### Taming the Boundless

#### Posted by David Corfield

In his article – The Invisible Link Between Mathematics and Theology, in *Perspectives on Science and Christian Faith*, Vol. 56, pp. 111-116 – Ladislav Kvasz argues for the thesis 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. Taking the first of these, we read

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 ofapeironhad a much broader meaning. It applied not only to that which was infinite, but also to everything that had no boundary (i.e. noperas), that was indefinite, vague or blurred. According to ancient scholarsapeironwas something lacking boundaries, lacking determination, and therefore uncertain. Mathematical study ofapeironwas impossible, mathematics being the science of the determined, definite and certain knowledge. That which had noperas, 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 ofapeironwas 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

apeironwas 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… (

De Docta Ignorantia, 1440, 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.

The fundamental differences between early modern mathematics and the mathematics of the Hellenistic period can be perhaps characterized as breaking of the boundaries of the unequivocally given and opening of the world of mathematics to the ambiguously given phenomena such as infinity, randomness, or motion. This is a fundamental change, perhaps the most important one since the discovery of proof and of the idea of an axiomatic system. And this fundamental change, this radical break towards modernity, is most likely linked with monotheistic theology.

We may wonder how infinity, chance, the unknown, space and motion have been treated mathematically in other cultures. The Wikipedia article on infinity has the Jains as the first to distinguish between different infinite sizes.

## Re: Taming the Boundless

Thanks for this extremely illuminating post.

I recall reading Stanley L. Jaki’s book The Road of Science and the Ways to God on the role of Christianity in the formation of modern empirical science. If I recall correctly, Jaki held that “law” in the sense of universal invariant was tied to monotheistic notions of the transcendental universal law given by God.

Joseph Needham judged this Judaeo-Christian-Islamic understanding of divine law to be one reason that the Chinese, in spite of their more highly developed culture in other respects, did not develop empirical science - they could not view nature as imbued with universal invariants because they hated the memory of their first emperor, a tyrant who propounded such laws with the sword.

Regards,

Mike Gogins