FOM: Weierstrass or Riemann? Limits of Arithmetization.
Robert S Tragesser
RTragesser at compuserve.com
Mon Dec 22 06:41:52 EST 1997
RE: Weierstrass or Riemann? Mathematically significance
Limits of Arithmetization.
There have been question of the geometric content
Dana Scott and Mark Steiner have pointed up senses
in which i has an essential geometric content.
FOM has been centered on the program of arithmeti-
zation. (See Hobson THEORY OF FUNCTIONS OF A REAL VARIABLE
for an account of how set theory was constantly re-envisioned
in ways that conceals its birth on the wrong side of
the blanket (in geometry).
There are two sense of arithmetization: reduction of
objects to natural numbers, reduction of mathematical
reasoning to arithmetical reasoning. (Reductive proof
theory which strives to stuff more and more mathematics
down into PA and PRA belongs to the last.)
Reference: Bottazzini, The Higher Calculus: A History.
"Thus by refuting the _global_ point of view
maintained by Riemann...Weierstrass comes to
conceive the study of analytic functions as a _local_ theory."
"Lie. . .[wrote] that it was entirely
because of Weierstrass and his school
[of arithmetization] that there was no serious
research in geometry in Germany."
Klein is quoted as pointing out that the
virtue of arithmetization is the logical sharpening
it permits/requires, but that is also its vice since
it rules out geometric reasoning which is limited
in the logical sharpening it tolerates.
Does not FOM owe an account of geometric
reasoning, perhaps an exact account of the sense
in which it is irreducible?
Can Klein's remark about geometric reasoning
having a limit to the logical sharpening it permits
be FOM-evaluated? Is there something deeper here than
some vague appeal to the essentially intuitive character
of geometric reasoning?
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