(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)Google Scholar 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), 137–159.CrossRefGoogle Scholar

(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)Google Scholar. 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.Google Scholar; or Bell, Eric T., The Development of Mathematics (New York, McGraw-Hill, 1945), pp. 425–433.Google Scholar

(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), 45–84Google Scholar; Cardwell, D. S. L., The Organization of Science in England (London, Heinemann, 1957)Google Scholar; Barker, Ernest, Universities in Great Britain (London, Student Christian Movement Press, 1931)Google Scholar; Ashby, Eric, Technology and the Academic (London, Macmillan, 1958)Google Scholar; Gillispie, Charles C., “English Ideas of the University in the Nineteenth Century”, in Clapp, M. (ed.) The Modern University (Ithaca, Cornell U., 1950).Google Scholar

(10) Merz, John T., A History of European Thought in the Nineteenth Century (London, Blackwell, 1896), vol. I, pp. 275–276.Google Scholar

(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. VIIIGoogle Scholar; Bell, Eric T., Men of Mathematics (New York, Simon and Schuster, 1937), chapter XXIGoogle Scholar; Baker, H. F., “Biographical Note”, in The Collected Papers of James Joseph Sylvester (Cambridge, The Press, 1912), vol. IVGoogle Scholar; MacMahon, P. A., James Joseph Sylvester, Proceedings of the Royal Society of London, LXIII (1898), IX–XXVGoogle Scholar; Franklin, Fabian, People and Problems (New York, Holt, 1908), pp. 11–27Google Scholar; Archibald, R. C., “Unpublished Letters of J. J. Sylvester”, Osiris, 1 (1936), 85–154CrossRefGoogle Scholar; SirBall, Robert, George Salmon, Proceedings of the London Mathematical Society, II, (1903–1904), xxii–xxviiGoogle Scholar; Baker, H. F., MacMahon, Percy Alexander, Journal of the London Mathematical Society, V (1930), 307–318CrossRefGoogle Scholar; Turnbull, H. W., “Edwin Bailey Elliott, 1851–1937”, Obituary Notices of Fellows of the Royal Society, II (1936–1937), 425–431Google Scholar; Todd, J. A., Grace, John Hilton, 1873–1958Google Scholar, Ibid, IV (1958), 93–97; Turnbull, H. W., Alfred Young, 1873–1941Google Scholar, Ibid, III (1941), 761–778; Aitken, A. C., Herbert Western Turnbull, 1885–1961Google Scholar, Ibid, VIII (1962), 149–158.

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

(14) Information on the Americans is found in the following: For Sylvester see footnote 12; for Franklin, , Who Was Who in AmericaGoogle Scholar; for Story, Poggendorf, J. C., Biographisch-literarisches Handwörterbuch (Leipzig-Berlin: various publishers, 1863–1953)Google Scholar; for Sylvester's students see Cajori, op. cit.Google Scholar; Smith, , op. cit.Google Scholar; Poggendorf, , op. cit.Google Scholar; and The Royal Society Catalogue of Scientific Papers; for White and Glenn, see Poggendorf, , op. cit.Google Scholar; Albert, A. A., Leonard Eugene Dickson, Bulletin of the American Mathematical Society, LXI (1955), 331–345CrossRefGoogle Scholar; “Leonard Eugene Dickson”, in American Mathematical Society Semicentennial Publication (New York 1938), vol. IGoogle Scholar; Bell, E. T., “Fifty Years of Algebra in America, 1888–1939”Google Scholar, Ibid. vol. II; for Dickson's students begin with Doctors of Philosophy (University of Chicago Publication XXXI (1931), n° 9).Google Scholar

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

(16) The biographical sources for the Germans come from among the following: Poggendorf, , op. cit.Google Scholar; Arndt, B., Wilhelm Franz Meyer zum Gedöchtnis, Jahresbericht der Deutschen Mathematiker-Vereinigung, XLV (1933), 99–113Google Scholar; Engel, F., Eduard StudyGoogle Scholar, Ibid. XL (1930), 133–156; Weiss, E., Eduard StudyGoogle Scholar, ibid. XLI (1932). For Weitzenböck see Coolidge, Julian L., A History of Geometrical Method (Oxford, Oxford University, 1940), p. 165Google Scholar; Newman, M. H. A., Hermann Weyl, 1885–1955, Biographical Memoirs of Fellows of the Royal Society, III (1957), 305–323CrossRefGoogle Scholar; Noether, Max, “Paul Gordan”, Mathematische Annalen, LXXV (1914), 1–45CrossRefGoogle Scholar; Weyl, Hermann, “Emmy Noether”, Scripta Mathematica, III (1935), 201–220Google Scholar; Blumenthal, O., “Hilberts Lebengeschichte”, in Hilbert, David, Gesammelte Abhandlungen (Berlin, Springer, 1935) vol. III.Google Scholar

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

* 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.Google Scholar for the various contemporary uses of Invariant Theory.