Cultures of proving

Social Studies of Science

Volumn 29, Issue 6, 1999, Pages 867-888

  Livingston, Eric ^a  

^a UNIVERSITY OF NEW ENGLAND   (Australia)

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EID: 0033249496     PISSN: 03063127     EISSN: None     Source Type: Journal    
DOI: 10.1177/030631299029006003     Document Type: Article

References (33)

1. This paper is based on the study of mathematics over a number of years and on extended observational research involving a range of techniques - for example, the material on the form of mathematical argumentation was first occasioned, in part, by the analysis of a videotaped session of theorem proving. The reference to anthropology is used here to stress the examination of lived practice as well as to highlight the distinction between cultural explanations and matters of nature or objective fact; it does not imply a commitment to conventionally understood ethnographic description, nor to any explicit research methodology as a means of substantiating real worldly claims otherwise not in evidence. Instead, the discovered praxeological character of cultural objects recommends those methods adequate to their further articulation and specification, a claim in many ways distinctive to ethnomethodological studies. Hopefully, the examples used in this paper are of sufficient clarity to direct attention to the phenomena that they are intended to help explicate.
2. See, for example, Abraham Seidenberg, 'Did Euclid's Elements, Book I, Develop Geometry Axiomatically?', Archive for History of Exact Sciences, Vol. 14 (1975), 263-95.
3. For example, Ludwig Wittgenstein, Philosophical Investigations (Oxford: Blackwell, 1953) and Wittgenstein, Remarks on the Foundations of Mathematics (Oxford: Blackwell, 1956).
4. For example, Ludwig Wittgenstein, Philosophical Investigations (Oxford: Blackwell, 1953) and Wittgenstein, Remarks on the Foundations of Mathematics (Oxford: Blackwell, 1956).
5. For example, Imre Lakatos, 'Proofs and Refutations', The British Journal for the Philosophy of Science, Vol. 14 (1963/4), 1-25, 120-39, 221-43, 296-342; David Bloor, Knowledge and Social Imagery (London: Routledge & Kegan Paul, 1976), esp. 74-140; and Donald MacKenzie, 'Slaying the Kraken: The Sociohistory of a Mathematical Proof, Social Studies of Science, Vol. 29, No. 1 (February 1999), 7-60.
6. For example, Imre Lakatos, 'Proofs and Refutations', The British Journal for the Philosophy of Science, Vol. 14 (1963/4), 1-25, 120-39, 221-43, 296-342; David Bloor, Knowledge and Social Imagery (London: Routledge & Kegan Paul, 1976), esp. 74-140; and Donald MacKenzie, 'Slaying the Kraken: The Sociohistory of a Mathematical Proof, Social Studies of Science, Vol. 29, No. 1 (February 1999), 7-60.
7. For example, Imre Lakatos, 'Proofs and Refutations', The British Journal for the Philosophy of Science, Vol. 14 (1963/4), 1-25, 120-39, 221-43, 296-342; David Bloor, Knowledge and Social Imagery (London: Routledge & Kegan Paul, 1976), esp. 74-140; and Donald MacKenzie, 'Slaying the Kraken: The Sociohistory of a Mathematical Proof, Social Studies of Science, Vol. 29, No. 1 (February 1999), 7-60.
8. See Marcia Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas (Pacific Grove, CA: Brooks/Cole, 1991) and Arthur B. Powell and Marilyn Frankenstein (eds), Ethnomathematics: Challenging Eurocentrism in Mathematics Education (Albany: State University of New York, 1997).
9. See Marcia Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas (Pacific Grove, CA: Brooks/Cole, 1991) and Arthur B. Powell and Marilyn Frankenstein (eds), Ethnomathematics: Challenging Eurocentrism in Mathematics Education (Albany: State University of New York, 1997).
10. Other than my The Ethnomethodological Foundations of Mathematics, I am aware of only one article attempting such a descriptive analysis: Bartel L. van der Waerden, 'How the Proof of Baudet's Conjecture was Found', in Leonid Minsky (ed.), Studies in Pure Mathematics (London & New York: Academic Press, 1971), 251-60. This article is written from a perspective, and with aims, different from the present paper, yet much of van der Waerden's discussion is compatible with the material presented here - the materiality and concreteness of mathematical reasoning, the substantiation of various categorizations and of the general argument through subsequent work, the 'fiddling' with proof-specific detail, and the piecing together of partial arguments. I thank Alan Bundy for referring me to this article.
11. An instructive review of the theory of Gestalt perception is given in Aron Gurwitsch, The Field of Consciousness (Pittsburgh, PA: Duquesne University, 1964), 87-153. The theme of mutually elaborating detail - 'Gestalt contextures' - is described there, a theme subsequently developed by Garfinkel in undergraduate lectures at UCLA as part of the analysis of social objects. The analogy between Gestalt figures such as Figure 1 and 'Gestalt switches' involving incommensurable perspectives has wide currency in social studies of science. However, the central point in the text is different: in that Figure 1 can be seen in two incompatible ways, the material figure itself does not determine its perception. The analogy for proofs is that a written mathematical argument is not, in itself, a proof; as described more fully in the text below, the proof associated with the written argument transcends the details of its written description.
12. See Árpád Szabó, trans. A.M. Ungar, The Beginnings of Greek Mathematics (Dordrecht, Holland: Reidel; Budapest: Akadémiai Kiadó, 1978), 192-93; and Wilbur Richard Knorr, The Evolution of the Euclidean Elements (Dordrecht, Holland: Reidel, 1975, 131-69). For the proof in Figure 2, see Szabó's presentation on p. 193.
13. See Árpád Szabó, trans. A.M. Ungar, The Beginnings of Greek Mathematics (Dordrecht, Holland: Reidel; Budapest: Akadémiai Kiadó, 1978), 192-93; and Wilbur Richard Knorr, The Evolution of the Euclidean Elements (Dordrecht, Holland: Reidel, 1975, 131-69). For the proof in Figure 2, see Szabó's presentation on p. 193.
14. The word 'shows' is, in fact, a reference to the gestalt of reasoning that the visual argument may be found to describe and, thus, does not refer to the self-sufficient details of a particular, isolated line of the visual display. When the visual argument is entertained as a demonstration of the general proposition, the generality of the first line of the diagram must be found as it bears on that general proposition; the word 'shows' is used in this instance as a gloss for the work of finding that generality.
15. 1.
16. 1.
17. Given a right triangle with sides a, √ 3a, and 2a, Figure 5 shows that the angle opposite the leg of triangle of length √ 3a is the angle of an equilateral triangle and, hence, that the sum of three such angles is an angle of 180° or a straight angle.
18. Jean-Pierre Serre, A Course in Arithmetic (NewYork: Springer-Verlag, 1973).
19. A deeper issue involving the self-referential, reflexive or incarnate character of representations of mathematical objects is involved: 'representations' of mathematical objects are used by provers to show forth properties of the objects so represented; through the exhibition of those properties, such 'representations' are seen to be adequate representations of those objects or, more precisely, representations adequate to the practices of proving properties of such objects. The development of this theme goes beyond the aim of this paper, which is to indicate the cultural character of the details of provers' work.
20. That a group treats a way of reasoning as universal reason - that reasoning of anyone and everyone is reasoning like that - does not make that reasoning reasonable, but indicates that, among that group, how they go about reasoning in that way does not seem to be questioned.
21. Once again, a claim is not being made that the demonstration does, in fact, concern Platonic entities 'really', but that the demonstration is treated and produced, in practical circumstances of proving, as a demonstration about such Platonic entities. Although the achievement of the demonstration - the witnessed proof - sustains the properties of mathematical objects as the properties of a transcendental domain of objects (that is, as objects about which such demonstrations can be made), the preservation of the transcendental appearance of mathematical objects is deeply embedded in the ways that provers go about their work. The last section of this paper, 'A Phenomenology of Mathematical Discovery', gives an indication of how this might be so.
22. Henry G. Liddell and Robert Scott, A Greek-English Lexicon, 9th edn revised and augmented by Henry Stuart (Oxford: Clarendon Press, 1940). Szabó discusses the early use of the word in The Beginnings of Greek Mathematics, op. cit. note 8, 185-96; I am indebted to Peter Toohey for his help with the Greek.
23. Henry G. Liddell and Robert Scott, A Greek-English Lexicon, 9th edn revised and augmented by Henry Stuart (Oxford: Clarendon Press, 1940). Szabó discusses the early use of the word in The Beginnings of Greek Mathematics, op. cit. note 8, 185-96; I am indebted to Peter Toohey for his help with the Greek.
24. Knorr, op. cit. note 8, 155. In Figures 7 and 8 below, the use of line segments to join groups of dots as a device of proving is from Knorr; the reader might wish to compare his proofs of the theorems with those given here.
25. See Leonard Eugene Dickson, History of the Theory of Number, Vol. 2 (New York: Chelsea, 1966), 165.
26. This formulation developed in dissertation research under the direction of, and as part of collaborative studies with, Harold Garfinkel. See Eric Livingston, The Ethnomethodological Foundations of Mathematics (London: Routledge & Kegan Paul, 1986) and Livingston, Making Sense of Ethnomethodology (London: Routledge & Kegan Paul, 1987).
27. This formulation developed in dissertation research under the direction of, and as part of collaborative studies with, Harold Garfinkel. See Eric Livingston, The Ethnomethodological Foundations of Mathematics (London: Routledge & Kegan Paul, 1986) and Livingston, Making Sense of Ethnomethodology (London: Routledge & Kegan Paul, 1987).
28. See note 6.
29. Lakatos, op. cit. note 4.
30. Imre Lakatos, 'Falsification and the Methodology of Scientific Research Programmes', in Lakatos and Alan Musgrave (eds), Criticism and the Growth of Knowledge (Cambridge: Cambridge University, 1970), 91-196.
31. George Polya, How to Solve It (Princeton, NJ: Princeton University Press, 1945).
32. In the following discussion, the word 'writing' will be used not only for physically writing or drawing, but for concrete reasoning that could be rendered in writing or drawing as well. The reference is to the materiality - the concrete physicality - of cultures of proving.
33. Eves, op. cit. note 10, formulates this argument as an illustration of the general method of homology.