Abstract
I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education.
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Notes
Following standard practice in Peirce scholarship, references to Reasoning and the Logic of Things (Peirce 1992b) are abbreviated RLT.
Peirce had systematic philosophical reasons for this, given his general theory of categories. Readers may consult his 1868 article “On a New List of Categories” (W 2, p. 49–59), as well as his 1903 lectures “On Phenomenology” (EP 2, p. 145–159) and “The Categories Defended” (EP 2, p. 160–178). Note that following standard practice in Peirce scholarship, references to Writings of Charles S. Peirce (1982–) are abbreviated W followed by volume number, while references to The Essential Peirce (1992–98) are abbreviated EP followed by volume number. For secondary literature introducing the system of categories, see Hookway 1985, p. 80–117, and Hausman 1993, p. 94–139.
Phyllis Chiasson proposes a triad of abilities for learning in general when she argues that “embedded within Peirce’s complete body of work is a design for thinking that provides a sturdy foundation for the development of three important learning capabilities. These capabilities are (1) the ability to identify, compare, and contrast qualities, (2) the ability to perform analyses, and (3) the ability to interpret the meaning of signs” (2005). My discussion here will be more circumscribed to mathematics, but it is compatible with Chiasson’s via Peirce’s systematic philosophy.
For an in-depth discussion of this course as related to Peirce’s philosophy of education see Anderson 2005.
It should not be inferred, however, that this was a narrow, technical course, since Peirce’s conceptions of logic, mathematics, and science were quite broad and inclusive. The part on science, for example, included questions on the relation of science to ethics and religion, as illustrated by Peirce’s description of its contents: “Exercises in sampling. Inductions. Extrapolations. Framing hypotheses. Analogies. The generalization of problems and methods. The art of asking questions. Precautions in considering moral and spiritual questions: the world not governed by blind law” (W 6, p. 11).
Following standard practice in Peirce scholarship, references to Collected Papers of Charles Sanders Peirce (Peirce 1932–58) are abbreviated CP followed by volume and paragraph numbers. For example, this reference is to volume 3, paragraphs 553–562.
The editors of Peirce’s Collected Papers refer the reader to Aristotle’s Metaphysics 1061a 28–1061 b 3 and 1061b 21–25.
See Benjamin Peirce 1870, p. 97.
See Chrystal 1883.
The following discussion on diagrams draws from a more extended explanation presented in Campos (2007).
Following standard practice in Peirce scholarship, references to The New Elements of Mathematics (Peirce 1976) are abbreviated NEM followed by volume number.
In this quotation, the square brackets indicate revisions by James Mills Peirce to render his brother Charles Peirce’s manuscript clearer. See Carolyn Eisele’s edited text and footnotes in NEM 2, p. 251.
For Peirce’s exposition of his view that “the very life of mathematical thinking consists in making experiments upon diagrams and the like and in observing the results,” see NEM 2, p. 345–346; 1894. For in-depth expositions of diagrammatic reasoning according to Peirce, see Kent 1997, Stjernfelt 2000, and Hoffman 2003 and 2004.
References to Peirce’s manuscripts are abbreviated as MS followed by their Robin Catalogue (1967) number.
I would like to emphasize, however, that a thorough discussion on the nature of the mind and of these intellectual “powers” according to Peirce is beyond the scope of my present concerns. Nevertheless, I should note that these “powers” are not meant to be any abstract entities somehow “contained” in the mind. They are rather “abilities” that are actualized in the performance of certain mental actions. They are instinctive capacities that must be cultivated and developed into “habits.” For example, the mathematical “imagination” is an ability that consists in being able to actually picture or imagine a mathematical “diagram”; that is, to create a “sign” representing a mathematical idea, where the idea is embodied in the sign. There is no abstract entity called the “imagination” apart from a habit actualized in the act of imagining. Likewise, there is no abstract power of concentration independent of the actual act of concentrating nor is there an abstract faculty of generalization separate from the act of generalizing. In what follows, I use the terms “powers,” “faculties,” and “abilities” in the sense of these skillful “habits” for performing a specific kind of action or for fulfilling certain semiotic functions.
References to Peirce’s letters are abbreviated as L followed by their Robin Catalogue (1967) number.
This example from geometry is my own, though it is similar to many of Peirce’s own examples. Alternative examples from arithmetic, on the other hand, can be found in Peirce’s own writings, especially his many manuscripts of arithmetic textbooks (see NEM 1). Carolyn Eisele (1979, p. 190–195) describes some of them. It is important to keep in mind that Peirce’s account of mathematical reasoning applies not only to geometry, but to mathematics in general, including arithmetic and algebra.
The following reconstruction has been presented in Campos (2009); however, the philosophical emphasis of the discussion has been shifted to address issues of mathematical education.
See Euclid 1956, vol. 1, p. 316–320.
Peirce would refer to this as the “theorematic” step in the demonstration. He considered his distinction between “theorematic” and “corollarial” reasoning to be his first real discovery about the methods of mathematical inquiry (NEM 4, p. 49; 1902). He means that some mathematical demonstrations involve truly ampliative reasoning—something new is discovered—while others only draw out deductively what is already contained in the statement of previous propositions, axioms, postulates, and definitions. For commentary, see Hintikka 1980, Ketner 1985, and Levy 1997.
On this point, see also Peirce’s March 1895 letter to Ginn & Co. (L 169), defending the content of the revised New Elements of Geometry, quoted at length in Eisele 1979, p. 188.
I elaborate at length on this point in Campos (2009).
Also consider Peirce’s view regarding the students’ use of diagrams and notations as aids to thinking: “The student must learn to use notations to think in, but he must not try to make the notation think for him, if he wished to push his reasonings far. Thinking is done by experimenting in the imagination. Notations are excellent things to experiment with; but still the experimentation requires intelligent supervision to come to much” (MS 94; quoted in Eisele 1979, p. 186).
I thank an anonymous reviewer for bringing this point to my attention.
Recall Peirce’s aforementioned view that “the meaning of a mathematical term or sign is its expression in [that] kind of signs [which mathematical reasoning manipulates]. For geometry, this [expression] is [in] a geometrical diagram” (NEM 2, p. 251).
As my colleague Jonathan Adler has suggested to me, this would also foster valuable training in imagining and proposing alternatives—different routes of reasoning, subtly different theses, and so on. This would address the problem of students’ argumentative skills being severely limited because of a weak ability to generate alternatives.
It must also be observed that practical applications are not necessarily reducible to empirical applications. For Peirce there are applications of mathematical concepts within mathematics itself (see, for instance, CP 1.82; 1896). But a full consideration of this point would take us far afield, so I will limit my discussion here to the relation between theory and practice understood as empirical application.
References
Anderson, D. (2005). Peirce and the art of reasoning. Studies in Philosophy and Education, 24, 277–289.
Campos, D. G. (2007). Peirce on the role of Poietic creation in mathematical reasoning. Transactions of the Charles S. Peirce Society, 43(3), 470–489.
Campos, D. G. (2009). Imagination, concentration, generalization: Peirce on the reasoning abilities of the mathematician. Transactions of the Charles S. Peirce Society, 45(2), 135–156.
Cellucci, C. (2002). Filosofia e matematica. Bari, Italy: Laterza.
Chiasson, P. (2005). Peirce’s design for thinking: An embedded philosophy of education. Educational Philosophy and Theory, 37(2), 207–226.
Chrystal, G. (1883). Mathematics. Encyclopaedia Britannica (9th ed.). Edinburgh: Adam & Charles Black.
De Waal, C. (2005). Why metaphysics needs logic and mathematics doesn’t: Mathematics, logic, and metaphysics in Peirce’s classification of the sciences. Transactions of the Charles S. Peirce Society, 41(2), 283–297.
Eisele, C. (1979). Peirce’s philosophy of mathematical education. In R. M. Martin (Ed.), Studies in the scientific and mathematical philosophy of Charles S. Peirce: Essays by Carolyn Eisele (pp. 177–200). The Hague: Mouton.
Euclid. (1956). The thirteen books of Euclid’s elements. Thomas Heath (Ed.). New York: Dover.
Hausman, C. (1993). Charles S. Peirce’s evolutionary philosophy. Cambridge: Cambridge University Press.
Hintikka, J. (1980). C. S. Peirce’s ‘First Real Discovery’ and its contemporary relevance. The Monist, 63, 304–315.
Hoffmann, M. (2003). Peirce’s ‘Diagrammatic Reasoning’ as a solution of the learning paradox. In G. Debrock (Ed.), Process pragmatism: Essays on a quiet philosophical revolution (pp. 123–143). Amsterdam: Rodopi.
Hoffmann, M. (2004). Diagrammatic reasoning as a tool of knowledge development and its pragmatic dimension. Foundations of Science, 9(3), 285–305.
Hookway, C. (1985). Peirce. London: Routledge.
Kent, B. (1997). The Interconnectedness of Peirce’s diagrammatic thought. In N. Houser, D. Roberts, & J. Van Evra (Eds.), Studies in the logic of Charles Sanders Peirce (pp. 445–459). Indianapolis: Indiana University Press.
Ketner, K. (1985). How Hintikka misunderstood Peirce’s account of theorematic reasoning. Transactions of the Charles S. Peirce Society, 21, 407–418.
Levy, S. H. (1997). Peirce’s Theorematic/Corollarial distinction and the interconnections between mathematics and logic. In N. Houser, D. Roberts, & J. Van Evra (Eds.), Studies in the logic of Charles Sanders Peirce (pp. 85–110). Indianapolis: Indiana University Press.
Peirce, B. (1870). Linear associative algebra. Washington City. Reprinted in American Journal of Mathematics, 4, 97–229 (1881).
Peirce, C. S. (1932–58). Collected papers of Charles Sanders Peirce. Vol 1–8. In P. Weiss, C. Hartshorne, & A. W. Burk (Eds.). Cambridge, Massachusetts: Harvard University Press. (Abbreviated CP.).
Peirce, C. S. (1976). The new elements of mathematics. Vol 1–4. In C. Eisele (Ed.). The Haghe, Netherlands: Mouton Publishers. (Abbreviated NEM.).
Peirce, C. S. (1982–). Writings of Charles S. Peirce: A chronological edition. Vol 1–6. Bloomington: Indiana University Press. (Abbreviated W.).
Peirce, C. S. (1992–98). The essential Peirce: Selected philosophical writings. Vol 1–2. In N. Houser et al. (Eds.). Bloomington: Indiana University Press. (Abbreviated EP).
Peirce, C. S. (1992b). Reasoning and the logic of things. In K. Ketner (Ed.). Cambridge, Massachusetts: Harvard University Press. (Abbreviated RLT.).
Robin, R. S. (1967). The annotated catalogue of the papers of Charles S. Peirce, Amherst: University of Massachusetts Press.
Rodrigues, C. (2007). Matemática como Ciência mais Geral: Forma da Experiência e Categorias. Cognitio-Estudos: Revista Eletrónica de Filosofia, 4(1), 37–59. <http://www.pucsp.br/pos/filosofia/Pragmatismo/cognitio_estudos/cog_estudos_v4n1/cog_est_v4n1_rodrigues_cassiano_terra.pdf> [Consulted: 07/07/2009.].
Steiner, M. (2003). Teaching elementary arithmetic through applications. In R. Curren (Ed.), A companion to the philosophy of education (pp. 354–364). Oxford: Blackwell.
Stjernfelt, F. (2000). Diagrams as centerpiece of a Peircean epistemology. Transactions of the Charles S. Peirce Society, 36(3), 357–384.
Wittgentstein, L. (1978). Remarks on the foundations of mathematics. In G. H. Von Wright, R. Rhees, & G. E. M. Anscombe (Eds.). Cambridge, MA: MIT Press.
Wu, H. (2009). What’s sophisticated about elementary mathematics? American Educator, 33(3), 4–14.
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Campos, D.G. Peirce’s Philosophy of Mathematical Education: Fostering Reasoning Abilities for Mathematical Inquiry. Stud Philos Educ 29, 421–439 (2010). https://doi.org/10.1007/s11217-010-9188-5
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DOI: https://doi.org/10.1007/s11217-010-9188-5