davidm » September 3rd, 2018, 7:36 pm wrote:Once again, let us be clear: "doubling the size of the universe" means: the distance between objects doubles. It does not mean that space is an object that doubles in size.

Ok. Now I agree with that. Earlier I had the impression you said the opposite, that the size of the universe got larger too. But perhaps that was the page you linked me to, so if that was not your intent at least it's all clear now.

Anyway ... I think this is a very interesting and enlightening example. Let's kick it around.

Let's work on the real line again. If you prefer 2 or 3 dimensions that's fine, it's the same argument. We have a Euclidean space and its integer lattice. On the line, our galaxies are at the integer points ..., -2, -1, 0, 1, 2, 3, ... extending in both directions.

Now your visualization is that we "stretch the line" so that the integer points are now labelled ..., -4, -2, 0, 2, 4, ... I hope you can see that while we can think of this as stretching the line; we can ALSO think of simply walking down the line and

repainting the address of each point. The point at 1 becomes the point at 2. The point at 2 becomes the point at 4. And so forth.

In other words

a real number is the address of a point on the real line. It is not the point itself, which has no name. The real numbers are a particular coordinate system on the real line. If we apply the transformation f(x) = 2x to each point on the real line, the resulting output set is still the real line. We can THINK of it as "stretching the line." But it's EQUALLY ACCURATE to say that we are merely

repainting the addresses. It's like the city council declaring that all the houses on your side of the street are getting a new address. You have to repaint the curb in front of your house, but nothing else in the universe changes. Only the label of a point changes.

How do you tell the difference between the before and after universe? Is the new line the "same" as the old one with just the coordinate system changed? Or has it somehow been "stretched" so that distances are now longer?

Well, since Einstein showed that

there is no preferred frame of reference in the universe, the question is meaningless. It's axiomatic in physics that changing the coordinate system does not change the world.

So when we apply the transformation f(x) = 2x to each point of our line-universe, it makes more sense to realize that all we've done is introduce a new coordinate system.

Nothing else has changed. We've just renamed each point, but everything's exactly where it was before. In the absence of a fixed background frame of reference, there is simply no way to distinguish the two cases.

In short, you cannot claim that applying a linear scaling factor to your coordinate system makes a material change in the real world. It doesn't, anymore than repainting the numeral on the curb in front of your house changes the location or nature of your house.

I will grant you that the distance has doubled between neighboring pairs of our distinguished points, our galaxies that were formerly at the integer positions and that are now at the even integers. But I'm not sure what that means or why it matters. I know it's important to your argument, but I don't understand why.

On the other hand our original one-unit measuring rod has doubled in size too, so the difference between neighboring galaxies is still one measuring rod. If everything in the universe stretches, your measuring rods stretch too. You can't tell the difference before and after. The before-universe and the after-universe look identical.

I'll leave it at that for now. I'd like to know your thoughts.