Friday, February 13, 2015

Modeling space

The obvious model of a Newtonian space is as the set of all triples (x,y,z) of real-numbered coordinates. But the model does not have isotropy that Newtonian space does. It has privileged directions, such as the x-axis, the y-axis and the z-axis. It has privileged coordinates such as (0,0,0). Of course, physical models generally do have properties that aren't found in what is modeled. If I build a model of the solar system out of fruit, the fact that some of the fruit is sweeter need not model any property of the solar system. If I make a model of an ethanol molecule out of sticks and balls, the balls that represent hydrogen atoms differ in their exact mass, and exhibit scratches, in a way that the hydrogen atoms do not.

Nonetheless, even though this is common to all modeling, there really is something a little unsatisfying when the mathematical model does this. Typically when we mathematically model something, we have to abstract or forget on both sides. On the side of what is modeled, the side of the world, we ignore aspects of the physical structure because otherwise things get too complicated. On the side of the model, we ignore aspects of the mathematical structure because they don't, as far as we know, correspond to anything in the physics. Wouldn't it be nice if we could abstract only on one side, that of the world? But some things that would be nice are not an option.

The above remarks do, I think, make Pythagoreanism less plausible. There seems to be structure in the mathematics that models the world that isn't found in the world. This makes it implausible that the world just is composed of the mathematics.

Michael Gonzalez said...

Newton leaves room for "physical space" and "physical time" which have to do with relative measures of these variables, and which can be considered isotropic. Sure, absolute space might have a privileged origin point and axes, but there is no expectation that our experience is in any way affected by this structure.

As for the problem with Pythagoreanism, isn't is some sort of quantifier error to say that, if the world is composed of mathematics, then it shouldn't be the case that the full structure of mathematical models is not instantiated in the actual world?

Alexander R Pruss said...

Here's what I was thinking about Pythagoreanism. Imagine a Pythagorean view of time on which points in space ARE triples of numbers. Then some point in space will be intrinsically special: it will be the one which is a triple each member of which equals zero. But this goes against the strong intuition that the points in space do not intrinsically differ in any significant way. I suspect a similar point can be made against any modern Pythagoreanism that equates the model with the modeled (Tegmark is the main figure here).

Michael Gonzalez said...

Wouldn't we have to give a more explicit definition of "special"? Special to whom? It wouldn't affect anything in physics. It could just be the physical center of the Universe. That might go against some versions of Relativity, but that's why you specified Newtonian mechanics from the beginning, right?