On a Geometric Foundation of Mathematics (Su una Fondazione Geometrica della Matematica)
DOI:
https://doi.org/10.23756/sp.v5i1.346Keywords:
Foundations of geometry, synthetic geometry, geometric foundation of numbersAbstract
Frege with Grundlagen der Arithmetik and Hilbert with Grundlagen der Geometrie are two outstanding figures that are attributed to a fundamental role in the arithmetization of mathematics. However, the latest writings of Frege, released posthumously, testify to his reflection on the nature of mathematics. In them Frege argues that mathematics is all about geometry and begins a theory that aims to define complex numbers geometrically. For this purpose he introduced a notion of identical relationships that tends to set up a geometric aspect ratio. In addition, Grundlagen der Geometrie can be given a radically different reading from that which emphasizes Hilbert's exclusive intention to found geometry on a purely formal axiomatic system. Several authors argue that by his work, and in particular through the arithmetic of the segments introduced in it, Hilbert wanted to emancipate the geometry from instruments outside her, such as numbers, finding them within a substantially synthetic geometry.Downloads
References
Anatriello G., Tortoriello F.S., Vincenzi G., On an assumption of geometric foundation of numbers. Int. J. Math. Edu. in Sci. and Tech. 2016; 47(3): 395-407.
Barwise, J, Etchemendy, J, Visual information and valid reasoning, In W. Zimmerman S. Cunningham (Eds.), Visualizingin teaching and learning mathematics (pp. 9-24).Washington,
DC: Mathematical Association of America, (1991).
Betti R., L’analisi logica dell’intuizione spaziale, tra apriorismo ed esperienza. In D. Hilbert, Fondamenti della geometria. Con i Supplementi di Paul Bernays. 2009 FrancoAngeli Milano
Boyer C.B., Descartes and the geometrization of algebra, Amer. Math. Monthly (1959)
Brown J.R., Philosophy of mathematics: A contemporary introduction to the world of proofs and pictures. New York: Routledge; 2008.
Corry L., The empiricist roots of Hilbert’s axiomatic approach. In V. F. Hendricks, et al. (Eds.) Proof theory. Dordrecht: Springer Netherlands; 2000. p.35-54.
Edwards L, Radford L, Arzarello F. (Eds.),Gestures and multimodality in the teaching and learning of mathematics. Special issue of Educational Studies in Mathematics. 2009; 70(2): 91-215.
Freudenthal, H. Zur Geschichte der Grundlagen der Geometrie. Nieuw Archief voor Wiskunde, 4(5), 105-142. http://math.unipa.it/~brig/sds/MATERIALI/MATEMATICA/ sitofondamenti/
Freudenthal%20nuovocarattereago2005.htm
Frege G., Scritti postumi (a cura di Eva Picardi), (trad Nachgelassene Schriften und Wissenschaftlicher Briefwechsel vol 1. Hamburg: Felix Meiner Verlag; 1969) Bibliopolis, Napoli, 1986.
Giaquinto M. Visual thinking in mathematics. Oxford: Oxford University Press; 2007.
Giovannini E.N. Bridging the gap between analytic and synthetic geometry: Hilbert's axiomatic approach. Synthese. 2016; (193): 31-70.
Hamami Y, Mumma J. Prolegomena to a cognitive investigation of Euclidean diagrammatic reasoning. J Log Lang Inf. 2013; 22(4): 421-448.
Hilbert D. Grundlagen der mathematik. Vol. 2. Berlin: Springer; 1943.
Kvasz L. Patterns of change: linguistic innovations in the development of classical mathematics. Basel: Birkhäuser, 2008.
Miller N. Euclid and his twentieth century rivals:diagrams in the logic of Euclidean geometry. Stanford: CSLI Publications; 2007.
Petri B., Schappacher, N. On arithmetization. In: The Shaping of Arithmetic after CF Gauss’s Disquisitiones Arithmeticae. Springer Berlin Heidelberg, 2007. p. 343-374.
Rivera F.D. Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues. Vol. 49. Springer Science Business Media, 2011
Rowe, D. The calm before the storm: Hilbert's early views on foundations. In V.F. Hendricks, et al. (Eds.). Proof theory. Dordrecht: Springer Netherlands; 2000. p.55-93.
Stillwell J. Ideal elements in Hilbert's Geometry. Perspectives on Science 2014; 22(1): 35-55.
Stillwell J. The four pillars of geometry. New York: Springer; 2005.
Vailati G. Sulla teoria delle proporzioni [On the theory of proportions]. In: Enriques E, editor. Questioni riguardanti le matematiche elementari - raccolte e coordinate da Federigo Enriques. Vol. I: Critica dei principii. Bologna: Zanichelli; 1912 p. 143-191.
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