There is a long-standing gap in the literature as to whether Gödelian incompleteness constitutes a challenge for Neo-Logicism, and if so how serious it is. In this paper, I articulate and address the challenge in detail. The Neo-Logicist project is to demonstrate the analyticity of arithmetic by deriving all its truths from logical principles and suitable definitions. The specific concern raised by Gödel’s first incompleteness theorem is that no single sound system of logic syntactically implies all arithmetical truths. I set (...) out some responses that initially seem appealing and explain why they are not compelling. The upshot is that Neo-Logicism either offers an epistemic route only to some truths of arithmetic; or that it has to move from a syntactic to a semantic notion of logical consequence, which risks undermining its epistemic goals. I conclude by considering Crispin Wright’s recent attempt to address Gödelian incompleteness, which I argue is not satisfactory. (shrink)

It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...) is an error like the one Plato mentions of thinking mathematics is about actions (of adding, constructing, extracting roots and so on). Mathematics is about eternal relations between universals, such as the ratio of the diagonal of a square to the side. Ethics too is about eternal verities, such as the equal worth of persons and just distributions. Mathematical and ethical verities do both constrain actions, such as the possibility of walking over the seven bridges of Königsberg once and once only or of justly discriminating between races. But they are not themselves about action. In principle, neither mathematical nor ethical verities are subject to historical forces or disagreement among tribes (though they can be better understood as time goes on). Plato is right: immersion in mathematics induces an understanding of the necessities underpinning reality, an understanding that is essential for distinguishing objective ethics from tribal custom. Equality, for example, is an abstract concept which is foundational for both mathematics and ethics. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

Philosophers are divided on whether the proof- or truth-theoretic approach to logic is more fruitful. The paper demonstrates the considerable explanatory power of a truth-based approach to logic by showing that and how it can provide (i) an explanatory characterization —both semantic and proof-theoretical—of logical inference, (ii) an explanatory criterion for logical constants and operators, (iii) an explanatory account of logic’s role (function) in knowledge, as well as explanations of (iv) the characteristic features of logic —formality, strong modal force, generality, (...) topic neutrality, basicness, and (quasi-)apriority, (v) the veridicality of logic and its applicability to science, (v) the normativity of logic, (vi) error, revision, and expansion in/of logic, and (vii) the relation between logic and mathematics. The high explanatory power of the truth-theoretic approach does not rule out an equal or even higher explanatory power of the proof-theoretic approach. But to the extent that the truth-theoretic approach is shown to be highly explanatory, it sets a standard for other approaches to logic, including the proof-theoretic approach. (shrink)

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating an account of numbers as properties of collections, work on the philosophy of absences, and recent (...) work in numerical cognition and ontogenetic studies. (shrink)

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

Logicism finds a prominent place in textbooks as one of the main alternatives in the foundations of mathematics, even though it lost much of its attraction from about 1950. Of course the neologicist trend has revitalized the movement on the basis of Hume’s Principle and Frege’s Theorem, but even so neologicism restricts itself to arithmetic and does not aim to account for all of mathematics. The present contribution does not focus on the classical logicism of Frege and Dedekind, nor on (...) the Russell-Carnap period, and also not on recent neologicism; its aim is to call attention to some forms of heritage from logicism that normally go quite unnoticed. In the 1920s, 1930s and 1940s, the logicist thesis became a stimulus for some deep innovations in the field of mathematical logic. One can argue, in particular, that two key ideas linked with formal semantics had their origins in the conception of logic associated with the logicist trend – the expansion of metamathematics brought about by Tarski, opening the way to model theory, and the insistence on the “full” set-theoretic semantics as “standard” for second-order logic. The paper proposes an analysis of those inheritances and argues that that logical theory ought to avoid some of their implications. (shrink)

In Der logische Aufbau der Welt, first published in 1928, Carnap aims to rationally reconstruct all objects of cognition by logico-definitional means. As a result, he intends to obtain a fully objective framework in which scientific discourse can take place. This is made possible by the novel method of ‘purely structural definite description’ of all scientifically relevant objects, which is first introduced in the Aufbau. Key to the attainment of this goal is the notion of ‘foundedness’, which Carnap presents as (...) a new basic notion of logic, in order to establish a link between the purely conventional world of logical and mathematical knowledge and the empirical world of knowledge of scientific objects. This idea experienced major criticism by Friedman (1999a,b) since he considers it to lead to the demolition of the boundary between those two worlds. In this essay, we want to defend foundedness against Friedman’s critique by arguing that its introduction is necessary within Carnap’s logicist world of thought to deal with a more fundamental problem: the demarcation of the empirical parts of the Aufbau. In the last section, we will give an outlook on the actual cause for the failure of the Aufbau, the lack of a principle to determine the truth of the instances of the basic relation in the Aufbau, and we will show how this can contribute to the explanation of Carnap’s future philosophical development and retrospective self-evaluation. This essay serves as a dense informal sketch for a later extensive formal treatment of this reading of foundedness and focuses on its implications for the interpretation of Carnap’s post-Aufbau development. (shrink)

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. (shrink)

In a recent article, 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 numerosities 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. Numerosities assignments differ from the standard assignment of size provided (...) by Cantor’s cardinality assignments. In this talk, I generalize some specific worries 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 talk has four main parts. The first takes a historical look at 19th-century attributions of equality of numbers in terms of one-one correlations 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 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 and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions. (shrink)

In Our Knowledge of the Internal World, Robert Stalnaker describes two opposed perspectives on the relation between the internal and the external. According to one, the internal world is taken as given and the external world as problematic, and according to the other, the external world is taken as given and the internal world as problematic. Analytic philosophy moved from the former to the latter, from problems of world-construction to problems of self-locating beliefs. I argue in this paper that these (...) problems are equivalent: both arise because experience and objective, external facts jointly underdetermine their relation. Both can be seen as a problem of expressive completeness; of the internal language in the former case, and of the non-indexical language in the second. (shrink)

Frege regarded Hume's Principle as insufficient for a logicist account of arithmetic, as it does not identify the numbers; it does not tell us which objects the numbers are. His solution, generally regarded as a failure, was to propose certain sets as the referents of numerical terms. I suggest instead that numbers are properties of pluralities, where these properties are treated as objects. Given this identification, the truth-conditions of the statements of arithmetic can be obtained from logical principles with the (...) help of definitions, just as the logicist thesis maintains. (shrink)