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The Last Invariant Theorists A sociological study of the collećtive biographies of mathematical specialiśts

  • Charles S. Fisher

When scientists account for changes in a theory or the theory's position within the world of the science, they usually do so in terms of the intellectual content of the theory. For example, the rise of quantum mechanics is attributed to the fact that quantum mechanics was able to solve problems which the preceding theories could not handle. This represents one way of describing occurrences within science and is a major component of the attitudes which scientists possess about their world. Another manner of viewing the changes which take place in science is that of the sociology of knowledge. From this perspective science is viewed as an activity carried on by the men who create the scientific ideas. The history of science is analyzed in terms of the-scientist-who-does-something-within-aparticular-social-context; that is the ideas are viewed as inseparable from the men who put them forward and the social environment in which they occur (1).

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(1) The view of the sociology of knowledge underlying this essay is more like that of Mannheim in Essays on the Sociology of Knowledge than in Ideology and Utopia. It will not be argued that the technical ideas of mathematicians can be correlated causally with such gross social variables as class, religion, nationality or economic interest. Such arguments are at best tenuous. At most it can be claimed that the milieu in which a man works is partially constitutive of the meaning which is given to his results. An example might be to compare the philosophical ‘science’ of mechanics as elaborated in medieval scholastic universities with the mechanics of today. The first subject is academic, whereas the second ranges from the rational mechanics of mathematicians to the drag strip of a hot-rodder. Cf. Clagett, M., The Science of Mechanics in the Middle Ages (Madison, U. of Wisconsin, 1959) and any lower-middle class American high school boy.

(2) Fisher, Charles, “The Death of a Mathematical Theory”, Archives for History of Exact Sciences, III (1966), 137159.

(3) Some obvious exceptions are game theory and computer sciences. In other fields one can find many subjects which become of interest to groups other than the scientists themselves, e.g. organic chemistry, atomic physics. The industrial and strategic needs of competing nations have elevated these specialities far above some of their close neighbours.

(4) A notable example is when Gödel proved the goal of establishing the consistency and completeness of mathematics as sought by Hilbert and his students was unattainable. Although this eliminated the major reason for Hilbert's work, he and others kept working on mathematical logic—there were many other questions they could try to answer. Mehlberg, H., “The Present Situation in the Philosophy of Mathematics”, Logic and Language (Dordrecht, Reidel, 1962). Essays in honor of R. Carnap.

(5) Hence any arguments to the effect that the theory lacks importance or intrinsic mathematical interest, is too hard or too computational, may become a matter of dispute between different groups of mathematicians. Therefore, any outside judge using these considerations as criteria for the theory's success or failure does no more than take sides in this argument—though, of course, from a distance. The only way that the argument is, in effect, settled is when one or another of the groups disappears.

(6) Scientists only occasionally think of the existence of schools and specialized students as having any bearing on the development of their discipline. They usually treat contributions as having some sort of intrinsic merit which thereby attracts the attention of others. It is only in the case of ascending, descending or embattled specialities that scientists become aware of the importance of creating an audience and a generation of successors.

(7) Mathematicians of other nationalities contributed important results to the Theory of Invariants, but on the whole they do not play a major role in the dialogue which centers about the theory's ‘death’.

(8) For a slightly different version of the development of the theory see Fisher, , op. cit.; or Bell, Eric T., The Development of Mathematics (New York, McGraw-Hill, 1945), pp. 425433.

(9) This sketch of the conditions in Britain is derived from college histories, biographies of mathematicians and the following general works: Bendavid, Joseph and Zloczower, Abraham, Universities and Academic Systems in Modern Societies, European Journal of Sociology, III (1963), 4584; Cardwell, D. S. L., The Organization of Science in England (London, Heinemann, 1957); Barker, Ernest, Universities in Great Britain (London, Student Christian Movement Press, 1931); Ashby, Eric, Technology and the Academic (London, Macmillan, 1958); Gillispie, Charles C., “English Ideas of the University in the Nineteenth Century”, in Clapp, M. (ed.) The Modern University (Ithaca, Cornell U., 1950).

(10) Merz, John T., A History of European Thought in the Nineteenth Century (London, Blackwell, 1896), vol. I, pp. 275276.

(11) This is much more emphasized in mathematics than in physics. Being less diverse than mathematics, it is more likely that successors to a professorship in physics will pursue somewhat similar areas. Whereas the discontinuities in mathematical interest are often radical. Hence a few physicists will create, just by the fact of concentration on the same problems, a more homogeneous community than will a few mathematicians who are likely to become interested in different problems.

(12) The sources of biographical material are: Forsyth, A. R., “A Biographical Note”, in The Collected Papers of Arthur Cayley (Cambridge, The Press, 1895), vol. VIII; Bell, Eric T., Men of Mathematics (New York, Simon and Schuster, 1937), chapter XXI; Baker, H. F., “Biographical Note”, in The Collected Papers of James Joseph Sylvester (Cambridge, The Press, 1912), vol. IV; MacMahon, P. A., James Joseph Sylvester, Proceedings of the Royal Society of London, LXIII (1898), IX–XXV; Franklin, Fabian, People and Problems (New York, Holt, 1908), pp. 1127; Archibald, R. C., “Unpublished Letters of J. J. Sylvester”, Osiris, 1 (1936), 85154; SirBall, Robert, George Salmon, Proceedings of the London Mathematical Society, II, (19031904), xxii–xxvii; Baker, H. F., MacMahon, Percy Alexander, Journal of the London Mathematical Society, V (1930), 307318; Turnbull, H. W., “Edwin Bailey Elliott, 1851–1937”, Obituary Notices of Fellows of the Royal Society, II (19361937), 425431; Todd, J. A., Grace, John Hilton, 18731958, Ibid, IV (1958), 93–97; Turnbull, H. W., Alfred Young, 18731941, Ibid, III (1941), 761–778; Aitken, A. C., Herbert Western Turnbull, 18851961, Ibid, VIII (1962), 149–158.

(13) For America see: Vesey, Laurence R., The Emergence of the American University (Chicago, University Press, 1965), chapter III; Ryan, W. Carson, Studies in Early Graduate Education (New York, Carnegie Foundation, 1939); Berelson, Bernard, Graduate Education in the United States (New York, McGraw-Hill, 1960); Cajori, Florian, The Teaching and History of Mathematics in the United States (Washington, D.C, Bureau of Education Circular No. 3. 1890); Smith, David E., A History of Mathematics in America before 1900 (Chicago, Mathematical Association of America, 1934).

(14) Information on the Americans is found in the following: For Sylvester see footnote 12; for Franklin, , Who Was Who in America; for Story, Poggendorf, J. C., Biographisch-literarisches Handwörterbuch (Leipzig-Berlin: various publishers, 18631953); for Sylvester's students see Cajori, op. cit.; Smith, , op. cit.; Poggendorf, , op. cit.; and The Royal Society Catalogue of Scientific Papers; for White and Glenn, see Poggendorf, , op. cit.; Albert, A. A., Leonard Eugene Dickson, Bulletin of the American Mathematical Society, LXI (1955), 331345; “Leonard Eugene Dickson”, in American Mathematical Society Semicentennial Publication (New York 1938), vol. I; Bell, E. T., “Fifty Years of Algebra in America, 1888–1939”, Ibid. vol. II; for Dickson's students begin with Doctors of Philosophy (University of Chicago Publication XXXI (1931), n° 9).

(15) The general sources on Germany are college histories and Ben-David, , op. cit.; Flexner, Abraham, Universities (New York, Oxford University, 1939); Kowalewski, Gerhard, Bestand und Wandel (Munich/Oldenbourg 1935).

(16) The biographical sources for the Germans come from among the following: Poggendorf, , op. cit.; Arndt, B., Wilhelm Franz Meyer zum Gedöchtnis, Jahresbericht der Deutschen Mathematiker-Vereinigung, XLV (1933), 99113; Engel, F., Eduard Study, Ibid. XL (1930), 133–156; Weiss, E., Eduard Study, ibid. XLI (1932). For Weitzenböck see Coolidge, Julian L., A History of Geometrical Method (Oxford, Oxford University, 1940), p. 165; Newman, M. H. A., Hermann Weyl, 1885–1955, Biographical Memoirs of Fellows of the Royal Society, III (1957), 305323; Noether, Max, “Paul Gordan”, Mathematische Annalen, LXXV (1914), 145; Weyl, Hermann, “Emmy Noether”, Scripta Mathematica, III (1935), 201220; Blumenthal, O., “Hilberts Lebengeschichte”, in Hilbert, David, Gesammelte Abhandlungen (Berlin, Springer, 1935) vol. III.

(17) Bell, , The Development of Mathematics, op. cit.

* The research for this paper was done under the sponsorship of the School of Nursing, University of California Medical Center, San Francisco. It was written and revised at Princeton on a National Science Foundation Post-doctoral Fellowship.

I am indebted to Anselm L. Strauss for his guidance and to Thomas S. Kuhn and David L. Krantz for their suggestions and critical comments.

(18) Fisher, , op. cit. for the various contemporary uses of Invariant Theory.

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  • EISSN: 1474-0583
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