Objects as Limits of Experience and the Notion of Horizon in Mathematical Theories

Stathis Livadas

Stathis Livadas University of Patras

Abstract

The present work is an attempt to bring attention to the application of several key ideas of Husserl's Krisis in the construction of certain mathematical theories that claim to be altenative nonstandard versions of the standard Zermelo-Fraenkel set theory. In general, these theories refute, at least semantically, the platonistic context of the Cantorian system and to one or the other degree are motivated by the notions of the lifeworld as the pregiven holistic field of experience and that of horizon as the boundary of human perceptions and the de facto constraint in reaching limit-idealizations. Moreover, I try to give convincing reasons for the existence of an ultimate constitutional "vacuum" of a subjective origin that is formally reflected in the application of a notion of actual infinity in dealing generally with the mathematical infinite.
Keywords: Actual infinity; alternative theory; field of expenence; horizon; life-world; limit-idealization; nonstandard

References

DROSSOS, A. C. (1989). Foundations of fuzzy sets: A nonstandard approach. In: Fuzzy Sets and Systems, 37: 287-307.
HUSSERL, E. (1992). Formale und Transzendentale Logik. Text nach Husserliana XVII. Hsg. E. Ströker. Hamburg: Felix Meiner.
— (1986). Die Phänomenologie und die Fundamente der Wissenschaften. Text nach Husserliana V. Hsg. K.-H. Lembeck. Hamburg: Felix Meiner.
— (1985). Texte zur Phänomenologie des inneren Zeitbewusstseins (1893-1917). Text nach Husserliana X. Hsg. R. Bernet. Hamburg: Felix Meiner.
— (1983). Ideas pertaining to a pure phenomenology and to a phenomenological Philosophy. First Book. Transl. F. Kersten. The Hague: M. Nijhoff.
— (1976). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch. Husserliana III/1. Hsg. Karl Schuhmann. Den Haag: M. Nijhoff.
— (1973). Ding und Raum: Vorlesungen. Husserliana XVI. Hsg. U. Claesges. Den Haag: M. Nijhoff.
— (1970). The Crisis of European Sciences and Transcendental Phenomenology. Transl. D. Carr. Evanston: Northwestern University Press.
LIVADAS, S. (2005). The Phenomenological Roots of Nonstandard Mathematics. In: Rom. Jour. of Information Science and Technology, 8 (2): 115-136.
LONGO, G. (2005). The Reasonable Effectiveness of Mathematics and its Cognitive Roots. In: Geometries of Nature, Living Systems and Human Cognition. L. Boi (ed.). Singapore: World Scientific, 35l-382.
NELSON, E. (1986). Predicative Arithmetic. Mathematical notes. Princeton: Princeton Univ. Press.
PETITOT, J. (1999). Morphological Eidetics for a Phenomenology of Perception. In: Naturalizing Phenomenology. J. Petitot et al, eds. Stanford: Stanford University Press, 330-372.
TZOUVARAS, A. (1998). Modeling Vagueness by Nonstandardness. In: Fuzzy Sets and Systems, 94: 385-396.
VAN ATTEN, M.; VAN DALEN, D.; and TIESZEN, R. (2002). Brower and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum. In: Philosophia Mathematica, 10 (3): 203-226.
VOPĚNKA, P. (1991). The Philosophical Foundations of Alternative Set Theory. In: Int. J. General Systems, 20: 115-126.
— (1979). Mathematics in the Alternative Set Theory. Teubner-Texte zur Mathematik. Leipzig: Teubner.
WEYL, H. (1991). The Continuum. Transl. Pollard, S. & Bole T. New-York: Dover.