Some interesting links
Here are some selected links to other interesting sites on the web related to Gödel’s Theorems, the foundations of mathematics, mathematical logic, philosophy and related subjects. These pages are designed to encourage researchers yet struggling to give concrete form to their intuitive vision. Choice of sites is thus influenced by a subjective assessment of their perceived levels of passion for, belief in, and commitment towards the subject they espouse, and is not dependent simply on their existing level of authority, or on any assessment of the scientific validity of their content.
Notes ► Random thoughts and extracts from on-going correspondence on issues of current interest in mathematics, mathematical logic, philosophy and related issues.
On formally undecidable propositions of Principia Mathematica and related systems I ►A translation of Gödel’s original paper, part of an HTML presentation by Siegfried. A creditable labour of love. The 1962 translation by B. Meltzer does not read as smoothly as does that by Professor Elliott Mendelson in ‘The Undecidable’ edited by Martin Davis and published in 1965. The trivial lack is more than compensated by the convenience of finger-tip reference through the web. The effort and achievement deserves acknowledgement and appreciation, even if one takes issue with the thesis underlying the motivations that are linked to the hosting of this work on the web.
A new approach to symbolic logic ► A web-page on logic and philosophy by Peter Corsellis. The focus is on the search for an alternative logical theory that better reflects our ability to think and talk about our thoughts and our language. Currently under revision “ ... as the existing version is hopelessly confused and incomplete. I start by addressing the question of what exists, and argue that all the popular answers to this question are true to some extent. I suggest that existence is a matter of degree, and that the more things a theory assumes to exist, the less it is true.”
Partial Realizations of Hilbert's Program ► This outstanding survey of the continuing consequences arising out of Hilberts bold assertion that infinitistic mathematics can be fully validated, by Stephen G. Simpson, was published in 1988 in the Journal of Symbolic Logic, volume 53, pages 349-363. It addresses the question : what part of infinitistic mathematics can be reduced to finitistic reasoning? It offers insightful new conclusions arising out of Partial Realizations of Hilbert's Program; such as that number-theoretic theorems which are provable using contour integration can generally also be proved ‘elementarily’, i.e. within primitive recursive Arithmetic. It gives an excellent overview of The Role of Reverse Mathematics, a highly developed research program whose purpose is to investigate the role of strong set existence axioms in ordinary mathematics. It concludes with visionary Answers to Some Possible Objections.
Links to some key resouces on the web ► A link to Link2Go’s logic resources.
Research groups in Logic and Theoretical Computer Science ► A web-site maintained by the Group for Mathematical Logic at Uppsala University, Sweden.
Is Arithmetic consistent? ► A fascinating discussion in the Set Theory and Foundations section of the MathPages, maintained by Kevin Brown, about doubting the consistency of Arithmetic itself; arising out of Andrew Wiles announcement of his proof of Fermat's Last Theorem, and the sceptical response by popular writer Marylin vos Savant questioning the proof's correctness. The discussion provides a useful insight into how a broad spectrum of researchers perceive Gödel’s Undecidability Theorems, and are influenced by his work and reasoning.
Minds, Machines, and Mathematics ► A provocative article where David J. Chalmers, commenting on Roger Penrose’s book Shadows of the Mind, suggests that the deepest flaw in all Gödelian arguments, essentially articulated by John Randolph Lucas in his 1961 article Minds, Machines and Gödel, lies in their assumption, at some point, that we know that we are fundamentally sound, i.e. consistent, despite occasional aberrant behavior. Chalmers concludes with a delightfully puckish aside by introducing a light-hearted - yet illuminating - categorisation of the philosophical inclinations of the scientific community based on their attitudes towards basic questions regarding the nature and source of ‘awareness’.
(Sorry for the paucity of published links. This site is under construction. Please bear with us.)
(Updated : 10/11/01 1:49:01 AM by email@example.com)