232 research outputs found
Predicativity and parametric polymorphism of Brouwerian implication
A common objection to the definition of intuitionistic implication in the
Proof Interpretation is that it is impredicative. I discuss the history of that
objection, argue that in Brouwer's writings predicativity of implication is
ensured through parametric polymorphism of functions on species, and compare
this construal with the alternative approaches to predicative implication of
Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Mathematical Logic: Proof Theory, Constructive Mathematics
[no abstract available
Topological Foundations of Cognitive Science
A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers:
** Topological Foundations of Cognitive Science, Barry Smith
** The Bounds of Axiomatisation, Graham White
** Rethinking Boundaries, Wojciech Zelaniec
** Sheaf Mereology and Space Cognition, Jean Petitot
** A Mereotopological Definition of 'Point', Carola Eschenbach
** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel
** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda
** Defining a 'Doughnut' Made Difficult, N .M. Gotts
** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts
** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi
** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki
A Universal Characterization of the Double Powerlocale
This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297{321.
The double powerlocale P(X) (found by composing, in either order,the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Locop; Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects P(X) comprising, respectively, the meet semilattice and join
semilattice homomorphisms. A key lemma shows that, for any locales X and Y , natural transformations from SX (the presheaf Loc
Kriesel and Wittgenstein
Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical
logician during a formative period when the subject was becoming
a sophisticated field at the crossing of mathematics and logic. Both with his
technical sophistication for his time and his dialectical engagement with mandates,
aspirations and goals, he inspired wide-ranging investigation in the metamathematics
of constructivity, proof theory and generalized recursion theory.
Kreisel's mathematics and interactions with colleagues and students have been
memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of
interpersonal conceptual interaction, Kreisel during his life time had extended
engagement with two celebrated logicians, the mathematical Kurt GĂśdel and
the philosophical Ludwig Wittgenstein. About GĂśdel, with modern mathematical
logic palpably emanating from his work, Kreisel has reflected and written
over a wide mathematical landscape. About Wittgenstein on the other hand,
with an early personal connection established Kreisel would return as if with
an anxiety of influence to their ways of thinking about logic and mathematics,
ever in a sort of dialectic interplay. In what follows we draw this out through
his published essaysâand one letterâboth to elicit aspects of influence in his
own terms and to set out a picture of Kreisel's evolving thinking about logic
and mathematics in comparative relief.Accepted manuscrip
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