Falling into ∞

That symbol

Lately everything I look at starts getting complicated in the same place. Not the algebra. Not the diagrams. $\infty$. That one symbol and suddenly you are allowed to go on forever, to cut forever, to add a next one that was not there. I keep thinking the world we walk around in does not have that. Does not have it at all.

Pythagoras said all is number. He meant the rationals, ratios of counting numbers. I am no longer sure he was wrong. $\sqrt{2}$ is not sitting in the room. Worse — wait — I am not sure the rationals are either. What might actually be there is only finite natural numbers. Things you can finish counting.

Because we saw them

But I have to stop. Mathematics is not the room. It is a bit like Plato. The straight line, the triangle, the square, the circle that we see are already wrong, because we saw them. Pencil. Marker. A stack of pixels. We looked, so what we looked at had to be made of something.

Euclid is honest about the other world. A point is that which has no part. A line is breadthless length. No size. No thickness. And then you may produce a finite straight line continuously in a straight line — always further, never forced to stop. There. Already. A first smuggling of infinity. The sketch on the table cannot do that. The sketch always ends. You still need the doodle. The doodle gives you the idea. Then you prove it, strictly, because the object was never the doodle. Abstract, perfect, simple. Nothing out here instantiates it. That is not a complaint. That is the whole point of having two rooms.

Every floor smuggles it

And look at how the perfect room even got its numbers. You start by counting. One stone, two, three. That is fine. To have the natural numbers — $\mathbb{N}$, the whole thing, as mathematicians mean it — you already promised that after every number there is another. No last one. Successor forever. That promise is already $\infty$. You have not even left counting and you have fallen in.

Then you want zero, and the other direction, and you get the integers $\mathbb{Z}$. Still no last one, both ways now.

Then ratios. $\mathbb{Q}$. A pair of integers, $p/q$. Between any two rationals there is another, and another. $1/n$ getting smaller without end. Not just an infinity of counting — an infinity of cutting. The rationals are already a swarm. Pythagoras’s “all is number” was already standing on that swarm, even before $\sqrt{2}$ showed up.

Then the line still has holes. The diagonal of the unit square is not a ratio. So you fill the holes. Each real number is not a digit. It is an endless object: a Cauchy sequence, rationals huddling closer forever; or a Dedekind cut, every rational thrown to one side or the other of an endless knife. $\mathbb{R}$ is not only an infinite set. Each point on that line is already an infinity.

So: naturals, rationals, reals. Every floor smuggles $\infty$. No wonder everything got complicated in the same place.

You held ink

Someone will say an irrational is easy. Two perpendicular unit segments, join the ends, hypotenuse $\sqrt{2}$. Euclid even proves it, Book I Proposition 47, the one from school, and in that room the diagonal of the unit square cannot be two integers in a ratio. The old wound in “all is number.” I used to think that wound settled it. Then I look at the paper. Those segments are not lines. Zoom in. Thickness. The diagonal too. You never held $\sqrt{2}$. You held ink. The proof is about objects that were never on the paper.

A king’s foot

So what is length, then, if I start going smaller. What does one metre even mean. Is there an absolute one, sitting somewhere, waiting to be copied.

Measurement used to be crude, a little ridiculous. Feet. The story we tell is a king’s foot as the standard. I still laugh. Historians can fuss over which king. The name itself is the joke: a body part, a particular body, turned into how long things are.

We got more serious. Since 1983 the metre is not a bar in a vault. It is how far light goes in vacuum in $1/299\,792\,458$ of a second. Length defined by time. Time defined by a caesium atom ticking. Still no metre-thing in the room. A recipe. Compare this with that. The ancient foot was a recipe too. Ours is cleaner. It is still a comparison.

Halfway, then puff

And the stick. Break it in half, the remainder in half, without end. I thought that as a child. Zeno already had it — to go anywhere you must first go halfway, and halfway of that, forever. Later someone says the infinite sum can add up to a finite length. A sentence in the perfect room. Is it true in this one.

I’ve heard of a smallest particle. If you try to break that — puff. Gone. Energy, maybe. “Cut” stops meaning anything. The universe seems to stop you taking things apart. Resists continuity. You go down in size and then the thing is not there. As if the world were saying: that is far enough.

So how do you define length at all. Count the smallest particles. And not just length. Any physical quantity. By measuring it. Or does it have a concept that stands apart from anyone ever measuring it. The more I push the more questions, they arrive faster than I can hold them.

You never approach

I’ve heard of the quantum. Packets. One portion at a time. Energy, at least, does not smear. So there is no continuum. So in the real world there is no $x \to x_{0}$. You never approach. You hop. The next hop is already the last hop, or there is no next hop.

Is distance continuous. Is time continuous.

Is the universe infinite. I’ve also heard that the atoms we can talk about — the observable universe — are finite. On the order of $10^{80}$. A number so large it feels like a joke and it is still a number. You could write it down in the perfect room. Lots. Still finite.

Still producing the line

I keep coming back to time. I keep feeling that the only infinity that might actually exist out here is $n \to \infty$ and $n$ is time, because time seems to keep walking forward and it does not look like it has an end. People say the universe is still expanding. Space getting more space. As if someone were still producing the line. Euclid’s second postulate, but as weather.

It is a strange place to land. We defined the metre by a slice of a second. Length leaned on time. And time is the one thing I can still imagine going on without running out.

I do not have the next sentence. I just cannot find $\infty$ in the room, except maybe that.