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One True Mathematics

  • Jane McDonnell
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Abstract

I have argued that the unreasonable effectiveness of mathematics in fundamental physics leads to a Pythagorean view of the world in which the underlying principles are structural, mathematical principles. This naturally raises one of the key questions which Wigner addressed in his investigation of the relationship between physics and mathematics: ‘What is mathematics?’ That is the question which I address in the next two chapters. I am particularly interested in exploring the implications of a Pythagorean view for one’s philosophy of mathematics. What should Pythagoreans believe about the nature of mathematics? What are the potential problems for this view? It seems clear that, for Pythagoreans, mathematics is necessarily true, eternal and unchanging. Pythagoreans should be realists about mathematics. They should regard the pursuit of mathematics as discovering facts about the world. They should look to mathematics as the ultimate arbiter of questions about existence and truth. It is the last point that I am mainly concerned with in this chapter: can mathematics serve as the ultimate arbiter of questions about existence and truth? More particularly, I investigate whether there is one background theory of mathematics in which any mathematical problem can be interpreted and (potentially) decided. If there is—as universalists believe—then the various disciplines of mathematics are part of one, true mathematical universe. If not—as held by pluralists—then there are alternative mathematical universes and mathematical statements may be true in some universes and false in others, not necessarily true. That would be a problem for Pythagoreanism.

Keywords

Mathematical Object Continuum Hypothesis Measurable Cardinal Incompleteness Theorem Inaccessible Cardinal 
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.

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Jane McDonnell
    • 1
  1. 1.Philosophy Department ClaytonVictoriaAustralia

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