Saturday, March 25, 2017

A Late Start on Lent

I did not receive the Ashes this year; and have no excuse, nor lessons to tell.  But it is never too late to … shut up;  so, no more posts until I am better instructed.

Wednesday, March 22, 2017

The Continuum: Mainstay or Menace? (erweitert)

The Continuum:  the original sin, from whose fecund loins
came all that is non-constructive in mathematics.
-- Anon.

Kronecker dismissed mathematical entities beyond the natural numbers as “Menschenwerk”.  An average practicing mathematician (who uses such entities all the time) may  agree with him to this extent:

(1)  Our intuitions about the natural numbers are clear and solid.   So long, indeed, as one deals only with some set of actual numbers (thus, a finite set), nothing especially surprising  or even all that interesting  turns up.  If we extend our horizon to the actual infinite of the set of all natural numbers, we meet some concepts that take getting used to (Hilbert's hotel):  but once we’ve done so, they seem natural enough.

(2) The rationals and negative integers  definitely, the algebraic numbers  probably, pretty much come along for the ride (that is, you can hardly exclude them once you’ve accepted N), and they still bring in no paradox – being, after all, of the same cardinality as the natural numbers themselves.  Though, a case could be made that these are not “entities” of the same standing as the integers, which in a sense we can hold in our hands (embodied in oranges, say), but rather abbreviations for operations on integers.  Thus, we cannot hold minus-two oranges in our hands; minus-two is not a thing, but a bookkeeping device. 

(3)  The real numbers, by contrast, are … a piece of work.  Maybe even Menschen-work, except that one could hardly imagine Menschen coming up with anything so intricate and even bizarre.  Their very cardinality baffles intuition  -- and the independence of the continuum hypothesis  shows that we are right to be baffled.  [Note:  The simple infinity of the integers already baffles *untutored* intuition;  but eventually you get the idea.  Click on the Label "Hilbert's Hotel" for further exemplification.  Whereas, the cardinality of the continuum is more like... Hilbert's Nightmare...] All sorts of queasy consequences arrive for simple quantification (cf. Quine re.  objectual vs. substitutional quantification).  They were invented (discovered?) for purposes of analysis, which in turn was developed largely for the sake of physics: but it now appears that physics (whether in its quantum cast, where Uncertainty provides a certain indissoluble granularity; or in the Wolframesque finite-automata approach) might not actually require, or afford, a continuum.

And yet standard mathematics speaks indeed ontologically of the reals, not merely pragmatically.  Thus for instance, Rudin’s standard text (Principles of Mathematical Analysis, 3rd edn. 1976, p. 8):
We now state the existence theorem [emphasis in original] which is the core of this chapter.
Theorem. There exists an ordered field R which has the least-upper-bound property.

The author then mentions that the proof actually constructs the Reals out of the Rationals.  This is, of course, the most solid sort of proof of all – not one of those Cantorian diagonalization thingies that has you winding up assenting to the Infinite Woodchuck, without ever quite knowing how you got into such a fix.  It gives you an actual recipe for the construction of these extended numbers, as concrete and explicit as for baking a cake.  And yet… all kinds of things can be thus “constructed”, at will, including items which presumably are not part of the furniture of the universe, in the sense that angels actually sit on them.


A roaring vote of confidence in the continuum  is voiced by the noted mathematician René Thom:

“God created the integers and the rest is the work of man.”  This maxim spoken by the algebraist Kronecker  reveals more about his past as a banker who grew rich through monetary speculation  than about his philosophical insight.  There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity.
-- “’Modern’ Mathematics: An Educational and Philosophic Error?”, in American Scientist (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 74.

That is in-your-face Platonism, with which, quâ Realism, we have no quarrel.  But the psychological claim seems dubious:  Our intuition of the continuum is probably no more than a vague notion of a smear (and not very infinite at that, neither going out nor going down).   And as for the ontology … When we first meet the Real numbers mathematically (that was the very first thing we did in first-year calculus, with the opening chapter of Spivak’s text), we conceive them as the completion of the rationals.  And such they are indeed:  only, with respect to the metric provided by the absolute value.   With a p-adic valuation, you get a different completion of the rationals, the p-adic numbers.   Lastly, the surreal numbers augment the continuum in yet a different unexpected direction.  (I have less than no intuition about any of this.)

The physicist Schrõdinger is less sure:

The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us.
-- Erwin Schrõdinger, “Causality and Wave Mechanics”, repr. in translation in: James R. Newman, ed. World of Mathematics (1956), p. 1059

And from an Intuitionist (close kin to a physicist):

This could be done  by seeing the continuum as something that is infinitely becoming, instead of already being.
-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 333

(Compare our old friend the actio/actum distinction.)
Might be fine for physics, doesn’t work for math.  ‘See’ it however you like; that uncompleted-account doesn’t jibe well with Cantor-style constructions.


One might say:  The continuum feels unproblematic enough, so long you take it for granted, as just some kind of smooth dense slippery thing, like mud.  Yet so soon as you pause to enquire more nearly, you are back in Saint Augustine’s predicament with regard to Time: “Quid est tempus? Si nemo a me quaerat, scio …”

Tuesday, March 21, 2017

The crown-bound brows of the Bard of Hibbing (expanded)

Well I recall, in 1965, listening to the new album “Highway 61 Revisited” in my parents’ basement;  thinking:  This is the real deal.   Unlike much of the pop music my generation has enjoyed, this will last;  people will still be listening to it,  fifty years from now.

Then the decades went by, his career veered into less interesting avenues, and his voice lost some of its timbre and zing; I no longer followed his activities.

Yet lo -- Who would have predicted that, half a century from 1965, not only would people still be listening,  but the old codger would still be touring.  A lot.  Amazing.

And now the Nobel Prize.   Kind of out of left field, but one of the committee members made the valid point:  that, going back as far as Homer, poetry has been meant to be performed, even sung.  And this, in many different cultures.

The rustic troubadour, in a lyrical mood

So:  A tip of the stetson to you, old man.

~  [The genre now shifts  from memoir  to sotie] ~

It is not for us to add any groat’s-worth of comment to his abundantly documented biography.   Yet we do take comfort in having been, apparently, the only HRNS [highly-respected news site] to document Ibn-Guthrie’s  brilliant but little-known 1965 concert in Oslo:

[Footnote] I just checked the link for this song:
Remarkably, it is still available -- most songs by the Prairie Skald  have been deleted or disabled on YouTube by the Copyright Police.   The Norwegian lyrics of this one  apparently protected it -- de minimis non curat Attila.   If this trend continues, by 2076, all Internet content will be in the Norwegian language.


Meanwhile, the snarky, fairly brainy site Boulevard Voltaire, is underwhelmed:

Nobel de littérature : aujourd’hui Bob Dylan, demain un « twittérateur »

They offer a political décryptage for the Committee’s choice, which I leave to your perusal.  That the choice might be politically motivated is not out of the question:  certainly this year’s choice for the (always highly politicized) Peace Prize makes no sense at all aside from such a perspective.  (And it’s not the obvious one -- nothing to do with the FARC really.  But my keyboard is running out of pixels, so you’ll have to figure it out for yourselves.)


Back to the blind bard of the Achaeans.
Whatever might have been their origin in oral performance, some folks have felt that such hit ditties as “The Iliad” and “The Odyssey” read pretty well on paper too.  Whereas Zimmerman’s lyrics, so displayed, are thin stuff.  He himself took them seriously as poetry; the lyrics were printed in full on the backs of his albums of circa 1965. 
I recall in the late ‘sixties, when (excellent) bootleg albums  were appearing (like the Basement Tapes), a book of poems came out, Tarantula.    Still very much a fan, I almost bought a copy, but, glancing at it first, was obliged to toss it aside.  (Lennon’s In his own Write was actually amusing by comparison.  Heck, I even enjoyed Ono’s Grapefruit -- now there’s a collecter’s item.)

It is no knock on his song-lyrics as lyrics  to say that they fail to impress on the printed page.   One of the finest lines of all time, from the world of music, goes:

Bom ba-bom bom,  ba-dang ba-dang dang,
ba-ding ba-dong ding   BLUE MOON.

No seriously, it’s great;  but you have to hear it, not read it.

Okay, that said:  We could still defend the Committee’s decision on the grounds that it recognizes the oeuvre, not sub specie printed poetry, like that of the modern eye-poets, but as a Gesamtkunstwerk, the music no more abstractable from than lyrics  than flesh from bones, or “The Godfather” from its soundtrack.

In any event, the Nobel committee was not the first to consider pop lyrics an integral part of the poetic canon, next to T.S. Eliot  and all the rest.  For, the anthology of American poetry published in 2000 by the prestigious Library of America, includes ditties from Tin Pan Alley (Ira Gerschwin, Lorenz Hart) and Delta Blues (Blind Lemon Jefferson  et alia).   And indeed, as I near the end of reading through both volumes, it may be affirmed that those offerings, while hardly standouts, are at least entertaining and worth reading  -- an evaluation that I would deny to (alas) a great deal that made it into these volumes, and that from  famous pens.

[Update 26 March 2017]  A judicious and enjoyable survey of the state of affairs, by David Orr, may be savored here:


Sunday, March 19, 2017

Singapore twistystich

A wa-ry cat   with a lopped-off tail
dashed   past  a plaster    lion
and disappeared
behind a forecourt wall.

-- Mark Abley, The Prodigal Tongue (2008), p. 62

Julian the Apostate mini-multistich

I folded the letter
into many squares,

than the other.

[--Gore Vidal, Julian (1964), p. 199]

Friday, March 17, 2017

Julian the Apostate multistich

“Do you think I would have minded  that?”
She turned  full on me,
and the  large  black  eyes       blazed
like obsidian
in the sun.

[--Gore Vidal, Julian (1964), p. 157]

Tuesday, March 14, 2017

Code Red

            Code -- Red!
            Rest in bed.
            Time to soothe
            your sleepy head.

{ for more about the lovely snow,
   gently press …    here …. }