Abstract
In his Cours de Philosophie positive, the nineteenth-century French philosopher, Auguste Comte, detailed a philosophy crucially dependent upon science and scientific thought.1 In his system, human comprehension and explanation of the natural world progressed through three stages: in the first, theological stage, divine will accounted for natural phenomena; in the next, metaphysical stage, interpretations of nature took an abstract, philosophical cast; and in the final, so-called “positive” stage, scientific truth characterized the world. According to Comte, the mind reached the positive stage by moving through a well-defined hierarchy of scientific thought. At its base lay mathematics, the most complex of the sciences in his view. Furthermore, since mathematics represented a body of scientific truths, it had already reached the stage of positive knowledge and, in fact, was historically the first science to attain this goal. Upon this base rested five sciences in ascending order of their historical achievement of the positive stage and thus, to Comte’s way of thinking, in order of their decreasing complexity and increasing generality. They were astronomy, physics, chemistry, biology, and, at the top, a new science for which Comte coined the term “sociology.” Of these, he held that all but the essentially brand-new science of sociology had reached the positive stage by the mid-nineteenth century. He thus took it as his task not only to establish this discipline in terms of scientific truths but also to achieve the synthesis of all of the positive sciences into one complete and unified positive philosophy.
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Notes
Auguste Comte, Cours de Philosophie positive, 6 vols. (Paris: Bachelier, 1830–1842).
Most noted among those who embraced as well as questioned many aspects of Comtian thought was John Stuart Mill. Even Thomas Huxley, in a blistering attack on positivism, allowed relative to Comte’s ideas that “[n]othing could be more interesting to a student of biology than to see the study of the biological sciences laid down as an essential part of the prolegomena of a new view of social phenomena. Nothing could be more satisfactory to a worshipper of the severe truthfulness of science than the attempt to dispense with all beliefs save such as could brave the light, and seek, rather than fear, criticism.” See Thomas Henry Huxley, “The Scientific Aspects of Positivism, ” Fortnightly Review 11 (1869): 653-670, on p. 653.
Joseph Henry to James Joseph Sylvester, 26 February, 1846, (Henry’s emphasis) in The Papers of Joseph Henry, ed. Nathan Reingold, Marc Rothenberg, et al., (Washington, D.C.: Smithsonian Institution Press, 1972-1992), 6: 380-381. The manuscript retained copy of this letter is held in the Henry Papers, Smithsonian Institution Archives.
James Joseph Sylvester to Joseph Henry, 12 April, 1846, in ibid., 6: 407-410. The original of this letter is also held in the Henry Papers, Smithsonian Institution Archives.
James Joseph Sylvester, “Presidential Address: Mathematics and Physics Section, ” Report of the Thirty-ninth Meeting of the BAAS Held at Exeter in August, 1869 (London: John Murray, 1870); revised and reprinted as “A Plea for the Mathematician, ” Nature 1 (1869-1870): 237-239, 260-263. See, also, James Joseph Sylvester, The Laws of Verse or Principles of Versification Exemplified in Metrical Translations: Together with an Annotated Reprint of the Inaugural Presidential Address to the Mathematical and Physical Section of the British Association at Exeter (London: Longmans, Green, and Co., 1870); and “Presidential Address to Section ‘A’ of the British Association, ” The Collected Mathematical Papers of James Joseph Sylvester, ed. H. F. Baker, 4 vols. (Cambridge: University Press, 1904, 1908, 1909, 1912; reprint ed., New York: Chelsea Publishing Co., 1973), 2: 650-661 (hereinafter abbreviated Math. Papers JJS). All specific page citations for Sylvester’s papers will refer to the latter, most accessible, source.
James Joseph Sylvester, “On an Application of the New Atomic Theory to the Graphical Representation of the Invariants and Covariants of Binary Quantics, —with Three Appendices;” American Journal of Mathematics 1 (1878): 64–125, or Math. Papers JJS, 3: 148-206, on p. 148.
Although there is as yet no full-scale biography of Sylvester, he has been the subject of many short biographical articles. See, for example, H. F. Baker, “Biographical Notice, ” in Math. Papers JJS, 4: xv-xxxvii. On Sylvester at the Johns Hopkins University, see Karen Hunger Parshall, “America’s First School of Mathematical Research: James Joseph Sylvester at The Johns Hopkins University 1876-1883, ” Archive for History of Exact Sciences 38 (1988): 153-196; and Karen Hunger Parshall and David E. Rowe, The Emergence of the American Mathematical Research Community 1876-1900: James Joseph Sylvester, Felix Klein, and Eliakim Hastings Moore (Providence: American Mathematical Society and London: London Mathematical Society, 1994), Chapters 2 and 3.
David Knight discusses the prestige associated with this honor within the British scientific community in The Age of Science: The Scientific World-view in the Nineteenth Century (Oxford: Basil Blackwell, 1986), pp. 128-132. See also Jack Morrell and Arnold Thackray, Gentlemen of Science: Early Years of the British Association for the Advancement of Science (Oxford: Clarendon Press; New York: Oxford University Press, 1981) for historical perspective on the organization, its members, and its influence.
Joan Richards approaches this speech from a different point of view in her book, Mathematical Visions: The Pursuit of Geometry in Victorian England (Boston: Academic Press, Inc., 1988), especially on pp. 134-136 and 163-166.
Sylvester, “Presidential Address to Section ‘A’ of the British Association, ” p. 652. (See note 5 above.)
Thomas Henry Huxley, “Scientific Education: Notes of an After-Dinner Speech, ” Macmillan’s Magazine 20 (1869): 177–184, on p. 182.
Ibid.
Sylvester, “Presidential Address, ” p. 654.
Ibid., p. 654. See Thomas Henry Huxley, “The Scientific Aspects of Positivism, ” p. 667.
Sylvester, “Presidential Address, ” p. 656. Gotthold Eisenstein’s researches were published in 1844 in “Allgemeine Auflösung der Gleichungen von den ersten vier Graden, ” Journal für die reine und angewandte Mathematik 27 (1844): 81-83. Here, Sylvester apparently forgot George Boole’s isolation of an invariant in 1841. See note 46 below.
Ibid., pp. 657-658.
Ibid., p. 654.
James Joseph Sylvester, “Algebraical Researches, Containing a Disquisition on Newton’s Rule for the Discovery of Imaginary Roots, and an Allied Rule Applicable to a Particular Class of Equations, Together with a Complete Invariantive Determination of the Character of the Roots of the General Equation of the Fifth Degree, &c, ” Philosophical Transactions of the Royal Society of London 154 (1864): 579–666, and Math. Papers JJS, 2: 376-479, on p. 419. Sylvester’s emphasis.
In 1863, just one year prior to the publication of Sylvester’s paper, Helmholtz systematized his findings in physiological acoustics in the book, Die Lehre von die Tonempfindungen als physiologische Grundlage für die Theorie der Musik (Braunschweig: F. Vieweg und Sohn, 1863).
Sylvester, The Laws of Verse. (See note 5 above for the full citation.) For example, relative to what Sylvester called the “principle of phonetic syzygy, ” he wrote “I conceive that the method of triadic analysis which I have employed in the genesis and distribution of the princi-ples of lyrical poetry, is founded on the nature of things and not on an arbitrary subjective rule of classification. I can honestly aver that I was not on the look-out for any such an arithmetical law” (p. 12). He continued: “Of one thing I have no doubt, which is, that when the analysis of principles which I have here faintly indicated has been carried to its full term, we shall be in possession of a system of rules, by which the criticism of the technical part at least of lyrical poetry may be reduced to the form of propositions capable of being logically argued and debated, and entirely removed from that indefinite region of taste which, like the so-called discretion of a judge, does not admit of being made the subject of rational discussion” (p. 12).
James Joseph Sylvester, “Address on Commemoration Day at Johns Hopkins University 22 February, 1877, ” Math. Papers JJS, 3: 72–87, on pp. 77-78.
Ibid., p. 78.
Ibid., p. 83.
Allen G. Debus, “Mathematics and Nature in the Chemical Texts of the Renaissance, ” Ambit 15 (1968): 1–28, on p. 3.
For a discussion of some of these developments, see Arnold Thackray, Atoms and Powers: An Essay on Newtonian Matter-Theory and the Development of Chemistry (Cambridge: Harvard University Press, 1970), Chapter 7, entitled “Quantified Chemistry: The Newtonian Vision, ” pp. 199-233.
On the complicated history of valency, see C. A. Russell, The History of Valency (Leicester: University Press, 1971); and J. R. Partington, A History of Chemistry, 4 vols. (London: The Macmillan Press, Ltd., 1970-1972), 4: 500-565.
Knight, Age of Science, pp. 157-158; and Partington, History of Chemistry, 4: 488 and 494.
Knight, Age of Science, p. 156.
Of course, others like Friedrich August Kekulé also developed notations around this time. See Partington, History of Chemistry, 4: 533-565ff; and Russell, History of Valency, pp. 92-107. For a general treatment of the high points of the linkage between graph theory and chemistry, see Robin J. Wilson, “Graph Theory and Chemistry, ” Colloquia Mathematica Societatis János Bolyai 18 (1976): 1146-1164. I thank Robin Wilson for referring me to this work as well as to his book cited in note 37 below.
Alexander Crum Brown, “On the Theory of Isomeric Compounds, ” Transactions of the Royal Society of Edinburgh 23 (1864): 707–718. See, also, Partington, History of Chemistry, 4: 552-553; and Russell, History of Valency, pp. 101-105.
Edward Frankland, Lecture Notes for Chemical Students: Embracing Mineral and Organic Chemistry (London: John van Voorst, 1866). Russell discusses the influence of this text in History of Valence, pp. 104-105.
Frankland, Lecture Notes, p. v. Kekulé had developed a graphical notation in which circles represented univalent atoms, but sausagelike shaded figures denoted 2-, 3-, and 4-valent atoms. Consult Partington, History of Chemistry, 4: 540 for a discussion of the highly idiosyncratic system.
Frankland, Lecture Notes, p. 17. Here, Frankland employed boldface type to indicate that the element so symbolized “is directly united with all the active bonds of the other elements or compound radicals following upon the same line” (pp. 16-17).
Ibid., p. 18.
Ibid., p. 24.
Ibid., p. 25.
Partington, History of Chemistry, 4: 552-553; Russell, History of Valency, pp. 92-100; and Norman Biggs, E. Keith Lloyd, and Robin J. Wilson, Graph Theory 1736-1936 (Oxford: Clarendon Press, 1976), pp. 56–60.
Clearly, Crum Brown had realized this as early as 1864 given the title of the paper in which he introduced his new symbolism. Frankland also explicitly singled out this advantage of the graphical notation in his Lecture Notes. As he put it, “[i]t is also of especial value in rendering evident the causes of isomerism in organic bodies” (p. 23).
Crum Brown, “Theory of Isomeric Compounds, ” pp. 707-718; and Biggs, Lloyd, and Wilson, Graph Theory, p. 60.
Arthur Cayley, “On the Theory of Analytical Forms Called Trees, ” Philosophical Magazine 13 (1857): 172–176, or Arthur Cayley and A. R. Forsyth, ed., The Collected Mathematical Papers of Arthur Cayley, 14 vols. (Cambridge: University Press, 1889-1898), 3: 242-246 (hereinafter cited as Math. Papers AC). All subsequent page citations refer to Math. Papers AC.
Arthur Cayley, “On the Analytical Forms Called Trees: Second Part, ” Philosophical Magazine 28 (1859): 374–378, or Math. Papers AC, 4: 112-115. Cayley gave the following recursive version of the formula for the number of trees φn with n nodes: (see p. 113). He also provided subsequent mathematical refinements of this formulation of φn. As his results showed, φn gets large quickly. For n = 1, 2, 3, 4, 5, 6, 7, 8, respectively, φn takes on the values 1, 1, 3, 13, 75, 541, 4683, and 47293.
Arthur Cayley, “On the Mathematical Theory of Isomers, ” Philosophical Magazine 47 (1874): 444–446, or Math. Papers AC, 9: 202-204.
Charles S. Gillispie, ed., The Dictionary of Scientific Biography, 16 vols. 2 supps. (New York: Charles Scribner’s Sons, 1970–1990), s.v. “Brown, Alexander Crum, ” by D. C. Goodman.
Arthur Cayley, “On the Analytical Forms Called Trees, with Application to the Theory of Chemical Combinations, ” Report of the British Association for the Advancement of Science: Bristol 1875 (London: John Murray, 1875), pp. 257–305, or Math. Papers AC, 9: 427-460, on p. 427.
Ibid., p. 428.
George Boole, “Exposition of a General Theory of Linear Transformations, ” Cambridge Mathematical Journal 3 (1841–1842): 1–20. See note 15 above.
In precise mathematical terminology, if L is a linear transformation of nonzero determinant Δ defined as in (1) which takes a binary form into then the expression I(a 0,a 1,…,a n) in the coefficients of f is an invariant provided for some positive integer k. The expression K(a 0,a 1, …,a n;x,y) in the coefficients and variables of f is called a covariant provided for some positive integer k.
Arthur Cayley, “On the Theory of Linear Transformations, ” Cambridge Mathematical Journal 4 (1845): 193–209, or Math. Papers AC, 1: 80-94; and Arthur Cayley, “On Linear Transformations, ” Cambridge and Dublin Mathematical Journal 1 (1846): 104-122, or Math. Papers AC, 1: 95-112. The quotation appears in the latter text on p. 95.
For discussions of the historical development of invariant theory both in Great Britain and on the Continent, see Tony Crilly, “The Rise of Cayley’s Invariant Theory (1841-1862), ” Historia Mathematica 13 (1986): 241-254; “The Decline of Cayley’s Invariant Theory (1863-1895), ” Historia Mathematica 15 (1988): 332-347; and Karen Hunger Parshall, “Toward a History of Nineteenth-Century Invariant Theory,” in The History of Modern Mathematics, ed. David E. Rowe and John McCleary, 2 vols. (Boston: Academic Press, 1989), 1: 157-206.
Arthur Cayley, “A Second Memoir upon Quantics, ” Philosophical Transactions of the Royal Society of London 146 (1856): 101–126, or Math. Papers AC, 2: 250-281. Cayley enumerates the complete system for the binary cubic on pp. 271-272.
See George Salmon, Lessons Introductory to the Modern Higher Algebra, 5th ed. (Dublin: Hodges, Figgis, & Co., 1885; reprint ed., Bronx, N.Y.: Chelsea Publishing Co., 1964), pp. 183-187 for the enumeration of the complete system for the binary cubic form. Indicative of the level of complexity of the theory associated with these computations, Salmon required the first 175 pages of his text to reach the point where he could carry out this enumeration.
These are listed in ibid., pp. 187-188.
Sylvester, “Address on Commemoration Day, ” p. 76.
Sylvester, “On an Application of the New Atomic Theory, ” p. 148. (See note 6 above.)
James Joseph Sylvester, “Chemistry and Algebra, ” Nature 17 (1877–1878): 284, or Math. Papers JJS, 3: 103-104, on p. 103.
Ibid. Sylvester’s emphasis.
Ibid., pp. 103-104. Sylvester’s emphasis.
Ibid., p. 104. Sylvester’s emphasis. Sylvester was initially confused on the issue of who developed the so-called chemicographs and attributed them to Kekulé (see ibid.). His error was pointed out to him by an editorial remark in Nature, and he subsequently acknowledged and corrected his mistake in the American Journal. See Nature 17 (1877-1878): 309; James Joseph Sylvester, “Historical Data concerning the Discovery of the Law of Valence, ” American Journal of Mathematics 1 (1878): 282; and “On an Application of the New Atomic Theory, ” pp. 172-173.
Sylvester dealt with this example in greater generality in “On an Application of the New Atomic Theory, ” pp. 164-165. Notice that this figure is a chemicograph because it represents a saturated system of atoms. Sylvester made no claims about the necessary existence in nature of all possible saturated systems.
Sylvester, “Chemistry and Algebra, ” p. 104.
See note 6 above.
Sylvester, “On an Application of the New Atomic Theory, ” p. 149.
Ibid., pp. 149-150. The diagram appears as figure 5 on p. 163.
Ibid., p. 165.
See, for example, Sylvester’s comments in ibid., pp. 165-166, relative to the possible physical implications of Hermite’s law, an important technical result in the theory of invariants due to the French mathematician, Charles Hermite.
For details on the establishment of the American Journal of Mathematics, see Parshall and Rowe, Emergence of the American Mathematical Research Community, Chapter 2.
Raymond C. Archibald, “Unpublished Letters of James Joseph Sylvester and Other New Information Concerning His Life and Work, ” Osiris 1 (1936): 85–154, on p. 134.
Ibid., p. 135.
William Kingdon Clifford, “Extract of a Letter to Mr. Sylvester from Prof. Clifford of University College, London, ” American Journal of Mathematics 1 (1878): 126–128, or “Remarks on the Chemico-Algebraical Theory, ” Mathematical Papers by William Kingdon Clifford, ed. Robert Tucker (London: Macmillan & Co., 1882; reprint ed., Bronx, N.Y.: Chelsea Publishing Company, 1968), pp. 255-257.
Ibid., p. 126.
See, for example, David M. Bishop, Group Theory and Chemistry (Oxford: Clarendon Press, 1973); and David S. Sconland, Molecular Symmetry: An Introduction to Group Theory and Its Uses in Chemistry (London: Van Nostrand, 1965).
James Joseph Sylvester to Daniel Coit Gilman, 7 September, 1878, Daniel C. Gilman Papers Ms. 1, Special Collections Division, Milton S. Eisenhower Library, The Johns Hopkins University. As always, I thank the University for permission to quote from its archives.
Edward Frankland, “Extract from a Letter of Dr. Frankland to Mr. Sylvester, ” American Journal of Mathematics 1 (1878): 345–349, on p. 346.
Ibid., p. 349. Sylvester’s paper on the atomic theory also attracted some critical attention in the United States. See, J. W. Mallet, “Some Remarks on a Passage in Professor Sylvester’s Paper on the Atomic Theory, ” American Journal of Mathematics 1 (1878): 277-281. Mallet was Professor of Chemistry at the University of Virginia.
Cayley briefly pursued his research in graph theory after 1875, but from an abstract point of view and not with an eye toward its application to chemistry. See Arthur Cayley, “A Theorem on Trees,” Quarterly Journal of Pure and Applied Mathematics 23 (1889): 376-378, or Math. Papers AC, 13: 26-28. Likewise, Sylvester returned to graphtheoretic researches for several months in 1889-1890 as a result of his contact with the Danish mathematician, Julius Petersen. See Gert Sabidussi, “Correspondence between Sylvester, Petersen, Hilbert and Klein on Invariants and the Factorization of Graphs 1889-1891, ” Discrete Mathematics, 100 (1992): 99-155. I thank Gert Sabidussi for providing me with a preprint of his edition of this correspondence.
Both George Salmon and Edwin Bailey Elliott tried to fill this lacuna in the literature by presenting a systematic treatment of the theory of invariants. See Salmon, Lessons Introductory to the Modern Higher Algebra; and Edwin Bailey Elliott, An Introduction to the Algebra of Quantics, 2d ed. (Oxford: University Press, 1913; reprint ed., Bronx, N.Y.: Chelsea Publishing Company, 1964).
Paul Gordan and W. Alexejeff, “Übereinstimmung der Formeln der Chemie und der Invariantentheorie, ” Zeitschrift für physicalischer Chemie 35 (1900): 610–633. See, also, Max Noether’s assessment of this work in “Paul Gordan, ” Mathematische Annalen 75 (1914): 1-41, on pp. 34-35.
On Sylvester’s idiosyncratic working style, see Parshall, “America’s First School of Mathematical Research, ” pp. 165-172; and Parshall and Rowe, Emergence of the American Mathematical Research Community, Chapters 2 and 3.
Sylvester, “On an Application of the New Atomic Theory, ” p. 166.
Biggs, Lloyd, and Wilson concur in this assessment in Graph Theory, p. 67.
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Parshall, K.H. (1997). Chemistry Through Invariant Theory?. In: Theerman, P.H., Parshall, K.H. (eds) Experiencing Nature. The University of Western Ontario Series in Philosophy of Science, vol 58. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5810-7_4
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