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The Foundations of Geometry by Peano’s School and Some Epistemological Considerations

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The Philosophers and Mathematics

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 43))

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Abstract

The aim of this paper is to individualize some contributions by Peano and his school (in particular Burali-Forti and Pieri) to the foundations of synthetic geometry. In particular, we propose: some remarks regarding Peano’s axiomatic foundation approach to geometry (1889) and some considerations on the fundamental role of geometrical calculus (Grassmann-Peano system) (1888).

This paper is a revision and an enlargement of the article: Freguglia, P., “Geometric calculus and Geometry Foundations”, in Giuseppe Peano between Mathematics and Logics (F. Skof ed.), Springer Verlag, Milano, Dordrecht, London, New York, 2011.

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Notes

  1. 1.

    See i.e. Henkin et al. (1959), Rosenfeld (1988)

  2. 2.

    See infra, for instance, Toretti (1978) and Magnani (2001)

  3. 3.

    See Veronese G. (1891) p. XVI

  4. 4.

    See Wylie C.R. (1964) p. 2.

  5. 5.

    We have taken into account Beth E. W and Tarski A. (1956), pp. 462–467), Robinson (1959) pp. 68–85, Royden (1959) pp. 86–96, Scott (1956) pp. 456–461, Tarski A. and Lindenbaum (1926) pp. 111–113.

  6. 6.

    That is, the geometrical concepts derive from immediate physical observations.

  7. 7.

    Typographically we find in the Peano’s text the symbol ε instead of the modern ∈.

  8. 8.

    See also Vailati (1892) pp. 71–75, (1895) pp. 75–78, 183–185.

  9. 9.

    See Palladino D. § 5 Appendix of Borga et al. (1985), pp. 244–250.

  10. 10.

    See Veronese G. (1891) Parte Prima, Libro I, Capitolo I, § 1, p. 209. See also Cantù P. (1999), infra.

  11. 11.

    Ibid, from p. 1.

  12. 12.

    See Peano G. “Recensione al volume Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare, Padova , 1891, pp. XLVIII-630”, Rivista di Matematica, 2 (1892), pp. 143–144. Actually in this review Peano blasts the Veronese’s treatise.

  13. 13.

    See Grassmann H. (1862) p. 415.

  14. 14.

    The Genocchi-Bellavitis letters are analyzed in Canepa, G., Freguglia, P., “Alcuni aspetti della corrispondenza Giusto Bellavitis-Angelo Genocchi”, Angelo Genocchi e i suoi interlocutori scientifici (edited by Conte, A. and Giacardi, L.), Deputazione subalpina di storia patria, Torino, 1991.

  15. 15.

    See Peano, G. (1888) pp. 110–111.

  16. 16.

    See Burali-Forti, C. (1926) pp. 5–6.

  17. 17.

    See Peano, G. (1888) p. 30.

  18. 18.

    Ibid. from p. 107.

  19. 19.

    See Bellavitis (1854) from p. 13.

  20. 20.

    See Peano G. (1888) p. 47

  21. 21.

    Ibid., p. 92.

  22. 22.

    Ibid. p. 95.

  23. 23.

    See Bottazzini (2000), infra.

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  1. Department of Information Engineering, Computer Science and Mathematics (DISIM), University of L’Aquila, L’Aquila, Italy

    Paolo Freguglia

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  1. Paolo Freguglia
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Correspondence to Paolo Freguglia .

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  1. Center for Philosophy of Science, University of Lisbon, Lisbon, Portugal

    Hassan Tahiri

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Freguglia, P. (2018). The Foundations of Geometry by Peano’s School and Some Epistemological Considerations. In: Tahiri, H. (eds) The Philosophers and Mathematics. Logic, Epistemology, and the Unity of Science, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-93733-5_5

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