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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.


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.

Infinity symbol April 8, 2009

Posted by David Pierce in Uncategorized.

Why is infinity denoted by a lemniscate?

Lemniscates (figure-eights)

In a recent talk in Ankara, Sasha Borovik used a photograph like those in his post Manifestation of Infinity. How does infinity appear in a picture of a ferry approaching a dock? One person in the audience suggested that a pair of tires on the side of the ferry formed the infinity symbol.

I speculate that the lemniscate is the simplest shape suggesting endlessness that will not be confused with the symbol for emptiness: the zero. That the zero is naturally a symbol for emptiness is suggested in the eighth Oxherding picture, Bull and Self Transcended:

[picture: an empty circle]

Whip, rope, person, and bull — all merge in No-Thing.
This heaven is so vast no message can stain it.
How may a snowflake exist in a raging fire?
Here are the footprints of the patriarchs.

Comment: Mediocrity is gone. Mind is clear of limitation. I seek no state of enlightenment. Neither do I remain where no enlightenment exists. Since I linger in neither condition, eyes cannot see me. If hundreds of birds strew my path with flowers, such praise would be meaningless.

(English text by Reps and Senzaki.)

How significant the lemniscate may be in the East, I do not know. I was able to find, on one yoga website, a suggestion to visualize a figure-eight while practicing Spinal Breath:

There are a variety of practices with awareness moving up and down the spine with the breath. One may do this practice between particular energy centers (chakras) or form different shapes of the visualized flow, including elliptical or a figure-eight…

The most straight forward, and yet completely effective method is to:

  • Imagine the breath flowing from the top of the head, down to the base of the spine on exhalation, and to
  • Imagine the flow coming from the base of the spine to the top of the head on inhalation.
  • This may be done lying down, or in a seated meditation posture.

One may simply experience the breath, or may be aware of a thin, milky white stream flowing in a straight line, up and down. This practice is very subtle when experienced at its depth, and can turn into a profoundly deep part of meditation practice.