Bolzano's Infinite Quantities

TL;DR

This paper revisits Bolzano's theory of infinite quantities, extending it into consistent mathematical structures that can serve as foundations for infinitesimal calculus, offering an alternative to Cantor's set theory.

Contribution

It develops a consistent extension of Bolzano's part-whole principle into algebraic structures usable in infinitesimal calculus, providing an alternative foundation to Cantor's set theory.

Findings

Constructed a linearly ordered ring of infinite quantities.

Developed a partially ordered ring including infinitesimals.

Structures can underpin infinitesimal calculus similar to Non-standard Analysis.

Abstract

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor's concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid's Axiom of the whole being greater than a part. Cantor's set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano's approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano's theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in Non-standard Analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent "cheap version" avoiding ultrafilters due to Terrence Tao.