Anna Bellomo

In this article, I compare the formulation and applications of the “principle of permanence of equivalent forms” due to George Peacock, to whom the principle is first attributed, with the formulation and applications due to Hermann Hankel, the German mathematician to whom the popularity of the principle is owed. Despite Hankel’s explicit references to Peacock and the British algebraic tradition more broadly, I argue that Hankel’s project and applications of the principle show a rather different interpretation of the latter than what is to be found in Peacock, Hankel’s source. The principle has been regarded as key for the emergence of a broadly formalist understanding of mathematics. My claim is that it is only in Hankel’s understanding of the principle, but not in Peacock’s, that such a formalist outlook can be detected.

Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent.

Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements.