Origins and Varieties of Logicism

On the Logico-Philosophical Foundations of Mathematics

Francesca Boccuni, Andrea Sereni, Francesca Boccuni, Andrea Sereni

• Éditeur: Routledge
• Année: 2021
• ISBN:
  9781000508123
• Édition: 1
• Sujet: Matemáticas
• Sous-sujet: Historia y filosofía de las matemáticas

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. Part I 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. Part 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, Part III 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 offers 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.

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Origins and Varieties of Logicism

An Overview

Francesca Boccuni and Andrea Sereni

DOI: 10.4324/9780429277894-1

WHAT IS LOGICISM?

The basic idea behind logicism in the philosophy of mathematics is disarmingly easy to state: mathematics is reducible to logic. Equally disarming, however, is the amount of subtleties, complexities, and variations that exploring that basic idea entails.

Logicism, as it was originally introduced or as it is still perceived today, is a foundational theory, and appeal to “reduction” is a pointer to its foundational role. How foundation and reduction are to be conceived of, however, is a vastly debated matter, combining issues in both the semantics, ontology, epistemology, and formal reconstruction of mathematics. Ideally, one may expect a foundation to roughly entail the following: unclear terms in a disputed area of discourse are explained, via definitions, on the basis of clearer or more familiar primitives, thus yielding clarity to the disputed concepts; such semantic reduction brings about an ontological reduction, for objects apparently referred to in, or anyway featuring in the subject-matter of, the target discourse are identified with objects referred to in the basic discourse, which are also taken to be more familiar or more easily characterized; finally, semantic and ontological reductions bring about epistemic reduction: disputed objects are now shown to be more easily accessible, and statements in the disputed discourse can be derived from statements in the basic language expanded with suitable definitions, thus inheriting part of the certainty the basic statements enjoy. Following Quine (1969), we can call the process of attaining clarity by reduction of obscure concepts to basic ones a conceptual study and the process of attaining certainty by deduction of disputed statements from basic ones a doctrinal study; to these, an ontological and an epistemological study clearly follow suit. A logicist foundation would then see in the language of mathematics (or some of its branches) the target disputed discourse, in some sort of logical objects those with which mathematical objects are to be identified, and logical vocabulary as the basic vocabulary with which, via definitions, we can recover the originally obscure concepts and objects, explain the content and truth-conditions of mathematical statements, and deduce such statements from purely logical ones.

This idealized picture, however, is hostage to a number of different variables. A standard presentation of logicism, mainly reflecting the views expounded by the German mathematician and philosopher Gottlob Frege, would roughly deliver the following outline when limited to arithmetic.

Despite its undeniable usefulness in applications, which gives us an indirect, inductive, and a posteriori warrant in its truth, arithmetic stands in need of a much more robust justification. Such justification will guarantee a number of alleged properties of arithmetical statements: that they are necessary, that they enjoy universal applicability and are not tied to particular applications, that they do not assert anything of particular material objects or their properties, that the reasons for holding them true make no appeal to empirical evidence, and that they have a form of objectivity that makes their truth independent of subjective human activities. An adequate semantic analysis of arithmetical statements characterizes numerical expressions as singular terms, whose semantic role is to denote individual, self-subsistent objects. These objects can be identified with particular logical objects (the ontological study). Frege considered these extensions of concepts, that is, loosely speaking, the sets of objects falling under (sortal) concepts. The existence of such logical objects is established by basic logical laws, delineating the essential vocabulary of our basic logical language. Via such vocabulary, through appropriate definitions, we can define the target arithmetical notions (the conceptual study): for instance, a certain law will characterize extensions; via extensions we can define the notion of cardinal number; other logical definitions will introduce the notions of following-in-a-series and immediate successor and allow definitions of individual numbers, beginning with the number 0; together these definitions will lead to defining in turn natural numbers as those finite cardinals that follow in the successor-series starting with 0. This will show how it is possible to derive all arithmetical truths from basic logical truths and definitions (the doctrinal study) once a proper formal language is set and legitimate inference rules are specified. Such derivation will also show that whatever epistemic access we have to logical objects and laws will transmit to mathematical objects and statements (the epistemological study). In an ideal scenario like the one just depicted, this access will be purely a priori and will show arithmetical statements to enjoy the same necessity and generality of logical statements: indeed, arithmetical statements will just turn out to be a sub-class (or a notational variant) of logical statements. Hence, they will be ipso facto provable by logic and definitions and thus analytic in Frege’s sense (cf. [GLA], §3).

Even without looking at the complexities such a picture requires when placed in Frege’s overall mathematical, semantical, and philosophical setting, it is easy to see from how many angles it can be attacked or simply modified.

To begin with, we need a clarification of what logic is, what its basic concepts and laws are, and how we acquire knowledge of them. Traditionally, logicism in the philosophy of mathematics has been developed within the setting of second-order logic. But second-order logic has come under fire from different charges, both formal—regarding, for instance, its incompleteness entailed by Gödel’s 1931 theorems—and philosophical—such as that leveled by Quine (1970), who suggested that second-order logic is not logic but rather set theory in disguise. Switching to first-order logic may provide some reassurance but will prevent recovering not just, first and foremost, arithmetic in the way Frege envisaged but also larger parts of mathematics—whereas ideally, logicism is meant to extend to all branches of mathematics. One can then either try to defend second-order logic from various accusations (Shapiro, 1991; Wright, 2007) or even resort to some alternative interpretations—like those provided by plural logic (Boolos, 1984, 1985)—or even to non-classical logics (cf. e.g., Tennant, 1987). Needless to say, the same concerns apply when one considers the variety of accounts of logical validity that will have to underpin the inferences to be allowed in a given formal system, as well as some of the principles to be endorsed (think, for a case in point, about the Axiom of Choice or various kinds of comprehension principles).

Even once such formal problems are tackled, many remain open. Logical (or, in different guises, set-theoretical) platonism, that is, the identification of mathematical objects with logical objects, raises several challenges. First of all, many would conceive of logical discourse as entirely topic neutral, and this would boost skepticism against the idea that the subject-matter of logic involves sui generis objects and against the idea (see Dummett 1973, ch. 4; Hale 1994, 1996) that numerical expressions are genuine singular terms. One can thus either try to account for the logical form of mathematical statements so that such expressions are interpreted differently—for example, as complex predicates (per some of Russell’s views)—or even adopt a background conception of logic as conventional in nature (as was the case for Carnap), with the side effect of threatening not just platonism, as intended, but also the objectivity of mathematics. More generally, the idea that the logical analysis of mathematical statements is to mirror faithfully their surface-grammar may be questioned and the identification of the grammatical and semantical roles of some mathematical expressions with those of singular terms jeopardized.¹ Also, the claim itself that the surface-grammar of those statements as it is adopted by standard platonist views is the correct one can be disputed (cf. e.g., Hofweber, 2005; Moltmann, 2013a, b).

1 This, in turn, would also jeopardize the semantical and theoretical import of Frege’s famous and controversial Context Principle (see Cariani, 2018; Dummett, 1995; Linnebo, 2009b; Picardi, 2009, 2017; Resnik, 1967), as well as the particular interpretation it receives in neo-logicists’ so-called Syntactic Priority Thesis (see Wright, 1983; Hale, 1987; MacBride, 2003).

Even if one accommodates logical objects, metaphysical questions about their nature come to the fore. The Fregean platonist view—at least as standardly presented—which locates such objects in a third realm additional to those of material objects and subjective representations and describes them as self-subsistent, mind-independent, abstract (aspatial, atemporal, acausal, possibly necessary existent) objects runs the risk of begging for more answers than it provides (cf. Dummett, 1991, ch. 18; Hale, 1988; Rosen, 2012). Qualms over our epistemic access to such objects have abounded, raising the suspicion that the so-called Integration Challenge (Peacocke, 1998) between semantics, ontology, and epistemology is hard to attain in mathematics, as paradigmatically emphasized by Benacerraf (1973).²

2 For the vast discussion on Benacerraf’s challenge and its relevance for platonism, see, for example, Field (1989), Introduction, Hale and Wright (2002), Liggins (2010), Linnebo (2006), Panza and Sereni (2013).

Concerns over the admissibility of abstract logical objects could be assuaged if one were able to provide some intuitive, immediate source of epistemic access to them, via clear, or unproblematic, or self-evident basic concepts. This points to another cradle of problems: definitions, which are the backbone of the conceptual reduction.

For one thing, as history has painfully taught us, naïve notions can be given apparently natural definitions and still engender paradoxes; Frege’s Basic Law V as a definition of the (supposedly) logical notion of extension or—equivalently—Cantor’s naïve notion of set or the notion of ordinal number all posed hardly surmountable challenges in this regard. Consistency clearly is a minimal requirement for a classical foundation of any branch of mathematics; how to ensure it is a complex matter. Even the discussion of where exactly the contradiction in Frege’s system should be located—as emphasized by, among other things, debates on the cause of such inconsistency (Boolos, 1993; Dummett, 1991, ch. 17, 1994), on the tenability of impredicative vs. predicative definitions (cf. e.g., Feferman, 2005), and on Russell’s Vicious Circle Principle (Chihara, 1973; Gödel, 1944; Jung, 1999)—opens up a range of diagnoses and solutions, including the possibility of modifying Frege’s formal system enough to individuate consistent sub-systems in which his own definitions can be redeemed (cf. e.g., Ferreira, 2018; Ferreira & Wehmeier, 2002; Heck, 1996).³

3 For a survey, see Burgess (2005).

Even when consistency is safeguarded, however, the nature and import of mathematical definitions, and definitions by abstraction in particular, maintain center stage. The Euclidean ideal of self-evidence is not easy to obtain, as the latter remains a highly problematic notion (see e.g., Jeshion, 2001, 2004; Shapiro, 2009). Nor it is uncontroversial, as many logicists have maintained (see e.g., Hale & Wright, 2000), to claim that definitions are able to afford us a priori mathematical knowledge. This may lead to seeking alternative ways of characterizing basic mathematical notions. One major example is given by the stipulation of sets of axioms, possibly themselves conceived of as implicit definitions of the primitive terms occurring in them. While such a move is often associated with other ...

Table des matières

• Cover Page
• Half Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents Page
• Preface and Acknowledgments Page
• 1 Origins and Varieties of Logicism: An Overview
• Part I Origins
• Part II Dangerous Liaisons
• Part III The Roads Ahead
• List of Contributors
• Index