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German Philosophy of Mathematics from Gauss to Hilbert
Published online by Cambridge University Press: 08 January 2010
Extract
Suppose we were to ask some students of philosophy to imagine a typical book of classical German philosophy and describe its general style and character, how might they reply? I suspect that they would answer somewhat as follows. The book would be long and heavy, it would be written in a complicated style which employed only very abstract terms, and it would be extremely difficult to understand. At all events a description of this kind does indeed fit many famous works of German philosophy. Let us take for example Hegel's Phenomenology of Spirit of 1807. The 1977 English translation published by Oxford has 591 pages, and as for style here is a typical passage:
Self-consciousness found the Thing to be like itself, and itself to be like a Thing; i.e., it is aware that it is in itself the objectively real world. It is no longer the immediate certainty of being all reality, but a certainty for which the immediate in general has the form of something superseded, so that the objectivity of the immediate still has only the value of something superficial, its inner being and essence being self-consciousness itself. The object, to which it is positively related, is therefore a self-consciousness. It is in the form of thinghood, i.e. it is independent; but it is certain that this independent object is for it not something alien, and thus it knows that it is in principle recognized by the object. It is Spirit which, in the duplication of its self-consciousness and in the independence of both, has the certainty of its unity with itself. This certainty has now to be raised to the level of truth; what holds good for it in principle, and in its inner certainty, has to enter into its consciousness and become explicit fork. (p. 211)
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References
Benacerraf, P. and Putnam, H. eds. Philosophy of Mathematics. Selected Readings, 2nd edn (Cambridge University Press, 1983).Google Scholar
Breger, H., ‘A Restoration that failed: Paul Finsler's Theory of Sets’, in Revolutions in Mathematics, ed. Gillies, D. (Oxford University Press, 1992), pp. 249–64.CrossRefGoogle Scholar
Chihara, C. S., Constructibility and Mathematical Existence (Oxford: Clarendon Press, 1990).Google Scholar
Dauben, J. W., Georg Cantor. His Mathematics and Philosophy of the Infinite (Cambridge MA: Harvard University Press, 1979).Google Scholar
Dauben, J. W., ‘Revolutions revisited’, in Revolutions in Mathematics, ed. Gillies, D. (Oxford University Press, 1992), pp. 72–82.CrossRefGoogle Scholar
Dedekind, R., Continuity and Irrational Numbers (1872). English translation in Essays on the Theory of Numbers (Dover, 1963).Google Scholar
Dedekind, R., Was sind und was sollen die Zahlen? (1888). English translation in Essays on the Theory of Numbers (Dover, 1963).Google Scholar
Frege, G., Begriffsschrift, Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879). English translation in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, MA: Harvard University Press, 1967), pp. 1–82.Google Scholar
Frege, G., The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Numbers (1884). English translation by J. L. Austin (Blackwell, 1968).Google Scholar
Gillies, D. A., Frege, Dedekind, and Peano on the Foundations of Arithmetic (Van Gorcum, 1982).Google Scholar
Gillies, D. A., ‘Review of Chihara’, British Journal for the Philosophy of Science, 43, (1990), pp 263–78.CrossRefGoogle Scholar
Hegel, G. W. F., Science of Logic, Vols I and II, (1812–1816), (Allen and Unwin, 1929).Google Scholar
Hilbert, D., Foundations of Geometry (1899). English translation of the 10th edn (Open Court, 1971).Google Scholar
Hilbert, D., On the Foundations of Logic and Arithmetic (1904). English translation in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Harvard University Press, 1967), pp. 129–38.Google Scholar
Kennedy, H. C., ‘Karl Marx and the Foundations of Differential Calculus’, Historia Mathematica, 4 (3) (1977), pp. 303–18.CrossRefGoogle Scholar
Kline, M., Mathematical Thought from Ancient to Modern Times (Oxford University Press, 1972).Google Scholar
Lakatos, I., (1966), Cauchy and the Continuum, in Worrall, J. and Currie, G.(eds), Imre Lakatos. Philosophical Papers. Vol. II (Cambridge University Press, 1978), pp. 43–60.Google Scholar
Marx, K., The Mathematical Manuscripts (c. 1881). English translation (New Park Publications, 1983).Google Scholar
Moore, G. H. and Garciadiego, A., ‘Burali-Forti's Paradox: A Reappraisal of Its Origins’, Historia Mathematica, 8(3) (1981), pp. 319–50.CrossRefGoogle Scholar
Peckhaus, V., ‘The Way of Logic into Mathematics’, Theoria-Segunda Época, 12(1), pp. 39–64.Google Scholar
Riemann, B., ‘On the Hypotheses which lie at the Foundations of Geometry’ (1854). English translation in D. E. Smith ed., A Source Book in Mathematics, Vol. II (Dover, 1959), pp. 411–25.Google Scholar
Russell, B., An Essay on the Foundations of Geometry (1897) (Routledge, Reprint, 1996).Google Scholar
Zermelo, E., ‘Investigations in the Foundations of Set Theory I’, (1908). English translation in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Harvard University Press, 1967), pp. 199–215.Google Scholar
Zheng, Y., The Revolution in Philosophy of Mathematics (Forthcoming).Google Scholar