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2 results

  • 6 - Arithmetic

  • from Part III - Where Are the Paradoxes?
  • Zach Weber, University of Otago, New Zealand
  • Book:
    Paradoxes and Inconsistent Mathematics
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 189-211

  • Summary

    This chapter follows the axiomatic development of elementaryarithmetic. Starting from Peano’s postulates, somefundamental properties of natural numbers are derived, concerningaddition, multiplication, and order. If further assumptions aboutmathematical induction are made, then famous results about primenumbers can be established, including the fundamental theorem ofarithmetic (unique prime factorization) and Euclid’s theoremabout the infinitude of primes. Some simple facts about even and oddnumbers lead to the original limitive diagonal theorem: that thesquare root of 2 is irrational. Features of working withoutcontraction are highlighted.

  • 7 - Kripke on Logicism, Wittgenstein, and De Re Beliefs about Numbers

  • Edited by Alan Berger, Brandeis University, Massachusetts
  • Book:
    Saul Kripke
    Published online:
    05 June 2012
    Print publication:
    06 June 2011, pp 160-176

  • Summary

    This chapter discusses the evolution of Saul Kripke's thinking about Wittgenstein. Philosophy is a struggle for clarity, and both Wittgenstein and Kripke have the power to get to the absolute nub of a problem, taking nothing for granted, especially the orthodoxies of academic philosophy. More significantly, the concept of a "buckstopper" takes account of the Wittgensteinian intuition that mathematical notation must be perspicuous to be mathematical. It is internal to the concept of a calculation that the calculation should have the power to persuade us to accept the result. Kripke notes himself that his proposal implies that the identity of the numbers is culturally dependent. Even a culture that recognized decimal numbers, but did not use our system of positional notation for the numbers would be using different numbers, and not just different numerals.