Infinity, Finitude

Here is another area where I find myself with mixed sympathies.

Plato seems to have regarded infinity — or what he called the Unlimited — as something bad. Aristotle argued that infinity exists only in potentiality and not in actuality, a view I find highly attractive. I think I encounter a world of seemingly infinite structure but only finite actualization.

Some time in the later Hellenistic period, notions of a radical spiritual infinity seem to have appeared in the West for the first time, associated with the rise of monotheism and the various trends now commonly called Gnostic. This kind of intensive rather than extensive infinity sometimes seems to be folded back on itself, evoking infinities of infinities and more. The most sophisticated development of a positive theological infinite in later Western antiquity occurred in the more religious rethinking of Greek philosophy by neoplatonists like Plotinus, Proclus, and Damascius.

In 14th century CE Latin Europe, Duns Scotus developed an influential theology that made infinity the principal attribute of God, in contrast to the pure Being favored by Aquinas. Giordano Bruno, burned at the stake in 1600, was a bombastic early defender of Copernican astronomy and notorious critic of established religion who espoused a curious hybrid of Lucretian atomistic materialism, neoplatonism, and magic. He proclaimed the physical existence of an infinity of worlds like our Earth.

Mathematical applications of infinity are a later development, mainly associated with Newton and Leibniz. Leibniz in particular enthusiastically endorsed a speculative reversal of Aristotle’s negative verdict on “actual infinity”. Nineteenth century mathematicians were embarrassed by this, and developed more rigorous reformulations of the calculus based on limits rather than actual infinity. The limit-based formulation is what is generally taught today. Cantor seemingly went in the opposite direction, developing infinities of infinities in pure mathematics. I believe there has been another reformulation of analysis using category theory that claims to equal the rigor of 19th century analysis while recovering an approach closer to that of Leibniz, which might be taken to refute an argument against infinity based solely on lack of rigor according to the standards of contemporary professional mathematicians. One might accept this and still prefer an Aristotelian interpretation of infinity as not applicable to actual things, though it is important to recall that for Aristotle, the actual is not all there is.

The philosophy of Spinoza and even more so Leibniz is permeated with a positive view of the infinite — both mathematical and theological — that in a more measured way was later also taken up by Hegel, who distinguished between a “bad” infinite that seems to have been an “actual” mathematical infinite having the form of an infinite regress, and a “good” infinite that I would gloss as having to do with the interpretation of life and all within it. Nietzsche’s Eternal Return seems to involve an infinite folding back on itself of a world of finite beings. (See also Bounty of Nature; Reason, Nature; Echoes of the Deed; Poetry and Mathematics.)

On the side of the finite, I am tremendously impressed with Aristotle’s affirmative development of what also in a more Kantian style might be termed a multi-faceted “dignity” of finite beings. While infinity may be inspiring or even intoxicating, I think we should be wary of the possibility that immoderate embrace of infinity may lead — even if unwittingly — to a devaluation of finite being, and ultimately of life. I also believe notions of infinite or unconditional power (see Strong Omnipotence; Occasionalism; Arbitrariness, Inflation) are prone to abuse. In any case, ethics is mainly concerned with finite things.