Saturday, December 10, 2016

Starting from zero

I tend to think of the origin of all things as a tension between One and Zero, that is, between Something and Nothing, or between Existence and Nonexistence.

In a sense, I start counting from zero.

To get to one from zero is an infinite leap, an unprecedented creative act by which existence comes into being. To reach one is to declare that something exists, that counting is possible.

Zero is a place where one can naturally start and remain for all eternity. It is complete in itself--add zero to zero, multiply by zero, and you're right back where you started. To reach one is, in the greatest of ironies, to reject uniformity, to create a chasm between the utterly self-sufficient zero and the utterly insufficient unit.

Indeed, in creating such a chasm, it is clear that what has been created is not one but two--a choice, between something and nothing. But two is not nearly so remarkable at one. The chasm between zero and one is infinite. The chasm between one and two is one itself.

One is insufficient. If we add one again, we get two, then three, then four... We tumble off into infinity, and we find that really this one generates something that cannot be contained. The leap was indeed infinite, and we see an infinite set emerge because of that.

But something really strange happens when we try to close this system. If we can go out from zero by adding one, we also want to be able to go back toward zero, by subtraction. Suppose we reach zero and decide to continue subtracting. Then we create a mirror image of the natural numbers. Perhaps there's nothing strange about this yet, until we consider that now we have an infinite set extending symmetrically in two directions. As a result, zero becomes arbitrary. There is no reason to think of zero is the "true middle" of this infinite set, precisely because it is infinite. One can simply move to a different "origin" and the two infinite branches proceeding from it are still equal in length.

This is the nature of Euclidean space: it has no center. In other words, a center (very suggestively called "origin" in mathematics) can be chosen arbitrarily. By convention we call this point "zero," but by a change of coordinates any point can be zero.

In geometry, then, the meaning of zero becomes distorted. It is merely a reference point, having no particular identity for the purposes of ontology. Indeed, no point in a geometric space possesses any such identity. Geometry is about relationships, without any center. Everything is relative.

Yet the existence of space is absolute. Either a thing exists or it doesn't. There is an infinite chasm between existence and nonexistence. Geometry has no infinite chasms. In geometry, everything is ultimately rather close to everything else, in an absolute sense; only in a relative sense can something be close or far away.

Kronecker is supposed to have said that God made the integers, and that all else was the work of man. I suppose the intuition for this comes from the fundamental difference between the study of number and the study of measure. For the latter, we take existence for granted. The construction of the real number line is a matter of "filling in the gaps" between points which are already imagined to have some spatiotemporal existence. But the construction of the set of natural numbers is something else entirely. It bursts into existence from nothingness. To measure the difference between existence and nonexistence is meaningless.

As moderns we laugh at the quaintness of a geocentric view of the universe. But I think we should try to be aware of what we might have lost in shedding the innocence of that view. Our universe no longer has any center, or rather we can choose one arbitrarily. Once upon a time space was every bit as real as number. Now it is wholly relativized. We might as well measure everything only in relation to ourselves.

But there is a point of reference far more absolute than we realize. When we envision our universe as nothing more than a space-time continuum, talking of a (geometric) origin becomes meaningless. It is only when we reflect upon its existence or nonexistence that we realize the true center. The true "origin," to which we must compare everything, is nonexistence.

How is it that the universe bursts into existence? How is this infinite chasm bridged? This is the fundamental question. The distance that everything around us traveled to get where it is now is a pitifully small question compared to the fundamental one.

The center of the universe does not lie geometrically in the center of our world, underneath the ground below us. Rather, it lies ontologically in the fires of hell--that is, in nonexistence. The Bible describes judgment as a fire that is never quenched. That is because fire obliterates flesh, and eternal destruction is the return to the center of the universe--to utter nothingness. There can be no greater torture than this. The sheer contemplation of ceasing to exist terrifies me more than words can express.

To exist is always to be away from this center. There is an infinite chasm between heaven and hell. Heaven means eternal existence, where one continually marvels at the fact of being, where there is infinite joy because there are infinite possibilities. In hell there are no possibilities.

What will the redemption of all things look like? Will it mean an end to the story, the end of time? Yet to imagine an end is to cut off all these infinite possibilities distinguish existence from nonexistence. It is zero that stays fixed forever; one, by contrast, can't help but generate infinite sets beyond itself. Heaven cannot be a place of eternal inactivity. It is not a place where all stories end.

God creates out of nothing. Even if space has no center, even if time itself has no real beginning, nevertheless the creation is the most fundamental fact of the universe. If we lose sight of this, we become disoriented. When the universe stops becoming a gift and is rather a meaningless background on top of which our lives are arbitrarily thrown, it is because we have lost the center. The center of all existence is nonexistence. Christ descended into hell, so that all might be raised to heaven.

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