Thoughts 'n' Things

  Rudy Rucker, Infinity and the Mind, p. 38:

(**) Just as a rock is already in the Universe, whether or not someone is handling it,  an idea is already in the Mindscape, whether or not someone is thinking it.

This is itself a pleasant thought, recalling the ditty about God-in-the-quad; but in actual fact – I don’t think so.

(So you see—I am not an uncritical Platonist.  Platonic heaven must be so gerrymandered, as to exclude such things as cheese doodles and Sponge Bob Squarepants.)

The actual universe has (for example) -- whatever geometry it has:  regardless of whether there are rational creatures capable of understanding it, let alone deriving it.  Likewise the landscape of math.  But particular formulations of physics, and perhaps even of math – matrix mechanics v. wave mechanics, Cauchy analysis vs. non-standard analysis – do not exist in complete independence from their proponents.  They are, one might say, propositions, not objects.  The objects (or patterns, or whatever they are)  exist  even in the absence of  a person to spout propositions about them; but the propositions require a proposer.  – Nothing specially abstract here; the same thing is true of rocks.  This rock exists independently of any finite mind, but: “There lies a rock” and “Behold that rock!” and “What a rock that is!” must come out of some actual someone’s mind or mouth.

            The unbridledly idealistic view in (**) conjures up a skyscape of untethered thought-balloons.  It is pleasant to contemplate, in a comic-strip sort of way, but not to be taken too seriously.  For one thing, unlike the situation with mathematical truths, where anyone at any place or time might discover them, there is no way for a rational creature in another galaxy or dimension to reach out and grab one of those thought-balloons by the tail;  he is required to blow his own bubbles.  Whereas the structures of mathematics are like fixed landmarks, which one encounters again and again, from different approaches.  For instance:  Yang-Mills gauge theories, discovered by the physics expedition; and connections on fibre-bundles, discovered by the math team; and lo, they meet in the middle.  Likewise group-theory.  Different body-parts of this have been grabbed onto by matrix theory, algebra (symmetries of solutions to equations), geometry (the Erlangen program), particle physics (glad you could get here; meet Sophus Lie), and in time it becomes clear that it’s all part of the same elephant.  Whether they come from physics, or mathematics, or computer science, two such explorers may not realise that they have come upon the same mountain, till they have circled around it a bit and compared notes.  And this happens repeatedly.  We may summarize in an epigram:  The mindscape of mathematics is a multidimensional torus:  whatever direction you set off in, you eventually wind up back at Hilbert’s Hotel.

It turns out that Shing-Tung Yau likes this montane metaphor as well.  Cf. The Shape of Inner Space (2010), p. 103:

A mathematical proof is a bit like climbing a mountain.

And he nicely outlines the Yang-Mills case (p. 290):

The physicist Chen Ning Yang was similarly astonished to find that the Yang-Mills equations, which describe the forces between particles, are rooted in gauge theories in physics  that bear striking resemblances to ideas in bundle theory, which mathematicians began developing three decades earlier, as Yang put it, “without reference to the physical world”.  When he asked the geometer S. S. Chern how it was possible that “mathematicians dream up these concepts out of nowhere,” Chern protested, “No, no.  These concepts were not dreamed up.  They were natural and real.”

            Contrast the case with “thoughts”.  Supposititious entities of the mindscape, even some popular thought-balloon, tethered to a billion different heads, need never be rediscoverable by another explorer, nor acknowledged as real should he simply be grabbed by the lapel by one of the thinkers, and treated to an exposition of same.  For example, the notion held dear by countless generations of schoolboys around the globe, of the uniquely funny nature of flatulence, will never appear among the gravely ellipsoidal thought-balloons of the solons of Fdrmrphlandia; even “funny”, for them, is not well-defined, and not particularly worth defining.

Now, probably Rucker meant to restrict the realm of “ideas” to just some of them.  Not, “Wouldn’t it be fun to dip Suzy’s pigtail into the inkwell!”, but things like “The square of the hypotenuse is equal to the sum of the squares on the other two sides.”  Fine; but careful, here.  The Pythagorean theorem has  as its basis  a fact about Euclidean geometry, in every possible world; just as Fermat’s Last Theorem expresses (in a possibly somewhat contingent and imperfect way) a fact about the natural numbers.   But a fact is not the same thing as an idea.  As a matter of fact, there is a coffee stain on this shirt; but “the idea of this coffee-stained shirt” is no strut or girder of God’s architectonics.  An idea concerning a fact of mathematics, in a finite mind,  may bear – must bear -- but an imperfect relation to the fact itself (‘fact’ here used broadly: it may refer to a wildly transfinite complexus of relations, some of them perhaps perceptible only to angels).   Most people’s ideas of mathematical truths bear as much relation to the truths themselves  as does a crayon scribble to the Sistine Chapel  which it might (based merely upon memory of a fleeting ill-lit glimpse) attempt to depict.  To posit that all truths of mathematics exist as Ideas in God’s mind, is logically allowable, but really adds nothing, and is in any case unknowable. To identify these truths with the neuronal states of the pitiful meat-wads sloshing around in our half-cracked crania, is to add nothing at all, but is rather to detract.

[Appendix]  Karl Kraus apparently entertained a notion of independent or pre-existent thoughts.  He speaks of someone being

von der Präformiertheit der Gedanken  überzeugt, und davon daß der schöpferische Mensch  nur ein erwähltes Gefäß ist; und davon, daß die Gedanken und die Gedichte da waren  vor den Dichtern und Denkern.

-- “Heine und die Folgen”, reprinted in J. Franzen, The Kraus Project, p. 88

The whole ‘meme’ idea (itself a meme) is similar -- not that the various Chiclet-thoughtlets were truly Platonically pre-existing, but that, once hatched, they lead a promiscuous existence, wandering into people’s minds  like pollen into our air-passages.

~

Footnotes from the 19^th century:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

For Frege,

In contrast to subjective ‘ideas’ (Vorstellungen), ‘thought’ was intended in an objective sense, rather like state of affairs, sharable among thinkers  and indeed independent of anyone thinking then.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 190

Good Question

One day they asked Babar, the elephant-king,

sage of the elephants, immensely wise,

"What is the meaning of snow bunnies?"

King Babar smiled, then broke into a grin, began chuckling, and laughed out loud.

-- then leaned forward, suddenly gentle.

"I'm glad you asked."

Consummatum est

Then Babar, the wise old elephant king,

went on a journey,

a long long journey,

to find the snow bunnies' ancestral home.

Long did he linger, far did he wander,

year followed year,

stretched into centuries,

and the planet grew old.

And then at last he opened his eyelids,

and lo! he was surrounded by snow bunnies,

row upon row of solemn still snow bunnies.

"Welcome, King Babar," they said.

Principles of Supernatural Selection

While our colleagues have been frittering away their energies on the vagaries of this and that contingent development in the visible world, we ourselves have, pursuant to a billion-euro grant from the IMF (Office of Unexplained Anomalies) undertaken to found an entirely new field:  Ekapsychic Adaptationism, or Supernatural Selection.  

To this end, we have assembled a crack team of researchers, including such luminaries of metanormal science as Bill, Chuck, Jim, and Bob, to aid us in p-p-penetrating  this inviting and (patulously) virginal  new frontier.

Thus for instance:  Why is Babar still with us, while Turgor, Twinkles, and Mr Moustache have gone extinct?  Something to do with DNA, perhaps? -- Nope, they don't have DNA, doofus;  they're imaginary.  (Sheesh). -- Some sort of string-theoretic quantum effect, mayhap? -- Wa-all, mebbe;  we're working on it.  But in the meantime, here (pursuant to the terms of the grant) are our preliminary findings.

* It helps to be rotund.

Witness:  Babar;  Dr Dolittle, Pooh-Bear, and (the popular romance of) Teddy Roosevelt.

* It helps to have a sidekick  very different from yourself.

As:  Don Quixote, Sherlock Holmes, the Lone Ranger.

* No matter how many things you have done in your career, you will be remembered, or conceptualized, for just a few.

Indeed,  never having done a dayyum thing  is no bar to iconic status.  (Witness:  Mr Peanut.)

(Just leave the Nobel Prize in the mailbox  K thx bai.)

Existence and Use

 Now let us return yet again, with our crude wooden mallet, and hammer anew on that empirical numerical square peg  which we are attempting to pound into a (so we hold) round hole, as round as a cow – the Platonic realm -- or as it might be, to embed it smoothly into Euclidean n-space.

I am not saying that numbers are ideas.  Or if they are that, they are that in addition, where they keep such scruffy company as the Idea of a Unicorn and the Idea of a Ham Sandwich. They are, rather, facts as hard as baseballs, packing quite as much wallop. (Also, admittedly, we are interested in the Pitcher …)

In pointing out the “reality” of numbers, we’re really doing much what we do when demonstrating the reality of a hammer, without venturing so much as a murmur as to what either of these things are “in themselves”.  Philosophy pretty much gave up on the ding an sich a long time ago, and the new quantum perspectives makes even ordinary objects seem more remote than ever.  The reality of a hammer, as against that of my imaginary rabbit friend,  consists in its experiential predictability: I feel its weight, you do too, we both agree it’s heavier than a walnut; it looks a certain way from this angle, another way from that, and these match up in a natural way, both visually and tactually, and so forth.  (Whereas you, with your confounded skepticism, scare away my imaginary rabbit-friend, and thus can never experience his charming and comforting presence.) The fact that we can also state that the hammer is right there, as opposed to another place, is nice: it’s one more fact about the hammer, but it’s not the essence of being real. In the case of a quantum particle, we cannot state that it is precisely there, though it’s somewhere around here for sure; but it does other things predictably enough  that we count it as part of the furniture of the universe, and not just some fluke.  It is likewise difficult to say “where” the Big Bang was when it popped, nor “where” our universe is now. The latter questions might make sense and they might not; we might be comfortably embedded in a multiverse (itself endowed with a metric), just a stone’s throw from Cosmos 87; or we might simply be, with nothing else to be relative to and hence no “where”.  But we are here, wherever “here” is; we are real. 

            Now, let us hammer away at this metaphor of math as a useful tool, and thus real.  (The point feels Wittgenstinian, though in general I am little in sympathy with his approach to math.)

            So.  A rock isn’t for anything.  Hence, unpoliced by pragmatics, rocks shade off into pebbles and sand and dust and molecules at one end, into boulders and cliffs and mountains and tectonic plates (or the entire Moon) at the other, with no clear boundaries.  (Unlike the boundaries among mathematical objects – numbers but others as well – which are sharp as a guillotine.)  A hammer can’t shade very far without ceasing to serve its defining function – it needs to serve to hammer things, to earn its keep – and thus ceasing to be a true hammer.

            Mathematical objects have this tool-like usefulness, and in that respect are more like hammers than like rocks.  And this does tend rather to tether them to this world, thus to make them more familiar, more acceptable to the Chamber of Commerce: for these are practical men, and like to know that a thing has a use, and hasn’t just been dreamt up in some pooftah parlor-game on a rainy afternoon.  Integers are useful for counting, Riemann manifolds for relativistic physics, and so forth.  But: cavete, gentlemen of the C-of-C:  they are not defined by these (human) uses. A toaster is so defined, but Mathematica thrones though we perish.  These objects exist prior to and independent of any such simian or homosapiensan applications – rather like rocks, the bare unappreciated rocks, which exist, in their original and ultimate solidity, prior to their being hewn into ashlars.

            One thing I am not claiming  is any special inevitability of this or that application of mathematical objects;  on this special subject I am comfortable with the nominalist position (the “hocus-pocus” position, vice “God’s truth”).  It is all too familiar, that this or that feature of physics, or of anything else, may be modeled (approached) by this particular mathematical formalism or that. Indeed, to make it all painfully plain:  Take the simplest application of the simplest mathematical object: the use of the natural numbers for enumeration.  The association is psychologically so tight, that some people doubtless imagine that N is defined by that application, and was even invented for that applicaton (by Homo sapiens or perhaps Neanderthal  underpaid mathematicians), the way a toaster was invented to toast bread, and is defined by this end.  If bread no longer existed, the toaster would, in a sense, cease to be. 

            Not so the natural numbers.  Fie, lest ye imagine, that they require our little home-improvement projects for their existence.  And as a (particularly smarting)  proof, consider this:  They are not even needed for enumeration.  (Yet mark:  Though seemingly to their discredit, this rather redounds to their greater glory.)  For:

            One could quite well enumerate things, and even perform simple arithmetic, without the slightest notion of integers, or indeed of counting.  You do it thus:

            First sculpt a monkey, and set it next the temple. Then (using your duplicator-zapper, left behind by those visitors from the Companion of Sirius), duplicate the statue and add another one just like it.  Put the resulting statue-grouping a bit behind the (unitary) first.  Now duplicate that, and add a monkey. Put this a bit behind the last; and so on.

            Now, if you want to figure out (say) how many children there are in your family, have them line up next to the rows of monkeys, beginning with the one in front, and have each child hold hands with one monkey.  Keep walkng back along the ranks until each child is paired with exactly one monkey. 

            You can perform addition and subtraction (and, more laboriously, multiplication) by similar purely physical means, having no more arithmétic notion of what you’re up to than do the Pirana.  To be sure:  Various arithmetical relations are implicit in this system, for instance “greater-than” means you have to walk farther along, away from the temple, till  you find a match.  But these relations need never, in practice, become conceptually explicit.  Like the Pirana, you can get by with no names for numbers as such, certainly no systematic numbering.  First, operation of the system, being purely physical, can be performed without words or symbols.  But suppose you do want names for each individual sculpture-row (as for: “Meet you at the (117)”.) Simply color the nearest group (the unit group) yellow, the next one purple, the next one green, the next one (say) black-and-white stripes, and then the next one pink polka-dots on an argent field; mark the next one simply with a little white flag; the next, with a happy-face; and “so” forth. No rhyme and no reason; none needed.  It covers the (finite) maximum group of statuary you have thus far needed to resort to.  Should the sets ever fail you, simply zap-dupe the hindermost and add a monkey, dubbing the new addition however you like.

            The fable is not idle.  Some of the skepticism about the “reality” – that is, the transcendental uniformity – of mathematical objects, may stem, I suspect, from a more justified skepicism about the God’s-truth view of applications of mathematical objects.  Thus, “Tensors are real because the were precisely what Einstein required for his field equations.”  Morris Kline (among others) saw more clearly: it is possible (or, in his view, actually likely) that tensors are not precisely what Einstein needed, and that something will supercede them.  Tensors had already been discovered (or, as the fools say: invented), and  as it turned out, they served him well enough.  Einstein notoriously lacked the mathematical tools ready to hand, when he came up with his intuitive take on gravitation.  He wandered down the hall to the math department, or maybe met some guy in a bar, and the guy says, Yo, Al, try this, always worked for me; and Al says, Good enough.  Sort of like going next door to borrow a hammer only they don’t have one but they have a paperweight and it will do.

Let’s take it a step further:  We built up the integers above, concretely, Peano-fashion, via the successor-function.  But having reached that height, we may throw away the ladder.  Notice that the hidden isometry between the size of the integer and the number of monkeys in the statuary that represents it, is logically unnecessary.  So, to save raw materials, we replace each statuary group by a simple stele, surmounted by the arbitrary symbol that was chosen to represent the group.  Counting then will amount to reciting a memorized list, “… Tyrolean hat, bowling ball, chartreuse lozenge, woodchuck rampant, dot, ….”  This is essentially what we settled down to with: “January, February, March….” (originally the list was etymologically more numerical, before those designations were ousted in favor of various godlings and tyrants.)

This, by way of a reply to Quine, who states, in his reply to Charles Parsons (Hahn & Schilpp, ed., p.401):

I prefer to say, with Benacerraf, simply that there are no natural numbers, and there is no need of them, since whatever purposes we might have used them for  can be served by any progression.

 

True enough – witness our monkey-statues.  After all, anything an actual person will ever have to count is finite -- a stock of two googolplex counters should suffice nicely.  And if one is trying to answer the (probably ill-posed and unanswerable) question, “What is an integer, really?”, then the observation is perhaps telling.  We ourselves have no interest in the quiditas, the “inner threeness” of the number three. A monkey trio, or a tricorner hat, will do just fine. But the “purposes we might have used them for” are by no means as meager as that dismissive phrase might sound – the way one might use a radio as a paperweight, or a statuette as a hammer.  You can’t settle the Riemann Hypothesis with nothing but poker-chips.

The uses to which numbers can be put, and the results we shall obtain with them, inhere in the structure of the set of integers itself.  It is universal, it is absolute, it is culture-free.  Numbers are Necessary, however you choose concretely to characterize them.  We may never get our hands on more than the trunk of this elephant, plus a couple of legs, but the elephant’s there, in his mighty totality.

Realism: What

[We choose a classic Lockean title-style for this chapter, and begin with a middle-of-the-road definition.]

Realism Defined

First, a plain-man's characterization of what we might call minimalist/core/plain-vanilla Realism:

Outside our heads  there is freestanding reality.  Only madmen and a scattering of constructivist philosophers doubt its existence.

-- Edward O. Wilson, Consilience (1998)

This view may also be referred to as Platonism (though cf. a special restriction of this term  to mathematics, outlined below):

One issue that has traditionally divided philosophers  is whether ther are abstract objects.  Nominalists have held that there are not;  realists (in a special sense of the word) or Platonists (as they have been called  to avoid the troubles of ‘realist’), have held that there are.

-- W.V.O. Quine, Word and Object (1960), p. 233

Now for the more careful distinctions of professional philosophers:

A. E. Taylor, Elements of Metaphysics (1903; page references to the University Paperback reprint), p. 67:

By Realism is meant the doctrine that the fundamental character of that which really is, as distinguished from that which is only imagined to be, is to be found in its independence of all relation to the experience of a subject.  What exists at all, the realist holds, exists equally  whether it is experienced or not.

Within this, he distinguishes (p. 68) two subtypes:

Agnostic Realism, while asserting the ultimate dependence of our experience upon a reality which exists independently of experience, denies that we have any knowledge of the nature of this independent reality.

Sic:  not “full knowledge”: any knowledge.  Which is absurd.

Contrast (p. 69):

Dogmatic Realism, of which Leibnitz and … Herbart are the most important representatives … while maintaining that real being is independent of experience, at the same time  holds that it is possible to have positive knowledge  not only of its existence, but of its nature.

Sic:  postive knowledge, but not necessesarily full knowledge.  And in this form I heartily subscribe, despite the invidious label conferred by its opponent Mr. Taylor.   As William James put it, in The Principles of Psychology (1890), vol. II, p. 634:

Reality exists as a plenum.  … But we can neither experience nor think this plenum.  What we experience, what comes before us, is a chaos of fragmentary impressions  interrupting each other;  what we think is an abstract system of hypothetical data and laws.

Full knowledge -- plenary knowledge of Reality in all its fullness -- not only in some rarefied ding-an-sich sense, but in the ordinary sense of the sciences -- is doubtless impossible in the case of anything so wildly complex as an acorn or a rock;  but in the case of something simple, like Hilbert Space (infinite-dimensional, it is true, but tamed by a norm that is based upon an inner product), we can perhaps come close to exhausting most of what can be known of it.

Taylor fancies he has refuted Realism in all its varieties, summarizing his triumph thus:

Produce any instance you please, we said to the realist, … and we will undertake to show that it derives its reality for you  from the very fact that it is not ultimately separable from the experience of a subject.

Note the crucial bait-and-switch!   He whisks Reality -- an ontological category, and a big deal -- under the thimble, and takes out -- “Reality for you”, a miserable psychological gewgaw of no general interest whatever.   This latter concept is, we readily concede, drenched in irreducible subjectivism;  no need to bother to argue the point.  And bearing, it may be, little relation to Reality, whether theoretically or empirically.

For:  Reality, in reality-for-X, is an incomplete symbol.

Thus for instance:   It is not possible to survey the roster of extant men, and finally lay your finger upon one specimen, the average man.  The Average Man is not a real, but an ideal, to which various actual men may approximate to one degree or another.  And Real-for-Joe-Blow is not a real, but a figment, which may reflect more or less of an actual reality, with greater or lesser distortion.

~

~  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.)

~         ~

~

A more carefully phrased importation of subjectivism into the debate  is provided by Michael Dummett, in “Truth” (1959), repr. in Truth and other enigmas (1978), p. 23f:

The claim … should be rejected by a realist, who might, and I think ought to, agree to the following weaker principle:  that a statement cannot be true unless it is in principle capable of being known to be true. […]  The anti-realist interprets ‘capable of being known’ to mean ‘capable of being known by us’, whereas the realist interprets it to mean ‘capable of being known by some hypothetical being  whose intellectual capacities and powers of observation  may exceed our own.’ … The issue between realism and anti-realism … is one of the most fundamental of all the problems of philosophy.

Thus here, clearly, we are once again confronted -- like it or not -- with the question of theism.

So:  if you are a full-bore, double-barrelled, two-seed-in-the-spirit, dyed-in-the-wool copper-bottomed no-holds-barred Cantorian Realist … (we pause for the roars of approval to subside) … even in merely so much as mathematical Platonism:  must you therefore, of logical necessity, be a theist in the sense of the Abrahamic tradition?

Perhaps not; certainly, we have not shown so.  (Nor do we so much as aim to show more that that.   At this point, Jews Muslims and Christians are all singing kumbaya together in one big tent.)   You could, in principle, construct for yourself an ontological halfway-house, with a non-omnipotent, though omniscient, Knower.   And who might this Wiser Being be?  Why -- none other than Babar, the Elephant King!

*

In these essays, we are concerned mainly with the form of Realism known (in a mathematical context) as Platonism, defined as

the theory that mathematics describes a realm or system of real and independently existing objects, whose nature is known to us through proof, but which are entities  over and above the proofs by whch we discover them.

-- Roger Scruton, Modern Philosophy (1994), p. 384

(This is fine, but don’t let’s squabble about “objects”; “patterns” or “truths” would do just as well.  We’re not interested in reifying anything, but merely in claiming that we’re not just making it all up arbitrarily.)

And:

… the subject matter of mathematics as realistically (i.e. platonistically) construed.

-- Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p. 35

 But for the record, there are other sorts:

Do we wish to say that there is a moral reality, which underpins our moral judgements  and guarantees their truth?  Some philosophers have argued for such a view (‘moral realism’).

-- Roger Scruton, Modern Philosophy (1994), p. 98

Scruton himself is far from being a moral relativist or nihilist, but seems not to feel forced to accept such moral realism, at least in the form of what we might call a moral Correspondence Theory (as opposed to the real existence of God and his laws).    Indeed, he compares (we shall call it) Aesthetic Realism:

It is obvious that St Paul’s Cathedral is beautiful, and the new Lloyd’s building  repulsive;  but is there some ‘aesthetic reality’ that makes these judgements true?

You readily see how problematic this is:  Saint Paul’s to a Wahhabi or an Iconoclast would lack appeal; and presumably the people who designed and paid for the Lloyd’s building  felt it had a certain something.   I still think a kind of case might be made out (which I’ll not pursue), that it’s not just all a matter of personal preference, and that de gustibus certe dispundandum est.   The argument would proceed by analogy with the much tighter case in mathematics.  Many things are obvious to the expert which are not obvious at all to the general public, and which indeed could never be made obvious to them no matter how hard you tried (sheaf theory, for starters).  And more tellingly,  the Man on the Clapham Omnibus will pronounce some things obvious (particularly in the area of probability, e.g. the Monty Hall problem) that are demonstrably mistaken (though some will understand the demonstration, and some will not).    We do not conclude from this that mathematical truth is just a matter of taste (“we” here referring of course to men of good sense, and excluding the Postmodernists).   Some are more qualified than others to grasp certain truths.    Quite possibly the same is true in the field of aesthetics, as regards, say, the Goldberg Variations, on the one hand, and “Who Let the Dogs Out”, on the other.   One would have to make out the case that those who exalt the latter and contemn the former have certain other things wrong with them as well.

[Update]   I have just come across an intriguing analysis relevant to the case of the “Lloyd’s building”, and that buttresses the Aesthetic Realist conjecture above, to the effect that it’s not all just a matter of mutually incomparable ‘tastes’, but that there are objectively different levels of aesthetic competence, owing to the importance of informed appreciation.  The analysis appears in a delightful essay, “The Gherkin”, by John D. Barrow, in his book 100 Essential Things You Didn’t Know You Didn’t Know (2008;  the book is better than its title).  The “gherkin” in question is a new building in the City of London, a.k.a. “the Pine Cone” for its pocked appearance:   “Prince Charles sees it as symptomatic of a rash of carbuncular towers on the face of London”.   Yet it won the Stirling Prize for architecture.  What gives?

It turns out the building is, from an engineering standpoint, quite ingeniously designed, and that principally with a view towards eco-friendliness.  Thus it narrows somewhat at the bottom, to defeat the wind-tunnel effect that annoys pedestrians; and tapers again towards the top, which “opens up more of the sky and reduces the dominating effect of the structure because you can’t see the top from close-by on the ground.”   Even the pocks have their point:  “They bring light and natural ventilation deep into the heart of the building”, saving greatly on energy costs.  And more details in this vein.

I’ve never seen it, but it sounds … beautiful …

~

Realism in psychology:

According to his friend and biographer, Freud “had a high and serious respect for the reality of psychological facts.  They were as real and concrete to him  as metals are to a metallurgist.” (Ernest Jones, Freud: Years of Maturity (1955), p. 432)

[continued here]

Further juicy quotations:

As a Platonist, he saw everything on earth as broken arcs, which merely suggested the perfect rounds above.

-- Louis Auchincloss, The Rector of Justin (1964), p. 92

Realists [with a capital R] are not the same thing as ‘realists’ in daily life, who are men who expect neither themselves nor others to be any better than they ought to be, and generally much worse.

-- Ernest Gellner, “The crisis in the humanities” (1964), collected in The Devil in Modern Philosophy (1974), p. 15

To mathematicians who study them, moduli schemes are just as real as the regular objects in the world.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xi

Our Friends the Integers

What, after all, is a natural number?  There are Frege’s version, Zermelo’s, and von Neumann’s, and countless further alternatives, all mutually incompatible, and equally correct. … There is no saying absolutely what numbers are;  there is only arithmetic.

W.V.O. Quine, “Ontological Relativity” (Journal of Philosophy, 1968)

Randbemerkung:  The repeated reference to the integers in these notes  might mislead the reader into imagining that I accord them the least importance, mathematically or ontologically (let alone theologically).  Not so.  I only recur to them on the rhetorical grounds that the arch-nominalist Kronecker gave them up for free.  Had he instead said:

            Die Mengen hat der liebe Gott gemacht; alles andere ist Menschenwerk

then we should have instead busied ourselves with the necessary Menschenwerk of building up the integers out of set theory in any of the usual ways, pointing out that this new epigram eventually commits you to the integers in any event.  And had he said (oh would that he had):

            Die offenen Mengen hat der Liebe Gott gemacht…. ,

then the touchstone would have been topology.  Yea, had he instead, like certain of our very ethereal contemporaries, posited rather Category Theory at the base, sets and numbers and all the rest to be developed out of that – well, I might personally have balked, because I don’t understand Category Theory.  Still, a donkey does not understand the Goldbach Conjecture, so intellectually that is no objection. 

            The infinities of the Creation  -- and a fortiori, of the Creator – are --- whatever they might be, we may never know, but in any event, nothing particularly to do with whatever a handful of our own species happens, at any particular instant, to latch onto.  The integers are one toenail in one foreleg (or is it hindleg) of the Infinite Elephant.  It matters not what an infinitessimal portion this may be, of that great beast (to Whom even Babar tips his hat), nor how ineptly we manhandle it:  what matters is that the Elephant is Real.  Praise Him!

            Anyhow, the integers are fine, but nothing special.  Frankly, in fact, the Riemann Hypothesis bores me to tears (mainly because I still cannot truly intuit its significance). The Poincaré Conjecture is much  more inciting – though, having been finally, after a century, proved in its entirety, it serves less well than the R.H. as an image of the blaue Blume.  Compare, indeed, the final chapter of Russell’s Introduction to Mathematical Philosophy, for a nice dissing of the integers.  Likewise Wittgenstein (Zettel 706): “Die Zahlen sind der Mathematik nicht fundamental.”