Retrieving the Mathematical Mission of the Continuum Concept from the Transfinitely Reductionist Debris of Cantor’s Paradise. Extended Abstract
International Journal of Pure and Applied Mathematics (forthcoming)
|Abstract||What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? The double-edged purpose of the present study is to save from the transfinite deadlock of higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum.|
|Keywords||the mathematical Continuum higher set theory post-Goedelian incompleteness scheme post-Turingean halting principle|
|External links||This entry has no external links. Add one.|
|Through your library||Configure|
Similar books and articles
Edward G. Belaga, From Traditional Set Theory – That of Cantor, Hilbert , Gödel, Cohen – to Its Necessary Quantum Extension.
Edward G. Belaga, Halfway Up To the Mathematical Inﬁnity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transﬁnite Design.
Kai Hauser (2002). Is Cantor's Continuum Problem Inherently Vague? Philosophia Mathematica 10 (3):257-285.
Mary Tiles (1989/2004). The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Dover Publications.
Anne Newstead (2001). Aristotle and Modern Mathematical Theories of the Continuum. In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
Matthew E. Moore (2007). The Genesis of the Peircean Continuum. Transactions of the Charles S. Peirce Society 43 (3):425 - 469.
Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.
Anne Newstead & Franklin James (2008). On the Reality of the Continuum. Philosophy 83 (01):117-28.
Christopher Menzel (1984). Cantor and the Burali-Forti Paradox. The Monist 67 (1):92-107.
Richard Schlegel (1965). The Problem of Infinite Matter in Steady-State Cosmology. Philosophy of Science 32 (1):21-31.
Richard Tieszen (1992). Kurt Godel and Phenomenology. Philosophy of Science 59 (2):176-194.
Gianluigi Oliveri (2006). Mathematics as a Quasi-Empirical Science. Foundations of Science 11 (1-2).
Added to index2011-03-15
Total downloads42 ( #26,949 of 548,977 )
Recent downloads (6 months)32 ( #1,261 of 548,977 )
How can I increase my downloads?