Summary
This paper deals with the influence exerted by Boole’s own work on differential equations on his creation of algebraic logic. The main traits of Boole’s methodology of logic, and the particular algorithms which he used in his 1847The Mathematical Analysis of Logic. ore first pointed out. An examination of the mathematical papeis which Boole wrote before the publication of the mentioned logical treatise shows that both the methodology leading to the production of his logic and the algorithms used in its development were repeatedly used by him in his earlier work in analysis.
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Notes
At the very beginning ofThe Mathematical Analysis of Logic (the reference is given in footnote 3 below) Boole stated that he was moved to resume his enquiries about an algebra of logic by the controversy between Hamilton and De Morgan. Besides having been the external stimulus which moved Boole to pursue his enquiries, the controversy may have had another kind of influence on Boole; it seems that several of the ideas interchanged in the controversy confirmed and clarified Boole’s own ideas, and that probably the latter stated his philosophy of mathematics in the introduction toThe Mathematical Analysis of Logic as a response to Hamilton’s ideas about the same subject.
Mary Everest Boole, Boole’s wife, referred in many places in her writings (which are cited in footnote 16 below) to the existence of a mystico-pedagogico-psychological background and aim of Boole’s logic. According to her, the equationx2 =x was the expression of a dualistic philosophy which claimed that the Eternal Unity of God could be reached from the contemplation of all opposite opinions and facts; (1 — jc) was the opposite of jc, and (1 –x) +x = 1. Such conception involved a psychological theory of the mind and had pedagogical implications. Several historians of logic have despised Mary Everest’s claim, but in fact, it seems that she was basically right. It would take too long to prove it here; I have examined Mary Everest’s claims in ‘A Study of the Genesis of Boolean Logic’, a doctoral dissertation submitted to the Graduate School of the University of Notre Dame. This dissertation was directed by Professor Michael J. Crowe. Among the psychological implications of Boole’s—probably existing—metaphysical conception was that God works directly through some hidden resort of the human mind, so that the latter does not need to be continuously conscious of its operations. In pedagogy, this idea materializes in the stress on the importance of using symbols which only at determinate steps of the argument need to have meaning, in the processes of learning and teaching. One interesting problem is to know the possible influence on Mary Everest (and Boole too?) of a symbolical school which developed in the early and mid-nineteenth century (for references to secondary literature on this theme, see footnote 46 below).
G. Boole,The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning (1847, Cambridge and London: repr. 1958, New York). Also in G. Boole,Studies in Logic and Probability (ed. R. Rhees: 1952, London), 49–124. This work is cited hereafter as‘MAU.
Neither the application of McLaurin expansions to obtain Boole’s elective functions, nor the calculus of solutions of the corresponding equations by means of the theory of linear differential equations, appears inMAL until the last two chapters. Nevertheless, the most important results of the first part of the book are based on the results obtained in those two last chapters. This fact shows that, if not in order of presentation, the theory of elective functions was first in order of procedure. But this theory was, on its hand, based in the theory of developments of functions. (See footnote 7 below for an explanation of the elective operators and functions.)
See the Introduction toMAL.
MAL, 17–18.
Boole stated the laws of logic from the contemplation of the operations performed by the mind when electing or classifying elements from a given universe. For instance, jc is a symbol which represents a selection of groups of individuals, namelyu andv. The result of an act of election is independent of the grouping of the individuals, so that we have the distributivity lawx(u + v) =xu +xv. In the same way, Boole inferred the commutativity lawxy(u) =yx(u) and the special lawx″ =x. The elective functions are functions the variables of which are precisely the elective operators, jc,y, . . . . He supposed that all the operative devices provided by ordinary algebra and analysis could be applied to these functions, with the only care of taking in account the irregularities provided by the conditionx″ =x.
This view that Boole’s logic wasapplied mathematics of a universal calculus of symbolssupports Russell’s view that ‘Pure mathematics was discovered by Boole’(Mysticism and Logic (1917, London; repr. 1963), 59), precisely because of the universal meaning attributed by Boole to the word ‘analysis’. Nevertheless, Russell’s assertion was too exclusive: even though it seems that Boole grasped by himself the existence of a universal calculus capable of embodying several known and possible fields, this idea was already at least implicitly contained in the works of the French mathematicians of the immediately prior period, and in those of Peacock, Herschel, Hamilton, Gregory, De Morgan and others. In mathematics it is often misleading to assign a particular discovery to a particular person. Russell himself retracted somewhat his comment on Boole discovering pure mathematics. (I owe the last information to Dr. Ivor Grattan-Guinness, who cites Russell’s letter inProc. Roy. Irish Acad., 57 (A) (1954–56), 28.)
See footnote 2 above.
See the last part of section 3 of this paper.
G. Boole, ‘Researches on the Theory of Analytical Transformations, with a Special Application to the Reduction of the General Equation of the Second Order’,Cambridge Mathematical Journal, 2 (1841), 64–73.
Ibid., 64.
Ibid., 65.
ibid.
Ibid.
M. Everest Boole.Collected Works (4 vols., ed. E.M. Cobham and E.S. Dummer: London, 1931), vol. 1,4,5,35.
G. Boole, ‘On Certain Theorems in the Calculus of Variations’,Cambridge Mathematical Journal, 2 (1841), 97–102.
Ibid., 97.
See, for instance,Cambridge Mathematical Journal, 2 (1841), 64–73 (p. 64); 2 (1841), 114–119 (p. 115); 3 (1843), 1–20 (p. 2); andCambridge and Dublin Mathematical Journal, 2 (1847), 7–12 (p. 7).
G.Boole (footnote 17), 99.
Ibid., 101.
Ibid., 97.
Ibid., 98.
Ibid.
Ibid., 101–102.
MAL, 49–50, 52, 64, 77.
It may be argued that what was arbitrary was precisely Boole’s methodological or philosophical assumptions, but this has been the way many (not to say most) scientific discoveries have taken place: by recourse to methodological and philosophical ideas extraneous to the field of direct investigation.
G. Boole (footnote 17), 98.
G. Boole, ‘On the Integration of Linear Differential Equations with Constant Coefficients’,Cambridge Mathematical Journal, 2 (1841), 114–119.
See footnotes 45–47 and text.
G. Boole (footnote 29), 115.
Ibid., 119.
G. Boole, ‘Analytical Geometry’,Cambridge Mathematical Journal, 2 (1841), 179–188.
G. Boole, ‘Analytical Geometry’,Cambridge Mathematical Journal, 2 (1841), 180,
G. Boole, ‘Analytical Geometry’,Cambridge Mathematical Journal, 2 (1841), 182.
G. Boole, ‘Exposition of a General Theory of Linear Transformations’,Cambridge Mathematical Journal, 3 (1843), 1–20, 106–119 (pp. 6–7).
G. Boole, ‘Exposition of a General Theory of Linear Transformations’,Cambridge Mathematical Journal, 3 (1843), 8.
MAL, 73.
MAL, 70.
G. Boole, ‘On a General Method in Analysis’,Philosophical Transactions of the Royal Society of London, 134 (1844), 225–282
G. Boole, ‘On a General Method in Analysis’,Philosophical Transactions of the Royal Society of London, 134 (1844), 225.
D.F. Gregory,Examples of the Processes of the Differential and Integral Calculus (1841, Cambridge and London).
D.F. Gregory,Examples of the Processes of the Differential and Integral Calculus (1841, Cambridge and London), 235.
D.F. Gregory,Examples of the Processes of the Differential and Integral Calculus (1841, Cambridge and London).
J. Herschel, ‘Considerations on Various Points in Analysis’,Philosophical Transactions of the Royal Society of London, 104 (1814), 440–468.
These important developments still await the historical study that they deserve, but interim information can be recovered from: S. Pincherle, ‘Pour la bibliographie de la théorie des opérations distributives’,Bibl. math., (2) 13 (1899), 13–18; his ‘Funktionale Operationen und Gleichungen’,Enc. math. Wiss., Bd. 2, Teil A (1903–21, Leipzig), 761–824; H. Burkhardt, ‘Entwicklungen nach oscillierenden Funktionen...’,Jber. Dtsch. Math.-Ver., 10, pt. 2 (1901–08), esp. eh. 13; E. Koppelman, ‘The Calculus of Operations and the Rise of Abstract Algebra’,Arch. Hist. Exact Sei., 8 (1972), 155–242; and I. Grattan-Guinness and J.R. Ravetz,Joseph Fourier 1768–1830 ... (1972), Cambridge, Mass.), esp. pp. 464–466.
G. Boole (footnote 40), 225.
MAL, 15–18.
G. Boole (footnote 40), 282.
G. Boole, ‘On the Inverse Calculus of Definite Integrals’,Cambridge Mathematical Journal, 4 (1845), 82–87 (p. 87).
G. Boole, ‘On the Equations of Laplace’s Functions’,Cambridge and Dublin Mathematical Journal, 1 (1846), 10–22 (p. 22).
G. Boole, ‘On a Certain Symbolical Equation’,Cambridge and Dublin Mathematical Journal, 2 (1847), 7–12 (p. 7).
G. Boole (footnote 51), 11.
Ibid., 11, 13,20.
MAL, 70–72.
G. Boole, ‘On the Theory of Developments’,Cambridge Mathematical Journal, 4 (1845), 214–223 (P. 219).
MAL, 60.
ibid.
Ibid., 60–61.
See especially footnotes 1 and 2 above.
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Departamento de Algebra y Fundamentos, Sección de Matemáticas, Universidad de Sevilla, Sevilla, Spain
Luis M. Laita
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University of Lausanne, Switzerland
James Gasser
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Laita, L.M. (2000). The Influence of Boole’s Search for a Universal Method in Analysis on the Creation of His Logic. In: Gasser, J. (eds) A Boole Anthology. Synthese Library, vol 291. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9385-4_3
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