What did you do in the Linguistics Wars, Daddy?

Awhile back I offered an essay in Mathematical Lexicography:

What is Mathematics?

It led off with this:

    First, a sociological/tautological non-definition:

  What is mathematics?  One proposal, made in desperation, is ‘what mathematicians do’.

  -- -- Ian Stewart,  How to Cut a Cake (2006), p. 27

That stab in fact fails to offer even the virtue of a tautology, since it isn’t even true, without the further qualification that it is what mathematicians do… when they are doing mathematics.  -- If you’d tried similarly, without qualification,  to define linguistics based simply on the activities of linguists (ex officio: faculty and students in the Linguistics Department) at Berkeley during the years I was there, you would conclude that the field consisted of:  fixing your Volkswagens[**];  eating Chinese food;  and dabbling in neighboring fields like psychology and philosophy (later all these fields hopped into the hot-tub together and were newly baptised as Cognitive Science) -- all this while studiously ignoring most of the work done in the previous centuries of philology and language-sciences.

[**]  Sprawling supine beneath one's VW Beetle or Bus  was known locally as the "Berkeley lotus position".

[Note:  The title of this post  alludes to an insightful, delightful sociohistory of the era,
Randy Allen Harris, The Linguistics Wars (1993)]



[For further reflections on Berkeleyana, click here.

For the place of linguistics among the sciences, here.

For sociohistorical reflections on other sciences, here.]

How I Spent the Rainy Day

            read read read read  (Inhale !)

            write write write write (Exhale !)

Slices of Time

Brian Greene monostich

(Not quite a monostich,

nor a haiku neither.

It is  what it is.)

Every moment is illuminated,

and every moment    remains illuminated.

Every moment  is.

-- Brian Greene, The Fabric of the Cosmos (2004), p. 141

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

~

I have  on occasion  fulminated against “Physics porn” -- the tawdry down-dumbing of physics for a popular audience -- not so much a matter of (necessary) simplification, as of appealing to their baser, sensationalistic instincts.

Brian Greene does not practice that. The books of his that I have read (The Elegant Universe, The Fabric of the Cosmos) -- with titles like the Parthenon, elegant but restrained -- are worthy examples of haute vulgarisation.   The matters he treats of  are difficult to explain, even to fellow-physicists (if they work in a different lab, and thus subscribe to a different groupthink); he does a laudable job of trying.  Moreover, the fellow writes really, really well -- at times even poetically (witness the above).

Nor is the cited haikustich  poetry merely.  The idea behind it, appears in the next paragraph  in plain prose:

Einstein said that the problem of The Now  worried him seriously.  He explained that the experience of the Now  means something special for man, something essentially different from the past and the future, but that this important difference  does not  and cannot occur within physics.

Albert Einstein,  worrying about The Now

That is not to suggest that the Now at all resembles a mathematical instant, as in the calculus and thus in classical physics.  Each perceived ‘moment’ of consciousness presumably corresponds to an integral over some interval.   The ‘density’ being integrated need have no determinate evaluation at a single point of the continuum, any more than does the Dirac delta.  The psychological present is more of a Nowabouts.

The same point, with a different metaphor:

The practically cognized present is no knife-edge, but a saddle-back, with a certain breadth of its own  on which we sit perched.

-- William James, The Principles of Psychology (1890), vol. I, p. 609

The present is simply the past’s ever-moving outer edge.

-- Hendrik Hertzberg in The New Yorker, 18 Dec 2006, p. 33

~

That the “now”, as such, has no special status in physics, is unsurprising, since it is relative to each observer.  That simultaneity (implicitly dependent thereon:  “Both A and B are happening now”) turns out to be untenable, was a shock introduced by Einstein; eventually you sort of learn to live with it.   The real scandal is that physics is likewise agnostic as to any distinction in principle between the future and the past.   And that thesis is cognitively and theologically abhorrent.

Feynman among others calmly accepted solutions to equations, whereby certain particles ran backwards in time.  At its grandest, as a prominent mathmatician-cosmologist writes, “the time-reverse of the universe  is just as much a solution of the dynamical equations” as the one we thought we lived in (Roger Penrose,  The Road to Reality (2004), p. 729).

For humans, the unidirectionality of time’s arrow  is as fundamental as the privileged status of the Now.  America’s premier 19^th-century psychologist, on remembering:

However the associationist may represent the present ideas as throning and arranging themselves, still, the spiritualist insists, he has in the end to admit that something, be it brain, be it ‘ideas’, be it ‘association’, knows past time as past, and fills it out with this or that event.

-- William James, The Principles of Psychology (1890), vol. I, p. 2

eppur’ si muove …

~

James crops up in the first paragraph of a New York Times op-ed concerning durée (subjective time) from 10 May 2015, by Gregory Hickok, a professor of cognitive science:

In 1890, the American psychologist William James  famously likened our conscious experience to the flow of a stream … the stream … of consciousness.

The professor then puts his predecessor in his place:

Recent research has shown that the ‘stream’ of consciousness  is, in fact, an illusion.  We actually perceive the world in rhythmic pulses rather than as continuous flow.

The cognitive scientist then climbs down a bit from the claim that this insight is “recent” -- “Some of the first hints of this new understanding came as early as the 1920s, when physiologists discovered brain waves.”

One suspects, however, that James would not slap his forehead upon being apprized of this newly-minted insight, since he cites “the Humian doctrine that our thought is composed of separate independent parts, and is not a sensibly continuous stream”.  (Hume wrote in the eighteenth century.)

The Case of Greece (updated)

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 ἑταίρα  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 17^th 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.

[Update 16 June 2015]   The above was originally posted back in January.   Blink;  five months go by;  and the headlines around the world  are still the same:  Bras de fer entre la Grèce et le FMI.    And now we have two new hippogriff words to toy with, of the sort that Europeans love to coin:  Brexit and Grexit -- British exit (from the EU) and Greek exit (from the Eurozone).  Greece, insolvent,  is trying to blackmail is creditors with a curious threat:  Lend us even more money or we won’t pay you back what we owe you.  Or in other words:  Throw good money after bad, or we’ll hold our breath till we’re blue in the face.  And the ultimate menace:  the dreaded GREXIT.

Re-reading the original essay, a riposte suggests itself:  Fine.  Tant mieux.  Rejoin your Balkan buddies.  Have a nice life.

 

[Update, 20 June 2015]  A pragmatic contrary view, from Lawrence Summers:

Make no mistake about the consequences of a breakdown. With an end to European support and consequent bank closures and credit problems, austerity will get far worse in Greece than it is today, and Greece will likely become a failed state, to the great detriment of all its people and their leadership. Once Greece fails as a state, Europe will collect far less debt repayment than it would with an orderly restructuring. And a massive northern out-migration of Greeks will strain national budgets throughout Europe, not to mention the challenges that will come as Russia achieves a presence in Greece. The IMF is looking at by far the largest nonpayment by a borrower in its history. True, there are good reasons to think enough foam has been placed on the runway to prevent financial contagion. Yet, this was asserted with respect to Long-Term Capital Management, subprime mortgages and the fall of Lehman Brothers.

http://www.washingtonpost.com/opinions/the-consequences-of-greeces-impending-breakdown/2015/06/20/9a4ab5d8-174d-11e5-89f3-61410da94eb1_story.html

Riposte:

Summers is an economist, and  on that turf  deserves all respect.  But:

(1) The introduction to his essay (not here quoted), compares the potential post-default events to the advent of World War I, the default being a kind of Sarajevo.    But -- wait, sanity check -- though much of post-WWII history, many nations of Europe were not using the Euro.  In fact, for most of that time, the Euro hadn’t even been invented.  Were Europeans walking around clad only in barrels?  Indeed, at press-time -- and this might surprise you, if all you are going on is Summers’ piece --  certain European countries  much more important than Greece  are still not members of the Eurozone:  Notably, Switzerland and the U.K. 

Nor are those countries  clamoring to join.  Indeed, Great Britain is on the brink of heading the opposite way, with a Brexit.

(2)  Has the accession of Eastern countries to the Eurozone or the E.U.  been a benefit to western Europe?  According to the Eurocrats, abundantly.  According to the citizens who have had to actually suffer through it, not so much.

(3)  Moral hazard.  (Bailing out Greece yet again  dangles poisonous carrots before Ireland, Spain, …)

(4)   Whether Summers sincerely has the best interests of America and Wesgtern Europe at heart, we do not know;  but he is certainly in tight with the big bankers.  And Summers’ argument really reduces to “too big to fail”.  We have seen, repeatedly, where that takes us. -- Sorry, banker-guys, time for a haircut.

That was the view from the Western side.  But what would be in store for Greece -- the equivalent of the first World War, as Summers would have it?  -- The closest recent historical was Argentina’s default on debt, in December 2001, described as “the biggest debt default in history”.

http://www.telegraph.co.uk/news/worldnews/southamerica/argentina/1366218/Argentina-makes-biggest-debt-default-in-history.html

This, in mid-depression.  It was followed by … a return to roaring growth.  Argentina did just fine out of it.  But a minority of private lenders -- the kind of people Summers hangs out with --  still want their pound of flesh, and have launched various lawsuits.

La Grèce au bord de la faillite, l'exemple argentin

http://www.medi1.com/infos/ch_eco/

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 ἑταίρα  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 17^th 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.

 

Consilience and Cognitive Science

In a recent essay, the polymath philosopher Ian Hacking, as the first and most fundamental of his list of “applications” of mathematics, puts “App 0:  Math Applied to Math”.

Why should there be so much ultimate connectedness …?  If we follow the cognitive scientists who think that there are distinct mental modules for arithmetical reasoning and for spacial reasoning, Descartes’ Geometry of 1637 is all the more astonishing.  This question needs a lot of philosophical work, right now.

-- Ian Hacking, “Why is there Philosophy of Mathematics at All?”, repr. in Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. 248

Well, here’s a tidbit towards that.

Hacking is right, that the Idols of the Neuroscientists make the success of Descartes’ breakthrough the more astonishing, as indeed they do any of the insights of mathematicians.  But that simply redounds to the discredit of the Neuroscientists.

Hacking happens to have chosen an instance of Consilience that neatly matches two pre-posited “modules” of the CogSci crowd.  (Such “modules” can be had for the asking, according to fancy;  there is no net, in that tennis-game.  -- Table-tennis, rather.  -- Right now, my “Stuff and Nonsense!” module is blinking red.)   But they would be rather more hard-pressed to come up with a separate “point-set topology module”, “abelian group theory module”, “number-theory module”, “operator theory modules”, “ring theory module”,  ET cetera ET cetera, to attempt to account all the sundry other intersubdisciplinary interilluminations which keep arising within mathematical practice -- deep connections among concepts that never fell within human ken, back when our geometry module, in collaboration with the arithmetical module, was calculating the angle and distance to the rampaging mastodon, yet which now yield such fruits as the beans Jack received in barter for a cow.

Perhaps the explanation lies well apart from the contingent biochemicalisms of the grey-matter blancmange in our brainpan, modular or otherwise.   Mayhap there is an actual unity -- a unity in variety, which we do but discover --

in bits and glimpses,

in pieces and snatches,

by dint of much study,

and by grace of Grace.

~

We are making a New Year’s Resolution, not to be so darned snarky.  (Having successfully shed nearly twenty pounds this past calendar year, we are ready to take on even greater challenges.)

Accordingly, so as to redeem our good standing among our Cognitive Scientist brethren (with whom we used to rub elbow-patches daily, back at UC Berkeley, in the permanent “temporary building” T-2), we herewith offer an extra-virgin olive branch:   preliminary results from a massive structural-neuro-longitudinal study undertaking by the Pataphysical Department of the World of Dr Justice (headquarters:  Geneva), abundantly funded by the grateful taxpayer.  With this we obtain direct insights into the child’s developing mind.   As our investigations prove, this comes pre-compartmented into the following empirically buttressed cubby-holes:

(1)  The “Everything is What it is and Not  Another Thing” foodstuff module.

The vegetables are bad enough, but when they touch the mashed potatoes -- Eeewwww.

(2)  The “Mine and Thine” module.

Though discernable in outline already in the early embryo, this module does not become activated until much later in development;  and in some specimens (B. Madoff, D. Trump) never.

Prior to activation -- much like the notocord that serves the embryo until the spinal column takes its place -- behavior is governed by the following, non-modular (“opportunist”) pragmatic maxim:

     “My bear.  My toy.  Mine.”

(3)  The “Girls Have Cooties” module.

For reasons not yet well understood, this module is activated only in immature males;  some genetic link to the Y chromosome is suspeced.  In any event, it is normally de-activated at puberty.

The "Idea" Idea

Much of the most important and vital work done in the last half-century  depends [not upon experiment or brute calculation, but] upon new ideas;  and new ideas are notoriously exceedingly difficult to grasp.

-- Louis J. Mordell, Reflections of a Mathematician (1959), p. 11

We previously stated that mathematics is best characterized as the science, not of number, but of structure (or of pattern -- at this level of generality, either term will do).   As MacLane phrases it:

This chapter introduces the idea of the formal  in terms of certain basic structures:  Set, transformation, group, order, and topology.  With Bourbaki, we hold that Mathematics deals with such “mother structures”.  Against the historical order, we hold that they arise directly from the basic stuff of Mathematics.

-- Saunders MacLane,  Mathematics:  Form and Function (1986), p. 7

That last bit, you will note, is unabashedly Platonist, counterposing contingent human praxis  to transcendent time-independent Truth.  (We discuss this contraposition here.)

But beyond that, or rather as an animating force within it,  and distinguishing mathematics from such structure- or pattern-centered enterprises as architecture or the plastic arts, is the central role of ideas. 

MacLane puts the matter well.  Re the derivation of Hamilton’s equations from Lagrange’s:

What appears as a trick is in fact an idea -- an idea which must have been clear to Hamilton when he did it.  But we claim that in general  most of the formal tricks appearing in Mathematics  are really ideas in disguise -- ideas presented as manipulations  because the manipulations can be made explicit, while the ideas are a bit nebulous.

-- Saunders MacLane,  Mathematics:  Form and Function (1986), p. 284

In a previous series of essays, we put forward certain particular “mother ideas”.  Here we reserve a meditation-space  for musing about “Ideas -- the very idea”.

~

Hadamard comments on Rodin’s testimony that, throughout the process of sculpting, he must keep the “global idea” in mind, even while working on the smallest details;  and that “this cannot be done without a very severe strain of thought.”

I do not feel that I have understood [a mathematical argument] as long as I do not succeed in grasping it in one global idea; and, unhappily, as with Rodin, this often requires a more or less painful exertion of thought.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 65

Hadamard scoffs at the account given by Souriau in his Théorie de l’Invention:  “Does the algebraist know what becomes of his ideas when he introduces them, in the form of signs, into his formulae?  Undoubtedly not,”  but just turns the crank of mechanical calculation.  Apparently Souriau never consulted an actual mathematician, says Hadamard:  The mathematician trusts his idea, his insight, his intuition, more than he does his calculations, which after all are not infrequently in error  (Hadamard confesses that he, like Poincaré, was but an indifferent numerical calculator):  If these clash, you first redo the calculations, before tossing overboard the Idea that motivated the whole thing.

~

Hadamard then makes an excursus  rather off the path of our principal inquiry;  yet we shall follow him a little ways.  He confronts the question of whether language be the key to thought;   and waxes indignant at those who, like Max Müller, dogmatically assert that, without language, thought itself must needs collapse:

I had a first hint of this when I read in Le Temps (1911):  “The idea cannot be conceived otherwise than through the word, and only exists by the word.”  My feeling was that the ideas of the man who wrote that  were of a poor quality.

-- -- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 66

The devotees of this position  point to the dual meaning of the early Greek word logos -- ‘word, language’ and ‘reason, thought’;  and would  by implication  deny  that our diminutive and prickly friend, the humble hedgehog, could really know One Big Thing  or even a little weentsy one.

Hadamard, by contrast, is virtually a militant in the opposite camp:  “I fully agree with Schopenhauer when he writes, ‘Thoughts die  the moment they are embodied in words.”  This even applies to algebraic symbolism:  too cumbersome to actually think with;  you mostly only use it when checking your work.

The Neothomist philosopher Etienne Gilson  seconds the opinion of his countryman:

Si un linguiste me dit que c’est notre langue qui modèle d’abord  le monde que nous pensons,  je sais qu’il ne me parle pas en linguiste, mais en philosophe, qui se dispenserait d’ailleurs de me donner aucune justification philosophique de son opinion.  Non seulement je ne sais pas si elle est vraie, mais je ne sais même pas pourquoi elle lui semble vraie.

-- Etienne Gilson, Linguistique et philosophie (1969), p. 51

A contemporary philosopher goes even further:  some ideas may be not only pre-linguistic, but even pre-conscious:

We may not be aware of our ideas.  An idea  in this sense  is a tendency to accept routes of thought ... that we may not recognize in ourselves, or even be able to articulate.

-- Simon Blackburn, Being Good (2001), p. 3.

The epigram "We may not be aware of our ideas" is deliberately paradoxical.  Blackburn means "idea", not in the sense of the completely conscious  "I have an idea, let's...", but of something like the often tacit metaphysical underpinnings of mentation and investigation, which we treated of earlier.  -- Blackburn extends this notion (in a way reminiscent of, but antedating, Freud):  "A permanent strand in Christian thought  is that we have no insight, or even lie to ourselves, about our heart's desires." (id., p. 30)

We close this excursus with an epigram of William Hamilton  which Hadamard quotes:

Speech is thus not the mother,

but the godmother of knowledge.

~

The reason such musings lie off our main track, is that we are largely uninterested in psychology, or thought-processes, or any of the hunches & hiccups that fallen Man is heir to  as he struggles to comprehend all that His hand hath made.  The philosophers among the penguins  have different cognitive quirks from ourselves, as they too strive to unravel the tragic mystery of all that lies above and beneath:  what matters is the Creation, and not the creatures, save as we matter to Him.
As for thought and language:  with Hadamard, we conceive that there are cognitive activities for which vocalization is neither required nor especially helpful:  say, playing Go, or basketball.

There is an epigram, variously ascribed, that has always fascinated me:

“How can I know what I think

 until I see what I say ?”

On the face of it, this would appear to be anecdotal evidence for the thought-needs-language thesis.  But upon nearer inspection, it might argue rather the opposite:  That thought rose or arose  from some wordless region of the self, and only became an object to critical consciousness after having been concretized (and perhaps partly simplified or falsified)  by transformation into words.

For us, the key question is to what extent an Idea -- one worthy of the majuscule -- can even be adequately expressed in our language.   Certainly the higher mathematics cannot be expressed in ordinary human language.  It has invented for itself a more or less arcane system of signs, obeying no human syntax;  you may, if you like, par abus de langage, call that too a “language”, but it is no natural human language, but rather an aide-mémoire cobbled together to express ideas that observe their own semantics, call that language or not.   Hadamard himself attests that human language does not serve him especially well, when he must express mathematical ideas.  Whenever he must hold forth on a mathematical topic, even one of his own devising  and thus, to him, abstractly clear as a bell, he must write out the text of his lecture beforehand, lest he be left gasping and groping for words.

 

There is another old adage, current among linguistic philosophers:

“Whatever can be meant

can be expressed.”

At this point  we hear the shade of that crusty critic of Le Temps, growling:  All that you mean, maybe.

~

Let us put the point even more starkly.  Ask Not  (we channel Kennedy here) whether our (necessarily human versions of) ideas  could be adequately communicated to some other rational species.  Ask whether the Idea, as pre-existent in Platonic paradise, has been adequately incarnated in us.

(There now swims within my vision  the image of a category-theoretic Universal Object, with arrows slanting downwards  this way and that, as in Blake’s great painting.)

~

This is becoming interesting.  Hoping that your appetite has been whetted as well, we link to a couple of math-related installments of the “Any Ideas?” series:

Induction

The free invention of structures

Actual Infinity

~

We have tried to outline a capitalized or pregnant sense of the everyday word idea, which in most contexts certainly does not bear such freight.  (“I’ve got an idea, let’s go get pizza.”)  There is, however, another sense, which is still scientific/intellectual, yet which bears no Platonic or foundational flavor:  what is sometimes called a “bright idea”.   A bright idea is what causes a light-bulb to appear over the cartoon character’s head.  And it does represent some genuine cleverness, though its success is by no means guaranteed (and in the case of Donald Duck, will almost certainly come to grief.)

This more powerful form of inductive construction  can be deduced rather simply from the older form.  The trick is to construct, not the sequence of values, but the sequence of partial functions…

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 145

A “trick” is to an idea  as tactics is to strategy. 

Similarly:

We could prove the inequality by a limit argument from the known inequality for finite sums, but the following reasoning involves a very interesting technical device.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 195

~

We have noted before  that, once you set out to focus on Ideas per se, you keep winding up back in mathematics -- if only because there are so many of them there.  Yet more:  In our own lifetime, math itself has spawned a subfield  whose task, it would seem, is precisely the study and development of Ideas -- for their own sake, almost, and beyond such practicalities as computing the area of the field of Farmer Brown (or rather, Farmer Enkidu, since this concern goes back to Babylonia and beyond) or even its offspring, geometry, or the handmaiden of that, the calculus, or …  This field is called Category Theory, which (as faithful readers of this tragic blog will already know)  I do not personally understand:  but do note, that a recent introduction to same (subtitled “A first introduction to categories” -- the style of the title is that of children’s books;  and God willing, someday toddlers will study this stuff), by Lawvere & Schanuel, is titled:

Conceptual Mathematics

C’est un titre astucieux.  For again (this is a phenomenon which we have treated, in these essays, under the label “faux-naïf”), on the surface this might seem to be one of those liberal-feelgood substitutions for the actual hard work of thought, meant to bolster the self-esteem of slow-learners;  whereas in actual fact, it points at concepts -- what underlies such relatively superficial activities as real analysis, point-set topology, algebraic geometry (you with me, kids?), and all the rest.

~

Lagniappe:

[Ernst Schröder] even distinguished a horse, the idea of a horse, the idea of the idea of a horse … and dwelt a little on the concept of a concept.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 165