What does it mean to be infinite?

Taking a break from my intensely political posts to write about mathematics again. I'm going to teach an honors class this fall which will deal with some advanced topics like set theory and cardinality, which deal with basic questions like, what is the size of a set?

I've been thinking about such questions ever since I ran into the possibility some years ago that math could have evolved without counting. You see, sets in mathematics are inherently discrete constructions. They consist of elements, each element being perfectly distinct from all the other elements. The size of a set can only be measured, then, by counting the number of distinct elements. You can tell if sets have the same size by matching elements together; if one set is bigger than another, you'll always have some elements left over in that set. Just think of boys and girls at a dance; if every boy has found a girl partner and there are still girls left over, obviously there are more girls than boys.

But counting isn't the only way we think of size. You can tell one object is "more" than another in several ways. Visually something could just be bigger than something else. One cup could have more water in it than another cup--just compare the height of the water in matching containers. One object could be heavier than another--just use a balance. None of these comparisons require actually coming up with numbers. "More" and "less" still have meaning even if there's nothing distinct about a given quantity.

I have written before about how the mere ability to distinguish two things can conceivably lead to an idea of discrete elements, thereby leading to counting, thereby leading to all of classical arithmetic, etc. But that seems pretty unnatural.

What I thought about doing instead was a thought experiment, built on the premise that mathematics develops with commerce. For this thought experiment we will want to assume that very different resources are being traded than what we are accustomed to. We count things because we have so many resources which can be conveniently counted--you know, cows, chickens, ears of corn, that sort of thing.

Imagine instead a civilization in which all of their resources consisted of relatively fluid objects--anything from water to goo. How might their mathematics evolve? I wager they would not be particularly concerned with counting, but that does not take away the need for comparison. Let's say people are trading goo. One person says, "I need enough goo to build a house." Another person says, "I need enough goo to build a pair of shoes." (These people are masters with goo, you see.) There is obviously a difference in the amount of goo these two people are asking for. The price of that goo will naturally be very different. I doubt that this civilization would use coins to trade with. Perhaps they would rather just give lumps of gold; as long the gold weighed enough, it would make no difference to them. Maybe we'll even assume that the eyesight of people in this civilization is not well-adjusted to seeing sharp boundaries between objects; for instance, they wouldn't naturally notice vertical and horizontal lines.

Obviously not all goo would be the same for these people. Some might be better suited for large structures, other goo might be better for flexible tools, other goo might be light and easy to wear as clothing. In any case, I'm still imagining a world in which it would be useful to trade some stuff for other stuff. This whole thought experiment would be pointless if there were no reason to compare things.

Markets functioning as they should (oh boy...), there ought to result some sort of equilibrium in prices. Certain amounts of red goo, for instance, should be generally held to be equivalent to a certain amount of green goo, where the amounts could be compared either by weight or by volume. Note that this might not even require the use of containers. It is only because we are so attuned to discrete amounts that we quibble over the slightest difference in amounts. But just as our own stores often have penny trays because they don't want to quibble over pennies, so these people would not quibble over the inherent uncertainty of dealing in quantities of goo. Indeed, they may very well be a happier society for not worrying so much whether or not they're getting their money's worth! Nevertheless, they could still have a fairly strong grasp on what amounts are fair, and what amounts are just way off.

Thus their only form of counting would be approximation, and unlike the ancient Greeks they would not take numbers so seriously as distinct entities. They probably would not have huge discussions about the ontology of numbers. This does not necessarily mean they could never develop any sort of mathematics. However, there would be some very serious obstacles.

For instance, their notion of equivalence would not be the same as our notion of equality. It would be fuzzy; all that would matter would be being "close enough." So for instance, we think of equality as being reflexive (a = a), symmetric (if a = b then b = a), and transitive (if a = b and b = c, then a = c). I'm sure this imaginary civilization would also think of equivalence as reflexive and symmetric, but transitivity might be a little difficult to accept. From experience, they might see how an amount A of red goo might be equivalent to an amount B of green goo, and that this might be equivalent to an amount C of blue goo; yet the amount A of red goo might not be quite right for the amount C of blue goo. A = B, and B = C, but A is not equal to C.

Moreover, while they would not be intuitively comfortable with counting, they could develop adding and subtraction, but again understanding these operations a little differently. For instance, as we understand it, if A = B, then A + C = B + C; if I add the same thing to both sides, surely the two sides are still the same. People in my imaginary civilization might, from personal experience, object to this claim. For instance, one could continually add the "same" amount to both sides on a scale, and eventually the result would be an imbalance, whereas before there was a balance. This is because the "same" amount is never exactly the same; so there's no meaningful sense in which just doing the same thing to both sides of a scale necessarily keeps the scale balanced. So every addition would have to be adjusted to re-balance the equation, so to speak, just as when you add something like sand to both sides of a balance you have to carefully adjust after the fact because of any errors you might have made.

The mathematicians of this imaginary civilization might come up with some rather clever axioms to suit the common experience of their people. For instance, with respect to equivalence, they might say the following:

1. Equivalence is reflexive: a = a for any a
2. Equivalence is symmetric: if a = b, then b = a
3. Equivalence is quasi-transitive: if a = b and b = c, then either a = c, or else if a is not equivalent to c, there exists a slight alteration to b which would make a = b but b not equivalent to c (or, by symmetry, there is a slight alteration to b which would make c = b but b not equivalent to a).

In with respect to addition, they might say, if a = b and c = d, then there is a slight alteration to a (or to c) which could be made so that a + c = b + d.

Quantities in this system would not be fixed and independent; rather, they would be fuzzy and interdependent. Their relationship to each other could be changed by certain operations, such as adding. The concept of "slight alteration" could, I suppose, be formalized by these mathematicians so that their mathematics would evolve into something akin to what we today think of as analysis: the art of approximation.

So questions about the size of sets would be very different for this civilization. It would be interesting to ask such people if they understand Zeno's paradox. The way Zeno's paradox is usually put is in very discrete terms: the hare can never catch up to the tortoise because it has infinitely many times to catch up to precisely the spot where the tortoise previously was. Without a firm, non-approximate understanding of equality, this paradox might appear incomprehensible. However, even without a firm notion of equality, it is still possible to understand that one could spend literally forever filling a certain container with goo: just make the quantity you add each time much smaller than the quantity before it. But it nevertheless appears implausible that these imaginary mathematicians could think of any such amount of goo as an infinite set, just because it's infinitely divisible. Indeed, we don't think of it as an infinite amount of goo, either; but our mathematics has a way of telling us that the three dimensional space occupied by the goo contains infinitely many points.

Indeed, set theory would make little to no sense to these imaginary mathematicians. The only concept of infinity that would make sense to them would be infinite in extension, or perhaps in weight (which seems more difficult to imagine). Imagine goo extending infinitely, so that you could swim through it forever without every stopping! Or water, if you prefer. Presumably these imaginary folks would need to drink something.

Given the development of our mathematics in the real world, we tend to think of the natural numbers (1,2,3,4,5,...) as the smallest infinite set you can have. If you want an infinite set, you have to at least have these elements. I have thought about this a lot, wondering if there is another possible conception of "infinite" having nothing to do with sets which would therefore introduce a competitor for our "natural" numbers. The thought experiment in this blog post has not introduced any such competitor. Rather, I think that the natural inclination of these imaginary mathematicians in my imaginary civilization would be to say that our natural numbers are virtually non-existent--if you were to try to draw them, they would take up no space. And indeed, our own measure theory accounts for this; taking the rather natural-feeling Lebesgue measure on the real number line, the natural numbers are a set of measure zero. To creatures used to thinking of size as a matter of extension in space, the natural numbers don't even exist, much less qualify as infinite.

There are many questions brought up in this thought experiment that I would love to think more about, but they will have to wait for another time.