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Part of the book series: Synthese Library ((SYLI,volume 306))

Abstract

An extension of ℝ (and *ℝ) with nilpotent infinitesimals (e.g. h ≠ 0 but h 2 = 0) is presented in order to obtain results similar to KockLawvere’s synthetic differential geometry [3], but in a classical and not intuitionistic context. The same extension can be used to add new infinitesimal points to spaces similar to Chen’s ones [1]. In the category of extended spaces we can develop differential geometry not only of usual manifolds but also of infinite dimensional spaces, without coordinates and with a strong geometric intuition, that is in a way that we will call “synthetic”.

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References

  1. K.T. Chen, On differentiable spaces, Categories in continuum physics, Lecture Notes in Mathematics 1174, Springer-Verlag (1982), 38–42.

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  2. A. Frölicher, A. Kriegl, Linear spaces and differentation theory, John Wiley & sons (1988).

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  3. A. Kock, Synthetic Differential Geometry, London Math. Soc. Lect. Note Series 51, Cambrige Univ. Press (1981).

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  4. R. Lavendhomme, Lecons de géométrie différentielle synthétique naïve, Ciaco (1987).

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  5. F.W. Lawvere, Categorical Dynamics, Topos theoretic Methods in Geometry, Aarhus, Various Publications Series no. 30 (1979), 1–28.

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© 2001 Springer Science+Business Media Dordrecht

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Giordano, P. (2001). Nilpotent Infinitesimals and Synthetic Differential Geometry in Classical Logic. In: Schuster, P., Berger, U., Osswald, H. (eds) Reuniting the Antipodes — Constructive and Nonstandard Views of the Continuum. Synthese Library, vol 306. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9757-9_7

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  • DOI: https://doi.org/10.1007/978-94-015-9757-9_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5885-0

  • Online ISBN: 978-94-015-9757-9

  • eBook Packages: Springer Book Archive

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