"Dimensionality 2" - Filzbach, Switzerland, 3/28/16

Dimensions are usually thought of as space-like. That means with think of something that has dimensions as occupying space.

So in order to have dimensions we have to have space.

What, then, is space? Well space takes up space...uh, I mean... space extends somewhere...that is, its size can be measured or estimated in all directions...OK, damn it, it has dimensions! 

So when we're talking about dimensions and space, we're talking about the same thing. (I'm leaving out time for the time being, but that will turn out not to be a problem, at least no more of a problem than time already is.)

So where did space/dimensions come from? The only answer that's possible is nowhere, which is a problem.

A model comes from mathematics.  We start with a dimensionless point (We can do this in math, but not in physics. That's why mathematicians aren't physicists). This dimensionless point doesn't exist except in the minds of a few mathematicians.

Then they start it moving. (How something that doesn't exist can move is just not asked.) When it moves it traces a line. A line has one dimension. This dimension didn't exist before the point moved, so it moved into space that didn't exist before either. It defines this space. It's a pretty meager space, because it has no width or height, so can't exist in physics anymore than the point can.

Next they take the line and move it sideways, creating a plane with another dimension. This plane still can't be real because it only has length and width, but zero height. However, in the minds of our fictional mathematicians (physicists tend to think of mathematicians as fictional anyway) a new dimension has been created.

At the next step, the plane moves up and traces a volume and a third dimension. Hooray! We have three dimensions-reality!

But wait, the mathematicians find another direction to move the volume that isn't right or left, back or forward, up or down, and they do this, thus creating a 4th spacial dimension and what they call a hyper-volume. We can't see this, but obviously if it's there, our 3 dimensional volume wasn't real either because it was zero in this 4th dimension. As we determined before, that means it didn't exist.

So where does one of these spaces/dimensionalities become real?  Sorry, but that's not clear at all.

It is tempting to say that it's real at three dimensions "as everybody and their mother knows" and those pesky mathematicians just don't know when to stop.

Unfortunately,,,and grudgingly...the physicists sidle over with the mathematicians, at least for a few dimensions more and tell us, actually, yes, Virginia, there is a 4th dimension, and a 5th and a 6th, but definitely no more than 10...or maybe 11...but definitely no more than that. (Unless that guy in the corner is right and there are actually 16 dimensions..but no more!).

All this time the mathematicians are happily creating branches of math that deal with infinite dimensions, hierarchies of infinite dimensions, and other monstrous concepts that, sadly, are all based on simple principles that even our measly three dimensions show are obviously valid.

Where does that leave us and our terra firma universe? A little like the coyote who hasn't noticed he's run out of mountain and is backpeddling on thin air.

Don't worry though. As Einstein pointed out, even if this world is an illusion, it's a persistent one.

OK, assuming there is space, where did the stuff that fills it up come from?

Next time; "The Essence of Physics, or Why Shit Does Stuff"

- Hugh





 

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