Thursday, January 29, 2015

Mélanges géopolitiques (neu bearbeitet)


Some recent essays of sociopolitical interest:


To update Marx:   History happens, first as tragedy, then as farce; then as slapstick re-runs on cable channels.

Even the principles did not realize for some time that it was leading to a World War.
Yet it marked a powerful and permanent turn, of the groaning millwheel of History;  and on that day, Clio laid aside her pen, and wept.

Lacrimae rerum


"The cabinet noir  introduced, in a curious way, the Open Diplomacy  advocated by the enthusiasts of the League of Nations."

The hyperconservatives’ public rhetoric may in many cases be simply ad usum minus habentium.

الدولة الاسلامية في العراق والشام

There is no more a logical contradiction between someone born by surrogacy  coming (upon mature reflection) to condemn the practice, than in the analogous case of someone born by rape, or prostitution, or incest, or bigamy, or A.I.D., or fructification by Zeus in the form of a swan, objecting to (as a general practice) rape, or prostitution, or incest, or bigamy, or A.I.D., or extra-Olympian dalliances by randy deities sub specie cygni.



Trigger warning:  This philological note may contain some very, very bad words.  You must be over 21 to read this.  No, make that 31.   91.   In fact -- Don’t read it at all.

As a native who returned, Thomas Frank combines, to advantage, the insights of the Visiting Martian (Toqueville, Dickens, Chesterton …) with those of the indigene.  And in richness of description, his book recalls that never-equaled masterpiece, John Gunther’s Inside USA.

Incredible though it may appear, the students at UC Santa Barbara face more deadly perils than the lapses from correctitude in The Great Gatsby:
www.latimes.com/local/lanow/la-me-ln-santa-barbara-deadly-shooting-20140524-story.html#navtype=outfit

A word on that fixed phrase, “rape or incest”, inscribed in stone like “peanut-butter and jelly” or “Laurel and Hardy”.   Sociologically, morally, the collocation makes perfect sense.  But practically, there is a certain redundancy …

The 1996 book It Takes a Village, by Barbara Feinman (writing as “Hillary Clinton”), was meant -- commendably -- to point to the need (which in practice, in America, had never been denied, until recently) for community involvement in child-rearing, beyond the autarky of the nuclear family.  Such was the unquestioned state of affairs during my own childhood…

A mittel-europäischer rationalist recalls “those golden, and, all in all very peaceful final decades of the colonial system”,  and adverts ad the hermeneuts …

An Irishman, a person of unrecorded nationality, and an individual from a nation we dare not name, walk into a bar …

Striking a blow  for freedom of adultery,
and the muzzling of the press.



If the geopolitical playing-field were momentarily tilted slightly differently, you’d have Congressmen standing on their chairs shouting for the inalienable rights of the freedom-loving Crimean People to determine their self-determination themselves.

But now a wildcard pops up out of the deck.   The current Justice Department, the one in office at this particular instant (and again, possibly after a wild night of ibogaine abuse, that part is unclear), lets it be known that it would prefer to stay in bed rather than defend the government’s side of the case, on this issue.   At which point a voice (of uncertain origin) bellows from the wings, that in that case, the Supreme Court lacks jurisdiction! Which means, the original litigant wins by default, just as though it were a Little League game where the Wellfleet Woodchucks failed to appear.

There is another, subtler layer to the story, namely that the ever-oily, ever-ingratiating Hollande -- who is by no means accustomed to laying bald facts out plain -- was, in this case, actually attempting, in his ham-handed way, to be politically-correct :  only, within a certain sociopolitical microclimate …

While France is absorbed in such insipid distractions as the latest entry in the palmarès of “The Wit and Wisdom of François Hollande”, some genuine events are unfolding in Françafrique, following in the wake of much bloodshed, and heralding more bloodshed to come

This item will sell like hot falafel!  Bulk-order Ur copies now !!!

It is depressing even to have to think about North Korea, the geopolitical equivalent of anal warts.

France’s practice of repeatedly doling out ransoms, while denying that it does so, has planted a tree with poison roots.  Here, perhaps, are some of its fruits.

Glurge alert!  The French President  goes groveling  before the fratricidal chaos of Africa.   If he needs a good whipping, why can't he go get it at Madame Sévère’s Bondage Basement, instead of dragging all France into his fantasies?

Mister Pecksniff  and Ted Cruz -- separated at birth?

Outsource the police state to charity:  The Swiss NGO Terre des Hommes lays a honey-trap…

“I don’t know which Mafia I dislike the most.  I’m leaning toward liking the Italian Mafia  because they are just immoral  and still believe in mother and child.  But the Art Mafia is immoral and, from what I can tell, they’ve stopped procreating.”

Tom announces on national television that he intends to kill John.   Then Tom is charged with murder, because, though his intended victim is still among the living, what Tom did was “just as bad”.  And if you imply otherwise, you are insulting the murder-victims community.

Wie eine Kultur  sich selbst auffrißt

ripouxblique

.

Bedtime for Bunnies





deep within the eve-ning-
time’s  the time for slee-ping.
lots of bunnies  brea-thing
in their burrow,  drea-ming




Tuesday, January 27, 2015

The Case of Greece


Greece has been in the headlines du jour -- in Europe, at any rate -- over what, on the surface, is a complex economic matter, a re-run of what we have seen before, involving  the IMF and a potential rééchelonnement of sovereign debt tranches …..
(ZZZzzzzzzzzzz …. Oh, have I lost you?)
What deeper waters this narrative might tap into, I have little idea, since I do not follow Greece;  taceo igitur;  the point here being merely to notice the oddly canted perceptual stance of Americans in regard to Greece, victims of distance and history.

What occasioned this reflection are the final chapters of Robert D. Kaplan’s outstanding travelogue/history, Balkan Ghosts (1993).
In this slender book, he quite consciously follows in the steps of Rebecca West (“Dame Rebecca”, as he gallantly denotes her), whose massive memoir of a trip to Yugoslavia on the eve of World War II, Black Lamb & Grey Falcon (published 1941) is among the truly great works of the twentieth century.   Knowing that he could not hope to outdo her magisterial historical survey, he treads more lightly but more widely, his final footsteps reaching as far as Greece, where he resided, with his wife, for seven years, during the reign (for “reign” it was) of a very strange figure indeed, Andreas Papandreou. 

The first thing you notice is the very presence of Greece in a book about the Balkans.  Granted -- once you come to think of it -- that country is indisputably geographically a part of the Balkan Peninsula;  Kaplan’s point is that it belongs with the other, Slavic or semi-Slavic countries of the region, spiritually and sociologically as well.
Very few Americans think of Greece in those terms, nor indeed in any terms at all except what we half-remember from school, limited to the Athenian Golden Age, several centuries B.C.  It is as though you were to try to conceive Germany in terms of what had been going on in the primeval forests of that time -- or America, as were it a continuation of its own prehistory of open plains, speckled here and there with miscellaneous blemmyes and buffalo, and otherwise largely empty.

It’s strange how little I knew of the tale that Kaplan tells.  It’s not as though I hadn’t yet come of age during the 1980s;  I did read newspapers.   But perhaps the reason for this ignorance had partly to do with the Western press, which had assimilated just one new Greek stereotype since the Age of Socrates:  the Medi-hippie world of “Never on Sunday” and “Zorba the Greek”.   Indeed, the English Wikipedia entry is astonishingly airbrushed:

    http://en.wikipedia.org/wiki/Andreas_Papandreou

I had to blink, to ascertain that this wasn’t an account of his more conventional father George.  (Someone seems to be curating his memory.)

Andreas knew well how to exploit the “Never on Sunday” sort of nonsense.  He appointed its star, Melina Mercouri (who played, one might say, a ταίρα  both onscreen and on the political stage) to his cabinet -- and re-appointed her again and again, while other underlings came and went.  It was a true Société du spectacle: 

In Papandreou’s name, Culture Minister Mercouri organized “human peace chains” around the Acropolis, even as Greek state companies were selling arms to both sides in the Iran-Iraq war, and to the two warring African states of Rwanda and Burundi.
-- Robert D. Kaplan, Balkan Ghosts (1993), p. 269

Likewise an eye-opener were Papandreou’s ties to a raft of morally quite unanchored chevaliers of terror:  PASOK and the November 17th Movement;  Abu-Nidal; Qaddafi’s hit-men, and on.  These, like such later ultraviolent inscrutable groups as Boko Haram and the central African Lord’s Resistance Army, cannot be understood terms of determinate ideology or even calculated, cynical self-interest;  it is not as though they have well-defined ends (whether good or bad) and merely overdo the means.   It seems to be more a matter of the morbido, and of metastatic narcissism.

None of this is part of the general American understanding of Greece.  “Cradle of Democracy” it must be for ever and aye.  For, our brains have only so much bandwidth.  Even in cases where party or interest do not preclude comprehension, we are apt, by acedia, to sink back into the Lay-Z-Boy of our early training and first impressions.  Cognitively, we settle for very little.



[Footnote]  Rather random but -- by way of counterbalancing the overall “down” of Balkan Ghosts as regards Eastern Europe:  there is a truly wonderful chamber ensemble called “Munich Artistrio”, though they seem to be Slovene.

Brahms (truly magisterial):
https://www.youtube.com/watch?v=N-L3UgoYvdA
(In particular the third movement, Adagio.)


Schubert (transcribed):
https://www.youtube.com/watch?v=nqv_-ll_LpE

[Update (alas) 9 III 15]  Greece threatens to flush its cisterns upon the rest of Europe:

http://www.lepoint.fr/monde/grece-le-ministre-de-la-defense-menace-d-inonder-l-europe-de-migrants-09-03-2015-1911427_24.php

Our notes upon this topic ici.
 

Monday, January 26, 2015

Minimalism in Mathematics (further updated)

A disclaimer:   What follows is not a substantive proposal, but a suggestive meditation, turning over this minute but multifaceted notion of “minimalism” and seeing how the light glints off.  It is neither better nor worse than a metaphor.

A couple of years ago,  a book-length treatment was published  that similarly plays with the notion of (in this case) “modernism”  -- which, like “minimalism”, is originally a term of the arts -- in relation to math:  Plato’s Ghost:  The Modernist Transformation of Mathematics, by Jeremy Gray.   To the extent that such an enterprise is worthwhile, it is in casting a bit of light from innovative angles, rather than deepening one’s understanding of math itself (though it did manage to get published by Princeton University Press):  it is more like a bull-session than a milestone.    Reviewing the book for American Scientist (Sept 2009), the mathematician Solomon Feferman sums up by quoting a remark by the historian Leo Corry, to the effect that
Extending the appellation modernism to mathematics … is like “shooting an arrow and then tracing a bull’s eye around it.”

Our own effort, in seeking resonances with the prior notion of minimalism, in mathematics, physics, and linguistics, is open to the same remark;  but it is what it is.


In the stylistic spirit of minimalism (and of that pointilliste Wittgenstein), we shall begin with a Delphic  epigram:

Logicism:  a kind of reductionist minimalism.

*

Considering that he took on the whole universe, in his methods  Newton was surprisingly Spartan.  Not only as regards “hypotheses non fingo”, but methodologically:

Newton consistently preferred Euclidean-style proofs.  He used his own calculus only where strictly necessary, and barred algebra from his treatise  entirely.
-- Leo Corry, “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008).

(Cf. a laborious non-analytic “elementary” proof in number theory.)
As fastidious as an Intuitionist!

*

Not a matter of method, let alone of taste, but sheer fact (albeit initially so counter-intuitive as to have been dubbed a "paradox"):
The Löwenheim-Skolem theorem: if a first-order theory has a model, then it has a countable model.
*

The attempts, lasting centuries, to do away with the Parallel Postulate by deriving it from the other Euclidean axioms, represent a remarkable early manifestation of the minimalist instinct.  Success would not have added to our fund of theorems about geometry, nor led to more perspicuous proofs.  The impulse was in part aesthetic.

A topic to explore:  the relation between abstraction in mathematics (an intellectual quality) and mathematical minimalism (which is not antecedently defined, but I have in mind the aesthetic, even spiritual side).

Contrast Finitism, Intuitionism, etc.:  Not Minimalism, but self-castration.

Zijn lange, magere  maar gespierde gestalte,  zijn scherp ascetische gelaatstrekken…


There is also a sterile sort of minimalism:  as, the replacement of the standard set of logical symbols AND, OR, NOT, by a single one --   NOR or  NAND (Sheffer’s stroke).  It led nowhere.

*

A variety of the Minimalist instinct  characteristic of abstract mathematics  is the notion of elegance.   Its role in mathematical practice (it has no purchase on mathematical fact) is reminiscent of, though practically distinct from, that of beauty in the practices of physics.

This, from a man with one foot firmly in either camp, math and physics:

The development of mathematics may seem to diverge from what it had been set up to achieve, namely  simply to reflect physical behavior.  Yet, in many instances, this drive for mathematical … elegance takes us to mathematical structures and concepts  which turn out to mirror the physical world in a much deeper and more broad-ranging way…

-- Roger Penrose,  The Road to Reality (2004), p. 60

(This is the "unreasonable effectiveness" motif.)

*

We earlier noticed what we called “the Dialectic of the Topological Enterprise” -- abstracting-away from rich familiar entities, extracting what seem the essentials, and seeing what happens.   The first step might seem Minimalist, but the consequence is an effusion and exfoliation of new spaces which meet the newly relaxed criteria, and which turn out to have an even richer riot of properties than we began with.   Per se, there is little in all this that might justify bringing in the aesthetically-tinged label of “Minimalist” (not a traditional term in mathematics; the closest you get is “abstract”):  but the aesthetic ethos is there, for all that.  Thus Shing-Tung Yau, The Shape of Inner Space (2010), p. 77:
We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

*

In the arts, Minimalism is a preference:  which, once adopted, is striven for.  In mathematics, you might like to keep things as simple as can possibly be:  but the mathematical facts seem to have a will of their own, at times.   Roger Penrose gives several instances of this, in The Road to Reality (2004).  For instance, with real functions, you can do pretty well as you like; but complex functions have a built-in naturalness.  You can try to define one on a given domain, but they have a mind of their own, and expand to their natural maximal domain by analytic continuation.   Thus, the larger set of numbers, the complex, spanned by the reals and the imaginaries, turn out to be in some sense more ‘real’ -- more round, more natural -- than the “reals” themselves.
Or again:   Suppose, once-bitten by the set-theoretic antinomies, you become twice-shy, and (p. 373)
adopt a rigidly conservative ‘constructivist’ approach, according to which a set is permitted only if there is a direct construction for enabling us to tell when an element belongs to the set.

(I picture this hypothetical constructivist as being played by Graham Chapman doing his officer’s shtick.)   But alas!  Penrose runs through the Turing/Cantor diagonal arguments and concludes (p. 376):
What this ultimately tells us is that, despite the hopes that one might have had for a position of ‘extreme conservatism’, in which the only acceptable sets would be the ones -- the recursive ones -- whose membership is determined by clear-cut computational rules, this viewpoint immediately drives us into having to consider sets that are non-recursive. … We are always driven to consider classes that do not belong to our previously allowed family of sets.

This is either a baffling, even a provoking mystery, or a simple consequence of what the Cantorian Realist indeed believes:  that these things are Out There, independent of ourselves (this might remind you of a certain Deity), and you can’t just methodologically sweep them away.   U B the judge.

(For a similar example applied to physics, click here.)

*

Pedagogical observation from a wise observer, who has been around the block:

Instead of the principle of maximal generality that is usual in mathematical books, the author has attempted to adhere to the principle of minimal generality,  according to which  every idea should first be clearly understood in the simplest situation;  only then can the method developed  be extended to more complicated cases.
-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), Preface to the second Russian edition

*

The nec plus ultra  of mathematical minimalism  is probably Category Theory -- which, however, I cannot elucidate, since I do not understand it.  It contains such things as the Forgetful Functor (this pops up in several introductory treatments, so it’s not as though I’m grasping at straws), which, given an algebraic group, “forgets” the group structure, leaving you with just a set  (excuse me: an element of the Category of Sets.)   Great -- die Gruppe ohne Eigenschaften.   The only way this even begins to seem to have a point  is if you then consider the adjoint functor, from sets to… free groups (these being a desolate Last Year at Marienbad landscape, again groups with the flavor removed).   Category theory looks at the bare bones common to many a different area of mathematics -- rather as though one were to study portraiture by looking at stick-figures.
(Actually, there is an analogy with the motif-index in folklore.  So, not knocking it here...)


~
On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.
-- Robert Kanigel, The Man who Knew Infinity, p. 204

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

~

From a logician:

The power-set operation has been interpreted  in the constructible hierarchy  as thinly as possible … We might be tempted to think of [the minimal model] as realizing a sort of contrary of the principle of plenitude -- a principle of paucity, if you will.     The principle of ontological parsimony … encourages some authors to eliminate individuals and un-well-founded classes.
-- Michael Potter, Set Theory and its Philosophy (2004) , p. 254



(All so difficult.  Why not relax with a mystery story instead?  Cool ones here: )

Snow Day


The druids are predicting eight hundred feet of snow. 
The bunnies make ready their burrow, with plenty of

            * hot chocolate
            * storybooks
            * pillows

The bunnies are looking forward to this day.

Saturday, January 24, 2015

Pugnaciously vacuous definitions


At the beginning of his preface to the introductory exposition Philosophy of Logic (1970),  W.V.O. Quine quotes Tweedledee:

“Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t.  That’s logic.”

That was a bit of nineteenth century donnish waggery (though it does give rather the flavor of modal logics and possible-world semantics  in our own day).

Quine then drolly goes on:

If pressed to supplement Tweedledee’s ostensive definition of logic  with a discursive definition of the same subject, I would say that logic is the systematic study of the logical truths.

We pause for Homeric laughter.
Anyone schooled in the lore of Russell’s paradox and impredicative definitions, will recognize that the professor is having his fun.

He then has some more:

Pressed further, I would say that a sentence is logically true  if all sentences with its grammatical structure are true.  Pressed further still, I would say to read this book.

(Ah!  ‘Tis a plaint  we ourselves have often made:  Buy my books!)



[Note:  Contrast the  equally terse but quite unselfreferential definition of logic in
R. Goldblatt, Topoi , 2nd edn. 1984:  “the study of the canons of deductive reasoning”.  Word for word, that’s very good.]


The jest is not without a subtle content, assuming as it tacitly does that there are such things as logical truths (a subject we see in a different light after Quine’s own extended attack on the analytic/synthetic distinction).   And it is quite in line with Quine’s celebrated existential bon mot:

    “To be is to be the value of a variable.”

To non-initiates, that will be mystifying;  to semi-initiates, cheeky; to familiar navigators of Quinespace, an ultra-compact allusion to his perennial concern with ontology in relation to quantification.

Put more pugnaciously:

Existence is -- what existential quantification expresses.
 -- W.V.O. Quine, “Existence and Quantification” 
~

The celebrated Cambridge Philosopher of Bland, G. E. Moore, wrote in his Principia Ethica:

If I am asked, “What is good?”, my answer is:  Good is good;  and that is the end of the matter.

That is:  If you have to ask, I can’t tell you.  (Even so, that pseudo-definition is preferable to the nihilist/reductionist dismissal by the Eliminative Materialists.)



Indeed, I shall now venture a definition  quite in Moore’s spirit:

Truth is … what is true (and known to God as such), quite apart from human-knowability, let alone provability (whether by finitistic, intuitionistic, or classical means).
~
From a college textbook:

A measure space consists of a set X equipped with two fundamental objects:
(1) a σ-algebra M of “measurable” sets, which is a non-empty collection of subsets of X closed under complements and countable unions and intersections.
(2) A measure μ: M -> [0, ∞] with the following defining property …
-- Elias Stein & Rami Shakarchi, Real Analysis (2005), p. 263

The bolded terms are there defined; but the term measurable sets is not -- the clause that follows reminds us of the definition of a σ-algebra instead.  The authors acknowledge their sleight-of-hand (effectively remedied by material elsewhere in the book) by placing the offending word in quotation marks. -- Note that this typographic care exemplifies the semantic Akribie of math writers, which we have elsewhere praised.




Not all math authors are as onomastically aware as those.  As:

Old-fashioned text-books  tend to start off with mystifying definition of these terms:  Euclid’s own definition,  “A point is that which has no part”, is a good example.  After a perfunctory discussion of these, the author clears his throat, begins a new chapter, and gets going with some concrete examples:  the definitions are mercifully forgotten.
-- Stephen Toulmin, The Philosophy of Science (1953), p. 72

~

To define -- literally, ‘delimit’ -- is by way of penning-in the definiendum with antecedently familiar landmarks.   To say that none such exist, is to refuse definition;  as in this classic hymn:

There is nothin' like a dame,
Nothin' -- in -  the -  world,
There is nothin' you can name
That is anything like a dame!
-- “South Pacific”



By this point  we have passed definitely from the realm of Oxbridge drollery to that of Joe on the Boat.  And here, indeed, we discover a whole subculture of definitional legerdemain, noticed in the delightful Dictionary of American Slang, by Wentworth & Flexner (1967).   In the Supplement we find this entry:

blivit   (n.)
   Anything unnecessary, confused, or annoying.  Lit. defined as “10 pounds of shit in a 5-pound bag.”  Orig. W.W. II Army use.  The word is seldom heard except when the speaker uses it in order to define it;  hence the word is actually a joke.


As a former harmless-drudge chez Merriam-Webster, I salute that as a gem of the lexicographic art.


A classic development of the pugnaciously uncooperative definition -- Lexicography with an Attitude -- is The Devil’s Dictionary, by Ambrose Bierce.  (Sample:  fork: an instrument for putting pieces of dead animals into the mouth.”)  Similarly impish was Hobbes’ definition of paradox:  “an opinion not yet generally received”.



A Pugnaciously Vacuous definition of the meaning of life  can be viewed here:
~

Quite other than such conscious humor, are cases like this:

Hamilton follows the Kantian notion of time closely in his “Essay on Algebra as the Science of Pure Time”.  Since the inner sense of time is more general than the outer sense of space, Hamilton concludes that algebra is a more general and fundamental branch of mathematics than geometry.
-- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 268

As the discoverer of quaternions, Hamilton has as much right as anyone to deliver himself of after-dinner remarks (a genre of public speaking to which he was particularly devoted) about the nature of algebra;  but this one is horse-hockey.
 

(Psychohistorical note:  Hamilton was for a time utterly immersed in Kant;  this characterization of algebra as the “Science of Pure Time” stems from psycho-philosophical exuberance, rather than algebraical expertise. 
A curious tentative parallel might be made with Hamilton’s countryman G.K. Chesteron, who likewise was given to flights of literary exuberance;  both were in marital situations requiring a great deal of self-sacrifice,  which they met with infinite patience;  and both were given to a kind of idealism  which some might diagnose as compensatory.
Okay, beyond our pay-grade.  Yet as Silvan Schweber affirms in his perceptive review [Isis, 1982] of this exemplary socioscientific biography:  “Hankins has eschewed giving psychoanalytical interpretations, [but] to anyone interested in the psychodynamics of creativity, William Rowan Hamilton presents a fascinating case study.”)

~

Another subcategory -- already bordered on by Hamilton’s epigram for the definition of algebra -- is formed by definitions which, while not vacuous, we might label Pugnaciously Perverse.   The poet Coleridge was (unfortunately) Hamilton’s philosophical mentor, even as regards what Science ought to be; and he defined that subject  thus:

“any chain of truths which are either absolutely certain, or necessarily true for the human mind, from the laws and constitution of the mind itself.  In neither case is our conviction derived, or capable of receiving any addition, from outward experience, or empirical data.”
-- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 268

(That first sentence does oddly prefigure the sort of prose  churned out by the truckload by the epigones of Donald Davidson.  -- Note too the anticipation of Post-Modernism.)


~

Another non-cooperative move in the orismological game, is pooh-poohing the very definiendum -- denying that there is anything coherent to define.  One Christian writer (C.S. Lewis or Hilaire Belloc, I forget which) once said testily, that the notion of “the Renaissance” was a will-o’-the-wisp, used  by secular writers to mean “whatever I like that happened in the fifteenth and sixteenth centuries.”

~

Composing a truly vacuous definition  is harder than you’d think.
Thus, consider the Euclidean definition of a point (“that which has no part”) which our philosopher friend scoffed at;  and let us phrase it even more egregiously:

point:  a point-like figure

That actually has cognitive content.  It means:  To visualize what is meant when geometers refer to a punctum (as opposed to a linea, etc.), think of something like a pencil-point.  Do not think of something like a dance-floor, or the cosmos, or an elephant.

And:

set:  a set of elements

Here the definition is so far from vacuous that someone could reasonably object that it is actually false, since it excludes the null-set.   (In this ‘definition’, the stress, so to speak, is on “elements”; “set” could be replaced by “bunch” or “passle” or “bucketload”.)

For indeed:

There is no direct circularity  if we presuppose sets in our study of sets (or induction in our study of induction), since the first occurrence of the word is in the metalanguage, the second in the object language.
-- Michael Potter, Set Theory and its Philosophy (2004) , p. 9


In further defense of impredicativity:

Impredicative definitions are necessary for ordinary mathematics, as they are unproblematic if one adopts a realist attitude about the objects defined -- realist in just the sense that the objecs exist in advance of the definitions, that they are picked out by the definitions, not created by them.  That imposes a substantial constraint on any acceptable philosophy of mathematics.
-- Shaughan Lavine, Understanding the Infinite (1994), p. 107

~

The grandfather of all tautological definitions is the one given by Yahweh in Exodus: 

~ ~ ~  I  Am   That  I  Am  ~ ~ ~

Yet at the same time, that is the best definition that could be given, since, in a common view of the Abrahamic religions, any limitative predication would be false.   (That view led in particular to the via negativa,  which insisted on the denial of all suggested predicates of the One.)

~

Another style of coyness with definition  is illustrated in the following, immediately after the authors have introduced Maxwell’s Equations  for E and H in free space:

At first we do not attempt to give physical meaning to these symbols.  We merely say:  let us assume that there exist physical quantities represented by symbols having the indicated properties, and see what these equations say about the quantities.  From the last two equations [to the effect that E and H -- whatever they might be -- have divergence zero], it is clear that E and H are solenoidal
-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 303

After a few pages of discussion, the authors summarize:

In a very real sense, therefore, these equations may be said to constitute a definition of E and H.
-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 306

Thus, back where we began.  The dangled definition in intuitive terms, is ultimately withheld:   No dessert until you finish your broccoli;  then -- Your broccoli was your dessert.

They subsequently reinforce the apophatic stance:

Physical meaning of E and H:  … The only real importance of the quantities is involved in the fact that they satisfy the field equations. … It seems most logical to go the whole way and treat E and H as defined by the field equations in all cases.  The commoner definitions can then be looked upon as mere picturizations. -- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 311

E & H:  They Are That They Are.


~

Lakatos offers his translation of an epigram from Poincaré:

    Mathematics is the art of giving the same name to different things.

Now, that is not vacuous (i.e., vacuously true, or anyhow only infinitessimally informative), since it is egregiously false;  but it is a witticism, not a blooper, since we all know that Poincaré was perhaps the leading mathematician of his time -- he has some cards up his sleeve, which he will slip out when it suits him.

For a series of essays on the art of definition, with especial reference to math,
try these:
http://worldofdrjustice.blogspot.com/search/label/definition



Thursday, January 22, 2015

Transfinite Cardinals: a Penguin Approach


Thesis:  There are Uncountably-Many penguins.

Proof:




.

Wednesday, January 21, 2015

Cream for your Coffee

[The following paragraph has just been added to our essay, "A New Proof of the Existence of Coffee-Cups".]


Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.
-- Roger Penrose,  The Road to Reality (2004), p. 508

And:

The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.
-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.


A complex Schrödinger equation,
after the Collapse of the Wave Packet


In for a penny, in for a pound.

Tuesday, January 20, 2015

Saint Anselm’s Proof of the Perfection of Penguins


Lemma.  Penguins have every excellence.
Proof.   Suppose there were some excellence which some one penguin  lacked. -- But that is absurd.   Whence the lemma.


Thus, using this result:  Create an ascending K-chain of Excellent Penguins, and use transfinite induction on P.   
Q.E.D.


“Zorn’s Penguin”:
The most perfect penguin of them all


[Biographical footnote:  Over brandy and cigars, Dr J was led to confess, that the proof was of his own devising, but that, out of modesty, he had attributed it to the saint.]

Monday, January 19, 2015

Charlie Hebdo: a Penguin Perspective








Je suis  Fluffie










Sunday, January 18, 2015

The Blind Men and the Billiard-Ball


We are all familiar with the parable of the Blind Men and the Elephant:   although it does suggest something about the limits of human perception (and, by parabolic implication, cognition), it is essentially a testimony to the complexity of elephants.**
Had the blind men been, rather, fondling a billiard-ball, they would have agreed on its characteristics.  From whatever angle the blindman came, he would conclude:  Smooth!  Round!***

[** Likewise characteristic is the excellence of elephants.
For proof and examples, click here.]


[*** If wise, each blindman would testify only to the local properties of the surface, such as the Gaussian curvature.
Additionally, even a sighted man, given free run all over the object, must needs refrain from asseverating, that what he felt and saw is all there is to the geometry -- the perceptual two-sphere might rather have been a cross-section of an unperceived hypersphere, of which it forms  but a negligeable part.]

Less obviously, such blindmen would concur as to the nature of the Stone–Čech compactification of any given set.  For, from whatever angle this grand object were approached, the verdict would be identical:  Maximal!  Universal!

~

Contrast the subject of Algebraic Topology.
I just finished reading one elementary introduction to this subject, and leafing through two others entitled Algebraic Topology  which, though scarcely elementary, claim the reassuring subtitle of An Introduction or A First Course.
The first labors long in the messy, mucking-about-with-triangles world of simplexes, as a lead-in to simplicial homology.
The second spurns these “rigid gadgets”,  and hews to the purer path of singular homology.
The third abjures homology altogether.



[Appendix]  Weiteres zur Elefanten-Mechanik:

It is well known that slowly-moving things obey classical mechanics.
-- Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 269

Corollary:
Elephants obey classical mechanics.