# Semantic Arithmetic: A Preface

*Agora*14 (1):149-156 (1995)

**Abstract**

SEMANTIC ARITHMETIC: A PREFACE
John Corcoran
Abstract
Number theory, or pure arithmetic, concerns the natural numbers themselves,
not the notation used, and in particular not the numerals. String
theory, or pure syntax, concems the numerals as strings of «uninterpreted»
characters without regard to the numbe~s they may be used to denote.
Number theory is purely arithmetic; string theory is purely syntactical... in
so far as the universe of discourse alone is considered. Semantic arithmetic
is a broad subject which begins when numerals are mentioned (not just
used) and mentioned as names of numbers (not just as syntactic objects).
Semantic arithmetic leads to many fascinating and surprising algorithms
and decision procedures; it reveals in a vivid way the experiential import of
mathematical propositions and the predictive power of mathematical knowledge;
it provides an interesting perspective for philosophical, historical, and
pedagogical studies of the growth of scientific knowledge and of the role
metalinguistic discourse in scientific thought.

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