FOM: Alternate histories for mathematical axioms
Joe Shipman
shipman at savera.com
Fri Apr 28 10:47:23 EDT 2000
The intuition of mathematicians has been influenced by historical
factors. I claim that if the history of physics and mathematics had
been a little different, then FOM might be very different.
Here is one case study. Please suggest others!
Consider the following set of physical intuitions:
I1) space is continuous and infinitely divisible
I2) space is homogenous (same everywhere)
I3) space is isotropic (same in all directions)
I4) matter is continuous and infinitely divisible
We learned approximately a century ago not to trust the last three of
these. However, we still use the first intuition as a ground for our
fundamental theories of physics, which represent space and time using
the mathematical continuum.
I4 leads easily to the axiom that a real-valued measure exists on the
continuum, and indeed on Euclidean 3-space. Unfortunately, with AC this
is seen to contradict I2 (Vitali, no countably-additive
translation-invariant measure). Modifying "infinitely divisible"
doesn't help, unless you sacrifice I3 (Banach/Tarski, no
finitely-additive translation-and-rotation invariant measure).
Proposition: If relativity had been developed before the atomic theory
was established, rather than after (which is logically quite possible,
it depends only on continuous physics like gravity and Maxwell's
equations), mathematicians would have preferred to sacrifice I2 and I3
rather than I4 and this would have led to the acceptance of RVM as an
axiom.
The resulting foundational developments could have been quite
different. For one thing, CH would not have been considered important,
bcause RVM easily refutes CH. The big question would be whether RVM was
independent of ZFC-type axioms (which would eventually be established
either by a proof that CH was consistent or a proof that RVM implied
Con(ZF)). Eventually, when the atomic theory of matter was established,
it would be realized that I4 was untrustworthy, and the whole
constructible universe would start to be taken seriously. The theory of
large cardinals could take a different path, with much more initial
acceptance for the small large cardinals because the consistency of
measurables would be more intuitively justified to many mathematicians.
etc. etc. etc.
Some other possible "alternate histories":
i) AD (Axiom of Determinacy) as an alternative to AC in a culture
obsessed with games or religious predestination
ii) A much earlier (rather than much later as above) establishment of
the atomic theory and the finite diameter of the Universe leading to a
complete rejection of the infinite (and, alas, probably much less
mathematical progress)
iii) A culture in which computation was developed much earlier (if
Babbage hadn't had his funding cut off...) might lead to a very
constructivistic or recursive mathematics and the widespread rejection
of arbitrary subsets.
iv) If medieval theologians had anticipated Cantor's discoveries and
developed a rigorous theory of levels of infinity, to the greater glory
of God, where would we be by now?
Please elaborate on these scenarios and suggest others!
-- Joe Shipman
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