Hostname: page-component-cb9f654ff-qc88w Total loading time: 0 Render date: 2025-08-27T04:36:19.280Z Has data issue: false hasContentIssue false
  • English
  • Français

‘A REMARKABLE ARTIFICE’: LAPLACE, POISSON AND MATHEMATICAL PURITY

Part of: General and miscellaneous specific topics Measure and integration Proof theory and constructive mathematics History of mathematics and mathematicians

Published online by Cambridge University Press:  24 July 2023

BRAM PEL
BRAM PEL*
Affiliation:
INDEPENDENT SCHOLAR
*
E-mail: brampel.pel@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In the early nineteenth century, a series of articles by Laplace and Poisson discussed the importance of ‘directness’ in mathematical methodology. In this thesis, we argue that their conception of a ‘direct’ proof is similar to the more widely contemplated notion of a ‘pure’ proof. More rigorous definitions of mathematical purity were proposed in recent publications by Arana and Detlefsen, as well as by Kahle and Pulcini: we compare Laplace and Poisson’s writings with these modern definitions of purity and show how the modern definitions fail to grasp some more nuanced aspects.

Keywords

mathematical puritytopical purityoperational puritypurity of methodLaplacePoissoncomplex substitution

MSC classification

Primary: 00A30: Philosophy of mathematics 01A50: 18th century 01A55: 19th century 03F20: Complexity of proofs 28-03: Historical (must also be assigned at least one classification number from Section 01)

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 4 , December 2024 , pp. 1018 - 1054
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

BIBLIOGRAPHY

Arana, A. (2014). Purity in arithmetic: Some formal and informal issues. In Link, G., editor. Formalism and Beyond: On the Nature of Mathematical Discourse. Boston–Berlin: De Gruyter, pp. 315–335.CrossRefGoogle Scholar
Arana, A. (2017). On the alleged simplicity of impure proof. In Kossak, R. and Ording, P., editors. Simplicity: Ideals of Practice in Mathematics and the Arts. Cham: Springer, pp. 205–226.CrossRefGoogle Scholar
Arana, A. (2023). Purity and explanation: Essentially linked? In Posy, C. and Ben-Menahem, Y., editors. Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Cham: Springer, Switzerland, pp. 25–39.CrossRefGoogle Scholar
Aristotle. (2007). Posterior Analytics. Available from: https://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=3314958 (accessed 28 June 2022).Google Scholar
Barany, M. J., Paumier, A.-S., & Lützen, J. (2017). From Nancy to Copenhagen to the world: The internationalization of Laurent Schwartz and his theory of distributions. Historia Mathematica, 44(4), 367–394.CrossRefGoogle Scholar
Bolzano, B. (1996). Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation. English translation of the 1817 German original in Ewald, W., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 Vols. Oxford: Oxford University Press.Google Scholar
Bottazzini, U., & Gray, J. J. (2013). Hidden Harmony: Geometric Fantasies. Cham: Springer.CrossRefGoogle Scholar
Bourbaki, N. (1950). The architecture of mathematics. The American Mathematical Monthly, 57(4), 221–232.CrossRefGoogle Scholar
Cauchy, A.-L. (2009). Oeuvres Complètes. Cambridge: Cambridge University Press.Google Scholar
Costabel, P. (2008). Poisson, Siméon-Denis. Complete Dictionary of Scientific Biography, 15, 480–490 Google Scholar
Dawson, J. W. Jr. (2015). Why Prove It Again? Alternative Proofs in Mathematical Practice. Boston: Birkhäuser CrossRefGoogle Scholar
Detlefsen, M. (2008). Purity as an ideal of proof. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 179–197.CrossRefGoogle Scholar
Detlefsen, M., & Arana, A. (2011). Purity of methods. Philosophers’ Imprint, 11, 1–20.Google Scholar
Ferreirós, J. (2016). Purity as a value in the German-speaking area. In Larvor, B., editor. Mathematical Cultures. Cham: Springer, pp. 215–234.CrossRefGoogle Scholar
Flament, D. (2014). Jean-Robert Argand (1768–1822) et la quantité imaginaire; Une ouverture par-delà la réalisation géométrique et l’imitation du réel. Paris: Images des Mathématiques.Google Scholar
Hallett, M. (2008). Reflections on the purity of method in hilbert’s grundlagen der geometrie. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford Academic, pp. 198–255.CrossRefGoogle Scholar
Hilbert, D. (2004). David Hilbert’s Lectures on the Foundations of Geometry 1891–1902. Heidelberg: Springer-Verlag. Edited by Michael Hallet and Ulrich Maljer.Google Scholar
Hipolito, I., & Kahle, R. (2019). Discussing Hilbert’s 24th problem. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 377(2140), 20180159.Google ScholarPubMed
Kahle, R., & Pulcini, G. (2018). Towards an operational view of purity. In Arazim, P. and Pávička, T., editors. The Logica Yearbook 2007. London: College Publications, pp. 125–138.Google Scholar
Lacroix, S. F. (1800). Traité Des différences et Des séries, Faisant Suite Au Traité du Calcul différentiel et du Calcul intégral. Quai des Augustins: Chez JBM Duprat, Libraire pour les Mathématiques.Google Scholar
Laplace, P.-S. (1785). Mémoire Sur les approximations des formules qui sont fonctions de tres grands nombres. Oeuvres Complètes, X, 209–291.Google Scholar
Laplace, P.-S. (1809). Mémoire Sur divers points d’analyse. Oeuvres Complètes, XIV, 178–214.Google Scholar
Laplace, P.-S. (1810). Mémoire Sur les approximations des formules qui sont fonctions de très grands nombres et Sur leur applications aux probabilités. Oeuvres Complètes, XII, 301–353.Google Scholar
Laplace, P.-S. (1811). Mémoire Sur les intégrales définies et leur application aux probabilités, et spécialementa la recherche du milieu qu’il faut choisir entre les résultats des observations. Memoires de l’Academie des Sciences de Paris, 11, 279–347.Google Scholar
Laplace, P.-S. (1811). Sur les intégrales définies. Bulletin de la Société philomathique de Paris, 43, 262–266.Google Scholar
Mancosu, P. (2008). Mathematical explanation: Why it matters. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 134–150.CrossRefGoogle Scholar
Mascheroni, L. (1790). Adnotationes Ad Calculum Integralem Euleri: In Quibus Nonnulla Problemata, Vol. 11. Ticini: Ex typographia Petri Galeatii.Google Scholar
Mercer, I. D. (2009). On Furstenberg’s proof of the infinitude of primes. The American Mathematical Monthly, 116(4), 355–356.CrossRefGoogle Scholar
Pincock, C. (2015). The unsolvability of the quintic: A case study in abstract mathematical explanation. Philosopher’s Imprint, 15(3), 1–19.Google Scholar
Poisson, S.-D. (1811). Sur les intégrales définies. Bulletin de la Société philomathique de Paris, 42, 243–252.Google Scholar
Steiner, M. (1978). Mathematics, explanation, and scientific knowledge. Noûs, 12, 17–28.CrossRefGoogle Scholar