The Great Conversation:
Infinity
By Gabriel Blanchard
Naturally enough for finite minds like ours, infinity is a tricky concept.
Like eternity, infinity is an idea we can articulate but which we never experience. It is also a little more difficult to define; eternity is related to time in particular, whereas infinity can be related to any thing or quality (one could, loosely, describe infinity as the genus of which eternity is a species). Even smallness can be related to infinity—something that has been infinitely divided is infinitesimal.
The literal meaning of infinity is “without end” or “without boundary,” which tends to imply space. The cosmos was thought to be spatially infinite for some time, between the collapse of the geocentric model and certain advances in astrophysics made in the twentieth century, and it is ironically as difficult to picture space as such simply stopping somewhere as it is to picture an infinite amount of space; what we tend to do is picture a finite universe located in some larger space, like nesting dolls. As for traversing space, the possibility of infinite speed was all but disproven by Einstein, whose study of physics determined that light itself has a determinate rate of travel.
Partly for these reasons, but also because they are by nature concerned with particulars, the physical sciences have done less with the idea of infinity than have disciplines like speculative philosophy and abstract mathematics. The latter, indeed, gives us the very concept of infinity: among other things, infinity expresses the fact that there is no such thing as “the highest number.” Many mathematical functions, such as asymptotes, spirals, and fractals, or the famous Möbius strip (a real physical object with only one side), lean on the strange boundary between finitude and infinity in their construction—obviously limited, and yet, in terms of measurement, incalculable. For that matter, incommensurable quantities and irrational numbers point in the same direction.
Homer composed the Odyssey; if we postulate an infinite period of time, with infinite circumstances and changes, the impossible thing is not to compose the Odyssey, at least once.
Mathematics and theoretical philosophy intersect in the region of probability theory, which itself has strange ties to infinity. The “infinite monkey theorem” states that, given infinite time, even random processes will eventually produce every intrinsically possible result; the name comes from the example of a monkey with a typewriter being given infinite time, in which case it would be statistically bound, after some interval, to produce a word-perfect copy of Shakespeare’s Hamlet. This is stood amusingly on its head by Tom Stoppard’s existentialist comedy Rosencrantz and Guildenstern Are Dead, in which a genuine and unweighted coin is flipped several hundred times and comes up heads every time.
Pure philosophy has made more extensive use of the idea of infinity, especially theological philosophy. The doctrine that God is infinite is common to nearly all Christian belief, and occurs in other religious traditions with some frequency. St. Thomas‘ interpretation of divine infinity rested on the idea that limitedness is relational, i.e., a thing is finite if it is related to something else that limits it; he used the example of form and matter, which mutually limit one another: form imposes one structure rather than another upon matter, and matter confers individual being upon its instantiation of form. God, being self-existent and eternal, is not inherently related to any other thing, and is therefore infinite. A generation later, John Duns Scotus gave a very different meaning to God’s infinity, using a new version of the ontological argument. Scotus reasoned that, because existence and infinity are not contradictory concepts, an infinite being is therefore inherently possible; from here, he argued (for reasons we have no time to explore) that this inherently possible thing was necessarily real. This version of the ontological argument has seen a surprising revival in more recent times, with some philosophers and mathematicians (such as Kurt Gödel, held by many to be as important a figure in the history of logic as Aristotle himself) finding it fully persuasive.
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If you liked this post, take a look at some of our other content here at the Journal, like these author profiles of Confucius, Boethius, and John Milton, or these “Great Conversation” pieces on the nature of language, the history of poetry, and the virtue of temperance. And be sure to check out our podcast, Anchored, released every Thursday and hosted by our founder Jeremy Tate.