Re: The relativity of simultaneity

Well, I can give you a fresh perspective on the mathematics of the relativity
of simultaneity, if you are interested:

We are in the midst of a renaissance in the historiography of set theory.
Above all, I recommend A. Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF
THE SET-THEORETIC 'PARADOXES,' but there's also Grattan-Guinness and
Ferreiros, discussed in the paper linked below.

Here is the central issue in the understanding of the relativity of
simultaneity:

Einstein used a mathematical approach which he called "practical geometry."
He thought the formulation of this point of view was his crowning achievement,

and thought very highly of the lecture in which you can read his discussion
of it, "Geometry and Experience." I recommend it.

Today this mathematical point of view is called constructivism or natural
mathematics, and in his day it had three branches: intuitionism, logicism and
formalism.

So you have to understand, first, that Einstein expressed the relativity of
simultaneity in practical geometry. I don't see any acknowledgment of that
in this chain of remarks. If you want to understand what he said, you have
to understand the issues which were important to him.

From Poincare (SCIENCE AND HYPOTHESIS), but also from the long tradition of
natural mathematics stretching back to Aristotle's concern over the "paradox"
of Zeno, he adopted the idea that all argumentation leads inevitably to
paradox. This is certainly the gist of the response to Cantorian set theory,
hence the fame of the supposed set-theoretic 'paradoxes.'

The most important result of this concern was the idea that there is no such
thing as logical content: arguments, if expressed in a certain way, can
approach logical content but can never actually contain it, because, again,
argumentation necessarily leads to paradox.

So, for practitioners of constructivism or practical geometry, the only way
out was a compromise: construct an argument, but make it contain the
constructivist idea. That idea is that mathematics is an inherent human
function.

For those interested in logical content, this is already so far afield that
eyes glaze over. And it was never seen to be relevant to relativity, because
no one was able to say how Einstein used practical geometry as a technique in
constructing an argument. "Geometry and Experience" was seen to be a bunch
of genial generalities with no relevance to relativity. Why were people
unable to understand where Einstein used "practical geometry"? Because, I
think, we share so much of constructivist mathematical thinking that we are
blind to its presence in arguments.

In any event, you ought to know that Einstein DID use practical geometry in
developing the relativity of simultaneity. Whatever else you may now argue
regarding the relativity of simultaneity, you can no longer ignore the
constructivist mathematics in it--that, now, HAS to be taken into account.
There is an historical sidetrack to this: Einstein's use of constructivist
math is in "disguised" form in the 1905 paper.

So, I think intentionally, he made it explicit in the "train experiment." If
you notice, the train experiment and the clock experiments are the same
experiment: they can be translated mechanically, one into the other.

Thus, the constructivist term Einstein inserted into the train experiment is
also present in the clock experiment.

Remember, that in doing this, he intentionally deprived the argument of
logical content, because he felt he had to do so. If YOU feel that one must
do so, this will not bother you. If you insist on logical content in your
argument, it will bother you A LOT. So it's really a matter of taste, and
not one for debate.

As the paper below says, at one stage of the argument, point M is said to
"naturally" (fallt zwar...zusammen" in the original German) coincide with
point M'. (By the way, close readers of this text--translators--have
realized that this was a conceptual anomaly: they treat it differently in the
French and Italian translations of RELATIVITY).

The logical problem with this notion is as follows:

1. if you drop the term, and M and M' coincide in traditional Euclidean
fashion, you are led to the contradiction of assuming two Cartesian
coordinate systems and deducing one such system (I leave the proof of this to
you).

So M and M' cannot coincide in a Euclidean way: that much cannot, I think, be
contested and no one has ever argued that they could so coincide and that
Einstein was saying that they did so coincide. At least, I haven't seen any
such contentions.

2. if you retain the term, you find that it is not part of the formulation
of the relativity of simultaneity. It is not a definition, an assumption, a
principle, a deduction or anything else. You will look in vain for the
logical role it plays in the argument. It doesn't play any at all. It
simply rattles around in the argument--a loose cannon on deck.

So what is it? It is what Einstein always meant it to be: it is an arbitrary
insertion into the argument, made necessary--according to his approach of
"practical geometry"--in order for the argument to avoid paradox. So
Einstein did exactly what he wanted to do. However, I think we have had an
unconscious prejudice against the lack of logical content, so we never wanted
to think that that's what he wanted to do, or did do. But that's not taking
Einstein seriously. I suggest you take him seriously--at least do him that
courtesy, if you are going to pay any attention to what he says.

By the way, are there any paradoxes? That is, was there anything for
Einstein to worry about? No. Not even Zeno's paradox has stood up to
analysis. The "logical" compulsion we feel with respect to the paradoxes so
far proposed, is an artifact of their construction--it's their rhetoric--it
is not a result of logical content in these arguments. They have none. Too
bad, because they are very seductive. But that's the way it is.

This is where the new set theory history is making all kinds of contributions.

Particularly Garciadiego is devastating with his care with respect to the
history and the terms, showing that Richard didn't even consider his argument
a paradox, that there is no Cantor paradox, no Russell paradox, and so on.
They are glib sleights of hand which do not stand up historically or
logically.

So you really have to do some more work understanding the history of math.

Another thing which is being revealed by new work into Einstein, is how
little he probed into contemporary set theory debates. He never criticized
anything Poincare said about those debates, although particularly Grattan-
Guinness is scathing in his discussion of Poincare. Einstein didn't really
know anything about the set theory which set him off on his mathematical
approach. Very remarkable, I think--very eye-opening.

Einstein is not alone in the sloppiness with which he approached the
mathematical foundation of his argument. The Fefermans and other
commentators are amazingly critical of Godel in their remarks in the
collected works, regarding his understanding of set theory debates.

We tend to think of these twentieth-century mandarins as close students, as
scrupulous thinkers. It turns out that they were slobs.

And of course, Cantor has been subjected to recent research which is even
more embarrassing for his work than the many longstanding critiques.

Again, there's more to the background of constructivism than the set theory
debates. And it has had an influence far beyond Einstein. You find it in
Darwin, Godel, Sraffa, really everywhere. It has stood in the way of logic
for a long long time.

Finally, you should consider where "natural" coincidence leaves us. If we
can't get to general relativity because of "natural" coincidence, then that
means that once again the Pythagorean theorem is at issue (it was a resolved
issue under general relativity, for reasons you know). Does the Pythagorean
theorem have logical content?

My feeling is, no. I think it also has a "natural" coincidence in it. But
where?

Ryskamp, John Henry, "Paradox, Natural Mathematics, Relativity and Twentieth-
Century Ideas" (June 17, 2008). Available at SSRN:
http://ssrn.com/abstract=897085






jem wrote:
If you want to know what my theory saysssss;
I suppose
[quoted text clipped - 21 lines]
assumptions is bogus, and that he is at least 20 years ahead of you in
the understanding of squares.

I know. I get sucked in too easily, although the challenge anymore
isn't to get him to become aware of his silliness - only to see how
trivial the facts he's forced to deny, can be made to be.

PD

--
Message posted via http://www.natscience.com

.

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