Tuesday, December 19, 2006

Zero vs. Nothing

Lately I've been reading about conceptions infinity, which is an important topic to natural philosophy and which Aristotle discusses in Book 3 of the Physics.

In any event, this is the reason I picked up David Foster Wallace's popular treatment of the mathematics of infinity (from Zeno up through Cantor). Wallace's writing is definitely mannered. He maintains a modern bias against Aristotle and in favor of the actuality of infinity, both of which points the book inadequately supports. (Sometime I'll have to do a full review.) Because of these flaws, the excellence of his explanation of the difference between zero and nothing is quite surprising:

It's a tricky difference [between the number 0 and the abstract word 'nothing'], but an important one. The Greeks' inability to see it was probably what kept them from being able to use 0 in their math, which cost them dearly. But 0 v. nothing is one of those abstract distinctions that's almost impossible to talk about directly; you more have to do it with examples. Imagine there's a certain math class, and in this class there's a fiendishly difficult 100-point midterm, and imagine that neither you nor I get even one point out of 100 on this exam. Except there's a difference: you are not in the class and didn't even take the exam, whereas I am and did. The fact that you received 0 points on the exam was thus irrelevant—your 0 means N/A, nothing—whereas my 0 is an actual zero. Or if you don't like that one, imagine that you and I are respectively female and male, both healthy 20-40 years of age, and we're both at the doctor's, and neither of us has had a menstrual period in the past ten weeks, in which case my total number of periods is nothing, whereas yours here is 0—and significant. End examples.

I suppose the difference can be summarized by noting that with zero, there is at least to start out with a possibility of having a something. Then of course the notion of possibility (vs. actuality) is critical to the whole notion of infinity....

David Foster Wallace, Everything and More: A Compact History of ∞ (New York: W.W. Norton and Company, 2003), 142.

Also of interest: Nothing Comes from Nothing


Anonymous said...

I ordered the book, to check whether Wallace uses a similar approach for explanation of actual infinity - it would be interesting.

Anonymous said...

OK, I'll bite. Why can't there be an actual infinity. I have Aristotle at home, but I am in India right now. I could make an argument myself, but I'm curious.

Lawrence Gage said...


I should have qualified my statement that the book doesn't support the claim of actual infinity: I wasn't able to find a straightforward argument to that effect in my reading of the book. It's not clear to me that Wallace understands what "actuality" actually means, but then this is a topic that I think requires further exploration.

It sounds like you have some thoughts on the subject of actual infinity. Would you mind sharing them here?

Anonymous, I'm horrible at reproducing arguments like this, and in fact am still grappling with Aristotle's discussion of infinity. While you're away from home, I recommend you read the freely available online version (Book III, chapter 4ff), e.g., here and here. The Hardie and Gaye translation is not the best, but it is good enough for most purposes.

I can say that Aristotle argues that infinity can only be potential: we can only potentially cut a rope, e.g., into an infinite number of parts, and we can potentially expand a coin collection, say, to infinity, but we can never actually complete these infinities. Implicit in his concept of actuality is realizability in the physical world.


Himself said...

(Supposedly I have a password, but it is not what I think it is, Hence, Anonymous.)

I've always figured that about actualized infinity. The universe cannot be infinitely old because then an infinite amount of time would already have passed. Essentially, the present would not have happened yet. But, respectable people (Bernie Clairvaux, iirc) argued to the contrary. I know Stanley Jaki mentions the unrealizability of infinity in passing fairly often. I figure if there is infinite space, there is an infinite amount of matter-energy in it and the gravitational potential in each direction would be infinite, which would play hell with local motion. Also, an infinite number of stars would make the night sky awfully bright. Well, they are too far away for their light to have reached us yet, supposedly. But then how do we know they are there. 'Tis late at night here in Chennai, and sleep beckons.

I've been enjoying this blog enormously. Refreshing to see a realist point of view.

CrimsonCatholic said...

I'm absolutely fascinated by the theological applications of the concept of infinity, so I'm glad to see someone with a common interest.

Here's a good summary on Nicholas of Cusa with a quick supplement

For the magisterial work on the philosophical history of divine infinity, I recommend Leo Sweeney, SJ, Divine Infinity in Greek and Medieval Thought. For some good examples of theological applications of Aristotelian infinity, consider Arkadi Choufrine, Gnosis, Theophany, Theosis. Ekkehard Muhlenberg was the pioneer on the philosophy of infinity in the work of Gregory of Nyssa, but later works have corrected him substantially. For Gregory, you might try Martin Laird, Gregory of Nyssa and the Grasp of Faith. Unfortunately, the scholars of Eastern authors didn't interact with Sweeney or with Cusanus, so the whole area is a bit opaque.

CrimsonCatholic said...

I should probably say also that the above examples demonstrate that Aristotle's position on the actual existence of infinity really had very little bearing on his concepts of infinity in the metaphysical context. In other words, I think Wallace is just plain wrong about what Aristotle's position entailing a cognitive blind spot in metaphysical concepts. He was simply dealing with exactly what he said he was: physics, not metaphysics. Aristotle appears to have been at best agnostic as to the abstract existence of infinity as far as I can tell.

Himself said...

Regarding "Divine infinity"

I've sometimes read comments that, considering the billions of humans past, present, and (presumably) future, and the trillions and trillions of alleged intelligent aliens throughout the universe, it is absurd to suppose that any alleged God could pay much attention to insignificant individuals like you, me, or that guy over there.

However, infinity divided by any finite number, no matter how large, is still infinity. So such a God could pay an infinite amount of attention to each and every individual - and still have a infinite amount left over.

Anonymous said...

A few quick thoughts:
     The concept of infinity is, in philosophically-“technical” terminology,” is a “being of reason,” i.e., it does not exist in the world external to our minds (minds are immaterial), i.e., it’s mode of existence is rarified, but it is still there because we are here discussing it. Also, infinity is not a number except in an analogically weak sense, so when one carries out Cantorian mathematics of infinity (say “dividing” infinity by any real number and still getting infinity), it’s the rules of that mathematics (which, by the way, are also beings of reason just like the rules of chess are beings of reason) that provide an “answer.” One must never forget mathematics itself is an abstraction from the real. Take the simplest, initial concept of the calculus: “the limit.” If anyone has taken a course in mathematical analysis, one understands the concept of infinity in forming the concept of limit... but, again, these concepts are beings of reason. They are useful for making our mathematical formalisms “work” as excellent descriptors of material entities and physical phenomena, but the latter are not the former, i.e., real beings have ontological import far, far beyond the abstracted mathematical formalisms we may (or may not) use to describe them. Finally, infinity is not a “concrete” real but a term that means “unbounded.” The concept is “fuzzy” in that sense.
     Regarding why there can not be an actual infinity: there are many good arguments why it’s impossible. Here are two simple ones: if there was an “infinite number” (i.e., unbounded) of any real being or beings, there’d literally be nowhere to turn: your own existence would be squished out of the way by those real beings. Infinite time? No, because this implies EVERY real being that was and is would already have been actualized—including the death/passing out of existence of those real beings... yet, here we are... and that must be explained. (Remember: time is a metric for understanding change.) Infinite space? No: space is a concept for understanding position, relation (including distance from), etc., etc., which are Aristotelian categories beyond the first category of substance. Space may be quantified and formalized into other types of beings of reason so that our feeble human minds may better grasp things, but space is not a real being.
     “Divine Infinity” is a whole, whole, whole other issue. It refers (roughly speaking) to the unbounded perfection of God. It’s presumptuous (to say the least) to assume God (who is Infinite Love Itself) would “not pay attention to insignificant creatures like ourselves.” These are theological issues that are better addressed by those disciplines rather than by the modern empirical sciences or mathematics... although philosophy can (and should!) assist in seeking understanding.