Meaning and Existence in Mathematics

ISBN-13: 9783709171158 / Angielski / Miękka / 2011 / 160 str.

Charles Castonguay

Opis

The take-over of the philosophy of mathematics by mathematical logic is not complete. The central problems examined in this book lie in the fringe area between the two, and by their very nature will no doubt continue to fall partly within the philosophical re- mainder. In seeking to treat these problems with a properly sober mixture of rhyme and reason, I have tried to keep philosophical jargon to a minimum and to avoid excessive mathematical compli- cation. The reader with a philosophical background should be familiar with the formal syntactico-semantical explications of proof and truth, especially if he wishes to linger on Chapter 1, after which it is easier philosophical sailing; while the mathematician need only know that to "explicate" a concept consists in clarifying a heretofore vague notion by proposing a clearer (sometimes formal) definition or formulation for it. More seriously, the interested mathematician will find occasional recourse to EDWARD'S Encyclopedia of Philos- ophy (cf. bibliography) highly rewarding. Sections 2. 5 and 2. 7 are of interest mainly to philosophers. The bibliography only contains works referred to in the text. References are made by giving the author's surname followed by the year of publication, the latter enclosed in parentheses. When the author referred to is obvious from the context, the surname is dropped, and even the year of publication or "ibid. " may be dropped when the same publication is referred to exclusively over the course of several paragraphs.

Spis treści

1. Extension and Intension.- 1.1 The Basic Doctrine.- 1.2 A Set-theoretic Formulation.- 1.3 Extension and Intension in Formalized Theories.- 1.4 Intension as Comprehension.- 1.5 Calculi of Extensions and Intensions.- 1.6 Extension and Intension of Theories.- 1.7 Intension as Connotation: Core Intension.- 1.8 Vagueness.- 1.9 Intensional Autonomy.- 2. Meaning.- 2.1 Correspondence and Coherence Views.- 2.2 Meaning as Intension/Extension.- 2.3 Meaning of Constructs in Mathematical Theories.- 2.4 Meaning in Formal Theories.- 2.5 C. I. Lewis on Meaning.- 2.6 Truth in Theory and Truth in Practice.- 2.7 Nonexistent Possibles.- 3. Existence.- 3.1 The Thesis that Existence is Consistency.- 3.2 Empiricist Notions of Existence.- 3.3 Objectivity and Evidence.- 3.4 A Seasoned Constructivism: Piaget’s Genetic Epistemology.- 3.5 Heuristics and Mathematical Existence.- 3.6 Style.- 3.7 Sets and the Semantics of Mathematics.- 3.8 Categories and the De-ontologization of Mathematics.- 4. Reduction.- 4.1 Reduction in Mathematics.- 4.2 Meaning-preserving Correspondences.- 4.3 Explanation v. Reduction.- 4.4 Ontological Commitment.- 4.5 Ontological Reduction.- Index of Names.- Partial List of Symbols.