March 12, 2006

• A Circle of Quiet

  From the Harvard Math Table page:

  “No Math table this week.
  We will reconvene next week on March 14 for
  a special Pi Day talk by Paul Bamberg.”

  

  Paul Bamberg

  Transcript of the movie “Proof”–

  Some friends of mine are in this band. They’re playing in a bar on Diversey, way down the bill, around… I said I’d be there. Great. They’re all in the math department. They’re good. They have this song called “i.” You’d like it. Lowercase i. They just stand there. They don’t play anything for three minutes. Imaginary number? It’s a math joke. You see why they’re way down the bill.

  From the April 2006 Notices of the American Mathematical Society, a
  footnote in a review by Juliette Kennedy (pdf) of Rebecca Goldstein’s Incompleteness:

  ⁴ There is a growing literature in the area of postmodern commentaries of [sic]
  Gödel’s theorems. For example, Régis Debray has used Gödel’s theorems
  to demonstrate the logical inconsistency of self-government. For a
  critical view of this and related developments, see Bricmont and
  Sokal’s Fashionable Nonsense [13]. For a more positive view see Michael
  Harris’s review of the latter, “I know what you mean!” [9]….

  [9] MICHAEL HARRIS, “I know what you mean!,” http://www.math.jussieu.fr/~harris/Iknow.pdf.
  [13] ALAN SOKAL and JEAN BRICMONT, Fashionable Nonsense, Picador, 1999.

  Following the trail marked by Ms. Kennedy, we find the following in Harris’s paper:

  “Their [Sokal's and Bricmont's] philosophy of mathematics, for instance,
  is summarized in the sentence ‘A mathematical constant like doesn’t
  change, even if the idea one has about it may change.’ ( p. 263). This
  claim, referring to a ‘crescendo of absurdity’ in Sokal’s original hoax
  in Social Text, is criticized by anthropologist Joan Fujimura, in an
  article translated for IS*. Most of Fujimura’s article consists of an
  astonishingly bland account of the history of non-euclidean geometry,
  in which she points out that the ratio of the circumference to the
  diameter depends on the metric. Sokal and Bricmont know this, and
  Fujimura’s remarks are about as helpful as FN’s** referral of Quine’s
  readers to Hume (p. 70). Anyway, Sokal explicitly referred to “Euclid’s
  pi”, presumably to avoid trivial objections like Fujimura’s — wasted
  effort on both sides.³² If one insists on making trivial objections,
  one might recall that the theorem
  that p is transcendental can be stated as follows: the homomorphism
  Q[X] –> R taking X to is injective.  In other words, can
  be identified algebraically with X, the variable par excellence.³³

  

  More interestingly, one can ask what kind of object was before the formal definition
  of real numbers. To assume the real numbers were there all along,
  waiting to be defined, is to adhere to a form of Platonism.³⁴  Dedekind wouldn’t have agreed.³⁵  In a debate marked by the accusation that
  postmodern writers deny the reality of the external world, it is a
  peculiar move, to say the least, to make mathematical Platonism a
  litmus test for rationality.³⁶ Not that it makes any more sense simply
  to declare Platonism out of bounds, like Lévy-Leblond, who calls
  Stephen Weinberg’s gloss on Sokal’s comment ‘une absurdité, tant il est
  clair que la signification d’un concept quelconque est évidemment
  affectée par sa mise en oeuvre dans un contexte nouveau!’³⁷ Now I find
  it hard to defend Platonism with a straight face, and I prefer to
  regard the formula

  

  as a creation rather than a discovery. But Platonism does correspond to
  the familiar experience that there is something about mathematics, and
  not just about other mathematicians, that precisely doesn’t let us get
  away with saying ‘évidemment’!³⁸

  ³² There are many circles in Euclid, but no pi, so I can’t think of any
  other reason for Sokal to have written ‘Euclid’s pi,’ unless this
  anachronism was an intentional part of the hoax.  Sokal’s full
  quotation was ‘the
  of Euclid and the G of Newton, formerly thought to be constant and
  universal, are now perceived in their ineluctable historicity.’ 
  But there is no need to invoke non-Euclidean geometry to perceive the
  historicity of the circle, or of pi: see Catherine Goldstein’s ‘L’un
  est l’autre: pour une histoire du cercle,’ in M. Serres, Elements d’histoire des sciences, Bordas, 1989, pp. 129-149.
  ³³ This is not mere sophistry: the construction of models over number
  fields actually uses arguments of this kind. A careless construction of
  the equations defining modular curves may make it appear that pi is
  included in their field of scalars.
  ³⁴ Unless you claim, like the present French Minister of Education [at
  the time of writing, i.e. 1999], that real numbers exist in nature,
  while imaginary numbers were invented by mathematicians. Thus
  would be a physical constant, like the mass of the electron, that can
  be determined experimentally with increasing accuracy, say by measuring
  physical circles with ever more sensitive rulers. This sort of position
  has not been welcomed by most French mathematicians.
  ³⁵ Cf. M. Kline, Mathematics The Loss of Certainty, p. 324.
  ³⁶ Compare Morris Hirsch’s remarks in BAMS April 94.
  ³⁷ IS*, p. 38, footnote 26. Weinberg’s remarks are
  contained in his article “Sokal’s Hoax,” in the New York Review of
  Books, August 8, 1996.
  ³⁸ Metaphors from virtual reality may help here.”

  * Earlier defined by Harris as “Impostures Scientifiques
  (IS), a collection of articles compiled or commissioned by Baudouin
  Jurdant and published simultaneously as an issue of the journal Alliage and as a book by La Découverte press.”
  ** Earlier defined by Harris as “Fashionable Nonsense (FN), the North American translation of Impostures Intellectuelles.”

  What is the moral of all this French noise?

  Perhaps that, in spite of the contemptible nonsense at last summer’s
  Mykonos conference on mathematics and narrative, stories do have an important role to play in mathematics — specifically, in the history of mathematics.

  Despite his disdain for Platonism, exemplified in his remarks on
  the noteworthy connection of pi with the zeta function in the formula
  given
  above, Harris has performed a valuable service to mathematics by
  pointing out the excellent historical work of Catherine Goldstein.   Ms.
  Goldstein has demonstrated that even a French nominalist can be a
  first-rate scholar.  Her essay on circles that Harris cites in a
  French version is also available in English, and will repay the study of
  those who, like Barry Mazur and other Harvard savants, are much too
  careless with the facts of history.  They should consult her
  “Stories of the Circle,” pp. 160-190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).

  For the historically-challenged mathematicians of Harvard, this
  essay would provide a valuable supplement to the upcoming “Pi Day” talk
  by Bamberg.

  For those who insist on limiting their attention to mathematics
  proper, and ignoring its history, a suitable Pi Day observance might include
  becoming familiar with various proofs of the formula, pictured
  above, that connects pi with the zeta function of 2.  For a survey,
  see Robin Chapman, Evaluating Zeta(2)
  (pdf).  Zeta functions in a much wider context will be discussed at next
  May’s politically correct “Women in Mathematics” program at Princeton, “Zeta Functions All the Way” (pdf).

  • 1:00 pm
  • Leave a comment