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This study connects Carnapian semantics for theoretical terms with eliminative structuralism in mathematics. Applying modal semantics to Peano arithmetic reveals universal structuralism and offers a simpler epistemology for arithmetic statements.

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Area of Science:

  • Philosophy of Mathematics
  • Formal Semantics
  • Epistemology

Background:

  • Exploring connections between Carnapian semantics and eliminative structuralism.
  • Interpreting Peano arithmetic using modal semantics of theoretical terms.

Purpose of the Study:

  • To demonstrate how applying modal semantics to Peano arithmetic yields universal structuralism.
  • To compare this approach with Hellman's modal structuralism and its epistemological implications.

Main Methods:

  • Utilizing the modal semantics of theoretical terms from Andreas (2010).
  • Interpreting the language of Peano arithmetic within this semantic framework.
  • Comparing the resulting universal structuralism with existing modal structuralist accounts.

Main Results:

  • The application results in a formal semantics of universal structuralism for Peano arithmetic.
  • This approach suggests that arithmetic statements express claims about all admissible interpretations of Peano axioms.
  • The proposed framework offers a more accessible epistemology for arithmetic statements compared to Hellman's modal structuralism.

Conclusions:

  • Carnapian semantics can be fruitfully applied to establish universal structuralism in the philosophy of mathematics.
  • This provides a formal semantic basis for understanding mathematical statements as general claims about structures.
  • The developed epistemology simplifies the understanding of mathematical knowledge within structuralist frameworks.