Some conversations never end. There are conversations with your relatives which somehow reappear at every gathering, where you rehash the same things you rehashed the last time, and the time before that, and which you’ll proceed to rehash again at the next one. There are useless, poorly run meetings at work which should have been emails. And there are conversations about the fundamental assumptions of mathematics which have been continuing since the great thinkers of Ancient Greece some 2,600 years ago.
That’s roughly when Aristotle introduced the idea of “potential infinity,” which is an infinity that does not exist but could exist. Think of it as the infinity of counting to the highest number. You’ll never reach the number, so the infinity doesn’t really exist, but you can approach the infinity through a process of counting, and counting, and counting some more. Potential infinity is distinct from actual infinity, which arose as a concrete mathematical concept primarily in the nineteenth century as part of work in set theory. That’s when the set of real numbers was defined as being infinite, not just potentially infinite, and when mathematicians recognized orders or sizes of infinity. Yes, some infinities are larger than others. Quanta Magazine is running a series of essays on the foundations of mathematics which addressed this period of development in some detail. In its most recent essay, “What Can We Gain by Losing Infinity,” a counterargument is explored. Instead of assuming the existence of infinity, these “ultrafinitests” assert infinity does not exist, or at least cannot be actualized (returning to the potential infinity of Aristotle’s day).
Infinity might not be one of those things you think about on a regular basis, although there is a better than average chance it is given you read this blog. It’s one of those abstract mathematical concepts which, like gravity, we are mostly content to allow to exist in the background and operate on top of it without giving much thought to how it influences the ways we interact with the world. Like zero, infinity is simply a default of our mathematical landscape. Most of us are taught some basics about how it operates and how to think about it in constrained contexts, like introductory calculus, and leave it there. I’m certainly not mathematician enough to approach the problem in a mathematically rigorous way – then again, most of the arguments cited in Quanta’s article are more philosophical or metaphysical than mathematical. The article caught my attention enough to write a post about it because the ultrafinitests raise the same sorts of questions I have whenever I think about the basis of integral and derivative calculus or the nature of limits, and because they echo the same arguments Zeno had with Aristotle.
If you haven’t already, I suggest you go back and read our post on Zeno’s paradoxes. We won’t rehash them specifically here, but you surely see the echoes. If you’re familiar with calculus, consider the Reimann Sum. In most introductory calculus courses, the notion of an integral is explained as finding the area under a curve by breaking the area up into smaller and smaller rectangles, with “infinitely” small widths, and heights equal to the height of the curve. That’s how I explain it to people, too. The problem is that, as one of Zeno’s paradoxes exemplifies, that should drive the area under the curve computationally to either zero or infinity. The way mathematics treats infinity and limits prevents that from happening and gives us the integral function, but there’s an intermediate step which is not immediately apparent from an intuitive understanding of what is meant by “zero” or “infinity.”
Mathematics is a strange field. It’s both a human system, something we invented and continue to invent, and something which seems to arise naturally and inevitably – hence the ongoing debate over whether new mathematical insights are “invented” or “discovered.” Gödel’s work in the twentieth century found the axiomatic basis of modern mathematics to be ultimately unprovable – that is, you must take certain things as assumptions and build from there. If the universe is finite, can there be infinity in mathematics? Does it matter? Should mathematics be tied to physical reality, or is mathematics a tool which can be applied so long as it is useful? Perhaps I am too steeped in the mathematics of infinity to fully appreciate the arguments of the purest ultrafinitests, but I do not see a need to do away with infinity entirely. A return to potential infinity rather than actual infinity, though, fits better with my personal intuition about the nature of zero and infinity. There’s probably no one answer to this problem (no matter how much we’re taught that math is the one subject in which there are definitive, correct answers), but it’s surely interesting to contemplate.
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