Benjamin Peirce (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeBenjamin PeirceFirst published Sat Feb 3, 2001; substantive revision Fri Aug 22, 2008Benjamin Peirce (b. April 4, 1809, d. October 6, 1880) was a professorat Harvard with interests in celestial mechanics, applications of planeand spherical trigonometry to navigation, number theory and algebra. Inmechanics, he helped to establish the (effects of the) orbit of Neptune(in relation to Uranus). In number theory, he proved that there is noodd perfect number with fewer than four distinct prime factors. Inalgebra, he published a comprehensive book on complex associativealgebras. Peirce is also of interest to philosophers because of hisremarks about the nature and necessity of mathematics. 1. Career2. Mathematics, mechanics and God3. Algebras and their philosophy4. The philosophy of necessityBibliographyOther Internet ResourcesRelated Entries1. CareerBorn in 1809, Peirce became a major figure in mathematics and thephysical sciences during a period when the U.S. was still a minorcountry in these areas (Hogan 1991). A student at Harvard College, hewas appointed tutor there in 1829. Two years later he became Professorof Mathematics in the University, a post which was changed in 1842 tocover astronomy also; he held it until his death in 1880. He played aprominent role in the development of the science curriculum of theuniversity, and also acted as College librarian for a time. However, hewas not a successful teacher, being impatient with students lackingstrong gifts; but he wrote some introductory textbooks in mathematics,and also a more advanced one in mechanics (Peirce 1855). Among hisother appointments, the most important one was Director of the U.S.Coast Survey from 1867 to 1874. Peirce also exercised influence throughhis children. By far the most prominent was Charles Sanders Peirce (1839–1914), who became aremarkable though maverick polymath, as mathematician, chemist,logician, historian, and many other activities. In addition, JamesMills (1834–1906) became in turn professor of mathematics at Harvard,Benjamin Mills (1844–1870) a mining engineer, and Herbert Henry Davis(1849–1916) a diplomat. However, Harvard professor Benjamin OsgoodPeirce (1854–1914), mathematician and physicist, was not a relative.Benjamin Peirce did not think of himself as a philosopher in anyacademic sense, yet his work manifests interests of this kind, in twodifferent ways. The first was related to his teaching. 2. Mathematics, mechanics and GodTo a degree unusually explicit in a mathematician of that time Peirceaffirmed his Christianity, seeing mathematics as study of God's work byGod's creatures. He rarely committed such sentiments to print; but ashort passage occurs in the textbook on mechanics previously mentioned,when considering the idea that the occurrence of perpetual motion innature would have proved destructive to human belief, in thespiritual origin of force and the necessity of a First Cause superiorto matter, and would have subjected the grand plans of Divinebenevolence 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 in1879, which James Peirce edited for posthumous publication (Peirce1881b). ‘Ideality’ connoted ‘ideal-ism’ asevident in certain knowledge, ‘pre-eminently the foundation ofthe mathematics’. His detailed account concentrated almostentirely upon cosmology and cosmogony with some geology (Petersen1955). He did not argue for his stance beyond some claims for existenceby design. 3. Algebras and their philosophyPeirce was primarily an algebraist in his mathematical style; forexample, he was enthusiastic for the cause of quaternions in mechanicsafter their introduction by W. R. Hamilton in the mid 1840s, and of thevarious traditions in mechanics he showed some favour for the‘analytical’ approach, where this adjective refers to thelinks to algebra. His best remembered publication was a treatment of‘linear associative algebras’, that is, all algebras inwhich the associative lawx(yz)=(xy)z was upheld.‘Linear’ did not carry the connotation of matrix theory,which was still being born in others' hands, but referred to the formof linear combination, such as: q = a + bi + cj + dkin the case of a quaternion q. Peirce wrote an extensivesurvey (Peirce 1870), determining the numbers of all algebras with fromtwo to six elements obeying also various other laws (Walsh 2000, ch.2). To two of those he gave names which have become durable:‘idempotent’, the law xm =x (for m≥2) which George Boole had introduced inthis form in his algebra of logic in 1847; and ‘nilpotent’,when xm = 0, for some m. Thehistory of the publication of this work is very unusual(Grattan-Guinness 1997). Peirce had presented some of his results from1867 onwards to the National Academy of Sciences, of which he had beenappointed a founder member four years earlier; but they could notafford to print it. Thus, in an initiative taken by Coast Survey staff,a lady without mathematical training but possessing a fine hand wasfound who could both read his ghastly script and write out the entiretext 12 pages at a time on lithograph stones. 100 copies were printed(Peirce 1870), and distributed world-wide to major mathematicians andprofessional colleagues. Eleven years later Charles, then at JohnsHopkins University, had the lithograph reprinted posthumously, withsome additional notes of his own, as a long paper in American journalof mathematics, which J.J. Sylvester had recently launched (Peirce1881a); it also came out in book form in the next year. This studyhelped mathematicians to recognise an aspect of the wide variety ofalgebras which could be examined; it also played a role in thedevelopment of model theory in the U.S. in the early 1900s. Enough workon it had been done by then for a book-length study to be written (Shaw1907). 4. The philosophy of necessityPeirce seems to have upheld his theological stance for all mathematics,and a little sign is evident in the dedication at its head: To my friends This work has been the pleasantestmathematical effort of my life. In no other have I seemed to myself tohave received so full a reward for my mental labor in the novelty andbreadth of the results. I presume that to the uninitiated the formulaewill appear cold and cheerless. But let it be remembered that, likeother mathematical formulae, they find their origin in the divinesource of all geometry. Whether I shall have the satisfaction of takingpart in their exposition, or whether that will remain for some moreprofound expositer, will be seen in the future (Peirce 1870,1).Peirce began with a philosophical statement of a different kind aboutmathematics which has become his best remembered single statement“Mathematics is the science that draws necessary conclusions” (Peirce1870, p. 1). What does ‘necessary’ denote? Perhaps he wasfollowing a tradition in algebra, upheld especially by Britons such asGeorge 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 agiven mathematical and/or physical situation) and claiming that itsform alone would deliver the consequences from the premises. In hisfirst draft of his text he wrote the rather more comprehensible“Mathematics is the science that draws inferences”, and in the seconddraft “Mathematics is the science that draws consequences”, though thelast word was altered to yield the enigmatic form involving‘necessary’ used in the book. The change is not justverbal; he must have realised that the earlier forms were notsufficient (they are satisfied by other sciences, for example), and soadded the crucial adjective. Certainly no whiff of modal logic was inhis air. His statement appears in the mathematical literature fairlyoften, but usually without explanation. One feature is clear, but oftenis not stressed. In all versions Peirce always used the active verb‘draws’: mathematics was concerned with the act of drawingconclusions, not with the theory of so acting, which belonged indisciplines 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 isrigidly 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; theyare 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 mathematicscould be used to analyse logic, not the vice versa relationship betweenthe two disciplines that Gottlob Frege was about to put forward forarithmetic, and which Bertrand Russell was optimistically to claim forall mathematics during the 1900s. Curiously, the third draftof the lithograph contains this contrary stance in “Mathematics, ashere defined, belongs to every enquiry; it is even a portion ofdeductive logic, to the laws of which it is rigidly subject”; but bycompletion he had changed his mind. Peirce's son Charles claimed tohave influenced his father in forming his definitive position, andfiercely upheld it himself; thereby he helped to forge a wide divisionbetween the algebraic logic which he was developing from the early1870s with his father, Boole and de Morgan as chief formativeinfluences, and the logicism (as it became called later) of Frege andRussell and also the ‘mathematical logic’ of Giuseppe Peanoand his school in Turin (Grattan-Guinness 1988). BibliographyThis list includes some valuable items not cited in the text. Primary SourcesPeirce Manuscripts: Houghton Library, Harvard University.1855. Physical and celestial mathematics, Boston: Little,Brown.1861. An elementary treatise on plane and sphericaltrigonometry, with their applications to navigation, surveying,heights, and distances, and spherical astronomy, and particularlyadapted to explaining the construction of Bowditch's navigator, and thenautical 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 ofChestnut 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” inAmerica, (I. Bernard Cohen, ed.), New York: Arno Press.[Photoreprints, including that of (Peirce 1881a).]Secondary SourcesArchibald, 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 associativealgebra and C.S. Peirce’, American mathematical monthly,34: 525–527.Kent, D. 2005. Benjamin Peirce and the promotion ofresearch-level mathematics in America: 1830–1880. DoctoralDissertation, University of Virginia.Grattan-Guinness, I. 1988. ‘Living together and living apart:on the interactions between mathematics and logics from the FrenchRevolution to the First World War’, South African journal ofphilosophy, 7/2: 73–82.Grattan-Guinness, I. 1997. ‘Benjamin Peirce's Linearassociative 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’, Historiamathematica, 18: 158–172.Hogan, E. 2008. Of the human heart. A biography of BenjaminPeirce, Bethlehem: Lehigh University press. King, M. 1881. (Ed.), Benjamin Peirce. A memorialcollection, Cambridge, Mass.: Rand, Avery. [Obituaries.]Novy, L. 1974, ‘Benjamin Peirce's concept of linear algebra’,Acta historiae rerum naturalium necnon technicarum (SpecialIssue), 7: 211–230.Peterson, S. R. 1955. ‘Benjamin Peirce: mathematician andphilosopher’, Journal of the history of ideas, 16:89–112.Pycior, H. 1979. ‘Benjamin Peirce's linear associativealgebra’, Isis, 70: 537–551.Schlote, K.-H. 1983. ‘Zur Geschichte der Algebrentheorie inPeirces “Linear Associative Algebra”’, Schriftenreihe derGeschichte der Naturwissenschaften, Technik und Medizin,20/1: 1–20.Shaw, J. B. 1907. Synopsis of linear associative algebra. Areport on its natural development and results reached to the presenttime, Washington.Walsh, A. 2000. ‘Relationships between logic and mathematicsin the works of Benjamin and Charles S. Peirce’, Ph. D. thesis,Middlesex University.Other Internet ResourcesThe MacTutor History of Mathematics Archive entry on PeircePhotos of Peirce at the MacTutor ArchiveRelated Entries Peirce, Charles Sanders Copyright © 2008 byIvor Grattan-Guinness<eggigg@ghcom.net>Alison Walsh<awalsh@mail.camre.ac.uk> |
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