So with several examples I’ve managed to prove what nobody really questioned, that it’s possible to imagine a complicated curve like the route of the New York Thruway and assign to all the points on it, or at least to the center line of the road, a unique number that no other point on the road has. And, more, it’s possible to assign these unique numbers in many different ways, from any lower bound we like to any upper bound we like. It’s a nice system, particularly if we’re short on numbers to tell us when we approach Loudonville.
But I’m feeling ambitious right now and want to see how ridiculously huge, positive or negative, a number I can assign to some point on the road. Since we’d measured distances from a reference point by miles before and got a range of about 500, or by millimeters and got a range of about 800,000,000, obviously we could get to any number, however big or small, just by measuring distance using the appropriate unit: lay megaparsecs or angstroms down on the Thruway, or even use some awkward or contrived units. I want to shoot for infinitely big numbers. I’ll start by dividing the road in two.
After all, there are two halves to the Thruway, a northern and a southern end, both arranged like upside-down u’s across the state. Instead of thinking of the center line of the whole Thruway, then, think of the center lines of the northern road and of the southern. They’re both about the same 496-mile length, but, it’d be remarkable if they were exactly the same length. Let’s suppose the northern belt is 497 miles, and the southern 495. Pretty naturally the northern belt we can give numbers from 0 to 497, based on how far they are from the south-eastern end of the road; similarly, the southern belt gets numbers from 0 to 495, from the same reference point.
The point at which I’m aiming is to imagine someone on the northern Thruway, confidently at some mile marker, and looking exactly to the left for the corresponding point on the southern, which will have a slightly different number, except at the zero point. It’s not quite that easy, unfortunately. Imagine there’s a stretch where the southern road stays straight, but the northern swerves out pretty sharply and comes back a bit. The point on the southern road directly to the left of the driver can shoot ahead as the northern road swerves out, and then go backward as the northern road swerves back again. That implies some points on the southern road being matched to three (or more!) points on the northern and we really don’t want that.
Still, last time around, we accepted the daft idea of picking up the imaginary road and stretching it out to a line. How about this time we imagine picking up the center lines of the northern and the southern Thruway roads, and stretching them out into half-circles instead? If we allow ourselves to do that, then not only does every point on the Smoothed Thruway North have only one point directly to its left on the Smoothed Thruway South, but we can even imagine neatly matching the points up. We can draw a line starting from the centers of the circle outward. It crosses both roads, at one point each. Whether on the Smoothed Thruway North or South, the matching point directly to the left (as long as one is facing the flow of traffic) is the other point on this center-starting line.
This means we can talk about the points on the Smoothed Thruway North using either the distance for the Smoothed North, or using the distance for the matching point on the Smoothed Thruway South. But since we could already number roads from any starting point to any ending point we liked, this may not seem like a major breakthrough.
Here’s the major breakthrough. Imagine drawing a new line, a straight one, horizontal, and stretching infinitely far east and west, that’s directly above the topmost point of the Smoothed Thruway North. I’m going to pretend the world is flat just long enough to do this, because I want the distances on this Great Northern Straight Road to be measured east-west from the point where it touches the northern arc, and I want those distances to be able to grow as big as possible, to grow infinitely large.
But remember that center-starting line which matched up the Smoothed North and Smoothed South. If it keeps growing out, it eventually touches the Great Northern Straight Road, and at just a single point. These points aren’t directly left of right of the matching points on the Smoothed Thruways, but they’re on this same radial line. And just as we could refer to points on the Smoothed Thruway North using its own mileage or the mileage of the matching point on the South road, we can also refer to them using the mileage of the Great Northern Straight Road.
The Great Northern Straight Road’s mileages run from negative infinity, way off to the west, up to zero where it touches the Smoothed North, and then keeps going up to a positive infinity, way off to the east. And look at the matching points on the Smoothed North, or the Smoothed South, if you prefer that view. Every positive number gets matched to someplace on the eastern half of the road, every negative number to someplace on the western half. The distance between these matched-up numbers isn’t constant; indeed, near the eastern and western ends of the Smoothed North or South, a few footsteps will cover the points matching ranges of thousands, millions, billions, as big a range as you can imagine.
Well, now that we know what our numbering of these points is, we can undo the smoothing out and restore the Smoothed North to the slightly more crooked path of the northern Thruway’s center line, and do the same for the Smoothed South. And now we’ve got it: we can put down ‘mileage’ markers of infinity — and minus infinity — on this road.
(You might have a nagging feeling that there’s something missing in this argument. There are some points I’ve glided over. I hope to cover the most important ones later on. Loose, conversational arguments like this are great ways to get into trouble, particularly when concepts like infinity are being discussed. I know I’m overlooking some big obvious objection right now; I just can’t think what it is. Well, onward to publish and then feel foolish.)