I am beginning to feel sorry for starting these threads, especially as they
are off topic. I was just
pointing out that there are good quality mathematicians (like R.Robinson,
Goro Shimura, J-P Serre,
Andre Weil and many, many others) whose "stamp of approval" would carry
almost instant acceptance
while other mathematicians (and I would include myself as a sort of applied
mathematician) really carry
no credibility to speak of.
I picked the Banach-Tarski Theorem (BTT) on spheres because it is so
mind-numbingly wierd that Robinson's work
makes it seem almost certainly unbelievable.
The result is that everyone commenting on the problem thinks that
"stretching" or "mapping" is what the BTT talks about.
No, no, no. The BTT is not done with mirrors, bent or otherwise. This is
an honest-to-set-theory result
about solid spheres in 3-space. This is not about the physical universe,
it is about mathematics.
Now, I am not a set theorist or analyst. However, I have one [readily
accessible] reference which mentions BTT:
"The World of Mathematics" by James E. Newman, Simon and Schuster, 1956.
See volume 3, page 1944-1945.
This 4 volume work has been printed numerous times in hardcover and
paperback and is quite a nice read for
high school students (and others of us) interested in mathematics.
The description of BTT given in [Newman] is pretty much as follows:
Take two solid 3 dimensional spheres S1 and S2 of unequal volume (i.e.,
different radii). The volumes can be as different as you wish -- Newman
suggests taking one the size of a pea and the other the size of the sun.
You can divide S1 into finitely many (n) disjoint pieces A1,A2,...,An and
S2 into finitely many disjoint pieces B1,B2,...,Bn such that (after
suitable rearranging, if needed), A1 is congruent to B1, A2 is congruent
to B2, and so on until An is congruent to Bn. [Note that the word
"congruent" applies -- not merely "similar".]
Just before this statement, Newman characterizes this construction (p.
1944) as follows:
"Two distinguished Polish mathematicians, Banach and Tarski, extended the
implications of Hausdorff's paradoxical theorem to three-dimensional space,
with results so astounding and unbelievable that their like may be found
nowhere else in the whole of mathematics. And the conclusions, though
rigorous and unimpeachable, are almost as incredible for the mathematician
as for the layman."
Again, Raphael Robinson showed that for the right two different volumes, n
= 5 (perhaps 4) is enough. I do not have an exact statement of Robinson's
result handy.
Please, no more silly discussion of y = 2x or y = 4x (or 4x/3) as
"criticisms" of this result. Multiplying by 2 or 4 or 4/3 stretchs
the image. Stretched things may be similar but they are not _congruent_.
BTT is not about "mapping."
Even trying to think about BTT tends to make people dizzy. ("Visualize
whirled peas.")
I hope this helps clarify one of our stranger 20th century results.
Next time, we can dipute the poetic content of the Picard theorems for
entire functions (an entire complex function which misses two complex
values in its range is a constant). This is technical enough that people
may have more trouble arguing about it.
Thanks,
Joth Tupper
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