On Dimension and Distance in Physics

On Dimension and Distance in Physics

Let's consider the plane. It's incredibly familiar. We have the horizontal x axis and the vertical y axis.

And any point in the plane needs 2 numbers to uniquely specify it, so we denote a point by (x,y) and we say the plane is 2-dimensional.

We can easily move between any 2 points. Everything is nice and smooth and of course we can define a distance using the Pythagorean theorem. All is good. We have a nice 2-dimentional space and we have a well defined distance between any 2 points.

But this is mathematics. Never confuse mathematics with physics. So what happens when we move to physics?

In mathematics it's ok to have a parameter specified by an infinite number of decimal places. But not in physics. Just a few decimal places are enough. In physics, if a theory can predict a parameter to 10 decimal places it's regarded as an incredibly successful theory.

So now let's replace our 2-dimensional plane with a set of discrete points.

We can think of a grid. But the grid is only used to generate the points, it does not imply any connectivity between the points.

Now our plane is very different. It looks 2D, but it's not. Because we can find a one-to-one mapping between the points and the positive integers. In fact, there many such mappings.

This means that any point is uniquely identified by 1 number. So our plane is 1-dimensional.

And what about distance? There is no clear definition of distance. So distance is not a fundamental concept.

This is immediately evident in physics. Take "spooky action at a distance" for example. There's nothing spooky about it. The problem is with our concept of distance, which nature does not respect. And if nature does not respect a concept then the concept is not worth much.

We need new thinking. We need new thinkers.

Content written and posted by Ken Abbott abbottsystems@gmail.com

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