People who know me well know that my first love was not poetry, but mathematics. In fact, I typically see my love of poetry in more or less mathematical terms, which may not be the best way to look at poetry -- indeed, I'm now persuaded that it is utterly inadequate -- but still, it's a link between me and the architects of medieval and Renaissance verse, not to mention the architects of medieval and Renaissance churches. So I dabble now and again in pointless problems. What is the length of the diagonal of a regular heptagon, with side 1? How many triangles will you get if you connect all the vertices of a regular polygon of n sides? I solved that last one many years ago, but I've forgotten the solution, which was kind of cute. I'll also perform pointless calculations in my head, sometimes when I have trouble sleeping, like squaring a four digit number, or coming up with the prime factorization of a four or five digit number, or calculating a square root with a method of approximation and exhaustion. I tease my students by telling them I can calculate, in a few seconds, the remainder, accurate to as many decimal places as they like, when their Social Security numbers are divided by 37. As I said, it's pointless, and there are calculating people who make what I do look like tic-tac-toe.
Still, I've been thinking about numbers lately. Perhaps it's the fine article by our good rivals at First Things, on the mathematics and the faith of that positivism-destroyer, Kurt Goedel. Let me give an example. We learned in high school that, on earth, the distance that a falling object will travel (in a vacuum, and try to get one of those) is given by the formula d = 16t(2), with distance measured in feet and time in seconds. The meaning of the formula is, more or less, that the acceleration of the object varies directly with the velocity; that the plot of the function giving the acceleration from the velocity will be a line.
Now, that intrigues me. First of all, the line is a mathematical object, an abstraction. It is not to be found in nature, strictly speaking, nor are points and circles, for that matter. I confess I am a Platonist when it comes to mathematical objects. I believe they have real existence. What I don't believe is that they exhaustively describe what goes on in the world. That puts me at odds, I know, with the modern allergy against metaphysics, an allergy which causes people to deny that mathematical objects have real existence, but to insist that everything in the world "obeys" these things; a conjunction of beliefs I find incoherent.
Anyway, when I think about it, I am compelled to conclude that every unit in that formula, d and t and the superscript 2 and the 16, is fraught with metaphysical and physical complications. Let's take the 16, for example. Of course everyone will readily concede that it is only an approximation. But I have two problems with it, even at that. The first is the assumption that the number, whatever it is, is invariable. I don't see that -- because, first, the object falling, and the earth itself, are not invariable; they are in continual change. But also, the gravitational force, whatever that is, is not the only force acting on the bodies approaching one another (the ball, let's say, and the earth). It is of course convenient for most purposes to ignore these other forces, but when the task is to determine with exactitude just what the rate of falling will be for such and such an object in a vacuum approaching such and such another object, I don't see that the other forces can be ruled out. Remember, what we want here is not an approximation, however useful that would be, but the actual law, with the actual specific constant (which we only approximate with 16). That's because we are doing more than applied physics here.
There's another problem, one that I've never seen addressed, though I admit I don't read up on these things. That's whether the actual number, whatever it is, can be specified. I divide all numbers into two groups: the expressible, and the inexpressible. All rational numbers, and many irrational numbers, are expressible in some finite form. That is, the sum of the series 1/x(2) ln x, with x going from 1 to infinity, is a particular and expressible number. The square root of 2 is an expressible number.
Now here's where the problem grows interesting. I conjecture that there is a qualitative difference between the number of expressible numbers and the number of inexpressible numbers. That is, each way we have of expressing a number is a discrete way, whether it involves a summation sign or a root sign or whatever. Let the number of these discrete ways be infinite. Still, they are what is called countably infinite, and the numbers we can express by their means will also be countably infinite. Now, all countably infinite sets are equal in size. This is counterintuitive, but provably true; there are no more rational numbers than there are integers; and I am conjecturing that there are no more expressible numbers than there are integers. If this is so, then the number of inexpressible numbers is uncountably infinite. Let us suppose you take the composite set of expressible and inexpressible numbers. What is the chance, if you pick a number at random, that you will pick one that can be expressed in some finite form? Well, the chance is, for all practical purposes, zero; it is lower than any number, no matter how much thinly greater than zero you want to propose; lower than a millionth, lower than a trillionth, lower than a zillionth.
The result of this is that that constant, which we approximate with the number 16, is really unknowable, undiscoverable. We believe that it exists, but we cannot tell what it is. Does it matter what it is? Yes, I believe; the smallest variation in that number will eventually produce a variant result. But note the metaphysical problem here -- one which people of faith will cheerfully embrace. An unknowable number is involved in a "law" which describes the behavior of matter. Whence this number, and whence this law? Matter alone is incapable of saying; mathematics too is incapable.
That's as I see it, anyway.