Adventures in Juvenile Lexicography

Katherine Nelson, in Keith Nelson, ed., Children’s Language (1978), p. 66,  offered the following glimpse into the orismological instincts of budding lexicographers.  Asked a “What is it?” question about the word tiger (“a large, fierce, flesh-eating animal (Panthera tigris)…” to you and me), the tots responded:

(1) “at the zoo”
(2) “animal”

(3) “it’s like a lion:

(4) “lives in the jungle and runs a lot”

(5) “animal with stripes and it eats a lot of things”

(6) “to run”

(7) “someone growls”

(8) “hair on its head”

I then polled our son (aet. su. 4 years,  3 months), who offered this:

   “It’s something that is big, and it eats people, and it runs around in the jungle.”

His assessment of her other examples:

apple: “it’s juicy; it’s big; it’s round”

car:  “Something that’s big, and not so tall -- it’s this tall [shows with his arms];  it can kill someone that stays in front of it, if it’s moving.”

coat: “It’s something that keeps you warm, is big and sort of smooth, and has little furry stuff”  [Note:  Our family was at that time facing an Edmonton winter]

bed:  “It has legs, or doesn’t, and it has a pillow, and it stands up on its posts”.
[Note: That first idea in the definiens, at once oddly precise and maddeningly vague, probably meant:  “Prototypically a bed has legs (the ones you see in books), but ours doesn’t” -- the family, indeed, then in exile and furniture-poor, slept on a mattress on the floor.]

Striking  is this repeated note of ‘big’, present in every definition except the last:  extending even to the humble apple -- even a baby is bigger than an apple, let alone a robust four-year-old.   But in light of the lad’s subsequent specialization in differential geometry, an explanatory hypothesis presents itself.  What may well have struck him was the apple’s unabashed convexity -- round, not like a thin dime, but round all around:  having everywhere positive and (roughly) constant Riemannian curvature, as he would no doubt rephrase the definition upon more mature reflection.  Such an apperception of an apple was indeed the Eureka moment of the founder of differential geometry, Carl Friedrich Gauss, as depicted in the movie “Die Vermessung der Welt”.

Das Lied des Idioten

Ah was ist das für ein schöner Ball
rot und rund wie ein Überall.
Gut, daß ihr ihn erschuft.
Ob der wohl kommt wenn man ruft?

What a bouncy, roundy ball!
Might it come, if I should call ?
Would I might grow very fat
so I could be   as round  as that !

[Aus dem Idiotischen, vom Herrn Doktor J. heraustraduziert.]

On What is Round (ter)

teres atque rotundus

 -- Horace  ('smooth to perfection, to perfection  round')

I saw Eternity the other night

like a great Ring of pure and endless light,

   All calm, as it was bright;

and round beneath it.  Time in hours, days years

   driven by the spheres

Like a vast shadow mov’d …

-- Henry Vaughan, “The World”

Could we but fill to harmony, and dwell

Simple as our thought, and as perfectible, […]

Grow to a radiant round love, and gear

Unfluctuant passion  for some perfect sphere.

-- Rupert Brooke, “Thoughts on the Shape of the Human Body” (1910)

In Vergleichende Anatomie der Engel (1825),  Fechner argued that the angels, as the most perfect beings, must be spherical, since the sphere is the most perfect form.

-- quoted in James R. Newman, ed. World of Mathematics (1956), p. 1152

We have discussed round squares (which do exist, despite everything you’ve ever read)

   http://worldofdrjustice.blogspot.com/2011/02/realism-what-continued.html

and round cows (which don’t, and more’s the pity)

   http://worldofdrjustice.blogspot.com/2010/12/how-now-round-cow.html

G.K. Chesterton wrote a whole book about the subject, called The Ball and the Cross.  As a Christian apologist,  he naturally reveres the Cross, but adds

           

            There must be some round earth to plant the Cross upon.

He put the case more tartly in Manalive:

“Science!” cried the stranger. “There is only one good thing science has ever discovered -- a good thing, good tidings of great joy -- that the world is round.”

It is difficult to find any passus  in which round  is used in dispraise.  The closest we can come is this:  Brian Greene,  in The Fabric of the Cosmos (2004), p. 294, tells us of “a famously caustic scientist  whose appreciation for symmetry led him to call his colleagues  spherical bastards:  because, he explained,  they were bastards any way you looked at them.”  Still, it sounds nicer than flat bastards.

Why did you do it.  And hearts.  And why was love so round.

-- J. P. Donleavy, The Ginger Man (1955)

~

Well-rounded (character, education) is another sterling term.  At the simplest level, it contrasts with Fachidiotie -- Merriam-Webster defines it using the terms “broad” and “comprehensive”.  But there is more to it than that -- more than the idea of a well-spread smattering,  jack-of-all-trades (and master of none).  There is the deeper education that all this learning and life experience has -- to borrow the language of topology -- rounded ‘round to form a compact surface.  As such, it gains in structural integrity -- much as a ping-pong ball, though made of inherently thin and flimsy material, shows great spherical strength.

~

Notions of ‘roundness’ crop up in mathematics as well, well beyond that of straightforward plane or three-dimensional geometry:  for instance, the “unit ball” in a normed linear space. 

The notion is generalized as that of convexity, which has various subtypes, and some surprisingly complex implications in functional analysis.   A lecture at the University of Alberta (Edmonton) in 1982 (Professor Lewis presiding) introduced an especially tasty flavor of this idea:

Definition:   A normed linear space is strictly convex (or ‘rotund’) iff the following holds:

||x|| < 1, ||y|| < 1,  =>  ||(x+y)/2|| < 1

He then added a historical observation:

This sort of thing was introduced by Clarkson in 1936, with  view to integration theorems for functions from the reals to such a space holding also for functions from Euclidean space to such a space.  Community interest switched to linear operators beginning around World War II;  and this remained the case until the late 1960s.  Now we’re back to rotundity.

~

We leave the last word to C.S. Lewis (English Literature in the Sixteenth Century, 1944):

"Columbus, a man of lofty mind, with missionary and scientific interests, had the original idea of acting on the age-old doctrine of the earth's rotundity, and sailing west to find the east ..."

[Update]  We still leave the last word to CSL, but from a different work, discussing medieval cosmology, and its explanation for the ‘natural’ orbiting-patterns of the celestial bodies:

A modern may ask  why a love for God  should lead to perpetual rotation. [It is because] the nearest approach to His eternal imobility, is eternal regular movement in the most perfect figure … the circle.

-- C.S. Lewis, Studies in Medieval and Renaissance Literature (1966), p. 51

 Does not Saint Thomas remark somewhere, that the most perfect shape, beneath the moon, is the belly of a penguin?  (One feels sure that he did;  yet I cannot  at present  lay my hand upon the passage.)

Q.E.D.

~

Further kudos to rotundity:

Dieser rasche Rundgang  durch Schuchardt’s mehr als 50 Jähre umspannende Wirksamkeit  zeigt, daß wir es tatsächlich mit einer  in sich geschlossenen, “runden” Lehre  zu tun haben:  das Bild des Kreises scheint mir am ehesten geeignet, Schuchardt’s Gedankenweben zu versinnbildlichen.

Leo Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2^nd edn. 1928), p. 6

~

Anyone who is up for 'another round', can find more here:

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

.

Scheinprobleme in der Pinguinologie

(I) Q:  Hamster-fanciers are all in a worrit -- which is the cutest li’l furball.  Do penguinologists face a similar problem?

A:  No.  We understand that, quâ  transitory incarnations of the Ur-Penguin in Platonic heaven, all penguins are cute-o-metrically equivalent.

(II) Q:  Which came first -- the penguin, or the egg?

A:  Why, the egg, obviously, as a local instantiation of the World-Egg, which in turn is subtended by the Perfect Platonic Sphere.

Then  from this  birthed-forth  an endless succession of Perfect Penguins.

Das Weltenei (artist's conception)

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

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

*

(III)  Q:  Do penguins have souls?

A:  Of course.   (Very round and nice ones.)

Counting Blessings

My honey sewed a button on.

Now there’s a button, where before there was none.

The button’s very happy !

He has a small, round job to do.

My honey sewed a button on.

She does nice things for me.

[Matthew 19:14]

~     ~     ~

[To read more nice things about round things, simply click here:

Round all around.

It’s so easy 2 do !]

On “Rounding Out”

The shortest and best way between two truths of the real domain  often passes through the imaginary one.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 123

We have noticed Quine’s grudging acceptance of the irrationals given his unquestioning acceptance of the rationals, a process we alluded to as “rounding out”.

But it is really not so much rounding out as filling out -- or rather, filling in: filling in the gaps between the rationals.  And the result is, unfortunately, not so rational as the rationals themselves.  The rationals -- that is, fractions -- are forced upon you by Nature already in nursery school:  How shall we divide these two cupcakes among the three children? (Answer:  Each gets two-thirds.)  But the Reals are (we admit this, despite our Realism) a bit unreal.  Full of all manner of set-theoretic paradox.  Inscrutable.  You can still work with them in practical terms, because the rationals, which are well understood, are, though no more numerous than the integers, dense in R, providing a sort of well-defined ladder or footbridge along which we may proceed.

Here in any event  is the testimony of a first-rate mathematician,  to the effect that the transition to the full reals  is essentially a forced move:

We shall show how to construct a complete ordered field  from a simple chain [Think:  the natural numbers].  This … proves that any contradiction inherent in the postulates for a complete ordered field -- that is, the real number system -- is latent in the postulates for a simple chain, which is a far less complicated structure  whose consistency is almost guaranteed by our intuition.

Note that we do not discuss the existence of the simple chain.  In spite of its intuitive simplicity, a simple chain carries within itself  the germs of all the difficulties in logic and mathematics;  we are obliged to take its existence as axiomatic.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  112

Such is the very axiom -- we need but two -- which we seized upon to begin this whole series of essays.   We quoted it in the form made famous by the Nominalist (and our otherwise-foe) Kronecker, which invokes the deity -- possibly casually or ironically, but perhaps more pertinantly than he knew:  for whence this “intuition” of which we are so sure?

~

The next step beyond the reals were named the “imaginary” numbers, and the name stuck.  Adjoin these, and you generate the Complex Plane.   And, unforced and arbitrary as this move might initially feel, it is on the Complex Plane  that you at last truly get a notion of a Natural Setting.  Everything just suddenly… works, and works better than you ever thought it could.  You have, all of a sudden, a Circle of Convergence -- sounds like something out of Lord of the Rings, and it is just as good.  Differentiable functions turn out to be perfectly smooth, and with a natural notion of their own domain.  (Try to define them on too small a region, and they will propagate themselves by analytic extension till they are nice and fat.)

On the complex plane, things are rounder.  (Note:  Round is good.) In R, a ‘ball’ is a line-segment, and a ‘sphere’ (the surface of a ball) is two points.  In C, they’re a disk and a circle respectively.  And you can round out or rather round off the complex plane yet further, by adjoining a single ‘point at infinity’, which is the limit of any ray pointing in any direction.  The plump, rotund result:  the Riemann Sphere.  This is homeomorphic to the surface of a penguin,  the world’s most perfect shape.

As Penrose puts it:

It is as though Nature had herself entrusted to these numbers  the operation of her universe.

(Again, note the theistic language which, all unbidden, surges forth at such a time, from even the driest of nibs.   It is a very early and natural theology, such as is depicted in that fine chapter of The Wind in the Willows, "The Piper at the Gates of Dawn".)

Another indication of the greater naturalness of the complex plane as a nursery for functions:  A real function may be C-infinity (infinitely differentiable) at a point, yet somehow “off” at this point, a fact revealed by the fact that its complex analogue is not there analytic.   Thus, as one writer put it, (complex) analytic functions are “smoother” than real functions.

[Example:  exp(-1/x), for x > 0; 0 at x = 0.  That last point is artificially “tacked on”, and in the complex picture, it shows.]

 

This Complex Plane  is a real find; it is not just a waystation to something better yet.  (David Berlinski calls complex numbers "instruments that providence had provided for the recovery of lost symmetries," a neatly postlapsarian formulation.)  There is very little beyond this, by way of fields suitable for the calculus -- certainly nothing that approaches the leap that the complex numbers represented beyond the reals.   There are the quaternions, which have their points, but are a very poor cousin indeed: the theory is poorer, not richer, for the extra generating elements, since the field of quaternions offers no analogue of holomorphic functions. ( “Quaternions have more or less dropped by the wayside.” -- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 325)

Then there are the octonions, for which no-one has ever found much of a use.  And there’s an end to it.

~

Other mathematical instances of “rounding out”:

*  The adjunction of zero to the natural numbers, and of the empty-set to the world of sets.  Both function exactly like their less spectral congeners.

And a nice aesthetic note -- both are represented by a round symbol: respectively, a goose egg, and a goose egg barre sinistre.

* There are various elaborate ways of constructing things out of other things, like a Stone-Cech compactification.  But in “taking the power set”, we just stand back and let it happen.  Again and again.  Yielding the “beth numbers”, and more infinities than most folks know what to do with.

* The mathematics of string theory adds extra tiny “compactified” spatial dimensions to the three of everyday experience; in these, you just go round and round.  But this isn’t rounding-out, really, since the large spatial dimensions may themselves be compact, in which any sufficiently long journey circles back on itself.  (“Compact” doesn’t mean “tiny”;  it’s a topological, not a metrical notion.)  Space could even be flat, yet finite -- thus having the topology of a three-torus.

~

Footnote:

It is well-known that it took mankind a long time to recognize zero as itself a number.  Less well known is that “not until modern times was unity considered a number” (D.E. Smith, History of Mathematics, vol. II, p. 26.)  Or that the negative numbers were long qualified as "false".

Compare the uncertainty over whether white qualifies as a “color”.  (And if it does, what about black, or grey?)

~

So where is Minimalism in all this?  Are we just tacking on turrets and wing-additions to some increasingly sprawling McMansion?

Not a bit of it.  The operative word here really is round.  For, round things are minimal surfaces -- indeed, the very simplest class of these -- in the sense of using-up a minimal area to enclose a prescribed volume.   Our purpose is, indeed, to group like with like, and to enclose them in some stable structure.  This is no multiplication of entities for their own sake -- the itchy-clutching witchfingers of insensately proliferating fractals, which is the very architecture of the dungeons of Hell.   In rounding out, the mathematician is seeking a coherent minimal structure to regiment and account for what he has hitherto seen:  one which, upon acquaintance, may become more intuitive than the partial structures initially encountered.  (The “upon acquaintance” part may of course require a bunch of Ph.D.’s and several hundred years.)

            And the things which we have seen, and which need explanation -- or at least for agencement into some larger and more natural whole -- do keep arising.  Connections are detected among them which cry out for elucidation.  So we ascend to a yet loftier bird’s-eye -- eagle-eye -- phoenix-eye view.  To arrive, it may be, eventually at Topos Theory, or the Lord of Hosts.

(For the latter, though note:  that ladder reaches only so high.  We quote the saint:

Remaneret igitur humanum genus, si sola rationis via ad Deum cognoscendum pateret, in maximis ignorantiae tenebris.
-- Thomas Aquinas,  Contra Gentiles, lib. 1 cap. 4 n. 4 )

~

The examples we gave were mathematical, merely for clarity.  But the principle of Rounding Out  applies to any field with structure.

These vague words ‘capable’ and ‘normal’  allow the grammarian scope for shaping his task to suit his convenience.  Seeking simplicity, he will round out and round off.

-- Quine, “Reply to Harmon”, in The Philosophy of W.V.O. Quine (1986)

~

Footnote re the irrationals:

Dedekind sttressed the distinction of category  between cut and number  in 1888; against the view of his friend Heinrich Weber  that “the irrational number is nothing other than the cut itself”, he explained that “as I prefer it, to creat something New distinct from the cut, to which the cut corresponds.  We have the right to grant ourselves such power of creation”,  and cuts corresponding to both rational and irrational numbers were examples.

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 87

A seemingly slight, even pedantic distinction;  but like many another such, it might have its point.   Cf. my astonished delight in junior high-school, upon meeting the distinction between  x (the thing itself) and ‘x’ (the name of x) -- already adequately foreshadowed in Alice in Wonderland, but encountered now in a new context.  Likewise the difference between  x and {x} (the singleton-set of x).

In the case of an algebraic number like √2, a simple number staring you in the face out of a hypotenuse  versus the infinite train of rational pilgrims (never quite arriving at their destination) of a Dedekind cut,  one is reminded of the variety of definitions of something so familiar as a tangent:  the slope of a curve (at a point); the closest linear approximation to the curve (at that point); versus the distressing definition in Loomis & Sternberg as an infinite equivalence-class of curves (through that point).

Realism: What (continued)

Scott Soames, Philosophical Analysis in the Twentieth Century (2003), p. 95, summarizes the realist commitments:

(1)  To the existence of everyday objects.

(2) To the existence of abstract objects (numbers and such).

(3) To every object of thought, which “must possess some kind of being, since otherwise we couldn’t think about it.”

Herewith our assessment of these commitments.

(1) To reject these would be churlish: a universe whose apparent bunnies and gumdrops, were but phantoms, would be the work of a devil, not of God.   We embrace the phenomenal world (beginning with the breast)  with the same uncomprehending acceptance with which we receive the Host.  We believe in the Real Presence of this chair in this room.

(2)  These are the most certain; only these survive the catastrophe of Brains-in-a-Vat.

(3)  These are much more problematic than numbers, or even differentiable 4-manifolds. The zoo of correspondents to every idea (already one is somewhat begging the question,  speaking of an “object of thought”) threatens to devolve into Meinongism – an ontological slum. 

            Certainly a mere expression – a string of words – does not in general denote or connote or (as Amos ‘n’ Andy might put it) jinote  anything at all.  “The square root of the present Pope of Islam’s antigravity mushroom non-unicorn”, though evocative enough, is referentially mere flatus vocis.  But there is no clear dichotomy.  Whether or not a thought corresponds to an expression (without which assurance we cannot even open the question of the expression’s potential reference or denotation) is not an all-or-nothing question.  Take the case of the much-maligned “present King of France”.  The expression suggests a clear idea; we know what sort of thing would fill the bill; upon enquiry we learn, with some disappointment, that as it happens, nothing does.  So far so good.

            Now take the equally widely execrated expression, “a round square”.  To the average man, this probably suggests nothing at all; it may even repel him, like a bad omelette;  and philosophers never tire of pronouncing it impossible, nay incoherent; but it did suggest something to me.  The idea was dim at first, but not vacuous for all that:  whatever object might fit the description, if any, would be round, for one thing, in some reasonable sense of this elastic term. And, it turns out, round squares do exist – not on our cul-de-sac, granted, but in R x R provided with the sup norm (║  ║∞).  Place four points equidistant from the origin, symmetric about both coordinate axes, and join with geodesics: the analogue of a square in this non-Euclidean space.  But in the sup norm, every point of that figure is equidistant from the origin, which is therefore the center of a circle (again in the relevant sense).  So now the image corresponding to the expression is quite precise; the object of thought, as real as a rocks.

*

In the chapter “On Realisms”, in An Experiment in Criticism (1961), C.S. Lewis writes:

The word realism has one meaning in logic, where its opposite is nominalism; and another in metaphysics, where its opposite is idealism.  In political language it has a third and somewhat debased meaning:  the attitudes we should call ‘cynical’ in our opponents  are called ‘realistic’ when our own side adopts them.  At present we are concerned with none of these, but only with realism and realistic  as terms of literary criticism.

(His discussion of this topic is bracing;  realistic, as commonly used, turns out to be something of a misnomer.

*

We close with a refreshingly clear and concise  thumbnail definition by Quine:

Unregenerate realism, the robust state of mind of the natural scientist.

Yo -- he means us!  --  Waiter, another beer!

How Now, Round Cow

[Note:  It has been maintained that Origen held that “the resurrected body will be spherical”. Henry Chadwick, The Early Church (1993), p. 106]

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is an old joke and a good one, retold here with some improvements.

~ ~ ~

United General Dairy wished to enhance milk production, so they called in an engineer, a physicist, and a mathematician.

The Engineer said, “No problem. Gimme a week.”

Good as his word, he showed up a week later  with an elaborate 3-D CAD prototype of an Advanced Magnetic Milk Extractor, consisting of a reverse rotor hardwired to an alternator (see diagram 12-c) routing through an ANSI-standard bolometric squinch, relying on either hex-nut variable findipulators, or (depending on parts availability) …

It was all very clever, but the humble dairymen couldn’t figure out how to work the thing, and figured they’d spend all their time in tech support instead of milking cows, which is what they liked to do.  So they sent the engineer on his way.

The physicist frowned, pondered a bit, then said:  “Doable. Fund me for a month.”

Thirty days later he returned, visibly pleased with himself.  “This is so much more elegant than what that engineer came up with.  A simple cylinder, 500 miles long.  Behold, gentlemen:  The Relativistic Linear Cow Accelerator!  Insert cow at one end, she emerges at the other, with (provably) every last lactic atom extracted, and placed into appropriate containers.”

Physicist launches a cow

Management was impressed, but inquired as to the cost.

“Ohh,” said the physicist with an airy wave of his hand. “A billion, a trillion, something in that range.  Ask Congress.”

Calculating that a pint of milk would have to retail at over a million dollars, management bade the physicist adieu, and turned to the mathematician.

The mathematician, however, did not turn to them.  He was… thinking about something.

Eventually they managed to snag his attention, and explained the problem.  The mathematician slowly nodded.  “It’s really a most intriguing problem… with ramifications in unexpected directions…. Allow me a year’s sabbatical, and I might have something for you.”

Management shrugged, and basically forgot all about him, until, at the stroke of noon, one year later to the day, the mathematician burst in, his moon-face beaming.

“Gentlemen, I have it.  Consider a spherical cow….”

* * *

As with many a good joke – those that make you smile instead of smirk or snicker – there is a theological dimension to this. 

Just what it is, is difficult to put into words – unless you are Chesterton, for whom it was (so literally) child’s play; and who put it thus:

            I find that most round things are nice,
            Particularly Eternity and a baby.

This says it all, but a footnote for mortals.  For you see, the thing about spherical cows is, they are so  ----- cowishly round, so… profoundly round, so – so round all about:  yea,

take them from this end     or take them from that,

they are

         round all around. …..

And this, indeed, is worth considering,

well merits our contemplation,

and our meditation,

through many an eternity afternoon …. 

~

Bonus poem:  Symmetry viewed by a mooncalf  (Rilke):

Ach was ist das für ein schöner Ball !
Rot und rund wie ein Überall.
Gut, dass ihr ihn erschuft.
Ob der wohl kommt wenn man ruft?

~

Good heavens... I Googled "how now round cow", which I'd fancied a basically new tweak of the traditional "how now brown cow", just to see if the search engine was updating its indexing of this site -- turns out there are already tons of sites that use this phrase.   Nothing new under the sun.


So:   TWoDrJ still rules the "humble woodchuck" universe, but is at present an also-ran in the lovely rotund world of Round Cows.

~~~~~~~~~
Addendum:

William Thurston, Three-Dimensional Geometry and Topology (1997), p. 103:

Just like the circle and the two-sphere, the three-sphere is very round.  But there are some beautiful, classical aspects to its roundness  that are not easy to guess from its lower-dimensional sisters.

The easiest way for a human to visualize the three-sphere is as the one-point compactification of ordinary three-dimensional Euclidean space (basically, you adjoin a point at infinity and define open sets as sets containing this point and with compact complement).   But, the author cautions,

this picture suffers from a loss of symmetry: [this compactification] is not as round as it should be.

 Pleasingly, a Google search on “not as round as it should be” brings up the Thurston quote as the very first hit, ahead several pages of more humdrum physical uses.

~~~~~~~~
Addenda:

Here is an actual unretouched photograph of a Spherical Cow:
     http://www.flickr.com/photos/unpredicable/68698656/

Let the welkin resound with the rotundity of round!
Here you can behold a genuine round square in captivity.