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Cantor's Grundlagen and the Paradoxes of Set Theory
"... This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motiva ..."
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This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motivated my first attempts to understand proper classes and the realm of transfinite numbers. I read a version of the paper at the APA Central Division meeting in Chicago in May, 1998. I thank Howard Stein, who provided valuable criticisms of an earlier draft, ranging from the correction of spelling mistakes, through important historical remarks, to the correction of a mathematical mistake, and Patricia Blanchette, who commented on the paper at the APA meeting and raised two challenging points which have led to improvements in this final version
RECOGNIZING VARIABLE SPATIAL ENVIRONMENTS —THE THEORY OF COGNITIVE PRISM Vom Fachbereich Mathematik und Informatik
"... To my parents ii Table of Contents Table of Contents iii Abstract xv Acknowledgements xvii ..."
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To my parents ii Table of Contents Table of Contents iii Abstract xv Acknowledgements xvii
1 Induction and Comparison
"... Frege proved an important result, concerning the relation of arithmetic to secondorder logic, that bears on several issues in linguistics. Frege’s Theorem illustrates the logic of relations like PRECEDES(x, y) and TALLER(x, y), while raising doubts about the idea that we understand sentences like ‘ ..."
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Frege proved an important result, concerning the relation of arithmetic to secondorder logic, that bears on several issues in linguistics. Frege’s Theorem illustrates the logic of relations like PRECEDES(x, y) and TALLER(x, y), while raising doubts about the idea that we understand sentences like ‘Carl is taller than Al ’ in terms of abstracta like heights and numbers. Abstract paraphrase can be useful—as when we say that Carl’s height exceeds Al’s—without reflecting semantic structure. Related points apply to causal relations, and even grammatical relations like DOMINATES(x, y). Perhaps surprisingly, Frege provides the resources needed to recursively characterize labelled expressions without characterizing them as sets. His theorem may also bear on questions about the meaning and acquisition of number words.
Coercion vs. Indeterminacy in Opaque Verbs
"... This paper is about the semantic analysis of opaque verbs such as seek and owe, which allow for unspecific readings of their indefinite objects. 1 One may be looking for a good car without there being any car that one is looking for; or, one may be looking for a good car in that a specific car exist ..."
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This paper is about the semantic analysis of opaque verbs such as seek and owe, which allow for unspecific readings of their indefinite objects. 1 One may be looking for a good car without there being any car that one is looking for; or, one may be looking for a good car in that a specific car exists that one is looking for. It thus appears that there are two interpretations of these verbs – a specific and an unspecific
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question