[FOM] Intuitions of Mass and Volume

[Harvey Friedman]

On 2/17/06 7:39 PM, "joeshipman at aol.com" <joeshipman at aol.com> wrote:

> The intuition that every subset of SPACE has an invariant "volume" is
> indeed shown to be incoherent by results that depend on the Axiom of
> Choice, but this is a different, and logically stronger, intuition than
> the intuition that every subset of a material object has a "mass".
> 

There is no intuition that "every subset of a material object has a mass".
There is arguably an intuition that "every subregion of a material object
has a mass". And this, basically because a "subregion" can be carved out to
form another object.

I submit that the right way to look at the relevant history is that the
distinction between a "subset" and a "subregion" gradually became
recognized, as set theory took hold as the accepted foundations of
mathematics. The first is an important mathematical notion, and the second
is, at least in naïve physics, an important physical notion.

Of course, it still isn't exactly clear what is meant by "subregion of a
material object", but the notion is sufficiently clear that one readily
accepts the idea in naïve physics that any subregion is at least Borel
measurable (and much much more). So there is, demonstrably, a unique
countably additive invariant measure on the regions.

By the way, I don't accept the validity of even the bare minimums of the
currently fashionable mathematical models of physical reality. That doesn't
mean that I reject them.

E.g., I don't accept the idea that there are infinitely "points in space",
or have been infinitely many "points in time". I don't reject these ideas
either.  

It's just that we don't know nearly enough about f.o.p (foundations of
physics) to make such determinations. We productively work with tentative
frameworks, and development them and extend them and patch flaws in them,
and derive predictions from them, and match them with experimental data,
etcetera.

Harvey Friedman