Naming the Infinite

Jorge Luis Borges, the intellectual's intellectual, begins his essay on the infinite, "Avatars of the Tortoise," with this:

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.

Of course the infinite would be worse than evil to an intellectual! It's a concept that evades all discourse: as soon as you name it, you are no longer discussing it.¹ The real infinite cannot be named. Think about what the word means: it means without borders (Latin, fines), boundaries, or form. But as soon as you say that, you've imposed a boundary: at very least you've separated "the infinite" from the strictly finite. How do we resolve this paradox?

When we speak of the infinite, it is usually in the context of quantity. But numbers themselves are forms. For example, we say the counting numbers are infinite, that is, they extend to infinity. But they, like the integers and rational numbers, are ordered. For the integers, at least there is a "next number"; the same cannot be said of the real numbers, since between any two real numbers is another number.² So the counting numbers and integers are not as "infinite" as the real numbers. Cantor says the real numbers have a larger cardinality than the integers, rationals, etc. There are innumerably more irrational numbers than rational.

And it's not just the size of number sets that shows there are bounds added or removed in passing between different sets of numbers. There are also degrees of generality. For example, in passing from counting numbers to integers, we gain negative numbers (and zero, but that is another, quite different matter): we incorporate into the numbers themselves the subtraction operation that is the inverse of the addition operation implicit in what we come to recognize as the positivity of the counting numbers. So now instead of taking away, I can talk of "giving" a negative quantity. So we've expanded our numbers beyond the verb (give/take) to handle giving both "positive quantities" and "negative quantities." We've expanded our concept of "number" to incorporate more of reality, to bound and in some sense homogenize within our numerical discourse what was once outside the boundaries and unbounded. Thus discover positive and negative numbers, we chip away at the boundless infinite.

But what if we generalize the give and take of the positive and negative numbers? What if we posit a continuum of giving and taking? So instead of just the two discrete operations, "give" (i.e., giving a positive) and "take" (i.e., giving a negative), we create a new continuum, a second dimension alongside or orthogonal to the continuum of the real numbers. So we recognize things in-between giving and taking, a kind of swinging back and forth between these positive and negative poles. One might say an oscillation. This is where "imaginary" (and then the complex) numbers come in, and they can help us keep track of the phase of an oscillation between giving and taking. We derive the full negative by multiplying together these parts, which are complex numbers.

With the successive removals of boundaries, it begins to become clear what mathematics has been doing the past couple centuries is to explore ways to better speak of the unspeakable, and to approximate the infinite. The task of modern mathematics is to name the infinite.

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3 Brown 1 Blue recently hosted an insightful video "What was Euclid really doing?." If really gives a sense of the physicality of ancient mathematics, very different from modern mathematics.

The difference between ancient and modern mathematics is the fundamental point of Jacob Klein's Greek Mathematical Thought And The Origin Of Algebra. With the rise of algebra, numbers lost their clear reference to physical reality, and become for the most part a self-enclosed set of symbols. That move muddles the clarity of how numbers relate to reality. As a result, it is less than clear how one conception of number relates to the other.

Leopold Kronecker's statement that "God made the integers, all else is the work of man" makes the basic point, while also providing an example of how so sedimentized the conceptions of number had become by the nineteenth century that he included negative numbers and zero as truly natural creations. On that latter point, the inclusion of zero even among the "natural numbers" is an example; zero is not a number (numbers count groups of things), but a place holder, a sign of a currently unfulfilled potentiality for things that can be numbered.

Where do the other sets of numbers come from? Rational numbers (fractions) the Greeks thought of as setting up a new unit. They knew of irrational numbers and connected them with the infinite, the unspeakable or unwritable as a finite fraction, which is why they're also called "surds."

People always wonder about imaginary numbers. But even before that, simple negative numbers are strange, and we do well to notice their oddity, unlike Kronecker. I have provided retrospective justifications for these above, and hopefully they make some sense.

It is interesting though that modern math has proven to be the backbone of modern physics³. It may not be an accident that at the same time modern physics has difficulty connecting to the everyday embodied experience of human beings. Relating these two is the great task that stands before anyone alert to the neo-gnostic chasm standing before the modern world and the doom it represents. I hope that this short discussion in some measure traces the outlines of a bridge between human life and some of the conceptual apparatus that dominates our technological civilization and makes possible its material wealth.

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Notes

1. Similarly, when we talk of "nothing," it's at least a concept and so is something. "Matter" too has many similarities with the infinite, which may be why mathematics is so useful for describing its activities.

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2. This is inexact, since between any two rational numbers is another rational number, even though the rationals and the integers have the same cardinality. More precisely, the integers and the rationals can be organized into a sequence, whereas the reals cannot. So the integers and rationals are countably infinite, but the reals are uncountably infinite.

3. "Modern" physics in contrast with Aristotelian physics/natural philosophy, not in contrast with Newtonian "classical" physics that is part of the former.

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Jorge Luis Borges, Labyrinths 1962, at 202.

Jacob Klein, Greek Mathematical Thought And The Origin Of Algebra 1968.