Abstract
For over one hundred years, the links between logic and mathematics have been so close that it is difficult to think of the one without the other. All the persons chiefly responsible for the development of symbolic logic were both mathematicians and logicians.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
Encylopedia Brittanica, 11th ed., s.v. “De Morgan, Augustus.”
See his wide-ranging contributions to the Penny Cyclopaedia, Notes and Queries, and, especially, the Athenaeum Even though his contributions to the Athenaeum were unsigned, it is possible to determine what they were because bound copies of them, initialed by De Morgan, are deposited at the University of London Library.
“On the Foundation of Algebra,” Transactions of the Cambridge Philosophical Society 7 (1841): 173–87; 7 (1842): 287–300; 8 (1844): 139–42; 8 (1847): 241–53.
In fact, only one of the 324 teaching notebooks deals solely with logic. This is Ms. 775/203, “On the Two Forms of the Universal Affirmative Proposition,” De Morgan Papers, University of London Library. A few other notebooks consider geometrical demonstration (see note 23).
“On the Study of Mathematics.”
De Morgan even uses this very language. See S5, 345 and De Morgan’s review of Herbert Spencer, The Classification of the Sciences, Athenaeum (21 May 1864): 709.
See note 1, Chapter Two.
See also De Morgan’s review of Sir William Hamilton, Lectures on Metaphysics and Logic, Athenaeum (24 November 1860): 706.
See also Notes and Queries (27 August 1864): 161.
See also De Morgan’s review of John Stuart Mill, An Examination of Sir William Hamilton’s Philosophy, Athenaeum (27 May 1865): 711; “Speech of Professor De Morgan, President,” Proceedings of the London Mathematical Society 1 (1866): 1–9.
Penny Cyclopaedia, Vol. XV (1839), s.v. “Mathematics,” 11ff.
G.C. Smith, ed., The Boole-De Morgan Correspondence 1842–1864 (Oxford: Clarendon Press, 1982).
For some other references to Boole by De Morgan, see his review of Boole’s Mathematical Analysis of Logic, Athenaeum (29 January 1848): 107–8;
his obituary of Boole, Macmillan’s Magazine (February, 1865): 279–80; “Budget of Paradoxes, No. XXIII,” Athenaeum (11 March 1865): 350;
De Morgan to Sir William Rowan Hamilton, 5 October 1852, in Robert Percival Graves, Life of Sir William Rowan Hamilton, Vol. III (Dublin: Hodges, Figgis & Co./London: Longmans, Green & Co., 1889), 421.
In the title of his detailed review of The Boole-De Morgan Correspondence, John Corcoran has gone so far as to use the apt phrase “Correspondence without Communication”; History and Philosphy of Logic 7 (1986): 65–75. For other reviews see those by Theodore Hailperin, Journal of Symbolic Logic 49 (1984): 657–59;
and Calvin Jongsma, Historia Mathematica 12 (1985): 186–90. In reading through De Morgan’s extensive correspondence with both Boole and Sir William Rowan Hamilton, one gets the impression that he never seriously studied their works. With his great variety of interests and heavy work schedule, he probably had neither the time nor the inclination to do so. Boole seems not to have studied De Morgan’s work carefully either. On 27 May 1852, he wrote Sir William Hamilton, “The peculiarities of Mr. De Morgan’s system of logic I have not made an object of study, and I do not feel competent therefore to pass any opinion upon the correctness or mutual consistency of his views.”
John Veitch, Memoir of Sir William Hamilton, Bart. (Edinburgh and London: William Blackwood and Sons, 1869), 344.
De Morgan to Boole, 28 November 1847, Correspondence, 25.
“Introductory Lecture” and Penny Cyclopaedia, Vol. XI (1838), s.v. “Geometry,” 151, 156.
In this we followed Ian Mueller, “Greek Mathematics,” 37–43.
These include The Connexion of Number and Magnitude: An Attempt to Explain the Fifth Book of Euclid (London: Taylor and Walton, 1836); “Short Supplementary Remarks on the First Six Books of Euclid’s Elements,” Companion to the Almanac for 1849, 5–20; “On Indirect Demonstration,” Philosophical Magazine 3 (1852): 435–38. Among his teaching notebooks in the University of London Library, see “Notions Preliminary to Geometry,” 775/202; “On the Two Forms of the Universal Affirmative Proposition,” 775/203; and 775/310 (untitled).
For De Morgan’s views on issues regarding the place of logic in the teaching of geometry, see the following articles from the Quarterly Journal of Education: “On Mathematical Instruction” 1 (1831): 264- 79; “On the Method of Teaching the Elements of Geometry, Part I” 6 (1833): 35–49; and “A Method of Teaching Geometry, No. II” 6 (1833): 237–51.
Penny Cyclopaedia, Vol. XV (1839), s.v. “Mathematics,” 12.
See especially the “Foundation of Algebra” series (n. 3).
“Calculus of Functions,” Encyclopaedia Metropolitana (1836), 306–92; and Penny Cyclopaedia, Vol. XVI (1840), s.v. “Operation.”
For the problems of interpreting negative and “impossible” numbers see Ernest Nagel, “‘Impossible Numbers’: A Chapter in the History of Modern Logic,” in Studies in the History of Ideas, ed. Columbia University Philosophy Department (New York: Columbia University Press, 1935), 429–74; Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972), 592–97, 626–32.
The Principles of Algebra, 2 vols. (London, 1796–99).
We will leave open the question of just which laws to retain. We must, for instance, exclude laws such as (x) ~ (x2 = -1), which hold for the reals and not for the imaginaries. Peacock seems to have had in mind algebraic identities, such as x(x + 1) = x2 + x.
For recent discussions of this period see Morris Kline, Mathematical Thought, 772–94; Harvey W. Becher, “Woodhouse, Babbage, Peacock and Modern Algebra,” Historia Mathematica 7 (1980): 389–400;
Daniel A. Clock, “A New British Concept of Algebra: 1825–1850” (diss., University of Wisconsin, 1964);
J.M. Dubbey, “Babbage, Peacock and Modern Algebra,” Historia Mathematica 4 (1977): 295–302;
Elaine Koppelman, “The Calculus of Operations and the Rise of Abstract Algebra,” Archive for History of Exact Sciences 8 (1971): 155–242;
Luboš Nový, Origins of Modern Algebra (Leyden: Noordhoff International Publishing, 1973);
Helena M. Pycior, “George Peacock and the British Origins of Symbolical Algebra,” Historia Mathematica 8 (1981): 23–45; “Historical Roots of Confusion among Beginning Algebra Students: A Newly Discovered Manuscript,” Mathematics Magazine 55 (1982): 150–56; “Early Criticism of the Symbolical Approach to Algebra,” Historia Mathematica 9 (1982): 392- 412; “Internalism, Externalism, and Beyond: 19th Century British Algebra,” Historia Mathematica 11 (1984): 424–41;
Joan L. Richards, “The Art and Science of British Algebra: A Study in the Perception of Mathematical Truth,” Historia Mathematica 7 (1980): 343–65.
A Treatise on Algebra: Vol. I., Arithmetical Algebra (Cambridge: Cambridge University Press, 1842); Vol. II., On Symbolical Algebra and its Applications to the Geometry of Position (Cambridge: Cambridge University Press, 1845); “Report on the Recent Progress and Present State of Certain Branches of Analysis,” Report of the Third Meeting of the British Association for the Advancement of Science (London: John Murray, 1834), 185–352. See also De Morgan’s review of Peacock’s Treatise in the Quarterly Journal of Education 9 (1835): 91- 110, 293–311.
“Report,” 194.
“Report,” 194–95.
“Report,” 198.
“Report,” 199.
“Report,” 200.
Trigonometry and Double Algebra (London: Taylor, Walton, and Maberly, 1849), 89. This will be refered to as “TDA.” For discussions of De Morgan’s work in algebra see Helena M. Pycior, “Augustus De Morgan’s Algebraic Work: The Three Stages,” Isis 74 (1983): 211–26;
Joan L. Richards, “Augustus De Morgan, The History of Mathematics, and the Foundations of Algebra,” Isis 78 (1987): 7–30;
and G.C. Smith, “De Morgan and the Laws of Algebra,” Centaurus 25 (1981): 50–70.
“On Infinity; and on the Sign of Equality,” Transactions of the Cambridge Philosophical Society 11 (1866): 145–89; “On Divergent Series and Various Points of Analysis Connected with Them,” Transactions of the Cambridge Philosophical Society 8 (1849): 182–203.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Merrill, D.D. (1990). Logic and Mathematics. In: Augustus De Morgan and the Logic of Relations. The New Synthese Historical Library, vol 38. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2047-7_7
Download citation
DOI: https://doi.org/10.1007/978-94-009-2047-7_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7418-6
Online ISBN: 978-94-009-2047-7
eBook Packages: Springer Book Archive