A thought for Shrove Tuesday

From a Jesuit:

We cannot find anything that does not originate in God.   Each tiny thing that we encounter, especially each human person, is directly related to the Godhead in all its glory.

-- James Schall, S.J., The Order of Things (2007), p.188

Some decades earlier, G.K. Chesterton put a similar thought more epigrammatically:

Man cannot love mortal things.

He can only love  immortal things --

   for an instant.

Oddly, although that phrasing wears the badge of Chestertonian love of paradox, this fragment seems -- paradoxically -- more intuitively convincing.  Father Schall’s pensée is nice, is pious, but might well be met with doubt.  With Chesterton, you suddenly just see it.

Die fröhliche Wissenschaft (lo gai saber)

I wish to make a serious argument about the relation of the cosmos, God, and the human purpose.

But I wish to do so in a certan lightsome mood.

-- James Schall, S.J., The Order of Things (2007), p. 60

Ein Universitätslehrer, der sein wenig anmutendes Spezialfach  reichlich mit Witzen zu würzen pflegt [to “sow it with jokes”].

-- Sigmund Freud, Der Witz und seine Beziehung zm Unbewussten (1905 ff)

These could serve as  motto for this site.

~

David Riesman (The Lonely Crowd, 1961)  laments the impairment of the American spirit of playfulness   at the hands of post-Puritan sobersides:  “It may be a long time before the damage done to play during the era depending on inner-direction  can be repaired.”

Richard Rorty writes (Philosophy and the Mirror of Nature, II.iv.1):

The spirit of playfulness which seemed about to enter philosophy around 1900  was, however, nipped in the bud.  Just as mathematics had inspired Plato to invent ‘philosophical thinking’, so serious-minded philosophers turned to mathematical logic  for rescue from the exuberant satire of their critics.  The paradigmatic figures in this attempt to recapture the mathematical spirit  were Husserl and Russell.

Here the via mathematica and the via jocosa are counterposed.   Whereas for us, they intertwine  like the rose and the briar.

Thus Gerald Holton, in a review (reprinted in The scientific imagination) of the work of Lewis Mumford:

I was … delighted with Mumford’s … acknowledgement that there is a subjective and qualitative side to the doing of science  which scientists hardly ever talk about, the “intellectual playfulness and aesthetic delight” in scientific work, which can be an enormously important component of scientific motivation.

“Playfulness” was one of the favorite words of the Romance philologist (and my former teacher) Yakov Malkiel -- though you would not have known it to look at that hard-working, ever-professorial man, twice over an exile.   This was evident, not only in his writing style (which, for better or worse, has influenced my own), but in his actual etymological practice.  Certain cruxes of etymology, which had resisted the usual attacks of sound-law philology, he would attempt to explain as the free creation of the human spirit: and indeed, such bodacious onomastic hippogriffs abso-blumin'-lutely do exist.  This, in contrast to the sobersides scholar who would attempt to lautgesetz back to some obscure figment of hypothetical Vulgar Latin, or else claim Celtic or Klingon substrate -- thus missing the joke.

E.g., 

On the subject of “Deviations from l’arbitraire du signe”:

There arises the possibility of elevating playfulness to the same kind of pedestal  as economy and clarity, particularly on the strength of linguistic developments where emotional coloration has been achieved  at the expense of economy, or clarity, or both.

-- Yakov Malkiel, “The Inflectional Paradigm”, in W. Lehmann & Y. Malkiel, Directions for Historical Linguistics (1968), p. 32

O mistress mine ....

It is key to the cognitive style of such thinkers as Richard Feynman;  and not accidentally, as retaining a streak of the child is helpful in leavening fact-impacted thinking.

Again Holton:

As Einstein himself once said, he succeeded  in good part  because he kept asking himself questions concerning space and time  which only children wonder about.

~

The original Provençal gai saber, which seems to lie at the center of a collection of near-equivalents (joyful wisdom, gai savoir), specifically  denoted the art of composing love-poetry in the then-contemporary style, that of the troubadours.  As that art has alas passed from the planet, felled by the effects of the Albigensian Crusade,  we use it (as did Nietzsche et alia) in a wider sense, denoting the desired confluence of homo sapiens and homo ludens.

*

Si cela vous parle,

savourez la série noire

en argot authentique d’Amérique :

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

*

We have a genuine philosophic eros.  Knowledge excites us.

-- James Schall, S.J., The Order of Things (2007), p. 22

Da nun  in meine Darstellung  mancherlei einfließen wird, was strengen Richtern  unwissenschaftlich erscheinen muß,  so möchte ich dieses  einigermaßen  durch das Zugeständnis entwaffnen, daß dem Ganzen  die Überschrift  Allotria gebühre …

-- Hugo Schuchardt, “Der Individualismus in der Sprachforschung”, in Leo Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2^nd edn. 1928), p. 422

~

The fact that language is used for communication  is no more intrinsic to it  than its use to tell jokes …

-- Norbert Hornstein, Logic as Grammar (1984), p. 119

So far, however, from exalting the raconteur’s art, that passage stems from a practitioner of the singularly humorless Chomskyan Government-and-Binding school;  and means, not to enrich our notion of language with that of humor, but to squeeze it dry  even of semantics.

Cf. Karl Jaberg, Spiel und Scherz in der Sprache (1930).

A Dive to the Depths (expanded)

A phrase you will often meet in higher mathematics, and almost nowhere else, is:

“a deep result”

O loveliest of monostichs, thou !

The very notion of what ‘deep’ means, in such a context, is itself deep;  indeed, too deep for me, at present.   This, owing to a crippling condition of mathematical oligophrenia.  --  which, however, I pray that time and diligence might partly palliate.  (For a glimpse into the terrible sufferings of mathematical oligophreniacs, click here, if you dare.)  Yet I am putting up this skeletal promissory-note of a post, so that there will be a space to scribble insights as they wake me in the night.

First off -- the term "deep" does not mean simply ‘difficult’; indeed, though such results lie in the depths, and are not to be had for the asking, once you have somehow managed to fish one up, it may seem clarity itself.  Nor does merely being difficult make anything deep.  Any humongous brute-force calculation falls into that category;  for a more-substantive example, consider the Four-Color Hypothesis, which people suspected should be deep, but the proof that changed "Hypothesis" to "Theorem"  is a combination of clever tricks and elbow-grease.  The response of the mathematical community was disappointment:  "So, it turns out it wasn't an interesting conjecture after all."  (Of course, it may yet prove to be "interesting" in our cognitive human sense; that awaits a proof of an entirely different kind.)

We may go further, and put forward that an overarching purpose of mathematical research is to reveal something previously difficult  as now simple, when seen in the right way.  Again and again this has happened in history, beginning with the replacement of finger-counting by symbols, and of clunky symbols like Roman numerals by decimals.  For a more recent example:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Nor is there any simple similarity or opposition between being “deep”  and using what are now called “elementary” methods in number theory (as opposed to analytic methods), e.g. Alte Selberg’s proof of the Prime Number Theorem by elementary methods, compared with earlier analytic proofs by Hadamard and others.  Typically, proofs that restrict themselves to “elementary” methods are harder than those that permit themselves a more capacious toolkit;  but whether they ever, or generally, gain depth via this austere discipline, I have no idea.

In the meantime, some related posts outside of a mathematical context  are these:

            On Depth and Breadth

            On Scope and Difficulty

As appetizers, try the following hors-d’œuvres platter -- to follow which, however, we have as yet prepared no meal (as with our early essays on the Realist vernacular, this is more by way of linguistic warm-up):

The connection between linear transformations  and bilinear functionals  goes quite a bit deeper …

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 38

The Riesz representation theorem depends on some of the deeper parts of the theory of measure and integration.

--George Simmons, Introduction to Topology and Modern Analysis (1963), p.

The analogy between cyclotomic fields  and fields formed from the points of finite order on elliptic curves   is very deep.

-- Neil Koblitz, 1993

The notion of Kan extensions is the deeper form of the basic constructions of adjoints.  We end with the observation that all concepts of category theory are Kan extensions.

-- Saunders MacLane, Categories for the Working Mathematician (1971; 2^nd ed. 1998), p. vii

The continued-fraction representation of real numbers is deeper than the decimal expansion.

-- Roger Penrose, 2004

There are deep ties between enumerative geometry and Ramanujan’s tau function.

Contrast:

The useful fact about products of projections  lies near the surface.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p.  47

In the same work, while acknowledging the possibility of a lattice-theoretic formulation of the subspaces of Hilbert space, he dismisses this possibility as “trite” (p. 22).

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

Much commoner than “trite” is trivial, which is virtually a terminus technicus of mathematical practice.  Let one quote stand for all:

One of the useful conclusions we can draw from Theorem 2 [to the effect that the norm of a Hermitian operator equals the supremum of its eigenvalues] is that the spectrum of a Hermitian operator  is not empty.  This is not a trivial conclusion.  We shall obtain the corresponding fact for normal operators  only after the application of a lot more relatively deep analysis.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 55

Psychosociological note:   Attempt to imagine the effect on our pumped-up math-major  freshman or sophomore brains, hearing dismissed as “trivial” (with a wave of the hand) propositions which, but a year before, we ourselves would not have begun to understand, and which indeed most of our countrymen will go peacefully to their graves without understanding.  (For more on the hubris involved, click here.)

Also note:  What counts as “trivial” is relative to where you stand.  Thus, in the very next sentence, Halmos adds:  “We hereby report that the spectrum of an arbitrary operator is also not empty;  since we shall have no occasion to make use of this fact, we shall not enter into its proof.”  The proof, one gathers, is more difficult still.  But when once you have mounted, and stand upon that summit, the fact that Hermitian operators in particular have eigenvalues, is trivial indeed.

Leave it to mathematics to recruit even the notion of triviality into some highly non-trivial constructions.   E.g.

A topological space over X is called a locally trivial fibration if every x in X has a neighborhood over which Y is trivial.

-- Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 129

And:

Sard’s Theorem … is … a highly non-trivial  theorem  which is elementary in the sense that it uses only the notion of a differentiable map.

-- Shlomo Sternberg, Lectures on Differential Geometry (1964)

The terms deep and elementary (here in the everday sense, and not the special number-theoretic meaning mentioned above) are not antonyms, but they do contrast:

The spectral theorem implies that every normal operator has a large supply of invariant subspaces;  this is classical  and can be considered elementary by now.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

This dependence of “depth-perception” upon experience accumulated with the passage of time, does not refute the notion of relative mathematical depth as ill-defined.   What is intuitive though difficult to put into words  is a notion of “deeper than” rather than of absolute depth.  To the giant, neither the pond nor the puddle appears deep;  but the pond is deeper  for all that.

~

Re the classification of simple algebras (“simple”, to be sure, in a certain technical sense, meaning roughly: incredibly complex and difficult):

The tools employed  are not deep; they are just, so to speak, linear algebra  raised to the nth power.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 122

~

Further attestations.

A pioneer of Hilbert space theory, expounding the then-contemporary state-of-the-art for a nonspecialist mathematical audience, particularly as regards dilations and extensions of operators:

There do not seem to be any conspicuous and challenging yes-or-no questions that serve to indicate the direction in which the search for new results might begin,  but I have faith.  There is depth in the subject;  the trouble is that the surface has not been explored enough  to show where the deepest parts lie.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Writes a premier algebraic-topologist:

There are geometric problems which require the use of the multiplicative structure of the topological invariants.  Such problems are deeper than those which can be solved by considering the additive structure alone.

Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 105

Re Lie algebras and a closed subgroup H of the full linear group:

It is furthermore true (and this is deeper) that these one-parameter subgroups  fill a neighborhood of the identity in H, and consequently generate H if H is connected.  …The converse part of the correspondence  involves a subtlety of the type that makes the study of Lie groups a quite sophisticated topic.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 118

Re the Hodge conjecture:

It arises deep within the subject, at a high level of abstraction;  and the only way to reach it  is by way of those layers of increasing abstraction.

-- Keith Devlin, The Millennium Problems (2002), p. 9

This is a truly adult depiction of depth: as lying deep within the layers of the enigmatic onion.  It is not a case where you can just swallow some peyote and see it all in a flash.  
(For more, compare:  The Ladder of Abstraction.)

Writes a philosopher:

The axioms are not logical truths ... Their truth is established by intuitions which lie too deep for proof, since all proof depends on them.

-- Roger Scruton, Modern Philosophy (1994), p. 392

An example from outside the field of mathematics -- though it is a mathematician who is writing this:

Just how a protein manages to organize itself in space, using only the sequence of its own amino acids, remains a mystery, perhaps the deepest in computational biology.

-- David Berlinski, “What Brings a World into Being” (2001), collected in:  The Deniable Darwin (2009), p. 243

~

Related vocabulary:

Related to the concept of depth (which focusses on the root of things) is that of richness (regarding the blossoms that bloom from this root).   Hadamard adopts this metaphor explicitly:

Application’s constant relation to theory  is the same as that of the leaf to the tree:  one supports the other, but the former feeds the latter.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 125

Further examples:

The algebra of composition of maps  resembles the algebra of multiplication of numbers,  but its interpretation  is much richer.

-- Lawvere & Schanuel, Conceptual Mathematics (1997), p. 11

… a recent, and surprising, theoretical advance by Winkler.  He shows that, for many … algebraically closed fields, the “free Skolemization” has a model companion.

-- Angus Macintyre, “Model Completeness”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 164

The concept of thickness [in graph theory] is the deep mathematical idea that underlies the recreational puzzle of Earth/Moon maps.
-- Ian Stewart,  How to Cut a Cake (2006), p. 126

~      ~      ~

Having convinced ourselves, by the examples of mathematics, that there is more to this assessment-word deep than an emotional or impressionistic grunt,  we look to some cases outside of mathematics where an idea has been similarly assessed.

Some discoveries provide answers to questions.   Others are so deep  that they cast questions in a new light,  showing that previous mysteries  were misperceived.

-- Brian Greene, Fabric of the Cosmos

We are not at home in the world, and this homelessness is a deep truth about our condition.

-- Roger Scruton, Modern Philosophy (1994), p. 464

T.S. Eliot affirms that what is past and what is present, even what might have been, indicate a present purpose.  This is a metaphysical point of great depth.

-- James Schall, S.J., The Order of Things (2007), p. 69

And, more prosaically, but no less tellingly for all that:

Although running Bain Capital required a lot more brains and savvy than playing roulette does -- a lot more brains and savvy than most of us could even pretend to possess -- the job was not conceptually deep.  Romney did not develop a model of the world from the business of private equity. … “He’s not a very notional leader,” [said] Romney’s campaign spokesman …

-- Louis Menand, “Money Pol”, The New Yorker (19 III 2012)

~      ~      ~

This is quite aside  from the path of mathematics, but -- it may be, that such depth is displayed in quite distantly allied regions:  all tracing back to Him, perhaps by some functorial construction.  In that spirit, this:

http://www.youtube.com/watch?v=s5xDEyuXaOY&feature=fvwrel

The final anguish  of the Asian bride  suggests the depth  of the Riemann Hypothesis.

The enigma of a woman’s heart,

finally espied  by a Private Eye,

for less than the price  of a Valentine …

This Rose

[Kindle]  [Nook]

~     ~     ~

Somewhat less far off the path …  Deep is indeed the term of art  in mathematics, antonymic to trivial.   Now compare, from another discipline, the word profound, in reference to Newton’s perplexing, little-known  philosophical-speculative opus:

That it is exclusively mystical  I do not believe -- that there is a mystical element  seems certain.  I hope that  one day  some profound student -- no one less will suffice -- will study this mass of papers.

-- E. Andrade, quoted in James Newman, ed. World of Mathematics (1956), p. 273

~ ~ ~

Above, we saw the distinction deep vs. difficult.  Here now even the latter concept is bilayered:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

In infinite dimensions, it is tedious but not difficult  to construct spaces  strictly  but not uniformly  convex.

-- Prof. Lewis, University of Alberta, course in Functional Analysis, 1982

The proof that Reidemesiter moves  and planar isotopy  suffice to get us from any one projection of a knot  to any other projection of that knot  is not particularly difficult;  however, it is technically involved.

-- Colin Adams, The Knot Book (1994), p. 15

And here the original distinction is made even sharper:  though they are not interchangeable, depth tracks with generality, which in turn tracks (on the plane of praxis -- of proof) with simplicity:

Re the Denjoy-Young-Saks Theorem on the derived numbers of functions:

As we would expect  in view of the great generality of the final statement of the theorem,  the proof due to Saks is of extreme simplicity.

-- F. Riez & B. Sz.-Nagy, Leçons d’analyse fonctionelle [references to the English translation, Functional Analysis, 1955], p. 17

~

Here a leading mathematician laments the shallowness of his understanding of something he himself proved (regarding representations of a Kac-Moody algebra, as it happens):

My proof of this result was technically quite involved.  I was able to explain how the Langlands dual group appeared, but even now, more than twenty years later, I still find mysterious why it appears.  I solved the problem, but it was ultimately unsatisfying to feel that something just appeared out of thin air.

-- Edward Frenkel, Love & Math (2013), p. 181

This is setting oneself high standards indeed.   Shakespeare probably did not lie awake o’ nights fretting how the devil he ever came to write Hamlet;  Mozart did not find the bread of pleasure at having written the Sonata in A  turning to ashes at the thought that it might have been dictated to his unconscious  by an angel.

~

A near-synonym of the math-word deep, but shorn of all irrelevant aesthetic echo, is:  “highly nontrivial”.   The term is decidedly commendatory, though to a layman it might sound like faint praise, as were one to dub one’s lady-love “seriously unugly”.  The expression may be extensionally impeccable, but ‘twould never pass in a sonnet.

 Further:

In set theory, a forcing extension in Cohen’s sense  is reminiscent of algebraic extensions of a field, but

… the forcing method is far more complex, both conceptually and technically, involving set-theoretic, combinatorial, topological, logical, and metamathematical aspects.

-- Joan Bagaria “Set Theory”,  in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 625

Although he does not use the term "deep" here, that expanded characterization  "complex, both conceptually and technically", especially the "conceptually" part, points in that direction.

~

One motive for Frege’s choice  was again generality:

Does not the ground of arithmetic lie deeper than that of all empirical knowledge, deeper even than that of geometry?

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 184

… [Cantor’s] remarks on functions of several variables (where the provability of theorems  was deepening the level of rigour in analysis)

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 223

 

.

On Sex and C. S. Lewis (updated)

I’m in the midst of reading an informative and entertaining volume by Edward Craig:  Philosophy: a Very Short Introduction (Oxford, 2002).  It engagingly tells me about some things which I know little about (e.g. Buddhism -- though here his exposition rather motivates me to know even less) and Nietzsche (whom I haven’t read  since I was a lad in lederhosen, but the reminder  led me to pull down  off the old shelf   those Goldmanns Gelbe Taschenbücher --  alas, in type too small for these old eyes).  But occasionally he writes of matters I do know something about, and one of these I wish to address:  his digs at C. S. Lewis.

Craig begins (p. 91) by alluding to Lewis marveling at the magnitude of a young man’s sex drive (not limited by periods of rut or oestrus), which, if it had its way, might “populate an entire village” (as indeed has apparently happened in the case of certain potentates of history, and occasionally  in our own day).   Craig then makes noises in favor of Nature, which, decoratively red in tooth and claw, “can be relied upon to ensure that this does not happen very often, to put it mildly.”

(Craig’s mild irony, which he puts forward as self-evident, requires some sympathetic interpretation if it is to make sense.   After all, something very much like that male hyperfecundity does regularly happen in some species, such as the elephant seal.  All Craig can really mean is that we can’t all be alpha-males  -- an arithmetic tautology  well within the grasp of C.S. Lewis.)

He concludes (p. 92):

            C. S. Lewis’s imagination was floating well clear of the facts.

(If you read that too hastily, you might miss the insinuation in that choice of word, imagination :  the image that epithet covertly  evokes  is that of sexual fantasizing, the frustrated dreams of potency by a celibate Oxford don. )

Well, let us be clear about what the facts actually are, before we go floating off.

Right off, if the author’s intention was to present a gleaming uncontroversial example of Natural Selection, versus a yahoo’s failure to appreciate it, it was a singularly unhappy choice.  Sex has always been a puzzle, precisely for students of Natural Selection, and Darwin himself was the first to notice this.   In a remarkable display of intellectual integrity, no sooner had Darwin achieved renown for his theory of Natural Selection, than he maintained that that very theory was inadequate in an important respect:  Sex just did not fit in with the established pattern of facts.  So far from dismissing this as a mere anomaly, he set up a second and parallel hypothesis of Sexual Selection.   (Sidenote:  Darwin’s reticence and self-critiques are exemplary in the history of public thought.  More typically, an intellectual or semi-intellectual comes up with a single Neat Idea (“soft power”!  “themata”! “the world is flat”!) and attempts to dine out on it for the rest of his career. )

In Darwin’s time, the puzzle was insoluble, since he lacked -- and was candidly aware that he lacked -- knowledge of the mechanism of the rise and maintenance of variation:  the world awaited Mendel, laboring obscurely in his monastery.   Once the algebra of genes was sorted out, for a while some scientists thought they had the key:   Sex dumps two separate sets of genes into the tombola, shuffles well, and voilà, all the variation you could ever wish for, a bonanza for Selection to work on.   That seems so sensible, many of us believed it for years, never giving it a thought.   So I was startled to learn that this account is now quite controversial -- having to do with such matters as the status of group-selection and the game-theoretic view of gene propagation.  The subject deserves a separate essay (or a separate book), but a thumbnail summary might go:  from the standpoint of the Selfish Gene, the world appears as it does to Donald Trump:  “I’m already perfect, just as I am !  Just make more clones of me-e-e-e-e-e ! ! !”

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

~

We thus leave the evolutionary mystery of sex to its able investigators, and return to C.S. Lewis.

Since Craig didn’t source his paraphrase, nor provide a googlable direct quotation, I had to hunt a bit.   And in riffling through a stack of CSL’s writings on my desk, in search of the passage in question, I was reminded that he actually wrote very little about sex itself (as opposed to love: beautifully in The Four Loves), so it is difficult to come up with passages illustrating the range of his thought.   Most of these volumes lack subject-indexes, alas;  and in those that have them, like the essay-collection God in the Dock, the word “sex” does not even appear.  I finally came upon the passage Craig was paraphrasing  in the chapter “Sexual Morality”, originally a wartime radio-talk, collected in the volume Mere Christianity.  Lewis continues the analogy with food, in a sort of parable:

Or take it another way.  You can get a large audience together for a strip-tease act -- that is, to watch a girl undress on the stage.  Now supposed you came to a country where you could fill a theatre by simply bringing a covered plate on to the stage  and then slowly lifting the cover so as to let every one see, just before the lights went out, that it contained a mutton chop or a bit of bacon, would you not think that, in that country, something had gone wrong with the appetite for food?

The flavor of that passage is probably enough for you to surmise that C.S. Lewis was neither a prude nor a suppressed lecher.   The whole chapter is wonderfully rich and subtle, and it seems Craig must have read it, yet he offers a bonehead summary:  “Before you castigate yourself a sinner and start bewailing the lost innocence of the human male…”   Lewis’ actual emphasis is quite otherwise:

The old Christian teachers said that if Man had never fallen, sexual pleasure, instead of being less than it is now, would actually have been greater. … Christianity is almost the only one of the great religions which thoroughly approves of the body … If anyone says that sex, in itself, is bad, Christianity contradicts him at once.

Lewis never hectors his audience about sinfulness and sex; in fact, reminiscing about his talks with the troops, he noted that he soon gave up even mentioning fornication in such a connection, since the troops themselves simply had no conception of sinfulness in that regard;  a much more fruitful subject was always “a sin that I myself had committed the previous week”.   Nor did he much regret this loss of subject-matter, for as he concludes his chapter:

Though I have had to speak at some length about sex, I want to make it as clear as I possibly can  that the centre of Christian morality is not here.  If anyone thinks that Christians regard unchastity as the supreme vice, he is quite wrong.  The sins of the flesh are bad, but they are the least bad of all sins.

Incidentally -- lest you imagine that Lewis is idiosyncratic on this subject, compare this, from a prominent Jesuit:

Pleasure itself is never wrong.

-- James Schall, S.J., The Order of Things (2007), p. 102

~

So.  I believe I have successfully defended C.S.Lewis against any suggestions that he was either a prude or a scold;  but what of his original suggestion that the sexual instinct in modern man has somehow gone wrong ?  That is indeed a startling thesis -- and utterly at variance with the  contemporary climate of right-thinking opinion that anything goes and that’s just dandy.  Gone wrong, not in the matter of excess of appetite (his parable about the village-populator was indeed not happily chosen, for the reason Craig notes), but almost the opposite -- in its perversion to ends having nothing to do with procreation.  And in this, I believe he was right -- even prescient, for matters have grown publically worse than in his day.   In his day there were no pressure-groups for the “transgendered”, nor for the sad souls who can find no psychosexual relief until they have their own leg amputated. 

By chance -- though not a long-shot, for the media brim with such things -- upon putting down the passage quoted at the beginning of this essay, I picked up a newspaper, and happened upon an article telling of a local man being sentenced to a long prison-term for possession of child pornography.  But what struck me was this detail:  When the police raided his home, they found over seventy-five thousand photographic images.  Now, surely even a naturalist who roams the globe documenting a variety of species in a panoply of ecotones, would not be so dedicated as to accumulate a (much more variegated) cache of that size.   For, all of this sad man’s photos presumably make the same, identical point.  See:  little girl;  no wee-wee.  Granted, every lad needs to learn that, and it’s surprising at first.   But why does he need to be told that, seventy-five thousand times?

Nor is the problem  that of just a few fringe deviants.  Such musically talentless divas as Madonna and (the pen recoils at even spilling its ink  to write this name) Lady Gaga,  have garnered their mass appeal -- among your sons and daughters and mine -- precisely by exploiting the lure of perversion, and the Satanic aspects of fallen sex. 

And even among the generality who do not indulge in such things, the fibre has somehow slackened, so that all we can do is mumble “Not that there’s anything wrong with that”.

But there really is something wrong with that;  and there is no limit to the depths to which depravity will go, if given a leash long enough:  In 2001, a German homosexual named Armin Meiwes posted an online ad for a well-built young male to volunteer to be slaughtered and eaten.  He received numerous replies to this tempting offer;  the lucky winner was a 43-year-old engineer.   Let Wiki tell the rest, in mouth-watering detail:

As is known from a videotape the two made when they met on 9 March 2001 in Meiwes's home in the small village of Rotenburg, Meiwes amputated Brandes' penis and the two men attempted to eat the penis together before Brandes was killed. Brandes had insisted that Meiwes attempt to bite his penis off. This did not work and ultimately, Meiwes used a knife to remove Brandes' penis. Brandes apparently tried to eat some of his own penis raw, but could not because it was too tough and, as he put it, "chewy". Meiwes then fried the penis in a pan with salt, pepper, wine and garlic; he then fried it with some of Brandes' fat but by then it was too burned to be consumed. He then chopped it up into chunks and fed it to his dog.

Which wine they had set aside to accompany this spicy delicacy is not recorded.

Bad enough;  but again, the supine impotence of the broader society before such atrocities must be noted.  Cannibalism was not even a crime in Germany (it being, after all, the traditional practice of those differently-cultured); nor could a straight case of murder be made out, since the entrée had voluntarily entered into this little intimacy between consenting adults.   They finally charged the gourmand with the archaic charge of “desecrating a corpse”.   This could not refer to the grotesque  and protracted emasculation described above -- the law was cool with that -- since the celebrant was then still living.   It was only because Herr Meiwes chopped up the rest of him and popped it in the pantry, that this ludicrous mini-charge could be brought.

C.S. Lewis was right on target.  There is indeed something amiss in the modern mechanism of sex.  Something is truly out of kilter, in a way that Darwin, despite his many merits, does not begin to explain.

That our philospher can dismiss him offhandedly is probably due, not to any deficiencies in Lewis' arguments, which are closely-reasoned always, but to the unspoken assumption that any out-and-out Christian must be an intellectual lightweight.   But a similar diagnosis was arrived at by the atheist Sigmund Freud:   "The sexual life of man is seriously disabled;  it sometimes makes the impression of being a function in process of becoming atrophied."  (Quoted in E. Jones, Freud, III 341)

(Note:  To keep your own healthy instincts from atrophying, simply click here:
  Dames, dames, dames )

[Update 31 May 2012]  Herr Meiwes is not without his acolytes:
http://worldofdrjustice.blogspot.com/2012/05/necrophilia-not-that-theres-anything.html
This is not Darwinian.  This is Satanic.

*

The detective versus an unnamed evil thing:

A tale of madness, and of the uncanny --

uncannily available for only ninety-nine cents:

Murphy Calls-in a Specialist

Available for Kindle or Nook

*

~

Re-reading, I notice another sidewise swipe at CSL:  Craig identifies him as a “popularist Christian theologian”.   (Sorry, professor:  You mess with my dawg, I mess with you.)  I am unsure what that “popularist” is supposed to imply -- is it a sly way of saying “pop”?  Is it a misspelling of "populist" (something Lewis emphatically was not)?  Or is it an awkwardly-worded reference to Lewis’s thoroughly honorable work -- much of it in the service of British troops in wartime -- in conveying some ideas of philosophy and of Christianity to the general public, much as Craig himself (in Oxford’s cleverly conceived and slickly marketed series of uniform Livre-de-poche-style mini-volumes or maxi-pamphlets) has done for philosophy, minus the Christianity?    Or can he possibly mean that Lewis ‘played to the populace’, watering solid doctrine down into milksop for the masses?  If that be the meaning, he has stood truth on its head.  -- In any event, Lewis never claimed to be a theologian, so far as I know  (In his public talk at Oxford, printed as "The Inner Circle", he describes himself merely as "a middle-aged moralist".)    He claimed to be a professor of literature, and he claimed to be a Christian -- a “mere” Christian, in fact.

~

But let us turn from these perhaps impalpable ethical/sociological observations, and return to the lab-last of exact and unforgiving Science.

Biologically, there is more to reproduction of  kind  than merely impregnation.    Of late, indeed, statistical medical investigations have increasingly stressed developmental (intra-uterine) environmental factors apart from strict Mendelian genetics -- an area that blurs the line between Nature and Nurture.   We shall briefly examine these, note their application to the species Homo sapiens, and conclude that, here as well, there are trends that raise Lewisian concerns.  Darwin, rather than puffing on his pipe and chortling “Just as I predicted !”, would probably recoil in horror.

The three basic stages in the reproductive cycle are:

            (1) Fertilization (impregnation)

            (2) Gestation (pregnancy)

            (3) Post-natal care.

This last, for familiarity and concision, we shall call simply “parenting”, although the behaviors it covers, in this species or that, may be remote from those conjured in any issues of Family Circle.   The mommy ichneumon fly, who unfortunately will be elsewhere engaged when her little ones emerge, thoughtfully provides for them after she is gone, by laying her eggs inside a paralyzed wasp, so that the hatchlings will have something to chow down on right from the get-go.

We have touched above on the paraphilias, in which activities are somehow powered by energy diverted from the sex-drive -- rather like stealing electricity from a power-line -- which do not lead to impregnation.   These need not be floridly deviant to qualify from an impartial Darwinian perspective:  simple contraception falls under this head.

[... tbd ?]

The Challenge of Darwinism, updated for Lent (and further updated for Advent)

In the post below, originally published during the Lenten season of this year, we locked horns with the Darwinist défi, from a mathematical/philosophical/theological point of view.  Herewith, by way of preface, a brief companion-piece, which abjures all those disciplines, and sticks to biology and sociology.

Characterizing Freud’s stance in 1908, in an article published in Mutterschutz  (“Defense of Motherhood”), his admiring biographer Jones writes:

The glorifying of monogamy  paralyzes the process of selection, which is the only hope of improving the human constitution, a hope which humanitarianism and hygiene  has already reduced to a minumum.

-- Ernest Jones, Freud: Years of Maturity (1955), p. 293

This passage may be difficult for today’s young people to understand.  By selection he means Darwinian Natural Selection, nowadays mostly known as “Evolution” -- Nature’s ruthless culling of the unfit.  Constitution here means ‘genome’.   Humanitarianism refers to welfare payments and the like, which provide survival -- and opportunities for further procreation -- to individuals who would, in an earlier age, have gone by the board without issue.   Hygiene refers to state-sponsored measures of Public Health,  which, reducing mortality across the population, again nurtures the frail along with the rugged.

Such thoughts are no longer publically expressed these days, let alone in such old-fashioned words.  In earlier decades, such qualms were mainstream:  though seldom was monogamy singled out  as a factor in devolution.  Yet the thrust is meet, and must be met.

Jones’ preceding paragraph  reports, from the same article, Freud’s view that “the prevailing sexual morality of civilization  is characterized by the transference of feminine demands  on to the sexual life of the man” (though on the next page he avers that “women suffer more than men from the prevailing morality” -- the apparent contradiction  is never addressed), which “leads to a double moral life, with evil consequences”.  In that context,  I at first assumed  that Freud was referring to garden-variety marital fidelity.  And that, surely, is immune from Darwinian attack:  the appalling consequences of agamy in our barrios and ghettoes  are apparent to all.  Now, I have not actually read Freud’s article -- unaccountably, that issue of Mutterschutz is missing from my shelves (probably one of the pesky neighborhood boys  has made off with it) -- but I venture to suggest that he may have had in mind an idea that grew to prominence in his later years, the years of Totem und Tabu:  the scenario of the alpha male, lording it over his harem, infusing their wombs with his übermensch übergenes, while his less-fit cousins and brothers circle, snarling, just outside of the firelight.

Now, if the highest pitch of your ambition  is to hunt the mastodon,  such a scheme might well work;  certainly, it works for walruses.  But if you set your sights higher -- on the Gospels, on Paradise Lost, on the Riemann Hypothesis -- you need a system maximally protective and nurturing of each individual offspring, since the monogamous couple’s offspring are so few;  a goal favored in that now each child has both a dedicated-mommy and dedicated-daddy, schooling him daily in Scripture and in analytic number theory.   This happy scenario  does not refute the truths of Darwinism, but simply sublates them, or transcends them:  for we, cognitively alert and free-willed beings, have other goals, not shared by the newts and toads, for whom Darwin is surely good enough.

*     *     *

~ Commercial break ~

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

We now return you to your regularly scheduled essay.

*     *     *

[And now to the original essay.]

~     ~      ~

It would ill become me, as if I were afraid of truth of any kind, to blame those who pursue secular facts, by means of the reason which God has given them, to their logical conclusions;  or to be angry with science  because religion is bound to take cognizance of its teaching.

-- John Henry Newman, Apologia pro Vita Sua (1864)

In Philosophy: a Very Short Introduction (2002), Edward Craig writes (p. 92):

 For Descartes, human reason was a faculty  given to us  and guaranteed by God … and that was why he could rely on it to tell us about the essential nature of mind and matter ... What if, instead , he had thought of it as a natural instrument which had developed because, and to the extent that [emphasis in original], it gave its possessors a competitive advantage over those without it?  Would he then have supposed that what it appeared to tell us on such matters could, with complete confidence, be taken as the truth?

This challenge is absolutely valid.  On this basis, we should, indeed, do as Descartes did, and call everything -- all our beliefs, absolutely everything -- into question.  Yet alas, now without the Cartesian certainty that our Idea of the Perfect must come from The Perfect, Himself.  Epistemologically, we are much humbled. We are sweaty, dust-caked survival machines -- that has been shown -- much like the cockroach and the rat.  Are we anything else as well ?

At this point, some despair, and take to drunkenness or drugs or Postmodernism (I cite these in descending order of worthlessness).

And yet, when we look about the landscape, we find something unexpected, unexplained, and entirely surprising:

  =>  ~ ~ ~  the Urysohn Metrization Theorem ~ ~ ~ <=

along with all the other truths of abstract mathematics: truths that are seen to be true, by anyone with wit enough to see them; truths not relativized to space or to time, to culture or to … species.  If the Humble Woodchuck should ever make it as far as point-set topology, he too will discover , and celebrate,  this truth.

Its survival value:   Zero.

What gives?

Well, we don’t know what gives, but something does.   We have a stark staring ideational miracle in our midst, something no-wise selected for, something simply True.

Now, it is no contradiction to Darwinism to posit that, quite independently of the brute world ever evolving and evolving, to no purpose and upon no plan, there might be a world apart, of Platonic forms, or celestial calliopes, or whatever you will.   What does contradict Darwinism -- fly in the face of it, knock its socks off -- is our repeated and reliable ability to discover this World of Forms, again and again struggling through jungles of calculation and conception, to arrive, from different directions, at the same ice-crowned mountain.  A World of Forms of no use at all  in escaping the sabre-tooth, whose bite is so fierce, or in hunting the mighty mastodon.

Though, let us hasten to add, among the facts that attest to the reality of these abstract truths, is that, in recent times, these have manifested themselves, not only in the clear understanding of minds suitably prepared, but in sheer physical engineering :  our spacecraft land where we want them to, and our nuclear power plants generate useful energy, owing to applications of our understanding of these Forms.

And if this be true, might it likewise be, that our insights and intuitions about our Maker, at first dimly conceived, but gradually taking form across the millennia, and finally developed to a state of great excellence by the finest minds in Christendom -- might it be that  these too  contain their share of truth?

~

Notice that, despite what we repeatedly read about the advance of science undermining the foundations of faith, we are now intellectually in a much better position to appreciate what it might mean for there to be a God who is the maker of

     visibilium omnium et invisibilium

-- all things visible and invisible -- than was Aquinas. 

[Update:  I had the sense, writing this, of committing a certain hardihood;  yet just now  happened upon the following passage, by a Jesuit of impeccable credentials:

Intellectual virtues may be an essential aspect of our existence, and prepare us for what is revealed.

-- James Schall, S.J., The Order of Things (2007), p. 198 ]

~

Note:  Despite possible appearances to the contrary, nothing on this blog should be taken as an attack upon Darwinism, or even a critique.  Charles Darwin himself is  for me  an intellectual and even a moral hero, who just gets better  the more you learn about him.  The whole field of evolutionary biology has been characterized by remarkably brilliant practitioners, who in addition are often philosophically alert.  The appearances-to-the-contrary stem from the accidental facts that,

            (a) You don’t need me to expound the findings of this fine science, since there are easily available volumes from writers far more professionally qualified, and who, by happy circumstance, are in many cases quite gifted writers, simply as writers:  Darwin himself, Steven Jay Gould, Richard Dawkins, H. Allen Orr, in writings too numerous to mention, along with excellent though lesser-known figures like D’Arcy Thompson, George Gaylord Simpson, J.B.S. Haldane, Ernst Mayr, Matt Ridley, Paul Colinvaux, Richard Fortey, George Williams, and John Maynard Smith.

            (b) With Darwinism, just as with any scientific idea in fashion, starting with atomism, magnetism, radiation, vortices, relativity, quantum mechanics, and even some mathematical ones like point-set topology and nonlinear dynamics, there have been some fanciful overreaching and stupid abuses  in attempts to apply these ideas to things to which they do not apply, mostly in the popular media.   With these I feel the need to break a lance, particularly when their agenda is to reduce those created in the image of God, to the level of cockroaches.  But the tick that feeds upon the rump of Pegasus, in no wise lessens the majesty of that noble steed.

~

A word on the … evolution (not by natural selection) of our ideas about science and about theology.

In both cases, if, naïvely, we thumb through some books and behold what our distant ancestors used to believe, we gape and gasp.   And in the perspective -- long regnant, but by now adequately criticized -- of the Whig Theory of History, those poor old dotards were simply clueless.

If, however, we examine the matter with more study and attention, we learn that the best minds of Antiquity and of the Middle Ages  were a lot smarter than our shallow understanding of their doctrines  would imply. 

For science, this post-Whig re-evaluation has been ably and painstakingly carried out by a distinguished body of historians and philosophers.  For theology -- I really don’t know;  but I do recall  that C.S. Lewis says somewhere that the old folk-images  tended to be held until some historical juncture causes them to be actually examined:  in which cases the Historical Church proved generally capable of refining its conceptions and language. Yet -- a subtle point, but crucial -- this replacement of the traditional image  is a dialectical sublation, and no mere negation.  Even from the standpoint of our modern and more nuanced understanding, the original idea, while dethroned, is yet in a sense  preserved -- aufgehoben.  Thus, the image of Jehovah as a tribal patriarch with a long white beard, sending down laws on stone tablets  and speaking, should occasion demand it, from a burning bush, remains yet a more faithful and more fruitful symbol, a more adequate gateway to eventual fuller comprehension, than would the alternative representation, as, say, a black-face minstrel, or a used-car salesman in a bow tie.  True, He is ultimately none of these three; but he participates of one of them  more than he does the others (proportionate to our understanding).

In science, it is the same.  You’ll hear it said that the Copernican heliocentric theory was resisted because

            (a) it “demoted the earth from its privileged position at the center”;

            (b) the Bible declares that Earth is stationary.

The efforts of the debunkers have decisively refuted (a) -- Craig himself has a good short summary of their findings -- but just a word about (b), since for some reason, philosophers of science are less eager to come to the rescue of the Bible.  True, it takes the Earth as stationary.  And so, by the bye, does any modern scientist in his right mind, when solving any practical problem such as the trajectory of a shell or a baseball:  the pitcher lobs it due south at x miles per hour;  the bat strikes it, traveling due north with such&such momentum; the wind is from the northeast at eighteen miles per hour, and the Earth… is stationary.  You do not factor in the sweep of this green planet round the Sun, nor Sol’s more stately wheeling along the rim of our spinning galaxy, nor that nebula’s proper motion through the celestial void, a vagrant among the observing unspeaking witness of the Fixed Stars -- which, as we now know, are not fixed in the least;  it’s just a manner of speaking.

~

A note on that last proviso, “proportionate to our understanding”.  This is actually a key idea.

By chance, I just now happened to re-read a most amusing poem from 1913, by Rupert Brooke, entitled “Heaven”.  Here he depicts fish, contemplating the afterlife:

somewhere, beyond Space and Time,

is wetter water, slimier slime!

and they imagine their unimaginable Maker,

immense, of fishy form and mind,

squamous, omnipotent, and kind

-- along with much more in this vein.

Now, this is satire, possibly even intended to be offensive;  yet let us take no offense.  Rather, the poem expresses a truth, from whatever base motivations.  For we are each, each species, but a small and finite part of the visible Creation;  which in turn is only an infinitessimal part, of the Creator.   We each receive, and cherish, such tatters and fragments of truth as we may.  And let Saint Francis be my witness: I doubt not, indeed, I do not doubt, that whither those fishes are destined, they shall find water  wetter than wet; that the celestial penguins shall throne upon ice-floes frozen to perfection, immutable; and that the sweet small vole, that dwelleth in the fields, shall … dwell there forever:  World without end.