Are Mathematical Theories Reducible to Non-analytic Foundations?
Axiomathes 23 (1):109-135 (2013)
| Abstract | In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process | |||||||||
| Keywords | Axiom of Choice Finitistic First-level idealization Gödel’s incompleteness theorems Individual-substrate Infinite Intentionality Second-level idealization Skolem-Löwenheim Theorem | |||||||||
| Categories | No categories specified (fix it) | |||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,705 |
| External links |
 |
| Through your library | Configure |
Similar books and articlesRaymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.
Alexander George (1985). Skolem and the Löwenheim-Skolem Theorem: A Case Study of the Philosophical Significance of Mathematical Results. History and Philosophy of Logic 6 (1):75-89.
Robert W. Batterman (2009). Idealization and Modeling. Synthese 169 (3):427 - 446.
Marek Zawadowski (1985). The Skolem-Löwenheim Theorem in Toposes. II. Studia Logica 44 (1):25 - 38.
Timothy Bays (2009). Skolem's Paradox. In Edward N. Zalta (ed.), Stanford Encyclopedia of Philosophy.
Jan Von Plato (2007). In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs. Bulletin of Symbolic Logic 13 (2):189-225.
Carlo Cellucci (1993). From Closed to Open Systems. In J. Czermak (ed.), Philosophy of Mathematics, pp. 206-220. Hölder-Pichler-Tempsky.
Robert J. Titiev (1998). Finiteness, Perception, and Two Contrasting Cases of Mathematical Idealization. Journal of Philosophical Research 23:81-94.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Peter Smith (2007). An Introduction to Gödel's Theorems. Cambridge University Press.
Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.
Analytics
Monthly downloads |
Added to index2012-02-02Total downloads9 ( #114,188 of 549,129 )Recent downloads (6 months)1 ( #63,397 of 549,129 )How can I increase my downloads? |
My notes
Discussion
