Abstract
The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of geometry. I will try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein.
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REFERENCES
Agoston, M.: 1976, Algebraic Topology, a First Course, Marcel Dekker Inc., New York.
Beltrami, E.: 1868, 'Saggio di interpetrazione della geometria Non-Euclidea', G. Mat. 6, 248–312.
Bronowski, J.: 1973, The Ascent of Man, BBC, London
Cayley, A.: 1859, 'A Sixth Memoir Upon Quantics', Phil. Trans. 149, 61–90.
Courant, R. and H. Robbins.: 1941, What is mathematics? Oxford University Press, New York, 1978.
Euclid: 1956, The Elements, Dover, New York.
Grattan-Guiness, I. ed: 1992, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, London.
Gray, J: 1979, Ideas of Space Euclidean, Non-Euclidean, and Relativistic, Clarendon Press, Oxford.
Klein, F.: 1928, Vorlesungen über nicht-euklidische Geometrie, Springer Verlag, Berlin.
Klein, F.: 1872, Vergleichende Betrachtungen über neuere geometrische Forschungen (Das Erlanger Program), A. Deichert, Erlangen.
Kline, M.: 1972, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York.
Lakatos, I.: 1970, 'Falsification and the Methodology of Scientific Research Programmes', in: The methodology of scientific research programmes. Philosophical Papers of Imre Lakatos Volume I, Cambridge University Press, Cambridge, 1978, pp. 8–101.
Lakatos, I.: 1976, Proofs and Refutations, Cambridge University Press, Cambridge.
Lobachevskij, N. I.: 1829, 'O nachalach geometrii', in: Polnoje sobranie sochinenij, GITTL, Leningrad, 1946.
Poincaré, H.: 1895, 'Analysis situs', J. École Polytechniques, Cahier 1, pp. 1–121.
Poincaré, H.: 1899, 'Complément á 'l'Analysis situs'', Rendiconti Circolo mat. Palermo, 13, 285–343.
Poincaré, H.: 1900, 'Second complément á 'l'Analysis situs'', Proc. London Math. Soc, 32, 277–308.
Poincaré, H.: 1902, 'Sur certaines surfaces algébriques, troisiéme complément á 'l'Analysis situs'', Bull. Sos. math. France, 30, 49–70.
Poincaré, H.: 1902, La Science et l'Hypothése. Paris, Flammarion.
Riemann, B.: 1851, Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse, Göttingen.
Rozenfel'd: 1976, Istorija neevklidovoj geometrii, Nauka, Moskva.
Širokov, P. A.: 1955, Kratkij ocherk geometrii Lobachevskogo, Nauka, Moskva, 1983.
Torretti, R.: 1978, Philosophy of Geometry from Riemann to Poincaré, D. Reidel Publishing Company, Dordrecht.
Wittgenstein, L.: 1921, Tractatus Logico-philosophcus. Routlege and Kegan Paul, London 1974.
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Kvasz, L. History of Geometry and the Development of the Form of its Language. Synthese 116, 141–186 (1998). https://doi.org/10.1023/A:1005008423734
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DOI: https://doi.org/10.1023/A:1005008423734