Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.

Heinrich Behmann (1891–1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved—independently of Löwenheim and Skolem’s earlier work—the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included.

In a recent article (Mancosu, 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign ‘sizes’ to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosity assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this paper I generalize some worries, raised by Richard Heck, emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The paper is centered around five main parts. The first (§3) takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part (§4) where I show that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP (or so I claim) and from which we can derive the axioms of second-order arithmetic. All the principles I present agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part (§5) connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part (§6) states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section I offer a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, §7 makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after.

Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a debate (on “fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, we begin in the fourth section the analytic work necessary for exploring various important claims about “purity,” “content,” and other relevant notions.

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the ‘size’ of A should be less than the ‘size’ of B (part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher).

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought.