## Monday, December 9, 2013

### How did we come to justifiedly believe that there are three dimensions?

I think a really interesting philosophy of science project would be to ask how it is that we came to have the then-justified belief that there are three dimensions of space? (I don't know that that belief is still justified now given the serious possibility that String Theory is true.) Do any of my readers know anything on the intellectual history of the tridimensionality of space? I don't even know when it was first proposed. Euclid would be my guess.

Austin said...

Hi Dr. Pruss,

I may be able to shed a little light based on a book on Relativity I've been reading. The history of belief in three dimensions may parallel the history of belief of four dimensions in the following way:

Hermann Minkowski and Albert Einstein used the following line of thought to argue for the existence of a fourth dimension-- time. The Theory of Special Relativity has predicted, and experimentation has confirmed, that observers in relative motion (in different inertial reference frames) will experience relatively different distance and time intervals. Not only that, but there is also an achieved loss of simultaneity. One observer might experience Event A before Event B, while another might experience Event B before Event A (note that this will only happen when the events are not causally connected). The only way for different observers to come to the same objective interval between events in time and space is to assume that they are actually experiencing four dimensions of "space-time" projected onto three dimensions. With this assumption, an objective distance between space-time events can be calculated the same regardless of the inertial reference frame. This Spacetime interval can be calculates as:

S^2 = X^2 + Y^2 + Z^2 - C^2*t^2

Where X, Y and Z are the distances along each spacial axis, t it the time difference, and C is the speed of light.

The book explains that idea with the following illustration. When someone holds up a pencil in a room full of people, it would not be odd for everyone to estimate a different length of the pencil, based on their perspective. Someone looking up the nose of the pencil may think it is short, while someone looking along the side may think it is very long. This is because we don't actually see the pencil as it is-- we see the three dimensional pencil as a two dimensional projection. Only when that third dimension is taken into account will multiple observers agree on the objective length of the pencil:

I suppose the existence of the third dimension could have been, or was, justified in that manner.

Alexander R Pruss said...

I agree: something like perspective is a nice way to justify the claim that there are at least three dimensions.

But how to justify the claim that there are at most three?