This chapter considers the idea that we have certainty in our basic arithmetic knowledge. The claim that arithmetical knowledge enjoys certainty cannot be extended to a similar claim about number theory “as a whole.” It is thus necessary to distinguish between elementary number theory and other, more advanced, levels in the study of numbers: algebraic number theory, analytic number theory, and perhaps set-theoretic number theory. The chapter begins by arguing that the axioms of Peano Arithmetic are true of counting numbers and describing some elements found in counting practices. It then offers an account of basic arithmetic and its certainty before discussing a model theory of arithmetic and the logic of mathematics. Finally, it asks whether elementary arithmetic, built on top of the practice of counting, should be classical arithmetic or intuitionistic arithmetic.
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