20 research outputs found
From axiomatization to generalizatrion of set theory
The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first
two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within
the mathematical approach, in the light of the significance of Cohen's Independence results
Poincaré's philosophy of mathematics
The primary concern of this thesis is to investigate
the explicit philosophy of mathematics in the work of
Henri Poincare. In particular, I argue that there is
a well-founded doctrine which grounds both Poincare's
negative thesis, which is based on constructivist
sentiments, and his positive thesis, via which he retains
a classical conception of the mathematical continuum.
The doctrine which does so is one which is founded on
the Kantian theory of synthetic a priori intuition.
I begin, therefore, by outlining Kant's theory of the
synthetic a priori, especially as it applies to mathematics.
Then, in the main body of the thesis, I explain how the
various central aspects of Poincare's philosophy of
mathematics - e.g. his theory of induction; his theory
of the continuum; his views on impredicativiti his
theory of meaning - must, in general, be seen as an
adaptation of Kant's position. My conclusion is that
not only is there a well-founded philosophical core to
Poincare's philosophy, but also that such a core provides
a viable alternative in contemporary debates in
the philosophy of mathematics. That is, Poincare's
theory, which is secured by his doctrine of a priori
intuitions, and which describes a position in between
the two extremes of an "anti-realist" strict constructivism
and a "realist" axiomatic set theory, may indeed be
true
Traditional logic and the early history of sets, 1854–1908
Ministerio de Educación y Ciencia (España
Ancient Logic and its Modern Interpretations: Proceedings of the Buffalo Symposium on Modernist Interpretations of Ancient Logic, 21 and 22 April, 1972
Articles by Ian Mueller, Ronald Zirin, Norman Kretzmann, John Corcoran, John Mulhern, Mary Mulhern,Josiah Gould, and others.
Topics: Aristotle's Syllogistic, Stoic Logic, Modern Research in Ancient Logic
Denotational proof languages
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.Includes bibliographical references (p. [417]-421).by Konstantinos Arkoudas.Ph.D