Who's Counting?

Anat Schechtman on non-quantitative notions of infinity

The German mathematician David Hilbert proposed a thought experiment about a hotel with an infinite number or rooms. Even at full capacity — its infinite rooms occupied by infinite guests — the hotel could always accommodate more. If, say, four new guests arrived, the hotel would move each of its existing guests to a room four numbers higher. The guest in room one would move to room five, room two to room six and so on, freeing up rooms one through four for the newcomers. Impressively, the hotel could even find space for an infinite number of additional guests. In this case it would move its existing guests to the double of their current room numbers (one into two, two into four, etc.), thus freeing up all the odd numbered rooms, an infinite amount of them. A clever system if we overlook the inevitably high turnover rate among the cleaning staff.

I describe Hilbert’s hotel to demonstrate two points. The first is that pondering infinity is entertaining, at least if you’re the type who enjoys brain teasers and paradoxes (and if you’re not, I probably lost you at “German mathematician”), and the second is that discussions of infinity often focus on quantity. Whether it’s the infinite years of eternity, the infinite space of our boundless universe, or the infinite amount of shrimp Red Lobster audaciously promised you could receive for only $20, we tend to think about infinity as a limitless amount of something.

But according to associate professor of philosophy Anat Schechtman there are multiple ways to conceive of infinity, and some have nothing to do with counting. Schechtman — who specializes in early modern philosophy, a period encompassing the 17^th and 18^th centuries that gave us influential thinkers like Descartes, Locke, and Kant — is writing a book on notions of infinity from this era and has uncovered several distinct philosophical approaches. Some will be familiar to contemporary sensibilities while others require us to stretch our imaginations.

A Scale Without Numbers

Even in the quantitative sense, infinity is not itself a number. It is more like an endlessness in the tallying process. If we start counting whole numbers at one, there is no point where we hit an upper limit. Go ahead and think of the largest number imaginable and I can always come up with an even larger number simply by adding one to yours. And if we pair something like units of time to the counting numbers, we run into a similar lack of limits. Our own lives may be finite, but there is no clear point where time itself runs out. We can always add another minute or hour or year, which is how we imagine eternity.

That’s my humble paraphrasing of how one of the philosophers Schechtman writes about – empiricist John Locke – posits infinity. If it’s not already apparent, Locke is firmly on team quantitative. Even when venturing into religious ideas of infinity, he remains grounded in numbers, explaining God’s purported omniscience in terms of an endless volume of things known, past, present, and future.

So, what is the alternative? According to Schechtman, we can try conceiving of infinity in non-quantitative terms by examining things that don’t lend themselves to counting but still manifest in greater and lesser degrees. Think about sensory properties, like color and taste, or more abstract qualities like goodness and love.

“We’re very familiar with mathematical scales and musical scales, but there are also scales that are not quantitative where we can still talk about ‘more’ or ‘less’ or even ‘maximum,’” Schechtman says. “It invites us to think about different kinds of scales and whether they allow for an infinite case or not. When we get away from infinite number or infinite space, our intuitions are less clear. Is there a maximal degree of love or can you always love something more than you do?”

It's certainly harder to determine whether something can be infinitely blue or infinitely sweet than it is to picture a line extending infinitely into space, but that hasn’t stopped philosophers from trying. And Schechtman has found historical examples arguing in favor of something like infinite blueness.

Mixing Disciplines

Another way to imagine infinity without counting is to look at the writings of Rene Descartes. To Descartes, quantities can be “indefinite” in that they have no obvious limit, but infinity is reserved for God, who exists in a higher level of being that is independent of finite entities like our embodied selves. Descartes sorts all of existence into a three-degree, non-numeric scale, with God at the top, humans and our fellow objects (“finite substances”) in the middle, and properties of those objects (“modes”) at the bottom. The lower two rungs are dependent on the rungs above them. Just as the roundness and redness (mode) of an apple (finite substance) don’t exist independently of the apple, so is all of human kind (finite substances that we are) dependent on God. In this case, Schechtman explains, infinity is just the maximum degree of the scale.

Got all that? Part of the challenge in understanding Descartes’ system — which Schechtman labels “ontic infinity” because of its focus on degrees of being — is that we currently live in a world where discussions of God and math have little overlap; at UT Austin these disciplines aren’t even housed in the same college. But this is a semi-recent development. In the early modern era, it wasn’t unusual for thinkers to be simultaneously mathematicians and philosophers. Schechtman herself has undergraduate degrees in both philosophy and mathematics, so she is perfectly at home with thinkers like Descartes.

“We’re at peak activity in mathematics and also peak activity in philosophy,” she says of the early modern era. “We have very high mathematical sophistication, more so than any time before, but also religion and theology and this idea of God as infinite is still taken very seriously. So, it’s an opportune time to think of how they reconcile these two different notions of infinity.”

But Schechtman stresses that a belief in God is by no means a requirement to understand more theologically driven ideas. “You can just be interested in infinity and see that there are various ways of thinking about it.”

Multiple Meanings

The appeal of exploring different ways of thinking holds true for philosophy in general, not just Schechtman’s particular era and subject. While the field is sometimes marketed as quest for meaning, the variety of viewpoints can be more interesting than the promise of a definitive answer to any question. In Schechtman’s undergraduate course on philosophy and the meaning of life, an underlying lesson is that there isn’t just one.

The same can be said for interpretations of philosophical texts, even those of the biggest names in the field.

“These texts are fascinating and rich and allow for many different approaches,” Schechtman says. Their richness also accounts for the occasional contradiction or disagreement between scholars who study the same period and pursue the same questions. “We are trying to settle what they say, but also there’s a sense in which the many different perspectives are kind of the point.”

On the quantitative side of things, the study of infinity is a big deal in mathematics, and not just for the imaginary managers of hypothetical hotels. Several significant advancements in the field have revolved around it. But Schechtman’s works shows that infinity also has enticements for those who want nothing to do with set theory.

“There’s something about us as humans that really attracts us to this notion. Descartes says that we can’t understand ourselves as finite without this contrast with the infinite,” Schechtman says. “It’s bound up somehow with our self-understanding and how we understand our relationship to this greater world we’re in.”