Sums, Numbers and Infinity: Collections in Bolzano’s Mathematics and Philosophy
Anna Bellomo
Abstract:
This dissertation contains a series of studies on 19th century philosophy of mathematics. The essays are linked together by two common threads: Bolzano's theory of collections on the one hand, and the emergence of modern sets and analysis in the 19th century on the other. Bolzano is often mentioned as an important figure for both developments, but sometimes the apparent similarity between his contributions and those of other thinkers is not sufficiently probed. Much of the work of this dissertation offers new interpretations of key aspects of Bolzano's writings to give a historically accurate and technically sound reassessment of Bolzano's contributions to mathematics and its philosophy in its own terms, obtained by careful textual analysis, and mathematical probing. This applies to the key notions of collections, natural numbers, measurable (i.e.~real) numbers, and infinity.
Chapter 2 is the one that focuses the most on Bolzano's theory of collections. Bolzano has different notions of collection: collection in general (Inbegriff), Reihe, Menge, Vielheit among others. Generally speaking, in his mathematics he usually appeals to Mengen in a way that has encouraged other scholars to interpret these as completely equivalent to the sets of set theory. Chapter 2 argues that Bolzano's Mengen though are markedly different from sets, for two reasons: first, they are not extensional in the sense of the extensionality axiom (although Vielheiten, a special kind of Mengen, are); second, they do not play the same foundational role as sets do. Ultimately this difference in function is insurmountable, because it stems from the fact that Bolzano does not extensionalise the notion of structure, whereas that is precisely the conceptual gain granted by set theory that allows for sets' foundational applications.
Chapter 3 is the first of the chapters that deal with Bolzano's mathematical objects, and we start with the most basic of those objects, namely, the natural numbers. The main focus of the chapter is to explain why, within Bolzano's conceptual approach to natural numbers, the question of how he measures the sizes of infinite collections needs to be rephrased. We argue that it is not possible to determine _the_ size of a given infinite collection of natural numbers, because this collection will always be given through a certain concept, and it is this concept that determines how the size of the collection is to be computed. This approach has the advantage of explaining how Bolzano's views on infinite collections of natural numbers evolve between the _Wissenschaftslehre_ (1837) and the _Paradoxien des Unendlichen_ (1851).
Chapter 4 considers Bolzano's most sophisticated number system, that of the measurable numbers. Ever since the relevant text first came to light, Bolzano's measurable numbers have been read as his attempt at giving a presentation of the real numbers. This argument has been made mostly by showing that Bolzano's presentation can be translated into a sequence-based presentation of the real numbers that strongly resembles Cantor's or Dedekind's (depending on how it is carried out). Whilst agreeing that Bolzano's measurable numbers ought to be seen as Bolzano's attempt at a rigorous presentation of the real numbers, we argue that, for it to be rigorous, it cannot be the sort of presentation that the sequence interpretations make it out to be. We also argue that sequence interpretations are motivated precisely by an effort to show that somehow Bolzano was `right all along', where being right boils down to anticipating a Cantor-style approach, even though such a reading introduces a host of mistakes in Bolzano's presentation that are simply not there.
With Chapter 5, the last one on Bolzano's mathematics, we shift our attention beyond number systems and on to Bolzano's `calculations of the infinite', as they appear in _Paradoxien_ §§29-33. Here we argue against what has been one of the mainstays of Bolzanian interpretations, namely the thought that the _Paradoxien_ contain an anticipation of Cantor's transfinite arithmetic. It was never Bolzano's intention to measure the size of infinite set-like collections, all he wanted was a principled way to compute with infinite sums. This new interpretation sheds light on passages from the _Paradoxien_ that are otherwise hard to make sense of, and it also allows us to defend Bolzano's arithmetic of the infinite as coherent.
Finally, Chapter 6 contains a comparison between a common 19th century understanding of how to extend mathematical concepts and domains, as exemplified by Dedekind, and a recent attempt at using model-theoretic notions to explain what domain extensions, and especially domain extensions via ideal elements, are supposed to do. I test each proposal against an array of prototypical cases of domain extension, including some from Dedekind's own mathematical work, and conclude that neither the modern proposal by Ken Manders (1989) nor the Dedekind-inspired proposal can offer a complete characterisation of domain extensions via ideal elements, although this negative result is insightful: it makes us realise that each criterion is meant to capture extensions that are meant to preserve different features of the domain we start from.