Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T10:29:33.003Z Has data issue: false hasContentIssue false
  • English
  • Français

Hegel’s ‘Bad Infinity’ as a Logical Problem

Published online by Cambridge University Press:  14 September 2016

Vojtěch Kolman
Vojtěch Kolman*
Affiliation:
Institute of Philosophy and Religious Studies, Charles University, Prague, Czech Republicvojtech.kolman@ff.cuni.cz
Get access

Abstract

The paper analyses the concept of ‘bad infinity’ in connection with Hegel’s critique of infinitesimal calculus and with the belittling of Hegel’s mathematical notions by the representatives of modern logic and the foundations of mathematics. The main line of argument draws on the observation that Hegel’s difference is only derivatively a mathematical one and is primarily of a broadly logico-epistemological nature. Because of this, the concept of bad infinity can be fruitfully utilized, by way of inversion, in an analysis of the conceptual shortcomings of the most prominent foundational attempts at dealing with infinite quanta, such as Cantor’s set theory and Hilbert’s axiomatism. As such, the paper is an attempt at reconstructing Hegel’s philosophy of mathematics and its role in his philosophical system and, more importantly, as a contribution to logic in the more general and radical sense of the word.

Type
Articles
Information
Hegel Bulletin , Volume 37 , Issue 2 , October 2016 , pp. 258 - 280
Copyright
© The Hegel Society of Great Britain 2016 

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.)

References

Berkeley, G. (1734), The Analyst or, a Discourse Addressed to an Infidel Mathematician. London: Tonson.Google Scholar
Bolzano, B. (2004), ‘Paradoxes of the Infinite’, in S. Russ (ed.), The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press.Google Scholar
Brandom, R. (1994), Making It Explicit, Reasoning, Representing, and Discursive Commitment . Cambridge MA: Harvard University Press.Google Scholar
Brouwer, L. E. J. (1907), Over de grondslagen der wiskunde. Amsterdam: Universiteit van Amsterdam.Google Scholar
Cantor, G. (1932), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo. Berlin: Springer.Google Scholar
Feferman, S. (1998), In the Light of Logic. Oxford: Oxford University Press.Google Scholar
Hegel, G. W. F. (1874), The Logic of Hegel, transl. from The Encyclopaedia of the Philosophical Sciences, trans. W. Wallace. Oxford: Clarendon Press.Google Scholar
Hegel, G. W. F. (1986a), Enzyklopädie der philosophischen Wissenschaften I. Frankfurt: Suhrkamp.Google Scholar
Hegel, G. W. F. (1986b), Jenaer Schriften 1801–1807. Frankfurt: Suhrkamp.Google Scholar
Hegel, G. W. F. (1986c), Wissenschaft der Logik I. Frankfurt: Suhrkamp.Google Scholar
Hegel, G. W. F. (2010), The Science of Logic, ed. G. di Giovanni. Cambridge: Cambridge University Press.Google Scholar
Kripke, S. (1982), Wittgenstein on Rules and Private Language. An Elementary Exposition. Cambridge MA: Harvard University Press.Google Scholar
Lacroix, A. (2000), ‘The Mathematical Infinite in Hegel’, Philosophical Forum 31:3–4: 298–327.Google Scholar
Lorenzen, P. (1962), Metamathematik. Mannheim: Bibliographisches Institut.Google Scholar
Pinkard, T. (1981), ‘Hegel’s Philosophy of Mathematics’, Philosophy and Phenomenological Research 41:4: 452–464.CrossRefGoogle Scholar
Plato (2005), ‘Letters’, in E. Hamilton and H. Cairns (eds.), The Collected Dialogues of Plato. Princeton: Princeton University Press.Google Scholar
Sellars, W. (1956), ‘Empiricism and the Philosophy of Mind’, in H. Feigl and M. Scriven (eds.), Minnesota Studies in the Philosophy of Science, vol. 1: The Foundations of Science and the Concepts of Psychology and Psychoanalysis. Minneapolis: University of Minnesota Press.Google Scholar
Stekeler-Weithofer, P. (1992), Hegels analytische Philosophie . Die Wissenschaft der Logik als kritische Theorie der Bedeutung. Paderborn: Schöningh.Google Scholar
Stekeler-Weithofer, P. (2005), Philosophie des Selbstbewußtseins . Hegels System als Formanalyse von Wissen und Autonomie. Frankfurt: Suhrkamp.Google Scholar
Stekeler-Weithofer, P. (2014), Hegels Phänomenologie des Geistes. Band 1: Gewissheit und Vernunft. Hamburg: Meiner.Google Scholar
Weyl, H. (1998), ‘The Current Epistemological Situation in Mathematics’, in P. Mancosu (ed.), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press.Google Scholar
Wittgenstein, L. (1954), Philosophical Investigations. Oxford: Blackwell.Google Scholar
Wittgenstein, L. (1956), Remarks on the Foundations of Mathematics, ed. G. H. von Wright and R. Rhees. Oxford: Blackwell.Google Scholar
Wolf, M. (1986), ‘Hegel und Cauchy. Eine Untersuchung zur Philosophie und Geschichte der Mathematik’, in R.-P. Horstmann and M. J. Petry (eds.), Hegels Philosophie der Natur. Veröffentlichungen der Internationalen Hegel-Vereinigung, Bd. 15. Stuttgart: Klett-Cotta.Google Scholar
Žižek, S. (2012), Less than Nothing. Hegel and the Shadow of Dialectical Materialism . London: Verso.Google Scholar