From Plato to Weil and beyond: Genericity through the history of mathematics

Authors

  • Renato Reis Leme Universidade Estadual de Campinas (UNICAMP)
  • Giorgio Venturi Universidade Estadual de Campinas (UNICAMP)

DOI:

https://doi.org/10.11606/issn.2447-2158.i10p140-158

Keywords:

Genericity, History of Algebraic Geometry, Foundations of Mathematics, History of Mathematics

Abstract

At the end of the 19th century, genericity took an important step toward mathematical analysis, due to the developments promoted by the Italian school of algebraic geometry. However, its origins can be traced back to ancient mathematics in the work of prominent philosophers and mathematicians, such as Plato and Euclid. In this article, we will try to show how a key notion in the structuralist turn of algebraic geometry evolved from a vague linguistic phenomenon and became a precise and fruitful mathematical concept.

Downloads

Download data is not yet available.

References

BELL, Eric Temple. The development of mathematics. 1992. New York: McGraw-Hill, 1945. p. 169.

BORGA, Marco et al. Logic and foundations of mathematics in Peano's school. Modern Logic, v. 3, n. 1, p. 18-44, 1992. p. 23.

BRILL, Alexander; NOETHER, Max. Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie. Mathematische Annalen, v. 7, n. 2, p. 269-310, 1874.

CASTELNUOVO, Guido. Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica. Rendiconti del circolo Matematico di Palermo, v. 7, n. 1, p. 89-110, 1893.

DIEUDONNÉ, Jean. History Algebraic Geometry: An Outline of the History and Development of Algebraic Geometry. 1985. Chapman & Hall. p. 1.

DIEUDONNÉ, Jean. The historical development of algebraic geometry. The American Mathematical Monthly, v. 79, n. 8, 1972. p. 67, 827-866.

FANO, Gino. Sulle varietà algebriche dello spazio a quattro dimensioni con un gruppo conti-nuo integrabile di trasformazioni proiettive in sé, Atti del R. Istituto Veneto di Scienze, Lettere e Arti, s. 7, 7, 1896, pp. 1069-1103.

FERREIRÓS, José. The architecture of modern mathematics. Oxford University Press, 2006. p. 4.

GOMBRICH, Ernst Han. The story of art. London: Phaidon, 1995.

GUILBAUD, S. Bourbaki et la fondation des maths modernes. 2015. Available in: <https://lejournal.cnrs.fr/articles/bourbaki-et-la-fondation-des-maths-modernes>. Accessed in: Oct. 29, 2020.

HEATH, Thomas Little. A History of Greek Mathematics, Volume I, From Thales to Euclid (first published in 1921 by the Clarendon Press, Oxford), 1981. Dover. New York. p. 176.

HEATH, Thomas Little. The thirteen books of Euclid's Elements. Cambridge University Press, 1968.

KRANTZ, Steven G. An episodic history of mathematics: Mathematical culture through problem solving. 2010. Maa. 2010. p. 151

LEBRUYN, L. Mumford’s treasure map. 2008. Available in: <http://www.neverendingbooks.org/mumfords-treasure-map>. Accessed in: Oct. 29, 2020.

LUCIANO, Erika; ROERO, Clara Silvia. Corrado Segre and his disciples: the Construction of an Inter-national Identity for the Italian School of Algebraic Geometry. From Classical to Modern Algebraic Geome-try. p. 93-241. Birkhäuser, Cham, 2016. p. 135.

MUMFORD, David. The red book of varieties and schemes. Lecture notes in mathematics, v. 1358, p. 14-01, 1996.

PLATO. Meno and Other Dialogues. 2005. Oxford University Press, 2005. pp. 115-121

SEVERI, Francesco. Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’suoi punti tripli apparenti. Rendiconti del Circolo Matematico di Palermo, v. 15, n. 1, p. 33-51, 1901.

SEVERI, Francesco. Sul principio della conservazione del numero. Rendiconti del Circolo Matematico di Palermo (1884-1940), v. 33, n. 1, p. 313-327, 1912.

THULLEN, Peter. Determinazione della serie di equivalenza individuata dal gruppo dei punti doppi impropri d’una superficie dell’S 4. Rendiconti del Circolo Matematico di Palermo (1884-1940), v. 59, n. 1, p. 256-260, 1935.

TURING, Alan M. The reform of mathematical notation and phraseology. The collected works of AM Turing: Mathematical logic, pp. 211-222, 1944. p. 211.

VAN DER WAERDEN, Bartel L. The foundation of algebraic geometry from Severi to André Weil. Archive for History of Exact Sciences, p. 171-180, 1971. p. 171.

VERONESE, Giuseppe. Fondamenti di geometria a più dimensioni ea più specie di unità rettilinee esposti in forma elementare: lezioni per la scuola di magistero in matematica. 1891. Tipografía del Seminario. Torino, 1891.

WEIL, André. Foundations of algebraic geometry. American Mathematical Soc., 1946.

Downloads

Published

2020-12-30

Issue

No. 10 (2020): Dossier “Arts, History of Sciences and Techniques: Interactions”

Section

Artigos

License

Copyright (c) 2020 Renato Reis Leme, Giorgio Venturi

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Authors who publish in this journal agree to the following terms:

  1. Authors retain the copyright and grant the journal the right to first publication, with the work simultaneously licensed under the Creative Commons Attribution License in the "Attribution-NonCommercial 4.0 International" (CC BY-NC 4.0) modality that allows sharing of the work with acknowledgment of authorship and initial publication in this magazine.
  2. Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publishing in institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
  3. Authors are allowed and encouraged to publish and distribute their work online (eg in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes, as well as increase impact and citation of the published work (See The Effect of Open Access).
  4. Any doubts or complaints about copyright must be directed to the Editorial Board or qualify and express themselves in accordance with the guidelines of the Committee on Publications Ethics (COPE).

 

How to Cite

Leme, R. R., & Venturi, G. (2020). From Plato to Weil and beyond: Genericity through the history of mathematics. Khronos, 10, 140-158. https://doi.org/10.11606/issn.2447-2158.i10p140-158

Funding data