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Mathematics in Mind

  • Ole Ravn
  • Ole Skovsmose
Chapter
Part of the History of Mathematics Education book series (HME)

Abstract

This chapter presents how, among others, Hume and Kant locate mathematics in the human apparatus of cognition. According to Hume, there exists two kind of knowledge, namely knowledge concerning “facts” derived from sense perception, and knowledge of “quantities or numbers.” While knowledge of facts is strictly empirical, knowledge concerning quantities of numbers is analytical. And analytical knowledge can be characterised as only concerning conceptual relationships.

To Kant, mathematics tells something about the way in which we experience the world, and not about the world as such. Mathematics applies to nature, but this is not due to any resemblance with nature. Mathematics fits nature because it represents how we, human beings, necessarily must experience nature. There is no ontological mathematics-nature unity, but there is a unity between mathematics and categories for human understanding. Thus Kant provides a radical relocation of mathematics. It was not any longer to be found in some eternal world of ideas, nor in nature, but in configurations provided by the human mind. Finally, the chapter addresses mathematics entities like Cantor’s Set, Sierpinski’s Triangle, and the Peano’s curve, which all constitute challenges for the Kant’s interpretation of mathematics.

Kant eliminated the possibility of scepticism with respect to mathematics and also with respect to the most fundamental statements in physics, like Newton’s laws.

Keywords

A posteriori A priori Cantor’s set Incomprehensible mathematics Mathematics as associations Mathematics as forms of understanding Mathematics as mental relations Peano’s curve Sierpinski’s triangle 

References

  1. Barile, M., & Weisstein, E. W. (2017). Cantor set. From MathWorld: A Wolfram Web Resource. http://mathworld.wolfram.com/CantorSet.html
  2. Cantor, G. (1883). Über unendliche, lineare Punktmannigfaltigkeiten V. Mathematische Annalen, 21, 545–591.CrossRefGoogle Scholar
  3. Hume, D. (1970). Enquiries concerning the understanding and concerning the principles of morals. Oxford, UK: Clarendon Press.Google Scholar
  4. Kant, I. (1911). Prolegomena til enhver fremtidig metafysik der skal kunne fremtræde some videnskab. Copenhagen, Denmark: Pios Boghandel.Google Scholar
  5. Kant, I. (1973). Critique of pure reason. (Translated by Norman Kemp Smith). London, UK: The MacMillan Press.Google Scholar
  6. Kant, I. (1997). What is enlightenment? Retrieved from http://web.mnstate.edu/gracyk/courses/web%20publishing/KantOnElightenment.htm
  7. Mill, J. S. (1970). A system of logic. London, UK: Longman.Google Scholar
  8. Weisstein, E. W. (2017a). Peano curve. From MathWorld: A Wolfram Web Resource. http://mathworld.wolfram.com/PeanoCurve.html
  9. Weisstein, E. W. (2017b). Sierpiński sieve. From MathWorld: A Wolfram Web Resource. http://mathworld.wolfram.com/SierpinskiSieve.html

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ole Ravn
    • 1
  • Ole Skovsmose
    • 1
    • 2
  1. 1.Department of Learning and PhilosophyAalborg UniversityAalborgDenmark
  2. 2.Department of Mathematics EducationState University of São Paulo, (Universidade Estadual Paulista, Unesp)São PauloBrazil

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