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- 1. Career
- 2. Mathematics, mechanics and God
- 3. Algebras and their philosophy
- 4. The philosophy of necessity
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would have proved destructive to human belief, in the spiritual origin of force and the necessity of a First Cause superior to matter, and would have subjected the grand plans of Divine benevolence to the will and caprice of man (Peirce 1855, 31).Peirce was more direct in a course of Lowell Lectures on ‘Ideality in the physical sciences’ delivered at Harvard in 1879, which James Peirce edited for posthumous publication (Peirce 1881b). ‘Ideality’ connoted ‘ideal-ism’ as evident in certain knowledge, ‘pre-eminently the foundation of the mathematics’. His detailed account concentrated almost entirely upon cosmology and cosmogony with some geology (Petersen 1955). He did not argue for his stance beyond some claims for existence by design.

in the case of a quaternionq = a + bi + cj + dk

To my friends This work has been the pleasantest mathematical effort of my life. In no other have I seemed to myself to have received so full a reward for my mental labor in the novelty and breadth of the results. I presume that to the uninitiated the formulae will appear cold and cheerless. But let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositer, will be seen in the future (Peirce 1870, 1).Peirce began with a philosophical statement of a different kind about mathematics which has become his best remembered single statement "Mathematics is the science that draws necessary conclusions" (Peirce 1870, p. 1). What does ‘necessary’ denote? Perhaps he was following a tradition in algebra, upheld especially by Britons such as George Peacock and Augustus De Morgan (a recipient of the lithograph), of distinguishing the ‘form’ of an algebra from its ‘matter’ (that is, an interpretation or application to a given mathematical and/or physical situation) and claiming that its form alone would deliver the consequences from the premises. In his first draft of his text he wrote the rather more comprehensible "Mathematics is the science that draws inferences", and in the second draft "Mathematics is the science that draws consequences", though the last word was altered to yield the enigmatic form involving ‘necessary’ used in the book. The change is not just verbal; he must have realised that the earlier forms were not sufficient (they are satisfied by other sciences, for example), and so added the crucial adjective. Certainly no whiff of modal logic was in his air. His statement appears in the mathematical literature fairly often, but usually without explanation. One feature is clear, but often is not stressed. In all versions Peirce always used the active verb ‘draws’: mathematics was concerned with the act of drawing conclusions, not with the theory of so acting, which belonged in disciplines such as logic. He continued:

Mathematics, as here defined, belongs to every enquiry; moral as well as physical. Even the rules of logic, by which it is rigidly bound could not be deduced without its aid (Peirce 1870, 3).In a lecture of the late 1870s he described his definition as

wider than the ordinary definitions. It is subjective; they are objective. This will include knowledge in all lines of research. Under this definition mathematics applies to every mode of enquiry (Peirce 1880, 377).Thus Peirce maintained the position asserted by Boole that mathematics could be used to analyse logic, not the vice versa relationship between the two disciplines that Gottlob Frege was about to put forward for arithmetic, and which Bertrand Russell was optimistically to claim for

- Peirce Manuscripts: Houghton Library, Harvard University.
- 1855.
*Physical and celestial mathematics*, Boston: Little, Brown. - 1861.
*An elementary treatise on plane and spherical trigonometry, with their applications to navigation, surveying, heights, and distances, and spherical astronomy, and particularly adapted to explaining the construction of Bowditch’s navigator, and the nautical almanac*, rev. ed., Boston: J. Munroe. - 1870.
*Linear associative algebra*, Washington (lithograph). - 1880. ‘The impossible in mathematics’, in Mrs. J. T. Sargent
(ed.),
*Sketches and reminiscences of the Radical Club of Chestnut St. Boston*, Boston : James R. Osgood, 376-379. - 1881a. ‘Linear associative algebra’,
*Amer. j. math.*, 4, 97-215. Also (C.S. Peirce, ed.)in book form, New York, 1882. [Printed version of Peirce 1870.] - 1881b.
*Ideality in the physical sciences*, (J. M. Peirce, ed.), Boston: Little, Brown. - 1980. Benjamin Peirce: "Father of Pure Mathematics" in America, (I. Bernard Cohen, ed.), New York: Arno Press. [Photoreprints, including that of (Peirce 1881a).]

- Archibald, R.C. 1925. [ed.], ‘Benjamin Peirce’,
*American mathematical monthly*, 32, 1-30; repr. Oberlin, Ohio.: Mathematical Association of America. - Archibald, R.C. 1927. ‘Benjamin Peirce’s linear
associative algebra and C.S. Peirce’,
*American mathematical monthly*, 34, 525-527. - Grattan-Guinness, I. 1988. ‘Living together and living
apart: on the interactions between mathematics and logics from the
French Revolution to the First World War’,
*South African journal of philosophy*, 7, no. 2, 73-82. - Grattan-Guinness, I. 1997. ‘Benjamin Peirce’s Linear
associative algebra (1870): new light on its preparation and
"publication"’,
*Annals of science*, 54, 597-606. - Hogan, E. 1991. ‘ "A proper spirit is abroad": Peirce,
Sylvester, Ward, and American mathematics’,
*Historia mathematica*, 18, 158-172. - King, M. 1881. (Ed.),
*Benjamin Peirce. A memorial collection*, Cambridge, Mass.: Rand, Avery. [Obituaries.] - Novy, L. ‘Benjamin Peirce’s concept of linear
algebra’,
*Acta historiae rerum naturalium necnon technicarum*(Special Issue) 7 (1974), 211-230. - Peterson, S. R. 1955. ‘Benjamin Peirce: mathematician and
philosopher’,
*Journal of the history of ideas*, 16, 89-112. - Pycior, H. 1979. ‘Benjamin Peirce’s linear associative
algebra’,
*Isis*, 70, 537-551. - Schlote, K.-H. 1983. ‘Zur Geschichte der Algebrentheorie in
Peirces "Linear Associative Algebra"’,
*Schriftenreihe der Geschichte der Naturwissenschaften*,*Technik und Medizin*, 20, no. 1, 1-20. - Shaw, J. B. 1907.
*Synopsis of linear associative algebra. A report on its natural development and results reached to the present time*, Washington. - Walsh, A. 2000. ‘Relationships between logic and mathematics in the works of Benjamin and Charles S. Peirce’, Ph. D. thesis, Middlesex University.

- The MacTutor History of Mathematics Archive entry on Peirce
- Photos of Peirce at the MacTutor Archive

Middlesex University at Enfield

and

Cambridge Regional College

*First published: February 3, 2001*

*Content last modified: February 3, 2001*