Infinitesimals and Other Idealizing Completions in Neo-Kantian Philosophy of Mathematics
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics.
|Keywords||Infinitesimals Limits Philosophy of mathematics Neo-Kantianism Hermann Cohen Natorp Cassirer Cantor Robinson|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Jeremy Heis (2011). Ernst Cassirer's Neo-Kantian Philosophy of Geometry. British Journal for the History of Philosophy 19 (4):759 - 794.
Judy Deane Saltzman (1981). Paul Natorp's Philosophy of Religion Within the Marburg Neo-Kantian Tradition. Olms.
Thomas Mormann (2005). Mathematical Metaphors in Natorp’s Neo-Kantian Epistemology and Philosophy of Science. In Falk Seeger, Johannes Lenard & Michael H. G. Hoffmann (eds.), Activity and Sign. Grounding Mathematical Education. Springer.
Jeremy Heis (2010). “Critical Philosophy Begins at the Very Point Where Logistic Leaves Off”: Cassirer's Response to Frege and Russell. Perspectives on Science 18 (4):383-408.
Carlo Proietti (2008). Natural Numbers and Infinitesimals: A Discussion Between Benno Kerry and Georg Cantor. History and Philosophy of Logic 29 (4):343-359.
Scott Edgar (2010). Hermann Cohen. Stanford Encyclopedia of Philosophy.
Johann-Peter Regelmann (1979). Die Stellung der Biologie in den Neukantianischen Systemen Von Ernst Cassirer Und Nicolai Hartmann. Acta Biotheoretica 28 (3):217-233.
Thora Ilin Bayer (2010). Ernst Cassirer. Ausgewählter Wissenschaftlicher Briefwechsel (Review). Journal of the History of Philosophy 48 (3):403-404.
Carleton B. Christensen (1999). What Does (the Young) Heidegger Mean by the Seinsfrage? Inquiry 42 (3 & 4):411 – 437.
Thomas Mormann (2008). Idealization in Cassirer's Philosophy of Mathematics. Philosophia Mathematica 16 (2):151 - 181.
Thomas Mormann (2009). Completions, Constructions, and Corollaries. In H. Pulte, G. Hanna & H.-J. Jahnke (eds.), Explanation and Proof in Mathematics: Philosophical and Educational Perspectives. Springer.
Lydia Patton (2008). Review: Munk (Ed), Hermann Cohen's Critical Idealism and Poma, Yearning for Form and Other Essays on Hermann Cohen's Thought. [REVIEW] European Journal of Philosophy 16 (1):142–148.
Massimo Ferrari (2009). Is Cassirer a Neo-Kantian Methodologically Speaking? In Rudolf A. Makkreel & Sebastian Luft (eds.), Neo-Kantianism in Contemporary Philosophy. Indiana University Press.
Lydia Patton (2010). Review: Makkreel and Luft (Eds), Neo-Kantianism in Contemporary Philosophy. [REVIEW] Philosophy in Review 30 (4):280-282.
Added to index2011-11-13
Total downloads83 ( #20,449 of 1,692,507 )
Recent downloads (6 months)4 ( #57,596 of 1,692,507 )
How can I increase my downloads?