Abstract
My purpose is to understand what “arithmetization of geometry” meant in the seventeenth century. I compare five proofs of the main proposition on geometrical proportion: two proofs in Euclid’s Elements (one for magnitudes, one for numbers), one proof in Antoine Arnauld’s New Elements of Geometry (1667) and two proofs in Bernard Lamy’s Elements of Geometry (2nd edn, 1695, 5th edn 1731). For each of these proofs, I examine the signs used both for magnitudes and for reasoning, using Peirce’s classification of signs. This examination clearly shows that in the seventeenth century geometry had undergone a process of arithmetization through the use of symbolization, and that the outcome of this process of arithmetization had a strong influence on proofs in mathematics.
The purpose of this paper is to understand the meaning of the expression “arithmetization of geometry” in the seventeenth century in the context of a new meaning for proof designed to enlighten and not just to convince. For this purpose, we compare five proofs of Thales’ proposition on geometrical proportion. Two of these proofs are from Euclid’s Elements (one for magnitudes, one for numbers), one in New elements of geometry (1667) by Antoine Arnauld and two in Elements of geometry (2nd edn. 1695 and 5th edn. 1731) by Bernard Lamy. In each case we use Peirce’s classification of signs to examine the use of magnitudes and their roles in reasoning.
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Notes
- 1.
Barbin, E., ‘The meanings of Mathematical Proof’, In Eves’ circles, The Mathematical Association of America, n°34, 1994, p. 44.
- 2.
Arnauld A., Nicole, P., La logique ou l’art de penser, Paris, PUF, 1965, p. 325.
- 3.
This classification is interesting to study mathematical writing and its understanding, but it is not often used. For instance, Fischbein (1993) uses psychological idea of mental images.
- 4.
Peirce, C. S., ‘New Elements’, 1904, The Essential Peirce, Selected Philosophical Writings. vol. 2, p. 307.
- 5.
Peirce, C. S. , ‘The Regenerated Logic’, 1896, Collected Papers of Charles Sanders Peirce, vol. 3, p. 434.
- 6.
Peirce, C. S., ‘A syllabus of certain Topics of Logic’, 1903, The Essential Peirce, Selected Philosophical Writings. Vol. 2, p. 292.
- 7.
Peirce, C. S., ‘Prolegomena for an Apology to Pragmatism’, 1906, The New Elements of Mathematics, vol. 4, pp. 315-316.
- 8.
Euclid, Elements, translated by Heath, vol. 2, second edition, Dover, pp. 216-217.
- 9.
Peirce, C. S., ‘On the algebra of logic : a contribution to the philosophy of notation’, 1885, quotation in Peirce, Écrits sur le signe, Éditions du Seuil, Paris, p.146.
- 10.
Euclid, Elements, translated by Heath, vol. 2, second edition, Dover, pp. 318-319.
- 11.
Peirce, C. S., ‘The art of reasoning’, 1895, quotation in Peirce, Écrits sur le signe, Éditions du Seuil, Paris, p. 151.
- 12.
Arnauld, A., Nouveaux éléments de géométrie, Savreux, Paris, 1667, p. 3.
- 13.
Arnauld, op. cit., p. 4.
- 14.
Arnauld, idem.
- 15.
Arnauld, op. cit ., p. 6.
- 16.
Arnauld, op. cit., p. 26.
- 17.
Peirce, C. S., ‘The short Logic’, 1893, quotation in Peirce, Écrits sur le signe, Éditions du Seuil, Paris, p. 153.
- 18.
Arnauld, op. cit., p. 32.
- 19.
Peirce, C. S., ‘On the algebra of logic: A contribution to the philosophy of notation, quotation in Peirce’, Écrits sur le signe, Éditions du Seuil, Paris, p. 146.
- 20.
Arnauld, op. cit., p. 39.
- 21.
Arnauld, op. cit., p. 40.
- 22.
Peirce, C. S., ‘Prolegomena for an Apology to Pragmatism’, 1906, The New Elements of Mathematics, vol. 4, p. 316.
- 23.
Arnauld, op. cit., p. 42.
- 24.
Peirce, op. cit., vol. 4, p. 531.
- 25.
Barbin, E., La révolution mathématique du XVIIe siècle, chapter VII, Ellipses, Paris.
- 26.
Lamy, B., Les éléments de géométrie ou de la mesure du corps, seconde edition, Pralard, Paris, p.124.
- 27.
idem.
- 28.
Peirce, C. S., ‘A syllabus of certain Topics of Logic’, 1902, The Essential Peirce. Selected Philosophical Writings, vol. 2, p. 273.
- 29.
Lamy, op. cit., pp. 124-125.
- 30.
Lamy, op. cit., pp. 128-129.
- 31.
Euclid, Elements, translation Heath, vol. 1, p. 379.
- 32.
Lamy, op. cit., p. 135.
- 33.
Lamy, op. cit., pp. 139-140.
- 34.
Lamy, op. cit., p. 140.
- 35.
Lamy, op. cit., p. 150.
- 36.
The French word « exposer » comes from the latin « exponere », it means to show or to exhibit. So, here Lamy introduces the word « exposant » to exhibit the ratio of two lines by an expression, which « shows » it.
- 37.
Lamy, Géométrie ou de la mesure de l’étendue, Nion, Paris, 1731, p. 153.
- 38.
Lamy, op. cit., p. 163.
- 39.
Barbin, E., “The historicity of the Notion of What is Obvious in Geometry’, in Using history to teach mathematics, Katz, V. ed., The Mathematical Association of America, Notes 51, 2000, pp. 89-98.
References
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Barbin, E. (2010). Evolving Geometric Proofs in the Seventeenth Century: From Icons to Symbols. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_16
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