Advertisement

Mathematics through Diagrams: Microscopes in Non-Standard and Smooth Analysis

  • Riccardo Dossena
  • Lorenzo Magnani
Part of the Studies in Computational Intelligence book series (SCI, volume 64)

Summary. Diagrams play an important role in the construction of mathematical concepts, mainly in (some) “limit” situations, like in the case of the mental representation of geometric tangent lines. They have many properties and can be viewed as particular epistemic mediators. Further, they are able to provide a better understanding of some mathematical concepts because they can be manipulated. In this paper we investigate how a particular kind of diagram (microscope) can serve to obtain two different and interesting visual representations of how a real function appears in small neighborhoods of its points.

Keywords

Mathematical Concept Tangent Line Intuitionistic Logic Nilpotent Element Standard Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Magnani, L.: Abduction, Reason, and Science. Processes of Discovery and Expla-nation. Kluwer Academic/Plenum Publishers, New York (2001)Google Scholar
  2. 2.
    Magnani, L.: Epistemic mediators and model-based discovery in science. In Magnani, L., Nersessian, N., eds.: Model-Based Reasoning: Science, Technol-ogy, Values, New York, Kluwer Academic/Plenum Publishers (2002) 305-329Google Scholar
  3. 3.
    Sullivan, K.A.: The teaching of elementary calculus using the non-standard approach. American Mathematical Monthly (1976) 370-375Google Scholar
  4. 4.
    Robinson, A.: Non-Standard Analysis. North Holland, Amsterdam (1966)zbMATHGoogle Scholar
  5. 5.
    Kock, A.: Synthetic Differential Geometry. LMS Lecture Notes Series 51. Cambridge University Press, Cambridge (1981)Google Scholar
  6. 6.
    Stroyan, K.D.: Uniform continuity and rates of growth of meromorphic func-tions. In Luxemburg, W.J., Robinson, A., eds.: Contributions to Non-Standard Analysis, Amsterdam, North-Holland (1972) 47-64CrossRefGoogle Scholar
  7. 7.
    Tall, D.: Elementary axioms and pictures for infinitesimal calculus. Bulletin of the IMA 18 (1982) 43-48zbMATHMathSciNetGoogle Scholar
  8. 8.
    Tall, D.: Natural and formal infinities. Educational Studies in Mathematics 48 (2001) 199-238CrossRefGoogle Scholar
  9. 9.
    Keisler, H.J.: Elementary Calculus. Prindle, Weber, and Schmidt, Boston (1976)zbMATHGoogle Scholar
  10. 10.
    Keisler, H.J.: Foundations of Infinitesimal Calculus. Prindle, Weber, and Schmidt, Boston (1976)zbMATHGoogle Scholar
  11. 11.
    Hurd, A.E., Loeb, P.A.: An Introduction to Nonstandard Real Analysis. Acad-emic Press, Orlando (1985)zbMATHGoogle Scholar
  12. 12.
    Goldblatt, R.: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Springer-Verlag, New York (1998)zbMATHGoogle Scholar
  13. 13.
    Stroyan, K.D., Luxemburg, W.A.J.: Introduction to the Theory of Infinitesimals. Academic Press, New York (1976)zbMATHGoogle Scholar
  14. 14.
    Bell, J.L.: A Primer of Infinitesimal Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  15. 15.
    Moerdijk, I., Reyes, G.E.: Models for Smooth Infinitesimal Analysis. Springer-Verlag, New York (1991)zbMATHGoogle Scholar
  16. 16.
    Browuer, L.E.J.: Intuitionism and formalism. In Benacerraf, P., Putnam, H., eds.: Philosophy of Mathematics. Selected Readings. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1964) 66-77Google Scholar
  17. 17.
    Dalen, D.V., ed.: Browuer’s Cambridge Lectures on Intuitionism. Cambridge University Press, Cambridge (1981)Google Scholar
  18. 18.
    Heyting, A.: Disputation. In Benacerraf, P., Putnam, H., eds.: Philosophy of Mathematics. Selected Readings. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1964) 55-65Google Scholar
  19. 19.
    Heyting, A.: The intuitionist foundations of mathematics. In Benacerraf, P., Putnam, H., eds.: Philosophy of Mathematics. Selected Readings. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1964) 42-54Google Scholar
  20. 20.
    Lavendhomme, R.: Basic Concepts of Synthetic Differential Geometry. Kluwer Academic Publisher, Dordrecht (1996)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Riccardo Dossena
    • 1
  • Lorenzo Magnani
    • 1
  1. 1.Department of PhilosophyUniversity of PaviaPaviaItaly

Personalised recommendations