Ordered Numerical Systems in Hilbert's "Grundlagen der Geometrie"
Abstract
Nello sviluppo della ricerca hanno avuto un ruolo determinante la scoperta di Hurwitz delle uniche quattro algebre di divisione normate e le idee maturate, già agli inizi della sua carriera, sulla finitezza degli strumenti da utilizzare in geometria, in particolare la rinuncia al concetto di continuità e la limitazione a sistemi numerici ordinati e numerabili. Nelle conclusioni dei Grundlagen, Hilbert rimarca l’importanza di dimostrare l’impossibilità delle assunzioni, tra cui l’impossibilità di determinare, attraverso solo relazioni geometriche di incidenza, dei sistemi numerici ordinati che abbiano le stesse proprietà dei quaternioni e degli ottonioni rispetto alle operazioni dell’aritmetica.
Keywords
Full Text:
PDFReferences
D. Albers e G.L. Alexanderson. Mathematical People: Profiles and Interviews. CRC Press, 2008.
G. Anatriello. On a geometric foundation of mathematics (Su una fonda-zione geometrica della matematica). Science & Philosophy, 5(1):91–108, 2017.
G. Anatriello, F. S. Tortoriello, e G. Vincenzi. On an assumption of geometric foundation of numbers. International Journal of Mathematical Education in Science and Technology, 47(3):395–407, 2016.
Anon. The foundations of geometry by D. Hilbert. The Mathematical Gazzette, 2(83):268–269, 1903.
Anon. Regeln für die deutsche Rechtschreibung nebst Wörterverzeichnis. Neue Bearbeitung. Weidmann Buchhandlung, Berlin, 1908.
J. Baez. The octonions. Bulletin of the American Mathematical Society, 39(2):145–205, 2002.
J. T. Baldwin. Model Theory and the Philosophy of Mathematical Practice. Cambridge University Press, 2018.
L. M. Blumenthal. A Modern View of Geometry. Dover Publications, 2017.
O. Blumenthal. Lebensgeschichte. In D. Hilbert, Gesammelte Abhandlungen III, Springer Verlag, 1935, pp. 388–429. Springer, 1935.
J. Boniface. Leopold kronecker’s conception of the foundations of mathema-tics. Philosophia Scientiæ. Travaux d’histoire et de philosophie des sciences, CS 5:143–156, 2005.
U. Bottazzini. Il flauto di Hilbert. Storia della matematica. UTET, 2005.
K. Conrad. Pfister’s Theorem on Sums of Squares Identity. 2009. Disponi-bile al sito www.math.uconn.edu/ kconrad/blurbs/linmultialg/pfister.pdf.
A. Conte. Sui “Fondamenti della Geometria” di David Hilbert. Notiziario dell’Unione Matematica Italiana, supplemento al n. 7, 1979.
J. H. Conway e D. A. Smith. On Quaternions and Octonions. CRC Press, 2003.
L. Corry. David Hilbert and the Axiomatization of Physics (1894-1905). Archive for History of Exact Sciences, 51(2):83–198, 1997.
L. Corry. Axiomatics, empiricism, and anschauung in hilbert’s conception of geometry: between arithmetic and general relativity. The architecture of modern mathematics: Essays in history and philosophy, pp. 133–156, 2006.
H. S. M. Coxeter. The Real Projective Plane. Springer New York, 1993.
M. L. Curtis e P. Place. Abstract Linear Algebra. Springer-Verlag, 1984.
H. D. Ebbinghaus et al. Numbers. Springer New York, 1991.
H. M. Edwards. Kronecker on the foundations of mathematics. In From Dedekind to Gödel. A cura di J. Hintikka, pp. 45–52. Springer, 1995.
H. M. Edwards. Essays in Constructive Mathematics. Springer New York, 2005.
Euclide. Tutte le opere. Bompiani, 2007.
H. G. Forder. Grundlagen der Geometry by D. Hilbert (7th ed.). The Mathematical Gazzette, 15(213):397–400, 1931.
G. Frege. Scritti postumi. Bibliopolis, 1986.
H. Freudenthal. Zur Geschichte der Grundlagen der Geometrie, zugleich eine Besprechung der 8. aufl. von Hilberts "Grundlagen der Geome-trie". Nieuw Arch. Wisk., (4):105–142, 1957. Trad. it. disponibile al sito http://math.unipa.it/ brig/sds/MATERIALI/MATEMATICA/sitofondamenti/ Freudenthal nuovocarattereago2005.htm.
H. Freudenthal. The main trends in the foundations of geometry in the 19th century. Studies in Logic and the Foundations of Mathematics, 44:613–621, 1966.
S. Gandon. Pasch entre Klein et Peano. Dialogue, 44(4):653–692, 2005.
E. N. Giovannini. Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach. Synthese, 193(1):31–70, 2016.
Paul Gordan. Beweis, dass jede covariante und invariante einer binären form eine ganze function mit numerischen coefficienten einer endlichen anza-hl solcher formen ist. Journal für die reine und angewandte Mathematik, 69:323–354, 1868.
T. H. Gronwall. David Hilbert, Grundlagen der Geometrie (4th ed.). Bulletin of the American Mathematical Society, 20(6):325–325, 1914.
M. Hallett e U. Majer. David Hilbert’s Lectures on the Foundations of Geometry 1891-1902. Springer Berlin Heidelberg, 2004.
G. B. Halsted. The Foundations of Geometry by D. Hilbert. Science, 16(399):307–308, 1902.
R. Hartshorne. Geometry: Euclid and Beyond. Springer, 2000.
Charles Hermite. Sur la fonction exponentielle. Compte Rendus, 77:18–24, 1873.
Gerhard Hessenberg. Beweis des desarguesschen satzes aus dem pascalschen. Mathematische Annalen, 61(2):161–172, 1905.
D. Hilbert. Über die invarianten Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Tesi di Dottorato di Ricerca, Universität Königsberg, 1885.
D. Hilbert. Über die Theorie der algebraischen Formen. Mathematische annalen, 36(4):473–534, 1890.
D. Hilbert. Projektive geometrie. Lectures notes for a course held during the year 1891. In David Hilbert’s Lectures on the Foundations of Geometry. A cura di M. Hallet e U. Majer, pp. 21–55. 1891.
D. Hilbert. Über die Transzendenz der Zahlen e und . Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, pp. 113–116, 1893.
D. Hilbert. Über die vollen Invariantensysteme. Mathematische Annalen, 42(3):313–373, 1893.
D. Hilbert. Grundlagen der Geometrie. Lectures notes for a course held during the year 1894 In David Hilbert’s Lectures on the Foundations of Geometry. A cura di M. Hallet e U. Majer, pp. 72–127. 1893.
D. Hilbert. Über die Zerlegung der Ideale eines Zahlkörpers in Primideale. Mathematische Annalen, 44:1–8, 1894.
D. Hilbert. Die Theorie der algebraischen Zahlkörper. Deutsche Mathematiker-Vereinigung, 1897.
D. Hilbert. The Theory of Algebraic Number Fields. Springer Berlin Heidelberg, [1897]-1998.
D. Hilbert. Grundlagen der Euklidischen Geometrie. Lectures notes for a course held during the year 1898. In David Hilbert’s Lectures on the Foundations of Geometry. A cura di M. Hallet e U. Majer, pp. 221–286. 1898.
D. Hilbert. Elemente der euklidischen geometrie. Lecture notes for a course held during the wintersemester 1898/99. In David Hilbert’s Lectures on the Foundations of Geometry. A cura di M. Hallett e U. Majer. 1899.
D. Hilbert. Grundlagen der Geometrie: Festschrift zur Feier der Enthül-lung des Gauss-Weber-Denkmals in Göttingen. Herausgegeben von dem Fest-Comitee. I. Theil. Teubner, 1899.
D. Hilbert. Les principes fondamentaux de la géométrie. Gauthier-Villars Paris, 1900.
D. Hilbert. Grundlagen der Geometrie. Zweite Auflage. Teubner, 1903.
D. Hilbert. Adolf hurwitz. Göttinger Nachrichten. Geschäftliche Mitteilungen, pp. 75–83, 1920.
D. Hilbert e S. Cohn-Vossen. Geometria intuitiva. P. Boringhieri, [1932]-1960.
D. Hilbert e F. Klein. Der Briefwechsel David Hilbert-Felix Klein: 1886-1918. Vandenhoeck und Ruprecht, 1985.
A. Hurwitz. Über die Zahlentheorie der Quaternionen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 313–340, 1896.
A. Hurwitz. Über die Komposition der Quadratischen Formen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 309–316, 1898.
I. Kleiner. From Numbers to Rings: The Early History of Ring Theory. Elemente der Mathematik, 53(1):18–35, 1998.
M. Koecher e R. Remmert. Real division algebra. In Numbers. A cura di H. D. Ebbinghaus et al. Springer Verlag, 1991.
F. Lindemann. Uber die Zahl . Mathematische Annalen, 20:213–225, 1882.
P. Mancosu. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, 1998.
C. Mangione e S. Bozzi. Storia della logica. CUEM, 2004.
C. McLarty. Hilbert on theology and its discontents. In Circles Distur-bed: the Interplay of Mathematics and Narrative, pp. 105–129. Princeton University Press, 2012.
H. Minkowski. Geometrie der Zahlen. Teubner, [1896]-1910.
F. R. Moulton. A simple non-desarguesian plane geometry. Transactions of the American Mathematical Society, 3(2):192–195, 1902.
P. Nabonnand. La théorie des Würfe de von Staudt. Une irruption de l’algèbre dans la géométrie pure. Archive for History of Exact Sciences, 62(3):201–242, 2008.
M. Noether. Paul Gordan. Mathematische Annalen, 75(1):1–41, 1914.
P. Odifreddi. Divertimento geometrico: le origini geometriche della logica da Euclide a Hilbert. Bollati Boringhieri, 2003.
M. Pasch. Vorlesungen Uber Neuere Geometrie. Teubner, 1882.
H. Poincaré. Les fondements de la géométrie. Journal de savants, pp. 252– 271, maggio 1902.
H. Poincaré. Les fondements de la géométrie. Bulletin des Sciences Mathématiques, pp. 249–272, settembre 1902.
C. Reid. Hilbert. Springer New York, 1996.
B. Rosenfeld. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer New York, 1988.
B. Rosenfeld e B. Wiebe. Geometry of Lie Groups. Springer US, 1997.
D. E. Rowe. The philosophical views of Klein and Hilbert. In The intersec-tion of history and mathematics. A cura di S. Chikara et al., pp. 187–202. Springer, 1994.
D. E. Rowe. The calm before the storm: Hilbert’s early views on foun-dations. In Proof theory. A cura di V. F. Hendricks, pp. 55–93. Springer, 2000.
D. E. Rowe. Hilbert’s early career: Encounters with allies and rivals. The Mathematical Intelligencer, 27(1):72–82, 2005
C. Segre. CGCV Staudt ed i suoi lavori In von Staudt, Geometria di posizione. A cura di M. Pieri. Bocca Editori, 1888.
C. Segre. Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici. Mathematische Annalen, 40(3):413–467, 1892.
F. Siacci. Degli invarianti e covarianti delle forme binarie ed in particolare di quelle di 3° e 4° grado. Annali di matematica pura ed applicata, tomo VII:73–140, 1865.
I. Smadja. Local axioms in disguise: Hilbert on Minkowski diagrams. Synthese, 186(1):315–370, 2012.
J. Sommer. Hilbert’s foundations of geometry. Bulletin of the AMS, 6(7):287–299, 1900.
J. Stillwell. The Four Pillars of Geometry. Springer New York, 2005.
J. Stillwell. Mathematics and Its History. Springer New York, 2010.
J. Stillwell. Ideal elements in Hilbert’s Geometry. Perspectives on Science, 22(1):35–55, 2014.
M. Toepell. On the origins of David Hilbert’s “Grundlagen der Geometrie”. Archive for History of Exact Sciences, 35(4):329–344, 1986.
M. Toepell. The origins and the further development of Hilbert’s "Grundlagen der Geometrie". Le Matematiche, 55(3):207–226, 2000.
R. Torretti. Philosophy of Geometry from Riemann to Poincaré. Springer Netherlands, 1978.
O. Veblen. A system of axioms for geometry. Transactions of the AMS, 5(3):343–384, 1904.
G. Venturi. A note on the introduction of Hilbert’s Grundlagen der Geometrie. Manuscrito, 40(2):5–17, 2017.
G. von Staudt. Geometrie der Lage. Verlag, 1847.
G. von Staudt. Beitrag zur Geometrie der Lage, volume I(1856), II(1857), III(1860). Verlag, 1856-1860.
B. L. Waerden. A history of algebra: from al-Khwarizm¯¯ı to Emmy Noether. Springer-Verlag, 1985.
C. Weibel. Survey of non-desarguesian planes. Notices of the AMS, 54(10):1294–1303, 2007.
H. Weyl. David Hilbert and his mathematical work. Bullettin of the AMS, 50(9):612–654, 1944.
DOI: http://dx.doi.org/10.23756/sp.v6i2.419
Refbacks
- There are currently no refbacks.
Copyright (c) 2018 Andrea Battocchio
This work is licensed under a Creative Commons Attribution 4.0 International License.
Science & Philosophy - Journal of Epistemology, Science and Philosophy. ISSN 2282-7757; eISSN 2282-7765.