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Philosophy of MathematicsStructure and Ontology$

Stewart Shapiro

Print publication date: 2000

Print ISBN-13: 9780195139303

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0195139305.001.0001

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Philosophy of Mathematics
Oxford University Press


  • Numbers . . . are known only by their laws, the laws
  • of arithmetic, so that any constructs obeying those
  • laws—certain sets, for instance—are eligible . . .
  • explications of number. Sets in turn are known
  • only by their laws, the laws of set theory . . . arithmetic
  • is all there is to number . . . there is no saying
  • absolutely what the numbers are; there is only
  • arithmetic.

Quine [1969, 44–45]

  • If in the consideration of a simply infinite system
  • . . . set in order by a transformation . . . we entirely
  • neglect the special character of the elements; simply
  • retaining their distinguishability and taking
  • into account only the relations to one another in
  • which they are placed by the order‐setting trans
  • formation . . . , then are these elements called
  • natural numbers or ordinal numbers or simply
  • numbers.

Dedekind [1888, §73]