Francesca Boccuni

This chapter argues against predicative analyses of plurality, which force plurals into the familiar mould of singular logic by turning an apparently plural term standing for several objects into a singular predicate standing for a concept or property. Michael Dummett enlists support from Fregean semantics in favour of a predicative analysis, but his arguments do not stand up, either as exegesis of Frege or on their own merits. As well as facing difficulties in eliminating plural content, predicative analyses are sunk by the equivocity objection: they misrepresent single English predicates as equivocal, by treating the predicate differently according as it combines with singular or plural arguments. Although George Boolos does not offer a predicative analysis, it is argued that his second-order treatment of plurals is also sunk by the equivocity objection. And Ian Rumfitt fails in his attempt to avoid the objection by modifying Boolos's scheme.

In this article, I investigate the philosophical issues connected with the consistent predicative fragment of Frege’s infamous Basic Law V that is presented in Heck (Hist Philos Log 17(1):209–220, 1996). This fragment of Frege’s Grundgesetze is philosophically disputable, since the predicative restriction it imposes on second-order comprehension leads to a strong revision of Frege’s assumptions on the Platonic existence of concepts as logical entities. According to Gödel (Russell’s mathematical logic. In: Schilpp PA (ed) The philosophy of Bertrand Russell. Northwestern University, Evanston/Chicago, pp 123–153, 1944; in Benacerraf and Putnam (eds) Philosophy of mathematics: selected readings. Cambridge University Press, Cambridge, 1983), predicativism, in fact, is taken to be committed to mathematical constructivism. In this paper, I am going to argue that, in order to justify Frege’s conceptual Platonism from a predicative perspective, a reassessment of Gödel’s dichotomy between impredicativity and predicativity is required. This is achieved by an investigation of Gödel’s objections to Russell’s vicious circle principle and its reformulation in terms of the Thesis of Arbitrary Reference by Martino (Topoi 20:65–77, 2001; Lupi, pecore e logica. In: Carrara M, Giaretta P (eds) Filosofia e logica. Rubettino, Catanzaro, pp 103–133, 2004). Finally, I also consider the consequences of this reformulation on Frege’s logicism.

Hume’s Principle (HP) does not determine the truth values of ‘mixed’ identity statements like ‘$ \#F $ = Caesar’. This is the Caesar Problem (CP). Still, neologicists such as Hale and Wright argue that (1) HP is a priori, and (2) HP introduces the pure sortal concept Number. We argue that Neologicism faces a Caesar-problem Problem (CPP): if neologicists solve CP by establishing that ‘$ \#F\neq $ Caesar’ is true, (1) and (2) cannot be retained simultaneously. We examine various responses neologicists might provide and show that they do not address CPP. We conclude that CP uncovers a fatal tension in Neologicism.

The aim of this Element is to provide an overview of abstractionism in the philosophy of mathematics. The authors distinguish between mathematical abstractionism, which interprets mathematical theories on the basis of abstraction principles, and philosophical abstractionism, which attributes a philosophical significance to mathematical abstractionism. They then survey the main semantic, ontological, and epistemological theses that are associated with philosophical abstractionism. Finally, the authors suggest that the most recent developments in the debate pull abstractionism in different directions.

Scottish Neologicism aims to found arithmetic on full second-order logic and Hume’s Principle, stating that the number of the Fs is identical with the number of the Gs if, and only if, there are as many Fs as Gs. However, Neologicism faces the problem of the logical ontology, according to which the underlying second-order logic involves ontological commitments. This paper addresses this issue by substituting second-order logic by Boolos’s plural logic, augmented by the Plural Frege Quantifier F modelled on Antonelli’s Frege Quantifier. The resulting theory (PHP) interprets second-order Peano arithmetic. Its ontological innocence is assessed: PHP offers an alternative that solves the problem of the logical ontology pervading Neologicism.

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula $$\phi (F)$$, with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA $$^2$$.

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system will be compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted.

This book offers a plurality of perspectives on the historical origins of logicism and on contemporary developments of logicist insights in philosophy of mathematics. It uniquely provides up-to-date research and novel interpretations on a variety of intertwined themes and historical figures related to different versions of logicism. The essays, written by prominent scholars, are divided into three thematic sections. The first section focuses on major authors like Frege, Dedekind, and Russell, providing a historical and theoretical exploration of such figures in the philosophical and mathematical milieu in which logicist views were first expounded. Section II sheds new light on the interconnections between these founding figures and a number of influential other traditions, represented by authors like Hilbert, Husserl, and Peano, as well as on the reconsideration of logicism by Carnap and the logical empiricists. Finally, the third section assesses the legacy of such authors and of logicist themes for contemporary philosophy of mathematics, offering new perspectives on highly debated topics--neo-logicism and its extension to accounts of ordinal numbers and set-theory, the comparison between neo-Fregean and neo-Dedekindian varieties of logicism, and the relation between logicist foundational issues and empirical research on numerical cognition--which define the prospects of logicism in the years to come. This book represents a comprehensive account of the development of logicism and its contemporary relevance for the logico-philosophical foundations of mathematics. It will be of interest to graduate students and researchers working in philosophy of mathematics, philosophy of logic, and the history of analytic philosophy.

Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a key neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter.

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of individuals rather than on the existence of concepts.

PG (Plural Grundgesetze) is a predicative monadic second-order system which exploits the notion of plural quantification and a few Fregean devices, among which a formulation of the infamous Basic Law V. It is shown that second-order Peano arithmetic can be derived in PG. I also investigate the philosophical issue of predicativism connected to PG. In particular, as predicativism about concepts seems rather un-Fregean, I analyse whether there is a way to make predicativism compatible with Frege’s logicism.

PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts.