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(for purposes here) "objects". An important component if David Lewis's modal realism is that anything can
coexist with anything, expressed initially in his unqualified principle of Recombination:
(R) For any objects (in any worlds), there is a world that contains any number of duplicates of
those objects.
Consider the following consequence of R:
(RC) For any cardinal number , it is possible that there are at least objects.
In the context of modal realism, RC entails:
(A*) For any cardinal number , there are at least objects.
But this is problematic for Lewis. Consider:
(SoA) There is a set of all objects.
In the context of ZFCU, it follows that (A*) is inconsistent with (SoA). Faced with such inconsistencies,
Lewis opted to abandon (RC) and, hence, Recombination. For, as properties are sets of objects and
propositions are sets of worlds for Lewis, many important properties and propositions will not exist if
there is no set of all objects and, hence, many of the applications of modal realism will fail.
But there is another option here. In a forthcoming paper I develop a modification ZFCU* of ZFCU that
accommodates the existence of "wide" sets, i.e., sets that, like (assuming (A*) and (SoA)) are too big
to have a definite cardinality but which have a definite rank. The key modifications are to Replacement
and Powerset. Say that a set is (mathematically) determinable if it is equipotent to some pure set. We
then our modified Replacement F* applies to sets that are either determinable or for which we have a
replacement mapping that is "bounded above" by rank on .
On the modified version PS* of Powerset, only the determinable subsets of a given set constitute a
further set. The motivation for this modification stems from Cantor's conception of the "absolute infinite",
i.e., the "size" that characterizes non-determinable collections. For Cantor, this "size" is an "absolute
quantitative maximum" that is subject to no mathematically definite increase. Accordingly, given PS*,
Cantor's theorem fails for wide sets. A final axiom further enforces the Cantorian intuition: Only
determinable sets are smaller than some other set. ZFCU* thus yields a rather different, "cylindrical"
picture of the cumulative hierarchy in which there is no mathematically definite increase in its "girth".
(ZFCU* is consistent relative to ZFCU + "There exists an inaccessible cardinal".)
Assuming the set of objects is wide, ZFCU* permits the construction of, even if not the full intuitive
power set of O, infinitely many complex sets of arbitrarily high rank over . The central focus of my
presentation will be to investigate the extent to which Lewis's program–with full Recombination–can be
restored if the modal realist adopts ZFCU*. Secondarily, I will address other implications of ZFCU* in
regard to the nature of the infinite, the structure of the comulative hierarchy, and the possibility of
absolutely general quantification.
Nicholas of Cusa's Methodology of the Infinite
Françoise Monnoyeur-Broitman (Linköping University)
During the mid-fifteenth century, Nicholas of Cusa, in his major work Of Learned Ignorance [1],
transformed the Aristotelian "potential" and "actual" infinity into a "negative" and "positive" infinity. In order
to explain these new features of the infinite, Cusa developed the method of the coïncidentia oppositorum.
In the context of what he coined as our "learned ignorance," he established a proportion between what
we know and what we do not know. First of all, Cusa's method does not treat the infinite as a logical
statement in the way the philosophers of the Middle Ages did. Their logical approach was based on
speculations upon the sophisms and paradoxes of the infinite to overcome the gap between language and
reality. For example, one of the most influential philosophers of that period, Gregory of Remini, came up
with a new logical expression of the infinite as "categorematical" or "syncategorematical" [2]. Secondly,
Cusa's perspective on the infinite was not quantitative as was Leibniz' calculus by the end of the
seventeenth century [3,4]. Instead of conducting a logical or quantitative analysis, Cusa identified various
orders of knowledge to define the infinite. According to his method of the coincidentia oppositorum, the
maximum infinite can be identified with the minimum infinite. Cusa thought that geometry was the best
science to use in order to get an understanding of the infinite by progressing through the different orders
of knowledge. He studied geometrical figures such as lines, triangles and circles, and demonstrated how
these figures can also be said to correspond to an infinite line, triangle, or circle. Cusa then applied this
model to the world, concluding that the relationship between the finite and infinite figures is analogous to
the relationship of the maximum infinite "to all things" [1].
In this presentation, I will examine how Cusa's method to reach the infinite through geometrical analysis
provides a unique standpoint for research on mathematical and philosophical infinity and the pursuit of
solutions to philosophical, mathematical and theological issues. In showing how Cusa's reasoning in his
methodology of the infinite differs from a logical analysis (Gregory of Remini) or calculation (Leibniz) of
the infinite, I will be able to underline a new way to manage the infinite. Finally, I will explain how the
different fields of geometry, theology, and philosophy interplay with each other to define the infinite.
References.
[1] Nicolaus Cusanus, Of learned Ignorance, Hyperion Press Inc : Wesport, (1954).
[2] Françoise Monnoyeur, Infini des mathematiciens, infini des philosophes, Editions Belin: Paris, 1st
Ed (1992).
[3] Françoise Monnoyeur, Infini des philosophes, infini des astronomes, Editions Belin: Paris, 1st Ed
(1995).
[4] Françoise Monnoyeur, Journal of the History of Philosophy 48 (4):527-528 (2010).
An Aristotelian Approach to Infinite Causal Sequences
Tamer Nawar (University of Cambridge)
Aristotle's antipathy to actual infinites (e.g., Physics 206b12-14) and his gnomic remark that nature flees
the infinite (Generation of Animals 715b15-17) are well known. Aristotle is often taken to rule out infinite
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History & Philosophy of Infinity: Cambridge, England
http://www.math.uni-hamburg.de/home/loewe/HiPhI/abstracts.html
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