This is a critical notice of Stewart Shapiro's 1997 book, Philosophy of Mathematics: Structure and Ontology.

[Robert Stalnaker] Saul Kripke made a convincing case that there are necessary truths that are knowable only a posteriori as well as contingent truths that are knowable a priori. A number of philosophers have used a two-dimensional model semantic apparatus to represent and clarify the phenomena that Kripke pointed to. According to this analysis, statements have truth-conditions in two different ways depending on whether one considers a possible world 'as actual' or 'as counterfactual' in determining the truth-value of the statement (...) relative to that possible world. There are no necessary a posteriori or contingent a priori propositions: rather, contingent a priori and necessary a posteriori statements are statements that are necessary when evaluated one way, and contingent when evaluated the other way. This paper distinguishes two ways that the two-dimensional framework can be interpreted, and argues that one of them gives the better account of what it means to 'consider a world as actual', but that it provides no support for any notion of purely conceptual a priori truth. /// [Thomas Baldwin] Two-dimensional possible world semantic theory suggests that Kripke's examples of the necessary a posteriori and contingent a priori should be handled by interpreting names as implicitly indexical. Like Stalnaker, I reject this account of names and accept that Kripke's examples have to be accommodated within a metasemantic theory. But whereas Stalnaker maintains that a metasemantic approach undermines the conception of a priori truth, I argue that it offers the opportunity to develop a conception of the a priori aspect of stipulations, conceived as linguistic performances. The resulting position accommodates Kripke's examples in a way which is both intrinsically plausible and fits with Kripke's actual discussion of them. (shrink)

Abstract:A number of philosophers of language have proposed that people do not have conceptual access to‘bare particulars’, or attribute‐free individuals (e.g. Wiggins, 1980). Individuals can only be picked out under some sortal, a concept which provides principles of individuation and identity. Many advocates of this view have argued thatobjectis not a genuine sortal concept. I will argue in this paper that a narrow sense of‘object’, namely the concept of any bounded, coherent, three‐dimensional physical object that moves as a whole (Spelke, (...) 1990) is a sortal for both infants and adults. Furthermore,objectmay be the infant's first sortal and more specific sortals such ascupanddogmay be acquired later in the first year of life. I will discuss the implications for infant categorization studies, trying to draw a conceptual distinction between a perceptual category and a sortal, and I will speculate on how a child may construct sortal concepts such ascupanddog. (shrink)

'Object' and 'criterion of identity' are philosophical terms of art whose application lies at a considerable theoretical remove from the surface phenomena of everyday linguistic usage. This partly explains their highly controversial status, for their point of application lies precisely where the concerns of linguists and philosophers of language merge with those of metaphysicians. This chapter explains the possession of determinate identity‐conditions. It argues that the distinction between 'abstract' and 'concrete' objects is itself a highly controversial one, and although it (...) has indeed been argued that this distinction turns ultimately upon differences between the criteria of identity governing objects of these two broad categories. It examines the role which criteria of identity play in our talk about objects of the least controversial varieties. The chapter concerns the problem of identity over time and the paradoxes to which identity criteria often appear to give rise when time is brought into the picture. (shrink)

A number of philosophers of language have proposed that people do not have conceptual access to‘bare particulars’, or attribute‐free individuals (e.g. Wiggins, 1980). Individuals can only be picked out under some sortal, a concept which provides principles of individuation and identity. Many advocates of this view have argued that object is not a genuine sortal concept. I will argue in this paper that a narrow sense of‘object’, namely the concept of any bounded, coherent, three‐dimensional physical object that moves as a (...) whole (Spelke, 1990) is a sortal for both infants and adults. Furthermore, object may be the infant's first sortal and more specific sortals such as cup and dog may be acquired later in the first year of life. I will discuss the implications for infant categorization studies, trying to draw a conceptual distinction between a perceptual category and a sortal, and I will speculate on how a child may construct sortal concepts such as cup and dog. (shrink)

_Identity and Discrimination_, originally published in 1990 and the first book by respected philosopher Timothy Williamson, is now reissued and updated with the inclusion of significant new material. Williamson here proposes an original and rigorous theory linking identity, a relation central to metaphysics, and indiscriminability, a relation central to epistemology.__ Updated and reissued edition of Williamson’s first publication, with the inclusion of significant new material Argues for an original cognitive account of the relation between identity and discrimination that has been (...) influential in the philosophy of perception Pioneers the use of epistemic logic to solve puzzles about indiscriminability Develops the application of techniques from mathematical logic to understand issues about identity over time and across possible worlds. (shrink)

The classic, influential essay in 'descriptive metaphysics' by the distinguished English philosopher.

[Robert Stalnaker] Saul Kripke made a convincing case that there are necessary truths that are knowable only a posteriori as well as contingent truths that are knowable a priori. A number of philosophers have used a two-dimensional model semantic apparatus to represent and clarify the phenomena that Kripke pointed to. According to this analysis, statements have truth-conditions in two different ways depending on whether one considers a possible world 'as actual' or 'as counterfactual' in determining the truth-value of the statement (...) relative to that possible world. There are no necessary a posteriori or contingent a priori propositions: rather, contingent a priori and necessary a posteriori statements are statements that are necessary when evaluated one way, and contingent when evaluated the other way. This paper distinguishes two ways that the two-dimensional framework can be interpreted, and argues that one of them gives the better account of what it means to 'consider a world as actual', but that it provides no support for any notion of purely conceptual a priori truth. /// [Thomas Baldwin] Two-dimensional possible world semantic theory suggests that Kripke's examples of the necessary a posteriori and contingent a priori should be handled by interpreting names as implicitly indexical. Like Stalnaker, I reject this account of names and accept that Kripke's examples have to be accommodated within a metasemantic theory. But whereas Stalnaker maintains that a metasemantic approach undermines the conception of a priori truth, I argue that it offers the opportunity to develop a conception of the a priori aspect of stipulations, conceived as linguistic performances. The resulting position accommodates Kripke's examples in a way which is both intrinsically plausible and fits with Kripke's actual discussion of them. (shrink)

I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) I rely on a distinction between 'basic' and 'constructed' structures. A conclusion is that ideas from the metaphysical tradition can be misleading when applied to the objects of modern mathematics. (shrink)

The book also includes a chronological table of significant events, select bibliography, succinct explanatory notes, and an index--all of which supply ...

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and (...) investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. (shrink)

I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which (...) it may successfully be applied. (shrink)

In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...

In this book, which thoroughly revises and greatly expands his classic work Sameness and Substance, David Wiggins retrieves and refurbishes in the light of twentieth-century logic and logical theory certain conceptions of identity, of substance and of persistence through change that philosophy inherits from its past. In this new version, he vindicates the absoluteness, necessity, determinateness and all or nothing character of identity against rival conceptions. He defends a form of essentialism that he calls individuative essentialism, and then a form (...) of realism that he calls conceptualist realism. In a final chapter he advocates a human being-based conception of the identity and individuation of persons, arguing that any satisfactory account of personal memory must make reference to the life of the rememberer himself. This important book will appeal to a wide range of readers in metaphysics, philosophical logic, and analytic philosophy. (shrink)

This volume collects a number of Perry's classic works on personal identity as well as four new pieces, 'The Two Faces of Identity', 'Persons and Information', 'Self-Notions and The Self' and 'The Sense of Identity'. Perry's Introduction puts his own work and that of others on the issues of identity and personal identity in the context of philosophical studies of mind and language over the past thirty years.

Die Grundlagen gehören zu den klassischen Texten der Sprachphilosophie, Logik und Mathematik. Frege stützt sein Programm einer Begründung von Arithmetik und Analysis auf reine Logik, indem er die natürlichen Zahlen als bestimmte Begriffsumfänge definiert. Die philosophische Fundierung des Fregeschen Ansatzes bilden erkenntnistheoretische und sprachphilosophische Analysen und Begriffserklärungen. Studienausgabe aufgrund der textkritisch herausgegebenen Jubiläumsausgabe (Centenarausgabe). Mit Einleitung, Anmerkungen, Literaturverzeichnis und Namenregister.

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...