Review of Suppes 1957 Proposals For Division by Zero

Abstract

We review the exposition of division by zero and the definition of total arithmetical functions in "Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. This book is a pedagogical introduction to first-order predicate calculus with logical, mathematical, physical and philosophical examples, some presented in exercises. It is notable for (i) presenting division by zero as a problem worthy of contemplation, (ii) considering five totalisations of real arithmetic, and (iii) making the observation that each of these solutions to "the problem of division by zero" has both advantages and disadvantages -- none of the proposals being fully satisfactory.

We classify totalisations by the number of non-real symbols they introduce, called their Extension Type. We compare Suppes' proposals for division by zero to more recent proposals. We find that all totalisations of Extension Type 0 are arbitrary, hence all non-arbitrary totalisations are of Extension Type at least 1. Totalisations of the differential and integral calculus have Extension Type at least 2. In particular, Meadows have Extension Type 1, Wheels have Extension Type 2, and Transreal numbers have Extension Type 3.

It appears that Suppes was the modern originator of the idea that all real numbers divided by zero are equal to zero. This has Extension Type 0 and is, therefore, arbitrary.