[FOM] Re: 206:On foundations of special relativistic kinematics 1

[Timothy Y. Chow]

I think a foundational treatment of special relativity would be
interesting, because as Harvey Friedman says it is supposedly all
worked out and clear as glass, yet there remain some puzzling features.

By puzzling features I don't mean Lorentz contraction and time dilation
and all those glamorous phenomena that confuse students and wow the
general public.  I mean, for example, the commonly-heard claim that
special relativity can be derived from just two postulates (the constancy
of the speed of light and the relativity principle), when in fact it is
clear that tons of other, unstated assumptions are needed as well.  These
include obvious assumptions such as the assumption that there are three
spatial dimensions, and subtler ones such as the assumption that space
satisfies certain basic geometric axioms (two points determine a line
and so forth).  More significantly, some assumptions with substantial
*physical* content sneak in the door when one starts discussing the
energy-momentum four-vector.  From some textbook treatments, or from
LEMG's comments in this thread, one can easily get the impression that
the physics of the energy-momentum four-vector is just one more direct
logical consequence of the two postulates of special relativity, whereas
it is clear to any non-physicist that certain presuppositions about the
concepts of energy and momentum are being tacitly introduced.

So I support Harvey Friedman's attempt to develop foundations for special
relativity.  I am not sure, however, that his initial attempt has gotten
off on the right foot, for a couple of reasons.

1. As LEMG pointed out, the apparently clear concept of an "observation"
or an "observer" turns out to be unexpectedly troublesome when you look
too closely at it.  For example, in special relativity, there is a
tendency to associate "observers" closely with "reference frames."  But a
reference frame typically is envisaged to extend indefinitely far out in
space, so that once one picks a reference frame it becomes meaningful to
talk of simultaneous events at widely separated locations.  But any real
observer cannot observe two widely separated points at the same time, so
what is going on here?  The difficulty I am hinting at here gets even more
troublesome (for a foundationalist) in general relativity.  I do not mean
to suggest that this problem cannot be solved, just that it is an example
of the kind of subtlety in the concept of an "observation" that makes me
wary of using it as a foundation.

2. I have a sneaking suspicion that somewhere in the logical development
of special relativity, one must smuggle in the assumption that special
relativity reduces to classical physics in the low-velocity limit.  Maybe
I'm wrong, and this can be derived purely as a consequence.  But if I'm
right, this may make it difficult to do foundations for special relativity
without first developing foundations for classical physics.

3. There's some work by philosophers that may indicate some stumbling
blocks with the idea of using observational equivalence as a foundation.
I'm not all that familiar with this work, but I believe that Carnap
attempted to develop science on the basis of an 'observation-language'
and ran into difficulties.  Also, there's a paper by Mackie entitled
"Three Steps Toward Absolutism" that I'm told shows that it's possible
to develop a physical theory containing an absolute reference frame
that is nevertheless observationally equivalent to special relativity.
But without studying it in detail, I'm not sure how relevant this
work is to Harvey Friedman's project.

Tim