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"Feferman, Solomon"
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PARSONS AND I
In the first part of this article, Feferman outlines his ‘conceptual structuralism’ and emphasizes broad similarities between Parsons’s and his own structuralist perspective on mathematics. However, Feferman also notices differences and makes two critical claims about any structuralism that focuses on the “ur-structures” of natural and real numbers: (1) it does not account for the manifold use of other important structures in modern mathematics and, correspondingly, (2) it does not explain the ubiquity of “individual [natural or real] numbers” in that use. In the second part, Feferman presents a summary of his reasons for the skepticism he has towards contemporary metamathematical investigations of set theory. That skepticism led him to reject the Continuum Problem as a definite mathematical one. He contrasts that attitude sharply to Parsons’s “great sympathy for the current explorations of higher set theory.”
Journal Article
And so on...: reasoning with infinite diagrams
This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a \"pre\" form of this thesis that every proof can be presented in everyday statements-only form.
Journal Article
In the light of logic
Solomon Feferman is one of the leading figures in the philosophy of mathematics. This volume brings together a selection of his most important recent writings, covering the relation between logic and mathematics, proof theory, objectivity and intentionality in mathematics, and key issues in the work of Gödel, Hilbert, and Turing. A number of the papers appeared originally in obscure places and are not well-known, and others are published here for the first time. All of the material has been revised and annotated to bring it up to date.
eBook
FOUNDATIONS OF UNLIMITED CATEGORY THEORY: WHAT REMAINS TO BE DONE
Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited” or “naive” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution.
Journal Article
AXIOMS FOR DETERMINATENESS AND TRUTH
A new formal theory DT of truth extending PA is introduced, whose language is that of PA together with one new unary predicate symbol T (x), for truth applied to Gödel numbers of suitable sentences in the extended language. Falsity of x, F(x), is defined as truth of the negation of x; then, the formula D(x) expressing that x is the number of a determinate meaningful sentence is defined as the disjunction of T(x) and F(x). The axioms of DT are those of PA extended by (I) full induction, (II) strong compositionality axioms for D, and (III) the recursive defining axioms for T relative to D. By (II) is meant that a sentence satisfies D if and only if all its parts satisfy D; this holds in a slightly modified form for conditional sentences. The main result is that DT has a standard model. As an improvement over earlier systems developed by the author, DT meets a number of leading criteria for formal theories of truth that have been proposed in the recent literature and comes closer to realizing the informal view that the domain of the truth predicate consists exactly of the determinate meaningful sentences.
Journal Article
Gödel, Nagel, Minds, and Machines
Feferman talks about the contrasting views on the possible significance of Godel's theorems for minds versus machines in the development of mathematics. In particular, Feferman argues that there is a fundamental equivocation involved in those assumptions that needs to be reexamined. In conclusion, it will lead to a new way of looking at how the mind may work in deriving mathematics which straddles the mechanist and anti-mechanist viewpoints.
Journal Article
Harmonious Logic: Craig's Interpolation Theorem and Its Descendants
Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for \"reasonable\" logics extending first-order logic within the framework of abstract model theory.
Journal Article
Reflecting on incompleteness
To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles . These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories . A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity , which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally. Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.
Journal Article