## The Invisible Dialog Between Mathematics and Theology April 29, 2009

Posted by dcorfield in Uncategorized.

An interesting paper by Ladislav Kvasz — The Invisible Dialog Between Mathematics and Theology, in Perspectives on Science and Christian Faith, Vol. 56, pp. 111-116.

The thesis of the paper is that monotheistic theology with its idea of the omniscient and omnipotent God, who created the world, influenced in an indirect way the process of this mathematicization. In separating ontology from epistemology, monotheistic theology opened the possibility to explain all the ambiguity connected to these phenomena as a result of human finitude and so to understand the phenomena themselves as unambiguous, and therefore accessible to mathematical description.

This thesis is explored through five notions: infinity, chance, the unknown, space and motion.

What we refer to today as infinite was in Antiquity subsumed under the notion of apeiron ($\alpha \pi \epsilon i \rho o \nu$).  Nevertheless, compared with our modern notion of infinity, the notion of apeiron had a much broader meaning. It applied not only to that which was infinite, but also to everything that had no boundary (i.e. no peras), that was indefinite, vague or blurred. According to ancient scholars apeiron was something lacking boundaries, lacking determination, and therefore uncertain. Mathematical study of apeiron was impossible, mathematics being the science of the determined, definite and certain knowledge. That which had no peras, could not be studied using the clear and sharp notions of mathematics.

Modern mathematics, in contrast to Antiquity, makes a distinction between infinite and indefinite. We consider the infinite, despite the fact that it has no end (finis), to be determined and unequivocal, and thus accessible to mathematical investigation. Be it an infinitely extended geometrical figure, an infinitely small quantity or an infinite set, we consider them as belonging to mathematics. The ancient notion of apeiron was thus divided into two notions: the notion of the infinite in a narrow sense, which became a part of mathematics, and the notion of the indefinite, which, as previously, has no place in mathematics.

So

While for the Ancients apeiron was a negative notion, associated with going astray and losing the way, for the medieval scholar the road to infinity became the road to God. God is an infinite being, but despite His infiniteness, He is absolutely perfect. As soon as the notion of infinity was applied to God, it lost its obscurity and ambiguity. Theology made the notion of infinity positive, luminous and unequivocal. All ambiguity and obscurity encountered in the notion of infinity was interpreted as the consequence of human finitude and imperfection alone. Infinity itself was interpreted as an absolutely clear and sharp notion, and therefore an ideal subject of mathematical investigation.

Evidence for the change from the Ancients is provided by Kvasz in his book Patterns of Change where he quotes Nicholas of Cusa on page 77:

It is already evident that there can be only one maximum and infinite thing. Moreover, since any two sides of any triangle cannot, if conjoined, be shorter than the third: it is evident that in the case of a triangle whose one side is infinite, the other two sides are not shorter. And because each part of what is infinite is infinite: for any triangle whose one side is infinite, the other sides must also be infinite. And since there cannot be more than one infinite thing, you understand transcendently that an infinite triangle cannot be composed of a plurality of lines, even though it is the greatest and truest triangle, incomposite and most simple… (Nicholas of Cusa 1440, p. 22) De Docta Ignorantia, trans. J. Hopkins.

It may seem odd to us that Nicholas could not imagine the limit as an isosceles triangle of fixed base is extended, but the point is that such a discussion of an infinitely large object would have been unthinkable for the Greeks.

1. JT - April 29, 2009

Just a bibliographic note, though it’s “Dialog” here and on the author’s publications page, it’s “Link” on the publisher’s site [1]. The publisher’s site also provides a nice PDF offprint.

2. Kip Sewell - May 16, 2009

I’ve enjoyed reading “A Dialogue on Infinity” and would like to comment on your post about “The Invisible Dialog Between Mathematics and Theology,” which raises an interesting distinction between apeiron and infinity. Apeiron was said to be of two sorts—the infinite and the indefinite; the indefinite, it was thought by the ancients, cannot be studied by mathematics since it is indeterminate, while the infinite can since it is determined. I’d like to add a wrinkle to these distinctions if I may.

It could be argued (as it is at http://www.scribd.com/doc/14703960/The-Case-Against-Infinity) that infinity is not really as determinate or as meaningful a notion as is often assumed. Instead, it could be argued that infinity only seems to be meaningful, but is actually a self-contradictory notion. That, of course, implies that infinity is as meaningless as a square circle or a four-sided triangle.

If so, the concept of infinity becomes incoherent unless it is reduced purely to its etymological sense as that which is simply “not finite,” which is to say that the concept of infinity becomes logically indistinguishable from zero (which is also “not finite”)—a use of the term “infinity” that no one would want to make use of in regular discourse. Otherwise, the term “infinity” as it is used in ordinary language simply becomes a misnomer for the indefinite—that is, whatever is finite but goes on “indefinitely.”

If this analysis is correct, and infinity is actually an otherwise meaningless notion, then all that is, is finite. It’s sometimes retorted that infinity can’t be meaningless because Cantor tamed the infinite. However, it could also be that Cantor’s “transfinite” mathematics hasn’t made infinity an intelligible notion at all, but instead has overturned the assumption of the ancient Greek scholars that apeiron cannot be studied mathematically. In other words, Cantor’s transfinite mathematics could really be applied as a mathematics of the indefinite, revealing degrees of indefiniteness rather than degrees of infinity, if such distinctions are really needed.

As for the theological implications, I can only suggest that if infinity is indeed meaningless in its usual senses, then the attributes of the divine would have to be finite (perhaps as apeiron—that is, indefinite—rather than infinite) in order to be meaningful in any literal sense.

I would like to know any thoughts you may have on this position.

Respectfully yours,

Kip Sewell

Kip Sewell - January 7, 2011

UPDATE: In the previous post I remarked that Cantor’s transfinite mathematics might be revised in the form of an indefinite mathematics. However, I now believe this to be incorrect for two reasons. First, there is a difference between the infinite as defined in transfinite mathematics and the concept of the indefinite. Transfinite mathematics is based on the idea that an infinite set is equal to an infinite subset (producing logical contradictions as I’ve pointed out in my article on the subject); but it is not at all clear that an indefinite set can be “equal” to an indefinite subset. Second,there is no need to attempt such a revision of transfinite mathematics; Dr. Shaughan Lavine’s system of finite mathematics uses indefinite sets instead of infinite sets and does so quite effectively (See his article, “Finite Mathematics” in Synthese, 1995). Hence, Lavine’s system makes a revision of transfinite mathematics superfluous for revealing the genuine nature of indefinite sets.

In addition, a revised copy of my own challenge to the notion of infinity can be found at PhilPapers.org for those who are interested. Here is the URL:
http://philpapers.org/archive/SEWTCA.2.pdf

Very Respectfully,

Kip Sewell

3. mariana - May 26, 2009

Probably an infinite quantity can be understood from both parts by considering the idea of a never ending entity.
On the other side since science seems to be without answers about the indefinite numbers, and there is a need for indefinite numbers for coherence I propose utilizing the fuzzy logic operators with fuzzy number, therefore you have completely indefinite number and completely acceptable mathematical operations.

At the very least, I hope that this type of proposed opinion exchange helps people get past the simplistic pigeonholing that all too often occurs when discussing science and religion—that religious people are “airheads and stubborn to science” and scientists are “cold materialists without a spiritual side.” I, for one, am a bit of both of these things.

4. Kip Sewell - May 27, 2009

Mariana’s idea of using “fuzzy logic operators with fuzzy number” to represent degrees or values of the indefinite is intriguing. It would be interesting to see examples of how this would work in practice.

5. mariana - May 27, 2009

Thanks kip, I am developing a fuzzy clustering module, it is indeed for linguistic stuff but it could be used in the future.

6. Aidan A. Kelly, Ph.D (Theology, GTU, 1980) - November 7, 2009

I’ve just stumbled across thia, and have not read your archives, but I’ve been contemplating the relations between the mathematics and theology of infinity for several decades now. Let me pose a few suggestions, hoping you haven’t already exhausted these concepts.

I suggest that:
there is not just one infinity, but an infinity of infinities, since Kantor demonstrated that there are at least two different infinities even in a simplified mathematical model;

human concepts, such as same and different, one and many, spirit and matter, simply do not apply to the infinite;

each infinity is itself the ultimate reality;

the ultimate reality is an infinite compassionate consciousness that is both male and female (yes, I am suggesting that gender, unlike matter energy, or spacetime, is an ultimate reality):

I look forward to seeing what responses I might get to this. I’ll be happy to explain the reasoning that led me to these hypotheses.

I’ll try following this as a blog, if I can figure out how.