical principles:

   (1) a certain uniformity or self-cohesion to nature

   (2) the reliability (not in a logical, but in a practical sense) of induction.

(There are others, of course, such as the usefulness of deduction;  but that applies in every world, not just in this contingent one.)

            Now, both these principles are of vexed and delicate application; as are, indeed, the Commandments.   Just as it is difficult to avoid situation ethics, I see no way around a kind of “situation metaphysics”.   The zoologist, using these principles, together with his discipline-specific knowledge (to the effect that animals generally come in species, not as singletons), concludes the presence of purple cows.  But here I must side with the formulation of the mathematician: there is at least one such cow, but perhaps no more.  For, nature’s uniformity depends on the domain.  Were an experiment ever to identify, say, a Higgs boson, we should conclude the existence of an emphatic plurality of Higgs bosons; that there should only be one in the universe, is inconceivable.  But biology is more mottled than that.  Given our collective prior acquaintance with hundreds of thousands of cows, none of which was ever purple, we would hypothesize here an extremely rare mutation for (bilateral) purple (bilateral because bilaterally symmetric animals generally continue to evolve with bilateral symmetry, pace the occasional narwhal).  Since a cow delivers but a single calf at a time, this one has no identical siblings, so it is probably unique.  Moreover, the bulls will likely shun her for her color, and she shall have no offspring.  The mutation dies with her.

Carnap attempts to avoid the assumption of a metaphysical principle, by declaring (1) to be “analytic” (i.e., more like “2+2 = 4” than like “animals tend to come in species” or “each elementary particle is invariable within its class”).  Popper comments (p.289) that “no such principle of uniformity can be analytic (except in a Pickwickian sense of ‘analytic’)."  In fact, we noticed, the principle is nothing remotely like analytic, having always constraints on its application, which in some domains may be severe.  And as for (2), I concur with Popper (p.292) that it is “a principle of a priori metaphysics.”  And none the worse for that.

[For further notes on ineluctable metaphysical underpinnings of the scientific enterprise, see here.]

Ontological Epigrams

Okhamians like Quine (for so I shall refer to him, out of deference to my former professor, so as to avoid the loathsome connotations of Nominalist) are concerned not to clutter up the ontology with extra entities like facts or meanings which parallel and move in unison with statements  without really adding anything useful.  Strawson (a reliable nutshell-stuffer) puts it in a nutshell  thus:

Facts are what statements (when true) state;  they are not what statements are about.

-- P.F.Strawson, “Truth” (1950; often anthologized)

~     ~     ~

Quantum superposition is ontological, not epistemological.  It is not that we don’t know what state the cat is in.

(Unfortunately I have forgotten who said this.  Mark Twain?)

~     ~     ~

The ontological touchstone for British analytic philosophy, whether of the Idealist or the Empiricist inclination, has ever been the Tree in the Quad.    For several thousand years, this noble oak has been under uninterrupted observation by a rota of pre-Chalcolithic druids, lest it pop out of being, unobserved.  What a shame that would be!  It was such a nice tree!

~     ~     ~

The first question of ontology is:  What is there?

The answer to this, as Quine put it: Everything.
(Yet adds the stern proviso:  But no more than that.)

The second is, of each thing:  What is it?

To answer this, we may revise good Bishop Butler thus.

=>  Everything is, what it is  --  along with several other things as well.

Thus: a diamond is:

* A costly gem, proverbial for hardness

* A face-centered cubic crystaline allotrope of carbon

* A girl’s best friend

* Forever.

Given this, the question, What is Being? (überhaupt or tout court), must be categorized as meta-ontological.

~     ~     ~

Meinong’s bizarre deviation(**) … was … jenseits von Sein und Nichtsein.  Oddly enough, I find this idea a good one, provided that we bolster it with Bentham’s theory of fictions.  Contextual definition, or what Bentham called paraphrasis, can enable us to talk very considerably and conveniently about putative objects  without footing an ontological bill.  It is a strictly legitimate way of making theories in which there is less than meets the eye.

--- Quine, “Existence and Quantification”

The concluding epigram of that paragraph  is particularly relished by Minimalists.

(** Note:  The reference is to that philosopher’s ontological exuberence,  and not to any possible undue familiarities with capybaras during off-hours.)

~     ~     ~

Thus far the apophthegmatic epitome.  If these have whetted your appetite, click here.

[Orthoepic update]  Since folks keep clicking on this post, I'll add this:
The fifty-dollar first word of this essay  refers to the Nominalist, William of Okham.  That last word is pronounced OCK-um.  But the adjective, Okhamian (which I possibly just made up, but on a proper model) would be pronounced ock-HAY-mee-an.
Try to work this into your next singles-bar conversation.  It's guaranteed.

~     ~     ~

« Ontology can be saved  with a sufficient ideology. »

-- Saul Kripke, “Is There a Problem about Substitutional Quantification?”,

in: Evans & McDowell, eds., Truth and Meaning (1976), p. 343

Not sure what that means, exactly;  but it has a lilt to it, like “Die Welt ist alles, was der Fall ist”;  of which, perhaps, it is a paraphrase.

Esse est percipi (redivivus ter)

“To be is to be perceived” -- Bishop Berkeley
[corresponding to the Latin   esse est percipi, pronounced ESS-ay est pare-KIP-ee.  Percipi is the passive infinitive, a delightful category, which does not exist in any of the other languages with which I am conversant.]

“We call a real dynamical variable whose eigenstates form a complete set   an observable.”  -- Paul Dirac

“To be  is to be the value of a variable.”  --  W.V.O. Quine

*

~

~  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.

(I am Bishop Berkeley, and I approved this message.)

~         ~

~

Three radically different takes on  being and seeing.

(Cf. "Seeing is believing" -- vs. Berkeley:  Being seen  is being.)

The first and most famous of the epigrams above  applies to the ontology of basic objects, such as coffee-cups -- about which no plain man has any doubts whatsoever.   (Quine's quip  will, however, frequently detain us further;  e.g. here.)

A related but different question  concerns things which, unlike a coffee-cup or its relativistic mass, you cannot directly perceive -- though someone else might.

Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. I, p. 17, re the views of G.E. Moore:

For things presented in space, but not things to be met with in space, to exist is to be perceived.  That is, afterimages .. and pains  can only exist when they are perceived or experienced.

Or, classically,  “Beauty is in the eye of the beholder”.

A psychological vice ontological version of Bekeley’s epigram, is the vernacular “Out of sight, out of mind.”  (Cf. French, "Si tu te tais, ni vu ni connu.")  Or, as Fielding wittily put it in Tom Jones,

De non apparentibus, et non existentibus,

eadem est ratio. -- In English, ‘When a woman is not seen to blush, she doth not blush at all.’ 

(Lawfolk interpret this Latin tag more prosaically:  “What is not juridically presented cannot be judicially decided.”   Spoilsports.)

~

A noted intuitionist philosopher, anent “the celebrated [loathsome, leprotic -- ed.] thesis that mathematical statements do not relate to an objective mathematical reality  existing independently of us”, writes that, on such a view, to be is to be conceived:

Unlike material objects,  mathematical objects are, on this thesis, creations of the human mind.  They are objects of thought, not merely in the sense that they can be thought about, but in the sense that their being is to be thought of [or rather, thought into existence -- ed.];  for them, esse est concipi.

-- Michael Dummett,  “ The Philosophical Basis of Intuitionistic Logic”, in: Truth and other enigmas (1978), p.  228

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

~

The above is really just a placeholder post, reminding me to someday address the deeper problem of observables.   This may never happen.  To begin with, even in the simplest quantum-mechanical case, an observable is an operator on Hilbert space, so you have to wrap your mind around that.  With some luck and some study, that might be doable.  But just now I encountered the following dismaying sentence:

In quantum field theory, unlike in quantum mechanics, position is not an observable.

I may well have retired to my hamster farm, before ever figuring such things out.

In the meantime, some further thoughts about ontology  here.

~

Bonus quote:

  How do you know you’re having fun  if there’s no one watching you have it?

  --Douglas Adams, The Restaurant at the end of the Universe (1980).

~

Visitors to the site have enquired about the pronunciation of the classic Latin phrase “esse est percipi”.   About this (although I used to be editor of pronunciation at Merriam-Webster, back in the day) I have so far had nothing to say.  Embarrassed, I messaged my colleague and comrade Dr. Keith Massey (officially out of comms, but reachable in the bush), as follows:

More than once, someone has searched the blog via
   pronounce esse est percipi
-- alas in vain.  In my head, I have always pronounced this pear-KIP-ee, but, given that the vowel is short, the Latin would be more like PEAR-kip-ee, no?
Or rather, the Vulgar Latin, since Latin itself, being quantitative, lacked phonemic lexical stress.  Correct so far?
A second question is how philosophers in various countries pronounce it -- anglicized, frenchified, etc.

Swifter than lightning, our colleague replied:

In classical Latin the C is always pronounced as a K. But we don't know when the thing started to change before E and I. I suspect it was already being realized regionally as something else in the Late Imperial period. And speakers of Latin dialects took to pronouncing their classical Latin as if it were their living register. And so, in France,  percipi was pronounced “persipee”. In Italy perchipee, and in Spanish pershipee (attested still in Ladino) and later, in Iberia as perthipee, and, yet later in the New World, as persipee.

Accent in Latin is a hotly debated topic. I personally follow the view that the classical accent is unknowable and therefore all we have to work from is the living dialects and the rules of Ecclesiastical Latin. And so, penultimate except in a few cases. Accent falls in the syllable before -ibus. But I'd say perCIPee.

As for Anglo-Latin, this is an exciting topic. There's some evidence that a Romance language survived in the Isles for a several centuries after the withdrawal of forces in 380. It apparently, from inscriptional evidence, turned long E into a long I, fecit --> feecit. The way we pronounce phrases like Habeas Corpus may actually be representative of the pronounciation of Anglo-Romance, rather than a later scholastic recasting.

Those further interested in this topic can consult the essay by Thomas Pyles, “The Pronunciation of Latin in English:  A Lexicographical Dilemma”, reprinted in his Selected essays on English usage (1979).  Executive summary:  The history is so tangled, it is now an unresolvable mess.

In his “Tempest in Teapot: Reform in Latin Pronunciation”, reprinted in the same volume, Pyles quotes an amusing epigram that alludes to what became of Latin in Spain, where original v and b merged into a bilabial fricative:

~ Felix natio, ubi vivere est bibere ~

[Variant: "o felix iberia, ubi vivere est bibere"]

For further adventures in pronunciation, click here. 

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

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

*

Distinguishing discontinuous state-reduction (which he dubs R) from linear evolution as per the Schrödinger equation -- i.e., the “collapse of the wave-function” upon “observation”, a mathematician remarks:

I do not mean to imply that the experimenter deliberately sets up a ‘measurement’ to achieve this.  … Nature herself is continually enacting R-process effects,  without any deliberate intentions on the part of an experimenter or any intervention by a ‘conscious observer’.

-- Roger Penrose,  The Road to Reality (2004), p. 593

Thus achieving the esse of percipi  ‘naturally’ (vacuously).

.

VISIBILIA VEL SENSIBILIA

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

Credo in unum Deum, Patrem omnipotentem,

factorem cœli et terrae, visibilium omnium et invisibilium.

-- The Creed

The theory of knowing  is modeled after what was supposed to take place in the act of vision.  …  A spectator theory of knowledge is the inevitable outcome.

-- John Dewey, The Quest for Certainty (1960)

Why anyway this fetish with *seeing* things?  There are some things which you can see, but which nobody else can; and you can’t touch them, and you can put your hand through them; and when you blink, they’re gone, or have shifted position. In such a case, we say: “You’re seeing things.”  These things are called “hallucinations”, or “muscae volitantes”, and most of them probably are.  If being “visible” is all you’ve got going for you, you’ll need something further on your resumé.

            Now try this thought-experiment.  In a universe (which might even be our own), there is a race of rational beings, known as the Blorks:  blind, deaf, immobile (and quite spherical), who nonetheless can sense objects around them with incredible precision, using a kind of echo-location, but only when those objects move.  (Vision in some animals is rather like this.)  Now, things don’t move much in their friction-ridden landscape, unless you nudge them.  And the only way the memberless Blorks have to nudge them, is via a biological analog to the technology currently being developed for quadriplegics:  you think an appropriate thought, and it has an effect on your computer, and thence, via robotic coupling, on the physical world.  Chez the Blorks, the action is direct, cutting out the middle-machine.  Now, in that world, the way to induce a certain movement of a body, is to discover (whether by calculation or direct insight) a solution to the equations of motion, and then to “think it aloud”.  Fortunately, hardwired into this species are the laws of dynamics, in their elegant Lagrangian form.  (We too have a few of these hardwired, though they don’t go very far, being practically exhausted in the infant’s experiment of repeatedly dropping her spoon.)  So, to calculate these trajectories  is virtually second-nature, rather like the abilities of an experienced driver, to calculate a merge.  (There are, to be sure, some drivers among us, who never manage this and similar skills, attempting to pass you on the right and then zip in and cut you off, in a way wholly inconsistent with present projected trajectories;  such I desire might become perfectly spherical, and then roll down some infinite inclined plane.)  If your solution is correct, the results are very real: the object moves as directed, and every detail of its motion registers on your echolocator, your nerves thrill at the inrush of sense-data, with a precision which beggars our own retinal amblyopia.  (If your solution is incorrect, God administers a mild – a very mild, not at all dangerous (for, Boshaft ist er nicht) – though admittedly rather sharply painful, electric shock.)  The boulder so moved may go crashing into other boulders, and if you miscalculated, it may roll right over you, thus reducing you from a ball to a disk. 

            It seems clear that, to these creatures, the equations of motion, and the mathematics of their solution, will have a lived reality, a concrete sensuousness, similar to  or exceeding  that of our own, dim, barely binocular judgment of distances, which we cannily calculate as best we can, before reaching for the coffee-cup.  (Dang. Spilled another one.)

            Upshot:  Discount sight.

            Problem to tackle next:  Not visibilia (trivial), but… observables.  Much more problematic.

[Update:  Sidelight:  Examples of detective work based on reasoning, not from sight, but from sound.
http://worldofdrjustice.blogspot.com/2016/03/a-locked-room-mystery-inside-out.html  ]

Credo (continued)

[continues this]

I see no reason why we should have less confidence in mathematical intuition, than in sense-perception.

          -- Kurt Gödel

            To grant necessary ontological status  only to the non-negative integers, as does Kronecker and his thought-mates, resembles the attitude which prevailed among geometers from Euclid down to Gauss and Lobachevsky, which considered geometry as described by Euclid  to be the only possible geometry.  It was a word with no plural, like “universe”; that both these terms are now pluralizable (the latter in a conjectural, the former in a now very precise sense) represents a triumph of the human spirit, which now knocks tankards with the Invisible, in a toast to Him who made us all, the math along with the meat.  The old geometers believed this because they fetishized the visible: the flat, drab world (well, pied with beauty, true, but drab compared with the full panoply of Riemann surfaces and Finsler space) whose mensuration is approximated so closely by Euclidean geometry  -- at least, that stretch of turf that lies visibly close to hand – that the actual deviations that do exist cannot be detected by ordinary, plain-man means; and those who scoff at the invisible infinite, are very plain men indeed.  And it was this servitude to the locally visible – which is  in particular  to say, to the contingent – which caused the finest minds to fritter fruitlessly after a derivation of the Parallel Postulate from the rest of the Euclidean axioms, a fiasco that lasted literally for over two thousand years, from antiquity down to the nineteenth century. Indeed, it was not until we became familiar with the “invisible” worlds  revealed to us by Lobachevsky, Riemann, Klein and Poincaré  that we became fully clear on the status of Euclid’s axioms, and the distinction between axiomatics and model theory.

            So, to argue concretely:  If the number one is real, then so is a half, for I give you half this pie. And if “one” is real then so must be the square root of two, as being the measure of the hypotenuse of the isosceles right triangle, by the inexorable evidence of the Pythagorean Theorem. (Likewise the square root of 5, 13, 17, etc., and thus their products.) And the square root of two turns out not to be a ratio of natural numbers. Now, Pythagorus himself shrank from this conclusion, and stigmatized the postulant entity as irrational; yet now we take them for granted.   And if third roots or electricity are real, then so are the imaginary numbers required for their description; and if matter is real, and with it atoms, then so is quantum mechanics: which mean that compact operators on Hilbert Space are real:  you cannot see them or touch them, but you can almost hear them, buzzing all around us…  It is a slippery slope (we might almost say, a declivity whose slope is infinite) when we admit the reality of the visible world: it quickly (“quickly”, considered sub specie aeternitatis) drags in all the invisibles wíth it.

            It is true, the atmosphere around those higher turrets is rather rarefied.  Sometimes we become light-headed, and wonder if we are not after all just making some of this stuff up.  Yet no sooner do we begin to doubt our senses – or rather, to put too much trust in our senses, and too little in our carefully nurtured sense of the unseen – than Nature coughs up some concrete correspondence with our most arcane designs.  That same Riemann hypothesis now seems to be in some strange harmony with the energy-levels of atoms.  And as for connections on fibre bundles – meet the gauge fields of particle physics, your twin, separated at birth!

*

            The question remains, whether mathematical entities are, so to speak, real in general, or only real within a particular reality: much as a planet might pursue its course, in our universe but not another.  Now, the examples of mathematical reality adduced thus far, have all been given a clean bill of health by our actual, particular universe.  Hilbert space is as much a part of our daily reality as are porpoises – quite as vibrant, almost as much fun, and much less likely to go extinct.  But the Continuum Hypothesis… ahh.  That’s another matter.  One would really like to have a better handle on that one.  There are models for set theory in which it is true, and models in which it is false.  This tends to make us acutely uncomfortable, since, unlike the logically equivalent but intuitively more ethereal Axiom of Choice, it’s the sort of thing where you feel there ought to be a plain fact of the matter.  Nevertheless, its status may be ultimately no worse than that of the parallel postulate, which is quite placidly and understandably true in some geometries, false in others, and indeed true in our own universe at appropriately small scales (here I mean, of course, not what happens unobserved at infinity, but such local effects as the sum of the angles of a triangle), while false at others.

[concluded here]