# Between discovery and justification

Department Mathematical communities

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## Abstract

This column is aforum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

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### References

- [1]The mathematics department may be the only spot on campus where belief in the reality of the external world is not only optional but frequently an annoying distraction.Google Scholar
- [2]
*Higher Superstition*, p. 261, note 9; Goldstein, “Quantum Philosophy: the Flight from Reason in Science” in Gross, Levitt, Lewis,*The Flight from Science and Reason*, pp. 119-125. These two books, together with*Fashionable Nonsense*(henceforth*FN*) by Sokal and Bricmont, first published in French as*Impostures intellectuelles*, and*A House Built on Sand*(N. Koertge, ed.) are the canonical texts of what passes for the pro-science camp of the Science Wars.Google Scholar - [3]Dürr, D., S. Goldstein and N. Zanghi, “Quantum equilibrium and the origin of absolute uncertainty,” J. Statistical Phys. 67 (1992): 843–907; also Dürr D., S. Goldstein and V. Zanghi, “Quantum Mechanics, randomness, and determinis- tic reality” Phys. Letters; 172 (1992): 6–12.CrossRefMATHGoogle Scholar
- [4]The avoidance of serious debate in Science Wars literature-on both sidesis one of the main themes of Yves Jeanneret’s
*L’affaire Sokal ou la querelle des impostures*(Presses Universitaires de France, 1998), the best book I’ve seen on the Sokal affair and its ramifications in France. This theme is also addressed in my unpublished review of*Fashionable Nonsense*, which can be viewed at http://www.math.jussieu.fr/~harris. Most of the present article was extracted from this review.Google Scholar - [5]I thank Josiane Olff-Nathan of GERSULP and Strasbourg mathematician Norbert Schappacher for providing this information.Google Scholar
- [6]
*Impostures scientifiques*(henceforth*IS)*, sous la direction de Baudouin Jurdant, Paris: Editions La DéFcouverte, 1998. Also published as a special issue of the journal*Alliage.*Google Scholar - [7]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
*ir*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.Google Scholar - [8]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 H- is included in their field of scalars.Google Scholar
- [9]Unless you claim, like the present French Minister of Education, that real numbers exist in nature, while imaginary numbers were invented by mathematicians. Thus tt 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.Google Scholar
- [10]Cf. M. Kline,
*Mathematics: The Loss of Certainty*, p. 324.Google Scholar - [11]Compare Morris Hirsch’s remarks in
*BAMS*, April 1994.Google Scholar - [12]
*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.Google Scholar - [13]Metaphors from virtual reality may help here.Google Scholar
- [14]
*Matière à penser*. Odile Jacob, 1989. For example, Connes writes, “Je pense que le mathématicien développe un ‘sens’, irréductible à la vue, à l’ouïe et au toucher, qui lui permet de percevoir une réalité out aussi contraignante mais beaucoup plus stable que la réalité physique, car non localisée dans l’espace-temps” (p. 49). In fact, the debate is even more complex, because Changeux often comes across as a social constructivist, though one who sees society as materialized in the human brain; thus he sees mathematical objects as “ ’représentations culturelles’ suscepti- bles de se propager, se fructifier et de proliférer et d’être transmises de cerveau à cerveau.“Google Scholar - [15]See Barry Mazur’s astonishing “Imagining Numbers (particularly V-15),” particu- larly for (among other things) an attempt to get beyond sterile ontological debates.Google Scholar
- [16]In public, at least. A course on founda- tions of mathematics is not a core re- quirement in any university with which I am familiar. One would think this fact would be of interest to sociologists of science, but I have not seen it addressed in the literature. As a graduate student at Harvard, I saw foundations actively discussed only in the graffiti in the men’s room on the second floor of the science library.Google Scholar
- [17]What they really think hardly matters. The “strong program” of Bloor and Barry Barnes is criticized at length in the philosophical “intermezzo” of
*FN*; but Sokal “agrees with nearly everything” in Kitcher’s attempt to “occupy middle ground” in Koertge,*op. cit.*[2]Google Scholar - [18]D. Bloor,
*Knowledge and Social Imagery*(U. of Chicago press, 1991), chapters 5-8; P. Kitcher,*The Nature of Mathematical Knowledge*(Oxford University Press, 1984).Google Scholar - [19]One might also say they share a common discourse. Thurston’s extended response to the Jaffe-Quinn article, in BAMS, April 1994, discussed below does refer to truth, but he seems more interested in knowledge and especially in understanding.Google Scholar
- [20]Chapter 3 of Kitcher’s
*op. cit.*[18] is devoted to a refutation of Kantian or Platonist intuition as a means to mathematical knowl edge, and what we mean when we use the word informally is presumably even less defensible. The quotation is from*Fashionable Nonsense*, pp. 143-44, note 183.Google Scholar - [21]As Barry Mazur reminded me, “interest” in this context is generally used as shorthand for an intellectual criterion, such as “enhancement of understanding” (see Thurston’s comments, quoted below). Such a criterion is by nature not well-defined, yet we have the sense that we know what it means.Google Scholar
- [22]Actually a “major mathematical discovery.” (The author obtained this information November 24, 1997 from Fredkin’s web page on Radnet. This web page has apparently been discontinued and the information regarding the prize has not been reconfirmed since 1997.)Google Scholar
- [23]This point is hardly novel; Lévy-Leblond says something similar in
*IS*(p. 39), and Dieudonné distinguished further between “mathématiques vides” and “mathématiques significatives” (quoted in Dominique Lambert, “L’incroyable efficacité des mathématiques,”*La Recherche*, janvier 1999, p. 50). Truth is also not what inter-ests Deleuze and Guattari: “ce n’est pas la vérité qui inspire la philosophie, mais des catégories comme celles d’Intéressant, de Remarquable ou d’Important qui décident de la réussite ou de l’échec.” This quotation is taken from*Qu’est-ce que la philosophie*? (1991), p. 80, one of the books subjected to extensive ridicule by Sokal and Bricmont.Google Scholar - [24]The debate provoked by the Jaffe-Quinn article is taken up in a recent article by Leo Corry, “The Origins of Eternal Truth in Modern Mathematics,”
*Science in Context*10 (1997), 297–342. Corry is so intent on developing his theme (that “the idea of eternal mathematical truth. . . has not itself been eternal “) that he completely misses the near absence of the word “truth” from the debate, claiming with no attempt at justification that “the eternal character of mathematical truth” was “implicit at the very least” in the Jaffe-Quinn proposal.MathSciNetGoogle Scholar - [25]Glimm goes on to say that mathematical truth is to be compared with the stronger standard of truth in science, “the agreement between theory and data.” The whole discussion can be found in
*Bulletin oftheAMS*, July 1993 and April 1994.Google Scholar - [26]There is a similar irony in the discussion of chaos in
*FN*(pp. 134-146). As in similar discussions in*The Intelligencer*, the non-post-modern position emphasizes the continued role of proofs in the theory of non-linear dynamical systems, insists on the determinism inherent in dynamical systems, and points to quotations from Maxwell and Poincaré to argue that no conceptual revolution has taken place. (See also*Higher Superstition*, p. 92 ff., and Bricmont’s contribution to*The Flight from Science and Reason.)*This is all reasonable (though I would need to be a his- torian to explain why a few late-19th-century quotations are hardly decisive in discussions of conceptual revolutions), but once again the interesting point has been missed. Namely, in the wake of chaos, computer modeling (“experimental mathematics ”) is being proposed as an alternative (or complementary) standard of justification to rigorous proof. This point is a cliché of the popular literature on chaos, and it is repeated in the article by Amy Dahan-Dalmedico and Dominique Pestre in*Impostures Scientifiques*(pp. 95-96).Google Scholar - [27]Thurston,
*op. cit.*[25]. Thurston’s comment referred to the computer-assisted proof of the Four Color Theorem, and echoes Deligne’s remarks on the same topic, quoted by Ruelle in*Chance and Chaos*, pp. 3-4.Google Scholar - [28]Coleman, “Manin’s proof of the Mordell conjecture over function fields,”
*L’Enseignement Math.*36 (1990), p. 393.MATHGoogle Scholar - [29]The text book is G. Cornell, J. H. Silverman, G. Stevens, eds.:
*Modular Forms and Fermat’s Last Theorem*, Springer-Verlag (1997). If I believe, or understand, or have some meaningful relation to the proof, it’s mainly because I have been collaborating with Richard Taylor to generalize parts of the argument to automorphic forms of higher dimension.Google Scholar - [30]John Horgan, “The Death of Proof,”
*Scientific American*, October 1993, pp. 92–103.Google Scholar - [31]As in David Bloor: “What if scientists need to believe a mythical and false ideology, because they would lose motivation without it?”
*Social Studies of Science*28/4 (August 1998), p. 658.Google Scholar - [32]As Thurston wrote, Wiles’s proof-still incomplete at the time- “helps illustrate how mathematics evolves by rather organic psychological and social processes.” Thurston,
*op. cit.*[25]Google Scholar - [33]Or so I would assume.Google Scholar

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