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).  The reals 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, 3^rd 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 Schrödinger 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 …”

~

Even in a universe which (like Wolfram’s) abjures the continuum, the continuum might turn out to be mathematically indispensible for its treatment.   Cf. the indispensible role of “imaginary” numbers in electromagneticsm or quantum mechanics, even though all observables must be real-valued.

Souvenirs d’apprentissage (bis)

I am currently reading the memoirs of André Weil, doyen of algebraic geometry, Souvenirs d’apprentissage.  Herewith a couple of notes.

The first thing you notice is that he is a very graceful writer when he so chooses.  You do not see this in his mathematical writings, which are straightforward and businesslike, when not (very occasionally) interrupted  by some dyspeptic outburst (we quoted one of them here).   And as he stresses the importance of hewing to original languages whenever possible (he himself was an admirable polyglot), we shall so hew here.

~

Weil prefaces his book with a tribute to his late wife:

Notre mariage a été de ceux qui font mentir La Rochefoucauld.  Fulsere vere candidi mihi soles…

For an elucidation of that Latin tag, I naturally turned to that fons sapientiae, Dr. Massey, who replied by return of post:

"Bright suns truly shone for me"

It's from Catullus Carmina 8, in which the poet is depressed after being dumped by his lover Lesbia.

~

Since Weil was born in 1906 (and ripened early), and the memoirs were not published until 1991, we sometimes get the benefit of the long view.   Alluding to the current Lake-Wobegone system of American puericulture, he remarks:

N’est-il pas étrange que l’émulation se trouve honnie à présent  comme ressort pédagogique, alors que l’esprit de compétition, dans presque tout les domaines, n’a peut-être jamais été si âpre qu’il l’est aujourd’hui ?

He also speaks somewhat dismissively of “the New Math” fad in schools, counterposing the value of a traditional grammatical education for training the mind:

Est-ce pure coïncidence  que l’Inde, avec Pânini, ait inventé la grammaire  avant d’inventer la numération décimale  et les nombres négatifs,  et que  par la suite  grammaire et algèbre aient pris  toutes deux  dans la civilisation médiévale de langue arabe  l’essor que l’on sait ?  Naguère on a cru préparer les petits enfants à l’étude des mathématiques  en les forçant à parler d’ensembles, de bijections, de nombres cardinaux  et de l’ensemble vide.  Peut-être n’y étais-je pas moins bien préparé par l’étude de l’analyse grammatical …

Later in the volume (p. 120) we read of another glancing yet important brush between linguistics and pure mathematics :  the ushering of structure, rather than number, to center stage:

Quant au choix du mot de structure, mes souvenirs sont en défaut;  mais à cette époque  il était déjà entré … dans le vocabulaire des linguistes, et je conservais des contacts avec ce milieu, et tout particulièrement avec Emile Benveniste …

*

Si cela vous parle,

savourez la série noire

en argot authentique d’Amérique :

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

*

~

André Weil was the brother of the better-known Simone Weil.  In the preface, he excuses himself for alluding to her but little, pleading that he has already said what he has to say, to her biographer.  But, recounting a stroll with some monks at Santo Domingo de Silos, he writes:

De leur conversation, au cours de la promenade rituelle dans le cloître, il m’est resté une phrase.  Comme il était question d’un saint au comportement quelque peu excentrique, l’un d’eux fit observer doucement, «Mais le christianisme est une folie» («el cristianismo es una locura»); ce propos, parfaitement orthodoxe, m’est souvent revenu à l’esprit au sujet de la vie de ma sœur.

~

Weil’s 1938 cri de cœur, “Science française”, which begins "J'en ai assez!", and which was refused publication at the time, is reprinted in the Œuvres scientifiques,  as was its eventual post-war airing.  But since neither version names names, nor does Wikipedia mention the incidents in question s.v. Jean Perrin,  here is a tidbit (p. 126):

L’une des cliques en question, et sans doute la plus puissante, avait  à sa tête  le physicien Jean Perrin, prix Nobel, … inventeur du C.N.R.S.  Non content des moyens importants dont il disposait déjà, il imagina de créer toute une hiérarchie de médailles  assorties de récompenses pécuniaires,  depuis la grande médaille d’or  jusqu’aux médaillettes … Il n’était pas difficile de soupçonner que la devise en serait:
«Nul n’aura de l’esprit   que nous  et nos amis.»

*

Pour d’autres friandises

de la confiserie 

du docteur Justice,

consultez:

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

*

I am reminded of a comment one writer made  on the much-ballyhoo’d creation of some new honor or other (it may have been the MacArthur):  “another lap in the meritocratic rat-race”.

~

All told, the book is mathematically disappointing.  I don’t mean that it should have been stuffed with equations.   But we do hope for some insight into mathematical ideation, such as is furnished by the memoirs of Hardy or of Hadamard.   For one thing, the field he helped to found, algebraic geometry, has the reputation of being one of the most ferociously abstract of all human endeavors.   It must be very different working in that field, or in topos theory, from solving the four-color problem or classifying finite simple groups.  But of this we get not an inkling. 

Above all,  Weil was long a core member of one of the most sociologically remarkable mathematical activities of all time:  the Bourbaki group, which labored collectively and published anonymously.   What was that like ?  

Apart from the pranks and in-jokes  characteristic of any working group, we are not given a glimpse.

So, frustrated at the reticence of one of its founding members,  I have no recourse but to quote the following satirical evaluation:

Named after a French general of widely admired stupidity, the Bourbaki was founded in the 1930’s  .. A committee was formed  and pedagogical improvements discussed.

This is the myth.  In all of French history, no mathematician of standing has ever concerned himself with the welfare of his students.  The Bourbaki was founded to amused the members of the Bourbaki.

To a man, these mathematicians believed that their first order of business was to correct, and, if possible, eliminate, the work of other mathematicians.

-- David Berlinski, Infinite Ascent (2005)

~

En fin de compte … Our attempt to experience mathematics more richly  by reading the memoirs of its practitioners, is like that of the gum-snapping beautician devouring the latest tabloids for the off-screen escapades of her favorite stars.  In both cases, we court a simulacrum of what is ultimately inaccessible.  Indeed, the beautician is  if anything  launched upon a more reasonable quest.   If you think that Lindsay Lohan is an interesting person, then her antics in the National Equirer should be satisfying, since that shallow cipher is little more than the sum of her antics.   Whereas the well-written, travelogue-y accounts of André Weil  gave no sense of what it is like to have his sort of towering mathematical mind -- they might have been written by anyone.  (Quine’s memoir, The Time of my Life, was disappointing in the same way.)

~

For another not-so-close encounter with algebraic geometry, via the man and not the math, click here:

http://worldofdrjustice.blogspot.com/2011/12/adventures-in-algebraic-geometry.html

~

André Weil’s teacher Jaques Hadamard, a major figure of number theory and cryptography, is best known to the lay public for his booklet Psychology of Invention in the Mathematical Field;  my father, no mathematician, but a typical subscriber to Eisenhower-era Scientific American, had it on his shelves, where I made its acquaintance in high school.  Spurred yet disappointed by Weil’s memoir, I ordered what I presumed to be the French original of this book, via InterLibrary Loan;  and in due course  it arrived at our local library.

Mais encore -- quelle déception !  For much the same reasons as Weil, Hadamard had fled (in 1940) to the United States, and indeed specifically to Princeton.  And it was there that he wrote that memoir, in English, which Princeton University Press brought out in 1945.

France just doesn’t know how to hold onto its mathematicians -- as Weil was already complaining in 1938.   And it was at Princeton that I made the acquaintance of the likewise-exiled French mathematician who earned a Fields medal for proving the Weil Conjectures, Pierre Deligne (we were fellow parents at the Princeton Friends School, and met to plan-out Math Day).   A distinguished intellectual genealogy, all very baronial -- but abroad.

~

For more from this pen, including a soon-to-be-released new title:

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

~ Afterword ~

I have from time to time -- fitfully, fretfully -- pecked away at some of the works of that French collectivity Bourbaki, without profit or enjoyment.   There is, then, a certain wry comfort in this assessment by their celebrated countryman René Thom:

No new theorem of any importance came out of the immese effort at systematization of Nicolas Bourbaki -- which in itself is not a true formalization, because Bourbaki uses a nonformalized metalanguage.

-- René Thom, “’Modern’ Mathematics (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 73

[Update]  The slim pickings of mathematical morsels in Weil’s memoir  are slightly supplemented by an anecdote in a book I just finished reading, the highly entertaining Genius in my Basement (2011), by Alex Masters.   (The genius alluded to  is not Weil  but group-theorist Simon Norton;  more elsewhere anon.)  The anthropologist Lévi-Srauss, he relates, baffled at what structure underlay Australian marriage taboos 

… went around New York … banging on the doors of mathematicians.  The first was dismissive:  “Mathematics has four operations, and marriage is not one of them.”  But the second was the young and brilliant André Weil… “When in doubt,” cried Mr. Weil, “look for the group!” and he bustled Lévi-Strauss off the street  into his study.  Within a few days, Weil had solved the problem.

(“Group” in the sense of Group Theory, of course;  though indeed sibs and clans can be relevant.)