When is the Task Force not the Task Force?

In May 2020, with the coronavirus still raging, President Trump said he was ready to “turn the page on all that”, and announced that the covid Task Force that had been led (apparently at the bottom of a mine-shaft somewhere) by the Vice-President (whose name  escapes me at present, and shall continue to do so in future) was being “disbanded”.  (In the French press reports of the anouncement, this verb was rendered démantelé -- ‘dismantled’, which sounds even more alarming.)  That offhand ad-lib caused consternation, so he “re-spoke” himself (verb ©WoDJ 2020, All Rights Reserved).  He now states that the Task Force will continue "indefinitely" -- albeit with substitute members and different mission. -- The young reporter who was obliged to recount all this on NPR, noticed a certain “philosophical question” here, whether this would be the “same” task force or not.

The philosophical problem alluded to  is that of Continuity of Identity, with many ramifications ancient and modern.  We have touched on some examples in these posts:

  https://worldofdrjustice.blogspot.com/search/label/continuity%20of%20identity

Meanwhile, herewith a handful of miscellaneous quotations, illustrating the variety of concerns:

Simon Blackburn (Think (1999), p. 127) quotes the joke about Irish axe that had been in the family for generations, the head having been replaced three times and the helve six; and asks whether Theseus’ ship, rebuilt plank by plank over the course of the voyage until no original molecule remained, was the “same” ship; and if so, what if someone had saved all the discarded planks and built a replica, this with all the original molecules, was that the “same” ship as well.

.

A modern example, with wry phrasing:

The truck had been a Renault back in the twenties, but had become, over time, a collection of replacement parts  cannibalized from every sort of machine.

-- Alan Furst, The Spies of Warsaw (2008), p. 146

And with a quite different topology:

The Romans who traversed the plains of Hungary  supposed that they passed several navigable rivers .. but there is reason to suspect that the winding stream of the Theiss  … might present itself in different places  under different names.

-- Edward Gibbon, The Decline and Fall of the Roman Empire (1776-1788)

Of current concern among Anglo-Saxon analytic philosophers:

Since these world-lines define the individuals we quantify over when we use modal logic (more accurately, when we ‘quantify into’ modal contexts), we do not have well-defined individuals at our disposal in any realistically quantified modal logic.

-- Jaakko Hintikka, “Quine on Who’s Who”, in Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 210

The Quinean position (on the "block universe" picture):

The space-time view helps one appreciate that there is no reason why my first and fifth decades should not, like my head and foot, count as parts of the same man, however dissimilar.  There need be no unchanging kernel  to constitute me the same man  in both decades,  any more than there need be  some peculiarly Quinian textural quality, common to the protoplasm of my head and feet;  though both are possible.

-- W.V.O. Quine, Word and Object (1960), p. 171

The “Idea” Idea (with an excursus on ideation and subvocalisation)

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.)

Voilà  le hic

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.

~

Ideation and subvocalisation

Hadamard then makes an excursus  rather off the path our our principle 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

Likewise, the behaviorist J.B. Watson says somewhere that “thinking is nothing but our talking to ourselves”.

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 them when checking your work.

The Dutch Intuitionist mathematician L.E.J. Brouwer is of similar mind:

De woorden van uw wiskundig  betoog zijn slechts de begeleiding van een woordloos wiskundig bouwen …

 

(Caption quotation from Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 38.)

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 noted Freudian psychiatrist agrees:

Every single thought, before formulation, has gone through a prior wordless state.

-- Otto Fenichel,  The Psychoanalytic Theory of Neurosis (1945), p. 46

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.  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 from some wordless region of the self, and only became an object to critical consciousness after having been concretized 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 astutieux.  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.

[Excelsior]   There is a vast philosophical literature (and a smaller, but still substantial, linguistic literature) concerning the relations between language and thought.   To rehearse this would be pointless;  to attempt to enrich it, quixotic.   Still we may feel our way forwards, and conceivably (eventually) contribute some minim of value, by taking as our paradigm area of Thought -- mathematics, rather than cats being on mats, and that sort of thing.   And Language as comprising, not only natural human languages, but any attempt at symbolic and communicable representation of Thought. 

(For this quest, I request:  God’s guidance and Grace.  Since, sine qua, non.)

An initial linguistic bridge is provided by our remarks above about the notion idea in the sense of ‘bright idea’.   A bright idea is no mere clothing of a perception;  it is closer to an invention.   And the key term it brings us up next to is:  insight.

[TBC?  Solâ gratiâ … ]

The Realist Vernacular

            What follows is neither proof nor argument, nor philosophy of any sort, but rather an exercise in sociolinguistics.  [And as such, a sort of sociological preparation for the thread announced here.]  The point is simply to exhibit a common way of talking among contemporary mathematicians, as well as some physicists and philosophers -- an easy style of conversation  that you would never imagine exists, if most of your acquaintance with science and its philosophy is mediated by figures like Daniel Dennett and Richard Dawkins.

            To cite evidence of theistic talk from the learned men of history, from antiquity through the nineteenth century, would be pointless, since the mode was well-nigh universal, at all times and in all realms.  Yet in our present day, most of the habitués of faculty clubs and coffee-houses  have managed to satisfy themselves, that all those who ever lived, in history, from Pythagorus  to the Einstein of “Der Herrgott würfelt nicht”, were, without exception,  imbeciles, and simply didn’t know what they were talking about, when they talked that way.  At last mankind has seen the light (or the darkness, rather);  we speak only of what is sensible and sniffable, like Donald Trump.  And who should be so rash as to cite the testimony of a mere Plato, or Galileo, or Cantor, or Gödel, against the brass certitude of so eminent a scholar as Sir Christopher Hitchens, Ph.D?

            Therefore I shall limit myself to recent citations.  A few examples from among many, snatched at random from  recent reading.

            Again, note:   I am not implying anything about the theology, per se, of the people here quoted, let alone suggesting that they never miss Mass.  Indeed a Dennett -- a roaring atheist -- can still indulge in such turns of phrases as "we can take advantage of the God's-eye perspective we have temporarily adopted" (the Devil can quote Scripture to his purpose).   So far, this is just corpus linguistics;  but more anon.

(1) Theistic language

 (a) mathematics

Arthur Koestler, The Act of Creation (1964):

Karl Friedrich Gauss described how he finally proved a theorem on which he had worked unsuccessfully for four years:  “At last, two days ago, I succeeded, not by dint of painful effort, but so to speak  by the grace of God.”

Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 219:

In one of the most cited discussions in his much-quoted book, Kuhn [1962] talks of scientific decision in terms of “conversion experience” and “faith”.

Richard de Millo et al., “Social Processes and Proofs”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 271:

The classical view does not require that an ordinary proof be accompanied by its formal counterpart; on the contrary, there are mathematically sound reasons for allowing the gods to formalize most of our arguments.

Similar language is often used in formulating the contemporary concept of a “hypertask”.  Cf. likewise

Shaughan Lavine, Understanding the Infinite (1994), p. 55:

            For Cantor … “countable” meant countable by God.

and similarly

Michael Potter, Set Theory and its Philosophy (2004), p. 250, re the plausibility of the Axiom of Choice:

… generalizing to the uncountable case  by appeal to the idea than an ideal being could achieve the choices required of him (or perhaps Him).

If we adopt some particular postulate system for abstract set theory, and agree that the criterion for accepting an intuitive set-theoretic argument  is that its analogue can be justified by the postuates, then we are  in efect  agreeing that the set of all intuitive sets  endowed with the membership relation  is a configuration satisfying the postulates.  We can have at best  intuitive reasons for believing this.  It is really an act of faith.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Gregory Chaitin, “Gödel’s Theorem and Information”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 306:

If God tells one how many different programs of size less than N halt, this can be expressed as an N-bit base-two numeral, and from it  one could eventually deduce  which of these programs half  and which do not.  An alternative divine revelation would be knowing that program of size less than N which takes longest to halt.

Robin Wilson, Four Colors Suffice (2002), p. 214, recounting a mathematician’s reaction to the immensely long and repetitive, unsurveyable, computer-aided proof of the Four Color Conjecture:

God wouldn’t let the theorem be proved by a method as terrible as that!

 (b) physics

Hermann Weyl, Symmetry (1952):

Contingency is an essential feature of the world.   Clarke  in his controversy with Leibniz  admitted the latter’s principle of sufficient reason, but added that the sufficient reason often lies in the mere will of God.  I think, here Leibniz the rationalist is definitely wrong, and Clarke [the theist] on the right track.  But it would have been more sincere to deny the principle of sufficient reason altogether, instead of making God responsible for all that is unreason in the world.

Abdus Salam (1988):

God created just two dimensions -- one of space and one of time. … At a later epoch, there was a phase transition to four dimensions, plus six internal ones.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 91, alludes to Roger Penrose’s paraphrase of cosmic censorship:  “God abhors a naked singularity.”   The (jocularly) theistic language is especially odd and noteworthy, since the epigram here being echoed -- “Nature abhors a vacuum” (Latin:  horror vacui) does not use it.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 126f.:

These laws [of physics] may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them  and does not now intervene…

That, then (for that author) leaves only the explanation of the settings of the parameters for initial conditions:

One possible answer is to say that God chose the initial configuration of the universe for reasons that we cannot hope to understand.  This would certainly have been within the power of an omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we could understand?

Simon Blackburn, Think (1999), p. 69, in a section called “A Scientific Model”:

Classical physics identifies the temperature of a gas with the mean kinetic energies of the molecules that compose it.  So in making hot gases, God has only one thing to fix…

Blackburn is a philosopher, not a physicist; as commonly in that community, he is writing in an “as if” mode;  but still, the choice of words is noteworthy.

John Gribbin & Martin Rees, quoted in Edward Harrison, Cosmology (2^nd edn. 2000), p. 489:  Absent the strong anthropic principle,

If there is a unique ‘theory of everything’, then … we would have to accept it as genuinely coincidental, or even providential, that the constants determined by high-energy physics happen to lie in the narrowly restricted range that allows complexity and consciousness to evolve…

(“Providential”… a lovely word…)

Paul Davies, The Goldilocks Enigma (2006), p. 3:

It appeared to Hoyle as if a superintellect had been ‘monkeying’ with the laws of physics. … Like the porridge in the tale of Goldilocks … the universe seems to be ‘just right’ for life.

Paul Davies, The Goldilocks Enigma (2006), p. 236:

Most theoretical physicists are Platonists in the way they conceptualize the laws of physics as precise mathematical relationships  possessing a real, independent existence…

Shing-Tung Yau, The Shape of Inner Space (2010), p. 101:

As Robert Greene puts it, “you’re trying to find the one metric given you by God.”

Michael Atiyah re string theory, quoted in Shing-Tung Yau, The Shape of Inner Space (2010), p. 292:

They’re onto something, obviously.  Whether that something is what God’s created for the universe  remains to be seen.  But if He didn’t do it for the universe, it must have been for something.

This statement is reminiscent of the Principle of Plenitude; and recalls Einstein’s celebrated quip, anent the possible failure of an experiment to confirm his prediction: "Da täte mir halt der liebe Gott leid; die Theorie stimmt doch."

Less seriously, but using a religious metaphor: J. L. Synge, Relativity:  The General Theory (1960), p.  ix:

It is to support Minkowski’s way of looking at relativity that I find myself pursuing the hard path of a missionary.

More serious is the following.  The author is discussing the vexed question of the Collapse of the Wave-Packet upon observation (related to:  If a tree falls in a forest, and no-one’s around, does it make a sound?  -- here rewritten in Oxford terms, replacing the forest by a quad), and quotes the old limerick:

Dear Sir, Your astonishment’s odd;

I am always about in the Quad.

And that’s why the tree

Will continue to be,

Since observed by Yours faithfully, God.

He comments (J. C. Polkinghorne, The Quantum World (1984), p. 67):

Divine reduction of wavepackets would be an overkill, since it would operate everywhere and always, forcing the electron each time to go through a definite slit.  The point about measurement is that it only occurs spasmodically.

Note the predicted empirical consequences of God in the Quad!   Then, between square brackets, the author adds:

This observation is in accord with the classic theological understanding of creation, which sees God as the ground and support of all that is (in our terms, the guarantor of the Schrödinger equation), but not as an object among objects (no collapser of wavepackets).

This aside is in fact central:  by the time he wrote this, Polkinghorne had quit his endowed Chair in physics to become a village vicar.

(c ) analytical philosophy

Michael Dummett, Truth and other enigmas (1978), p. 15, discussing character as (it may be, untested) hidden propensities:

If B still wishes to maintain the necessity of ‘Either Jones was brave or he was not’, he will have to old  either that there must be some fact of the sort to which we usually appeal … or else that there is some fact of an extraordinary kind, perhaps known only to God.

Dummett’s use of the term, to characterise the viewpoint of a hypothetical philosopher inclined to Realism, is so to speak opaque, not representing his own view, which is rather (id, p. 150) that “only a philosophically quite naïve person would adopt  realist view of statements about character”.  Only such, or Saint Peter.

(2) Realist language

Arthur Koestler, The Act of Creation (1964):

Gauss is reported to have said: “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”

Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 35:

When topological groups are found in nature [emphasis added], they  are generally not given abstractly as a set G with a composition law and a topology, but concretely, as a group of transformations…

This is not really anymore outrageously realist than saying “When a set of six objects is found in nature…”   (say, the familiar six-pack; although the one at my side is already down to five, and it’s not even noon).

And again, p. 157:

Covering spaces very often “occur in nature”:  that is, one comes across them spontaneously, while studying entirely different problems.

Shaughan Lavine, Understanding the Infinite (1994), p. 160: 

We seem to have nontrivial intuitions concerning the infinite, going far beyond simple things like Extensionality, Pairing, or even Power Set.

Shing-Tung Yau, The Shape of Inner Space (2010), p. ix:

The strength of this discipline [i.e., mathematics] lies not simply in its ability to explain physical reality…, because to a mathematician, mathematics is reality.

If all you know of math is elementary arithmetic, this statement may lack punch.   But in the upper reaches of set theory, topology and so on, it embraces a world of miracles and of monsters.

~ ~ ~

It is important to note, that this is the way Realists talk en famille.  The talk is casual, often not literal, yet is meant in some serious sense.  It is not to be compared with the tawdry pseudo-theological, pseudo-mystical sort of gibberish that popular authors (and even some serious ones, yielding no doubt to the Satanic promptings of the marketing department) use to gin up their pap for the masses -- like “God particle” for the freaking Higgs boson.

It may be objected (it will be objected;  it has been objected) that such expressions, in a modern mouth, are a mere façon de parler.  To which we reply (with Whorf), that a façon de parler tends to cohere with a façon de penser. 

Nor is it a refutation of the point here made, to adduce agnostic pseudepigrapha from any of the gentlemen here quoted.   We are each a walking contradiction;  we contain multitudes.   C.S. Lewis himself  confessed that he tended to be a cranky agnostic at dawn, but a theist later, when he’d had tea and was more himself.

Once again:   The point here is not to assert that so-and-so among our near contemporaries  is or is not a believer.  Indeed it will strengthen my eventual case, if many of them are not in fact believers, yet find themselves attracted, or guided, or driven, to Realist or Theistic language, whether for convenience, or (in the case of Cantor, Gödel, and Einstein) something deeper.

So, a very modest initial move, a sort of pawn to king’s four.  (Only later -- much later -- shall we see if we can capture Satan’s Queen.)   So far we hold simply, that occasional use of Realist or Theist language, in serious discourse, is not  in and of itself  diagnostic for Trisomy 21.

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

 

Philosophy Porn

“…that hardy revengeful spirit which earlier writers had regarded as not inconsistent with the Christian profession.”

-- Ward & Waller, eds. The Cambridge History of English Literature, vol. I:  From the Beginnings to the Cycles of Romance (1907), p. 229.

It is one of the strengths of The New Yorker -- a magazine I have enjoyed weekly for the past half-century almost -- not simply to spray you with a bunch of articles, but to categorize these:  “Our Far-Flung Correspondents”, “A Critic at Large”, and so forth.  What then was my delight to see, in the current issue, an entry in the much less frequent category “Annals of Ideas”!  (Cf. our own ongoing series here.)  Specifically -- very promising indeed -- : “How to Be Good:  A philosopher’s quest for moral truth.” 

So far so good;  though qualms arise when we turn to the actual article, and find the summary:

An Oxford philosopher thinks he can distill all morality into a formula.  Is he right?

Already you can detect that telltale whiff of intellectual pornography, for those alert to such things.   (Predecessors skewered here and here.)  

Still, the reporter would seem to be well-qualified to tackle this task, since, as the bio blurb informs us, “She is working on a book about extreme virtue.”  (“Extreme virtue” is not something that you or I might aspire to, but it apparently is practiced by certain adepts.)

Derek Parfit, we learn, is “thought by many to be the most original moral philosopher in the English-speaking world”;  and though, to anyone of apostolic bent, “originality” is not really the prime ideal to be striven for in this area -- there is many a pop guru for that -- that was no doubt meant as a compliment.  And upon learning that “he became increasingly disturbed by how many people believed that there was no such thing as objective moral truth” (by “people”, the reporter probably means “professors”, but perhaps not), I was prepared to embrace a fellow warrior.

It turns out, however, that the man is a self-proclaimed atheist.  Well, fair enough;  we have alluded to  G.K. Chesterton and  C.S. Lewis, their admiring portraits of atheists doggedly devoted to the pursuit of truth, and saluted one ourselves.   God speed him on his quest, by all means;  but indeed, beginning from that premise, he is going to require some deep ideas indeed.

And these, alas, we never meet.   We learn instead that “He believed that he had little native talent for photography, but that by working hard at it he would be able to produce, in his lifetime, a few good pictures.” (Whether he succeeded, we do not know, as none are shown.)  We hear of his soul-wrenching travails:  “When he was in America, he was compelled to procure his own food.”  That saga has a happy ending, however, as we learn his inveterate favorite order at Thai restaurants (wait for it…. Curry!)  Once he went out -- the wag! -- and “bought three identical black suits” -- thus rivaling Arthur “Two-Sheds” Jackson for sheer loveable eccentricity. Uncharacteristically for The New Yorker, the article becomes downright fawning, as when it reports, with a straight face, that “he learned that Reasons and Persons” (this, not a prayer or a poem, but a full-length book) “was being memorized and chanted … by novice monks at a monastery in Tibet.”

At this point, you’re supposed to be moaning softly, lips parted, “I Want His Ba-by…”

-- Well, bushwah.  In the old days of the New Yorker humorous end-of-column fillers, this would have appeared under the headline “Chants We Doubt Ever Got Chanted by Legions of Tibetan Monks”.

Our present point is this.   It is only once a major thinker’s reputation had been established and understood, that such sartorial and gustatorial trivia  could be of any possible interest.   Those unable to understand Relativity -- whether the Special, or the General Theory, in either its tensorial or its later coordinate-free notation -- can still sigh over Einstein’s sweatshirts and shock of hair.  But… Parfit?  Surely few of the magazine’s readers can have pored through his books;  and the article has so far given us no real idea of what distinguishes him as a thinker.  Towards the end of the article, we are finally given a glimpse;  drums roll, lights dim -- but it is not encouraging:

Then, at last, he was in a position to propose his top-of-the-mountain formula, which he called  the Triple Theory:

An act is wrong just when such acts are disallowed by some principle that is optimific, uniquely universally willable, and not reasonably rejectable.

(Try chanting that.) -- Dumbstruck at this insight, the Oxonian gentleman, in the very act of ravishing and then eating his grandmother, suddenly leaves off, and becomes a vegetarian.  ( Orthoëpic nota bene:  “Optimific” is not a typo;  it is the way you pronounce optimistic when your mouth is full of digestive biscuits.)

Whatever this all may mean, the stakes are high:

With his Triple Theory, Parfit believed that he had achieved convergence between three of the main schools of moral thought, but even this didn’t satisfy him.

(Stout fellow.) -- This is not philosophy, this is sciamachy, as when (in “The Day the Earth Stood Still”) the alien and the professor scribble dueling equations onto a blackboard  to show that they are each really really smart.

Of the rest of our philosopher’s thought, we are given barely a spoonful.  Sample:

            I am now inclined to believe that time’s passage is an illusion.

(Grant me leave to doubt the optimificity of that.)  -- This… delusion, is anyhow not “original”, having previously been put forth by certain crippled physicists.  At any rate he adds:  “I strongly want time’s passage to be an illusion.”   Now, that is telling.   He strongly wants to deny that framework without which free will, and purpose, and morality, fall to the ground.  To this creature we are to turn, for moral guidance?  Why not the Marquis de Sade?  -- And, do tell us, Professor:  Are you now inclined to believe that your much-gazed-upon navel  is an illusion as well ???

“Top-of-the-mountain” -- “Triple Theory” -- This is the language of People magazine, or of Oprah.  And indeed, it directly develops, that like these, which cater to the self-caressing, victim-licking, pruriently empathetic downswirl of our current culture, the present article (nestling in the venerable New Yorker like a maggot in meat) is interested in its fifteen-minute-famous philosopher, not as a thinker, but as a freak -- someone marginally more pitiful than our flabby selves.   Our very first introduction to the man is in the caption to his (ghoulish and overblown) photograph:

Derek Parfit has few memories of his past … a fact that he attributes to an inability to form mental images.

  And:

Parfit’s words … even in speech, all have a similar timbre -- it is difficult to distinguish them.

And: 

He had become, he realized, what psychiatrists call institutionalized.

And:

He had transient global amnesia … He didn’t remember getting married.  He didn’t remember having written his book.

And -- lest you imagine that a marriage, at least, lets him off a certain fashionable hook, for which Oxbridge has long been notorious -- we get some pulpy, purple prose describing his man-crush on Bernard Willams,  a noted adulterer and philosopher:

“Bernard was standing on the roof of King’s College with a kind of haughty, British, aristocratic look --“

(ooh!  ravish me! may I lick your tweeds?)

“-- you know, master of all he surveys, and all of Cambridge was shown below in the distance.

(How greatly do the townsfolk resemble ants!)

“And Derek said, ‘Isn’t he wonderful?’  I’ve seen that only once before, with a picture of Rudolph Nureyev.”

(And Nureyev, as we know, sure knew how to fill out a codpiece!)

However, mere bromance these days is not enough to get you on Oprah;  you also need to weep.  And so he does:

Williams died in 2003.  Even years later, Parfit would tell people over and over again how he had loved him.  He would break down in tears when he thought of how he had never been able to get Williams…

(into bed?  No : )

… to see what he saw about the truth, and how he never would.

The professed concern for “ideas” was just a sham -- a fig leaf or g-string for the reporter’s real concern, which is dish.  And when her subject has been squeezed for such juice as he may afford, she moves on to other quarry, like that new paragon for narcissists, the brainy-but-beautiful exemplar  familiar from the movies and Dan Brown:  in the present case, the author “well known … for writing The Skeptical Feminist” (already we can picture the jacket photo, in a get-up that, in better decades, would have brought a blush to the cheek of a tart, but which now is quite chic for professors), who

was teaching philosophy of science at the Open University.  She was very beautiful and very feminine.

Feminist, and feminine -- get it?  She’s got it all!  (Whether that combo would appeal to any actual guy, we leave to the reader;  but it is perfect for admiring yourself in a mirror.)

Artistically, this is tripe.  Morally, it is pornography.  And philosophically -- it is nothing,  just nothing at all.

*

Fortunately, a sort of antidote to all that is to be found between of covers of this same issue:  a very readable essay by Louis Menand, “Dwight Macdonald vs. middlebrow”.   The puff-piece on the curry-eating philosopher is an example of middlebrow, which Macdonald would have lanced.

Macdonald was an interesting character, characteristic of his place and time, straddling politics and the arts as now so few do.   He seems in general little remembered today -- I sought a bio in the Merriam-Webster Encyclopedia of Literature, but none was to be found; space being dedicated in preference to a different Macdonald, a certain Sarah Lawrence poetess, of whom we learn:  “Almost all the poems concern freakish people who have undgone amputation -- either physical or symbolic -- of a body part.”  -- Ach, back in the Oprahoid slime-pit, where we began.

[Update:  Menand has edited a collection of Macdonald's essays.  Reviewed -- albeit dyspeptically -- here.]
[Update, 23 October 2011:  Reviewed more genially here.]
[Update, 26 November 2011] A very informative review.

[Update 17 III 2012] Menand is an acute and wide-ranging critic, though something of a skeptic as regards the authenticity of the so-called "Lost Sonnet" of Saint Augustine -- while still conceding that it may well be "the greatest sonnet -- correction, the greatest poem -- ever written"  (at least, in the original Latin ...)

~     ~     ~

Palinode.

Have I been too harsh?

Anent the miscreants, not nearly harsh enough.  But any unintentional carom effect upon the honor of the grand old weekly, is to be regretted.

Let us therefore express our profound conviction, that the piece in question was approved for publication  during the August doldrums when all the senior staff were away in Maine.   It was approved by the doorman, and probably written by some homeless guy, who is still laughing.

Anyhow, as antidote, here is where I say very nice things about some very fine New Yorker writers:    here and here and here and here and here.

The point here is not to take exception to one particular thinker.  It is possible that the fellow has somewhere penned a splendid line or two (though the present sample seems sorely lacking in optimificalositaciosity).   The point is the degraded --and, here, degrading way, in which the media goes at such stuff.   It pretends to care, but does not care, what the thinker thinks, but only what she wears, or what he drinks.   In the same way:  Excellent work has been done in quantum field theory, but justice is by no means done to it by the standard presentations of physicists stumbling around with their little lanterns, in quest of the “God particle” (technically known as the Freaking Higgs Boson).

~     ~     ~

[Update 10 Jan 2012]
A photographer can make anyone look weird with the right (wrong) choice of light and lenses.
Usually they don't bother to freakify philosophers, but one such contemptible attempt can be viewed here:
http://blog.talkingphilosophy.com/?p=4071
Note: aldaily.com dignified this swill by featuring it as today's sole recommendation in the category "Essays and Opinions".

~     ~     ~

You can read a recent review of Parfit, by the ever-judicious Philip Kitcher, here.   Mercifully, it has nothing in common with the New Yorker takedown-cum-puff-piece.  A couple of excerpts:

On the face of it, the thought that morality should have a “supreme principle” is puzzling, for it is not obvious that our ethical life can be subsumed under any single formula. Some people worry about the idea that physical theory can be distilled into some mighty equation, taken to be the core of a “theory of everything,” but the field of ethics appears even less susceptible to such spectacular unification. If ten commandments are unable to suffice, how can we hope to manage with one?

Although the characterization of wrongness appears to involve three different tests, he claims that the three criteria give the same results—acts debarred by one would be ruled out by the other two. The point of exploring and offering all three is to reveal an unexpected convergence among three major traditions in moral thought. In an image he employs repeatedly, the history of ethical theory is described as a series of attempts to climb a mountain from different sides—and now that we are closer to the peak, it is possible to recognize that these are attempts on the same mountain and that the routes are coming together.

The three traditions of thought whose convergence Parfit wishes to affirm are consequentialism, Kantianism, and contractualism.

~     ~     ~

[Update 22 January 2012]

I have just begun to read an unusually well-written introduction to ethics, Simon Blackburn’s Being Good (2001).  And despite the New Yorker’s accolades to Parfit as “the most original moral philosopher in the English-speaking world”, Blackburn does not feel constrained to mention the blighter, neither in the index nor the bibliography.  

[Update May 2014]  In his peripatetic scamper through the capitals of world philosophy, in a valiant search for something illuminating to say about, literally, Nothing (“Why is there something rather than nothing,” is the topic of the book), the philosophically-trained science journalist Jim Holt arranges to go interview Derek Parfit.  Yet, though the reporter must travel all the way to England to do this, Parfit imposes a restriction on the interview  which newsman should be subjected to, and which even Ben Ladin did not have the chutzpah to demand:

He added that, since he was very slow in formulating his thoughts, he would prefer not to be quoted verbatim.  Instead, he would try to answer any questions I had about his written work  with a “yes” or “no” or some other brief response.

-- Jim Holt, Why does the World Exist? (2012), p. 223

(Sounds like a Turing Test.)   And this, note, for an interview broadcast live, where indeed the pressure of maintaining a snappy airtime  often trivializes discussion, and where the subject is his own work, which he has had a lifetime to think about, and approached by a deferential amateur in an attitude of near-reverence.

Accordingly, nothing of interest stems from the empty exercise of the interview itself;  undaunted, our author tricks out his report with quotations from Parfit’s previously published writings (something he could have done from home).   We are reminded of Edmund Morris’s desperate expedient, in his abortion of a biography of Ronald Reagan, where, confronted by the blank mind and depthless shallows of his subject, he filled things in with the imaginings of fiction.