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Abstract: This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König’s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

Abstract: A semantics for quantified modal logic is presented that is based on Kleene’s notion of realizability. This semantics generalizes Flagg’s 1985 construction of a model of a modal version of Church’s Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church’s Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra’s generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott’s graph model (for the untyped lambda calculus) which witnesses the failure of the stability of nonidentity.

This article surveys recent literature by Parsons, McGee, Shapiro and otherson the significance of categoricity arguments in the philosophy of mathematics.After discussing whether categoricity arguments are sufficient to securereference to mathematical structures up to isomorphism, we assess what exactlyis achieved by recent `internal' renditions of the famous categoricityarguments for arithmetic and set theory.

This paper sets out a predicative response to the Russell-Myhill paradox ofpropositions within the framework of Church's intensional logic. A predicativeresponse places restrictions on the full comprehension schema, which assertsthat every formula determines a higher-order entity. In addition to motivatingthe restriction on the comprehension schema from intuitions about the stabilityof reference, this paper contains a consistency proof for the predicativeresponse to the Russell-Myhill paradox. The models used to establish thisconsistency also model other axioms of Church's intensional logic that havebeen criticized by Parsons and Klement: this, it turns out, is due to resourceswhich also permit an interpretation of a fragment of Gallin's intensionallogic. Finally, the relation between the predicative response to theRussell-Myhill paradox of propositions and the Russell paradox of sets isdiscussed, and it is shown that the predicative conception of set induced bythis predicative intensional logic allows one to respond to the Wehmeierproblem of many non-extensions.

Abstract: Frege’sGrundgesetzewas one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of theGrundgesetzeformed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of theGrundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of theGrundgesetzewhich defines a model of all the axioms of Zermelo–Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel’s constructible universe of sets and to Kripke and Platek’s idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen’s fine structure theory). The axioms of theGrundgesetzeare examples ofabstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.

Abstract: Frege’s theorem says that second-order Peano arithmetic is interpretable in Hume’s Principle and full impredicative comprehension. Hume’s Principle is one example of anabstraction principle, while another paradigmatic example is Basic Law V from Frege’sGrundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic (cf. Theorem 3.2).

Abstract: Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (Parsons, 1990; Parsons, 2008, sec. 49; McGee, 1997; Lavine, 1999; Väänänen & Wang, 2014). Another great enterprise in contemporary philosophy of mathematics has been Wright’s and Hale’s project of founding mathematics on abstraction principles (Hale & Wright, 2001; Cook, 2007). In Walsh (2012), it was noted that one traditional abstraction principle, namely Hume’s Principle, had a certain relative categoricity property, which here we termnatural relative categoricity. In this paper, we show that most other abstraction principles arenotnaturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to (i) stability-like acceptability criteria on abstraction principles (cf. Cook, 2012), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli (2010b) and Fine (2002), and (iii) supervaluational ideas coming out of the work of Hodes (1984, 1990, 1991).

Abstract: A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here, an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics.

With the recent surge of college protests against various forms of economic, political, social, and racial injustice, there have been persistent and pernicious reactions from other students, administrators and public figures that function to undermine the emancipatory impulses animating these demonstrations. The reactions are often justified under the banners of tolerance, chastising students to listen instead of protest. This article, focusing on Marcuse’s concepts of repressive toleration and counterrevolution, evaluates the reactionary responses to these events, as well as the critical potential of this fledgling student sensibility, a burgeoning refusal represented by protest events at American universities. We maintain that many of the calls for tolerance are actually demands for silence and belong to a wider counterrevolutionary phase of late capitalism observed by Marcuse. Bedrock liberties are dialectically inverted whereby speech and toleration are repressively deployed against demands for justice. This article concludes by arguing that it is crucial to the success of this resurgent sensibility for justice—and progress toward a radical socialist movement that coincides with the emancipatory vision of Herbert Marcuse—that the counterrevolutionary character of the responses are demystified. [ABSTRACT FROM PUBLISHER]