Mathematics without Numbers

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters-one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

The Oxford handbook of philosophy of mathematics and logic

Part 1 Introduction. Part 2 Infinity, mathematics' persistent suitor: incommensurable lengths, irrational numbers Newton and Leibniz go forward, and faith will come to you vibrating strings infinity spurned infinity embraced. Part 3 Sets of points: infinite sizes infinite orders integration absolute versus transfinite paradoxes. Part 4 What are sets?: Russell Cantor appendix A - letter from Cantor to Jourdain, 9 July 1904 appendix B - on an elementary question of set theory. Part 5 The axiomatization of set theory: the axiom of choice the axiom of replacement definiteness and Skolem's paradox Zermelo go forward, and faith will come to you. Part 6 Knowing the infinite: what do we know? what can we know? getting from here to there. Part 7 Leaps of faith: intuition physics modality second-order logic. Part 8 From here to infinity: who needs self-evidence? picturing the infinite the finite mathematics of indefinitely large size the theory of zillions. Part 9 Extrapolations: natural models many models one model or many? sets and classes natural axioms second thoughts schematic and generalizable variables.

Understanding the infinite

30 hours Syntax and semantics of higher order logic. Expressivity of first and higher order logic: how to formalize finiteness. Finite models. Between first and second order: monadic logic, fixpoint logics, predicative logics. Henkin semantics and completeness. Proof theory of higher order logic. Higher order logic in mathematics: how to formalize Peano and Dedekind. Higher order logic in computer science: finite models and complexity, type systems, proofs–as–programs and computational theorem proving. The course is somewhat advanced and assumes the contents of the course Applied Logic.

Higher order logic

In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether (a particular form of) second-order logic can count as logic in some (further to be specified) proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not have the “epistemic innocence” which would be needed for, e.g., foundational programmes in (the philosophy of) mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense.

Second-order logic : ontological and epistemological problems

Publisher Summary This chapter explains set theory and higher-order logic. Several mutual applications of set theory and higher-order logic are developed. Second-order logic is used to discover the standard models of Zermelo–Fraenkel set theory. Consideration of standard models leads to the introduction of new systems of set theory; one of these systems is applied in finding a definition of truth for higher-order sentences and Zerrnelo–Fraenkel set theory with individuals is given a philosophical justification as logically true within higher-order logic. The chapter considers three well-known first-order theories. The first, called “Peano's arithmetic,” has the non-logical constants. The second theory, called “the theory of real closed fields” has the non-logical constants. The last principle is called the “continuity schema” and plays a role analogous to that of the induction schema.

Set Theory and Higher-Order Logic

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Set Theory and Higher-Order Logic