Minimalism in Physics (updated)

Truth is ever to be found in simplicity,

and not in the multiplicity and confusion of things.

-- Isaac Newton

[Note: 
In using the term “minimalism” in evaluating styles of physics, I am importing  into the philosophy of science  a term mostly associated with the arts.  The move may or may not be fruitful.  But here is a well-known parallel move, from Gerald “Mr. Themata” Holton, in The scientific imagination (1978), p. 7:

I have proposed … thematic analysis (a term familiar from somewhat related uses in anthropology, art criticism, musicology, and other fields).  ]

The whole enterprise of physics, from ancient times to today, is itself  in one sense  Minimalist, in that it seeks to sweep aside the riot of epiphenomena, and to discover underlying laws, from which that riot derives.  It does not so much banish the fullness of reality, as bracket it:  we hope later to derive much of it back, in an explanatory manner.  (Note:  Not quite the same thing as Reductionism.)

To some extent, all rational inquiry ‘minimizes’ -- idealizes, works with toy models, etc.   Some more than others -- economics notoriously so, tossing out so much bathwater that sometimes the baby goes missing.   Others roll up the sleeves of their labcoats and go in elbow-deep to the mess of reality, little bothering with economy or philosophy.   Chemists, in particular, seem perfectly content to potter about in labs, to discover stuff, invent stuff, patent stuff.  They literally multiply entities, in that they invent chemicals that weren’t there before.   Whether they thereby multiply them beyond necessity (mustard gas, thalidomide,  napalm, LSD) is a matter of individual taste.  But certainly the ethos is anything but austere.

And likewise, for the most part -- biology, geology, astronomy, engineering, what have you. 

But modern physics  raises parsimony to a central tenet, almost the prime purpose of the whole enterprise as currently understood.  This development being by now taken for granted among those of the guild, it may not be apparent how odd this really is.

The goal for some time has been the “Theory of Everything”.  This certainly sounds like a Maximalist program:  but really it is not.   For the knights who pursue this grail do not actually intend to explain any of the things that real people care about   and that motivated the enterprise of physics in the first place:  why the sky is blue, why snowflakes are the way they are, why clouds are shaped that way, what lightning is all about, why airplanes can fly…  (Purported explanations of these things exist, but the ones I’ve heard seem all fallacious.)   Instead, they want to wrap their arms around a passel of abstractions, so complex as to leave the plain man -- nay, any but the professional physicist -- behind many decades ago, and show that, at a still deeper level, they are all but facets of One Big Thing.  (Hedgehog physics, we might dub this.)  This is Minimalist, and ferociously so.

It may be objected:  All that is nothing but plain reductionism, which is simply to say:  Science.  No call to drag in an arts-related term like “Minimalism”.  -- But I believe there is an aesthetic dimension -- seldom mentioned in journal articles, though over-emphasized in popular writing -- which lies outside the bare logical necessities.  As,

“There shouldn’t be laws of physics,” Strominger maintains. “There should be just one law, and it ought to be the nicest law around.”

(Quoted in Shing-Tung Yau, The Shape of Inner Space (2010), p.  14.)

Gerald Holten, characterizing the attitude of Einstein (The scientific imagination, p. 281):

At stake was nothing less than finding the most economical, simple, formal principles, the barest bones of nature’s frame, cleansed of everything that is ad hoc and redundant.

In his own personal life, the legendary simplicity of the man was an integral part of this reaching for the barest minimum on which the world rests.

*

Central to the program is the “unification” of the various fundamental forces -- meaning, showing them to be symmetry-broken castoffs of an original single Force.    An analogy in evolutionary biology is explaining various related species  as having descended under various environmental pressures  from a common progenitor.   Only -- in physics, the enterprise is far more audacious than this analogy would suggest, if all you are thinking of are the breeds of dog, or the various canine species, or even the various land-mammals.   The forces are so fundamentally different in their phenomenology, that the task is more like tracing the common descent of the penguin, the echinoderm, and the paramecium.   Or even the mastodon, the gnat-swarm (considered as a sort of collective entity), and the sand-dune.   A tall order.

The reason so hubristic a program could come into existence despite the odds, is that it had an early success:  Maxwell’s unification of electricity and magnetism, back in the nineteenth century, truly a monument of the human intellect.  Now, later analysis has suggested that this success was something of a lucky fluke:  in the four macroscopic dimensions in which we reside, electricity and magnetism are both expressed by a vector.  You can not only analogize these, the one to the other, but calculate with them in the ordinary way -- say, forming their cross-product to get the Poynting vector.  In higher dimensions, electricity would be a vector and magnetism would be a tensor, and they would not play so nicely together.

The next success along these lines was far spookier:  the unification of electromagnetism with the “weak force”, into an unassuming-sounding entity called electroweak.   Now, this is far more bizarre than it seems. Electricity and magnetism were always rather like Batman and Robin, typically showing up together in the lab.  Whereas the weak “force” seems, to my untutored mind, like a force in some Pickwickian sense, like the  “force” of a metaphor in a poem.  A thing more different than electrostatic attraction or repulsion  can scarcely be imagined:  it deals neither in repulsion nor attraction, but rather in a handful of obscure and scarcely explicable processes such as beta decay.   Even to have conceived the project of their unification  was an act of extraordinary intellectual audacity;  the eventual success is, well, beyond any but specialist comprehension.

This new composite entity, this hippogriff, the electroweak, was subsequently unified with the “strong force”, yielding the hyperweak of today’s Standard Model.   The next -- and long elusive -- step, is the unification of that with gravity.   Now, to your average toddler, the natural analogy would be rather between electrostatic and gravitation attraction -- both, in their simple nonrelativistic forms, central forces obeying an inverse-square law    But your average toddler, like your average Nobel-Prize-winner-in-anything-but-Physics, would be mistaken.   And so the torch has passed to an ever-more-esoteric brotherhood, in particular  the magi of String Theory:  pale, spectral beings, who neither eat nor defecate, and whose results -- well, they do not as yet have anything so vulgar as actual verifiable physical results, mind you, but they do have theories, and conference papers, incomprehensible to all but the magi.  That does not mean they are on the wrong track;  perhaps they, and they alone, are on the right track, in which case, the more fools we.

*

A startling development in some corners of recent physics  is an actual ‘Maximalism’ -- basically, the catastrophic breakdown of any parsimonious project, yet not taken as a reductio ad absurdum of reductionism itself, but rather embraced, by amor fati (a fancy name for making the best of a bad bargain).   No longer can one really -- nor does one aspire to -- explain anything, since everything that might exist, does exist, and the (now uninteresting) facts of the matter in our own neck of the woods  can be chalked up to Selection Effect.  (We satirized this Rabelaisian Fay ce que voudras  here).   

Templeton-Prize winner Paul Davies,  in The Goldilocks Enigma (2006), p. 264, takes rather understated notice of this:

The disadvantage of the multiverse theory is that it invokes an overabundance of entities, most of which could never be observed, even in principle.  This profligacy strikes many people as an extravagant way to explain bio-friendliness.

Likewise, though for different reasons, the earlier Many-Worlds school (or cabal) of quantum theory,  in which entities -- again, entire universes in this case -- are multiplied, not simply beyond necessity, but beyond common decency.

The ethos of all this is atheistic -- a-anything, really.  It is perhaps no accident that Hugh Everett, an early pioneer of many-worlds, was (in Wiki’s words) “a committed atheist".  Or that the thélémisme of the distinguished hexagonal/pentagonal humanist  was taken up with gusto  by the diabolist Aleister Crowley, the stench of whose cinders may occasionally bother your nostrils, whenever a high wind blows up from Hell.

[Update 27 III 12]  Freeman Dyson in the current NYRB, reviewing a book by Margaret Wertheim about eccentric amateurs:

String cosmology is different. String cosmology is a part of theoretical physics that has become detached from experiments. String cosmologists are free to imagine universes and multiverses, guided by intuition and aesthetic judgment alone. Their creations must be logically consistent and mathematically elegant, but they are otherwise unconstrained. That is why Wertheim found the official string cosmology conference disconcertingly similar to the unofficial Natural Philosophy conference. The insiders and the outsiders seem to be following the same rules. Both groups are telling stories of imagined worlds, and neither has an assured way of deciding who is right. If the title Physics on the Fringe fits the natural philosophers, the same title also fits the string cosmologists.

[Note:  Dyson -- a notably fair man -- has long been a fixture of the Institute for Advanced Studies in Princeton; and the IAS, in recent years,  has been premier in string theory.  So Dyson's assessment here  is by no means that of an envious outsider.]

On the extra profusion of different string theories, a mathematician remarks dryly,

It was hardly an idea calculated to appeal to a man with a taste for desert landscapes  … There are more than 10^500 versions of string theory  lounging indolently about.

-- David Berlinski,  The Deniable Darwin (2009), p. 532-3

*

It will sometimes not be obvious, which proposals are Minimalist in spirit.  Thus, imagine some wretched Nominalist, who balks at the infinite, and proposes that the numbers needed for physics  are finite -- specifically, the field of integers mod a prime p (necessarily quite large, to accord with observation).   Finite’s gotta be simpler, more minimal, than infinite, right?  Roger Penrose retorts (The Road to Reality (2004), p. 359):

A physical theory which depends fundamentally upon some absurdly enormous prime number  would be a far more complicated (and improbable) theory than one that is able to depend upon a simple notion of infinity.

More precisely:   The problem is not essentially that the number is so large, but rather, with infinitely many primes to choose from, the choice would seem arbitrary:  in much the same way that the omniscient computer in  The Hitchhiker’s Guide to the Galaxy reveals, quite disappointingly, that the Meaning of Life is … “26”.

*

Differing from a theory-wide programatic theoretical minimalism, is a kind of personal cognitive-epistemological economy, described by Gerald Holton, The scientific imagination (1978), p. 158:

Fermi ordered the overwhelming and vast amount of knowledge  into a set of very few principles and ‘cases’, which allowed him to understand almost any new problem as an example of one of about seven primitive or primary physical situations.  Fermi would return throughout his career  to a listing or digest of the chief ideas in physics, which he had made when he first organized the field for himself as a young student.

Our own thoughts about such Leading Ideas in other areas, may be surveyed here.

*

A curious philosophico-cosmogonic anticipation of the TOE vs. Landscape divide  goes back several hundred years:

Leibniz … assure que la perfection de Dieu ne lui permettait pas de procéder d’autre manière que de la meilleure … mais Thomas d’Aquin sait que, créant du fini, un Dieu infini pouvait librement créer un nombre illimité d’univers différents, tous bons  et chacun commençant de manière différente.

-- Etienne Gilson, Linguistique et philosophie (1969), p.  163

And indeed, though Leibniz coinvented the calculus, we must say that, here, from a mathematical standpoint, it is Saint Thomas who is closer to the target.

Bonus quote:

Willem de Sitter found an exact solution to Einstein’s field equation … having no matter at all. … Why should we be interested in such a universe?  Because the real universe is of rather low density …

-- J. Richard Gott, The Cosmic Web (2016), p. 16

.

Thoughts 'n' Things

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

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

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

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

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

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

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

A mathematical proof is a bit like climbing a mountain.

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

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

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

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

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

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

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

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

~

Footnotes from the 19^th century:

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

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

For Frege,

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

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

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 "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”.

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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

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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

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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

 

Dominionism

A welcome addition to the political vocabulary is dominionism -- welcome because, rather than denoting an incoherent movement (like Democrats or Tea Party), it denotes an actual idea:  roughly, that (extrapolating from the OT advice to “have dominion over the fish of the sea, and over the fowl of the air”) everyone is to render unto God that which is God’s, and that which is Caesar’s … well, God now gets that too -- you render unto your pastors.

The term has been criticized as being an outsider’s coinage, rather than used by those in that current themselves.  But in this, it is just one of a long tradition of terms so minted, which eventually became coin of the realm:  Quakers, Tories, suffragette, Impressionist, originally derogatory (and much moreso than the dominionist, which has the ring of the King James), but eventually adopted as a badge by those so dubbed.    It is possible that the term Tea-bagger may follow this trajectory, since we need something to fill out the proportion Democratic Party : Democrat :: Tea Party : ??? .
The new term Islamists, apparently an outsider’s coinage (borrowed into Arabic as islâmiyyûn), is undergoing this process as we speak.  It has been pointed out that an analogous counter-coinage, Christianist, could legitimately be applied to the dominionists -- sauce for the gander, sauce for the goose.  Or you could call them “sharia Christians”.   These are they who, when Christ said “My kingdom is not of this world,” figured he was just pulling their leg.

G.K. Chesterton, in the course of his biography of Chaucer -- or rather, in his book entitled Chaucer (1927), in which he discusses all his favorite subjects, with an occasional nod towards that worthy bard -- mentions a celebrated medieval predecessor:

The only Lollard doctrine that was ever properly defined or denounced [was] the doctrine that ‘dominion is founded on Grace’:  that is, that nobody but a good Catholic purified by the sacraments  has any political rights or powers.

The word ‘Catholic’ here is misleading; Lollardy preceded the Reformation, so it really just means ‘Christian’.   Dominionism has never been characteristic of the Catholic Church, least of all in our own day;  indeed, the notion is characteristically Protestant, as witness the circumstances of the foundation of the Church of England.

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An older approximate-synonym of dominionism is theocracy.   Now, C.S. Lewis wrote some celebrated passages denouncing this;  which however I could not relocate in the essay collections on my shelf.  A Google search, and voilà (counterintuitively, in Reflections on the Psalms).  Oddly, I found it here:

http://richarddawkins.net/articles/1244-a-quote-against-theocracy

And since the Richard Dawkins folks had the courage and decency to quote CSL in extenso on their site,  I shall return the favor by linking to them here.  (Odd factoid: In the month before my adult baptism, I was reading both Mere Christianity and The Blind Watchmaker -- a chapter of one, then a chapter of the other.  Both excellent books.)

Further information here:
http://en.wikipedia.org/wiki/Dominion_Theology

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Linguistic footnote:

In the course of seeking that CSL theocracy quote, I stumbled upon a parallel, of his own coinage, to the neologistic distinction between  Islamism (bad)  vs.  Islam (good).  It is from the essay De futilitate (reprinted in the collection Christian Reflections):

This cosmic futility  [Dr J note:  cf. Steven Weinberg, “"The more the universe seems comprehensible, the more it seems pointless."] is concealed from the masses by popular Evolutionism.  Speaking to a scientifically trained audience  I need not labour the point that popular Evolutionism is something quite different from Evolution as the biologists understand it.  Biological evolution is a theory about how organisms change … As J.B.S. Haldane says, in evolution, progress is the exception and degeneration the rule.  Popular Evolutionism ignores this.  For it, ‘Evolution’ simply means ‘improvement’.  And it is not confined to organisms, but is applied also to moral qualities…

Earlier, Chesterton has some choice passages, quite along these lines.

Note:  Both CSL and GKC  essentially accepted Darwinism as Darwin himself stated it.

And another instance of “pejorative -ism” from CSL:

I distinguish sharply between the noble discipline called History  and the fatal pseudo-philosophy called Historicism.

-- C.S. Lewis, “Modern Man and his Categories of Thought” [unpublished MS, 1946], printed in Present Concerns (ed. Hooper, 1986)

(Actually we could draw the lines finer:  History is what-all happens; Historiography is the scholarly study and exposition of same.)

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A sociological fact that has puzzled me is that conservative Christians are often more ecologically indifferent than one would logically imagine for those who see the flora and fauna of this Earth as literally God’s handicraft.

In Consilience (1998, p. 278), Edward Wilson speaks of “the clash of two opposing human self-images”, and this tricky word dominion crops up :

The first is the naturalistic self-image, which holds that we are confined to a razor-thin biosphere  within which a thousand imaginable hells are possible  but only one paradise.

The competing self-image is the exemptionalist view.  In this conception, our species exists apart from the natural world  and holds dominion over it.  We are exempt from the iron laws of ecology that bind other species.

This “exemptionalism” is further related to the long-standing theme of American exceptionalism.

Approximate synonym of dominionism:  Caesaro-papism.  As, in a somewhat extended use, Ernest Gellner referred to "the Soviet caesaro-papist system" ( Plough, Sword, and Book (1998), p. 231).