[FOM] Choice of new axioms 1
drago at unina.it
Sun Feb 12 18:39:03 EST 2006
Joe Shipman wrote:
It is my belief that it is something of a
historical accident that the axioms of set theory arose in their
current form. The discovery of the Banach-Tarski paradox was shocking,
but because the notion that matter was infinitely divisible had already
been called into question by the atomic theory, it was not a fatal blow
to physical intuition, and the result was instead that
non-Lebesgue-measurable sets were deprecated as unphysical.
I claim the theory of general relativity could have been discovered
BEFORE the atomic theory was verified (since before the very late
1800's there was no direct evidence of elementary particles and the
only evidence for the atomic theory was the indirect evidence of fixed
mass ratios of chemical compunds).
Your historical reconstruction does not correspond to what I know. The
infinite divisibility of matter - an assumption drawn from the great
relevance gained by infinitesimals in science - was the dominating
hypothesis before Lavoisier; so that his chemical revolution is called also
the "post-poned revolution". H. Guerlac: "Quantification in Chemistry" Isis
52 (1961) 206 gives this history where innovative chemists have been led by
the methodological principles: "The matter is not divisible to infinity".
At the late 19th century the atomic hypothesis did not have evidence; rather
several scientists fought it vigorously as being a metaphysical hypothesis.
It was experimentally proved not before 1909 when errin by means of an
ultra-microscope revealed the brownian motion of molecules.
Rather, first Janaud and Appel in around 1886 stressed that a function
likely as Dirichelet function cannot be applied by theoretical physics.
About your axiom, I submit a question: what about the basic "theorem" of the
calculus of variations, very relevant for physical applications: "Any
positive function whose integral on the closed interval 0,1 is null, is
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