Frege, Dedekind, and the Epistemology of Arithmetic

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.Peer reviewe

Linguistic realism in mathematical epistemology.

One project in the epistemology of mathematics is to find a defensible account of what passes for mathematical knowledge. This study contributes to this project by examining philosophical theories of mathematics governed by certain basic assumptions. Foremost amongst these is the "linguistic realism" of the title. Roughly put, this is the view that the semantics of mathematical sentences should be taken at face value. Two approaches to mathematics are considered, realist and fictionalist. Mathematical realism affirms the existence of mathematical objects, taking much of what passes for mathematical knowledge as knowledge of such things. It faces the challenge of explaining how such knowledge is possible. The main strategies here are to appeal to the faculty of reason, to a faculty of intuition or to the faculty of sense perception. Recent examples of each strategy are considered and it is argued that the prospects for a satisfactory mathematical realism are limited. Mathematical fictionalism does not affirm the existence of mathematical objects, claiming that mathematics is, or should be considered to be, a form of pretence. It faces the challenge of explaining how a form of pretence can discharge the roles mathematics has in empirical applications. Strategies here are to argue that mathematics is an eliminable convenience or, acknowledging that this may not be the case, that the roles played by mathematics in empirical applications are played in similar contexts by acknowledged forms of pretence. It is argued that the first strategy is not promising but that there is a version of the second that can be defended against objections. In closing, consequences of the conclusions reached are explored and directions for future research indicated

What structuralism could not be

Frege's arithmetical-platonism is glossed as the first step in developing the thesis; however, it remains silent on the subject of structures in mathematics: the obvious examples being groups and rings, lattices and topologies. The structuralist objects to this silence, also questioning the sufficiency of Fregean platonism is answering a number of problems: e.g. Benacerraf's Twin Puzzles of Epistemic and Referential Access. The development of structuralism as a philosophical position, based on the slogan 'All mathematics is structural' collapses: there is no single coherent account which remains faithful to the tenets of structuralism and solves the puzzles of platonism. This prompts the adoption of a more modest structuralism, the aim of which is not to solve the problems facing arithmetical-platonism, but merely to give an account of the 'obviously structural areas of mathematics'. Modest strucmralism should complement an account of mathematical systems; here, Frege's platonism fulfils that role, which then constrains and shapes the development of this modest structuralism. Three alternatives are considered; a substitutional account, an account based on a modification of Dummett's theory of thin reference and a modified from of in re structuralism. This split level analysis of mathematics leads to an investigation of the robustness of the truth predicate over the two classes of mathematical statement. Focussing on the framework set out in Wright's Truth and Objectivity, a third type of statement is identified in the literature: Hilbert's formal statements. The following thesis arises: formal statements concern no special subject matter, and are merely minimally truth apt; the statements of structural mathematics form a subdiscourse - identified by the similarity of the logical grammar - displaying cognitive command. Thirdly, the statements of mathematics which concern systems form a subdiscourse which has both cognitive command and width of cosmological role. The extensions of mathematical concepts are such that best practice on the part of mathematicians either tracks or determines that extension - at least in simple cases. Examining the notions of response dependence leads to considerations of indefinite extensibility and intuitionism. The conclusion drawn is that discourse about structures and mathematical systems are response dependent but that this does not give rise to any revisionary arguments contra intuitionism

Introduction to Knowledge, Number and Reality. Encounters with the Work of Keith Hossack

The Introduction to "Knowledge, Number and Reality. Encounters with the Work of Keith Hossack" provides an overview over Hossack's work and the contributions to the volume

Variables

Variables is a project at the intersection of the philosophies of language and logic. Frege, in the Begriffsschrift, crystalized the modern notion of formal logic through the first fully successful characterization of the behaviour of quantifiers. In Variables, I suggest that the logical tradition we have inherited from Frege is importantly flawed, and that Frege's move from treating quantifiers as noun phrases bearing word-world connection to sentential operators in the guise of second-order predicates leaves us both philosophically and technically wanting