60. Infinity and Continuity

Danie Strauss

Thursday, October 12, 2006 - 16.00u - 18.00u - VUB - Room D.1.08

Abstract: The most basic awareness of infinity is given in the intuition of one, another one, and so on indefinitely, endlessly - traditionally designated as the potential infinite. Already within Greek philosophy and mathematics this extensive infinite was turned 'inwards' - in relation to an extended continuum that is infinitely divisible - entailing that wholeness (totality) implies infinite divisibility (still merely in the sense of the potential infinite).

Although Greek philosophy and mathematics contemplated what was designated as the actual infinite particularly Aristotle argued against the actual infinite - but his arguments turned out to be defining features of the actual infinite. What is in addition remarkable is that Aristotle specified two criteria for continuity - infinite divisibility and taking every point of division twice (as end-point and as starting-point) - on the basis of rejection the actual infinite while the Cantor-Dedekind definition of continuity adheres to the same criteria on the basis of accepting the actual infinite. In order to clarify this apparent paradoxical state of affairs an alternative understanding of the mentioned two kinds of infinity is required.

This will lead us to reconsider the relationship between multiplicity (discreteness) and wholeness (continuity) attempting to avoid the historical extremes of arithmeticism and holism. It is indeed amazing that the entire history of mathematics only explored the following two options: either reduce space to number or reduce number to space. The obvious third option was never examined: accept both the uniqueness (i.e. irreducibility) of number and space as well as their mutual interconnectedness.

The following issues will receive attention in the presentation:

(i) the impasse of logicism;

(ii) the nature of the whole-parts relation;

(iii) tacit assumptions of axiomatic set theory;

(iv) ordinality and cardinality;

(v) the idea of ontic modes (aspects - Bernays, Gödel, Dooyeweerd);

(vi) basic concepts of mathematics;

(vii) a deepened understanding of infinity; an "as if" approach to the actual infinite (as regulative hypothesis);

(viii) the circularity present in the relation between Hilbert and Kant in respect of the actual infinite;

(ix) mathematics and logic;

(x) the circularity in modern set theoretic attempts to arithmetize mathematics completely;

(xi) solving the apparent contradiction in the Aristotelian and Cantor-Dedekind understanding of continuity.

About the speaker

Danie Strauss studied mathematics and philosophy at the University of the Free State and completed his "doctoraalexamen" and promotion at the Free University of Amsterdam (the latter on the distinction between "Begrip en Idee" - 1973). During the past 30 years his focus was on the philosophical foundations of the natural sciences and social sciences. During this period he has published 15 books and about 200 articles. Most recently two books appeared at Peter Lang: (i) Paradigmen in Mathematik, Physik und Biologie und ihre philosophische Wurzeln 2005 (216 pp.); (ii) Reintegrating Social Theory - Reflecting upon human society and the discipline of sociology 2006 (300 pp.). Currently he is working on a new book: Is Philosophy the Discipline of Disciplines? (destined to appear in 2007). He served as Head of the Department of Philosophy for 20 years and for one term (between 1997 and 2002) as Dean of the Faculty of Humanities. He is currently still attached to the Dean's office. He is the General Editor of the Collected Works of the Dutch philosopher Herman Dooyeweerd (during the past decade 12 Volumes appeared).