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Matheology § 038
Posted:
Jun 14, 2012 2:03 AM
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Matheology § 038
One of the most often heard arguments in favour of transfinite set
theory is the completeness requirement of |R and real functions. It is
not true.
Well, then tell me, Herr Professor Doktor Mueckenheim, how do you
solve the equation
ih partial(du/dt) = H(u) [JR, Matheology § 022, sci.logic June 13,
2012]
In general, yes, most real numbers lack names, and we cannot
effectively distinguish them (in the usual story). [AS, Matheology §
022, sci.logic June 13, 2012]
Numbers are free creations of human mind. They serve as a means to
easen and to sharpen the perception of the differences of things.
Zahlen sind freie Schöpfungen des menschlichen Geistes, sie dienen
als ein Mittel, um die Verschiedenheit der Dinge leichter und schärfer
aufzufassen."
[Richard Dedekind: "Was sind und was sollen die Zahlen?" 1887, 8.
Aufl. Vieweg, Braunschweig 1960, p. III]
But if not even the numbers can be distinguished, what are they good
for? Real numbers that cannot be distinguished will not complete
mathematics, they will not make the real axis continuous, they cannot
guarantee that every polynomial has its zeros.
All real numbers that ever can appear in mathematical calculations
have finite names (at least the definition of the problem like: "find
the fourth root of 16") and belong to a countable set. Therefore
uncountably many unreal reals are good for nothing.
I am convinced that the platonism which underlies Cantorian set theory
is utterly unsatisfactory as a philosophy of our subject [...]
platonism is the medieval metaphysics of mathematics; surely we can do
better. [S. Feferman: "Infinity in Mathematics: Is Cantor
Necessary?"]
Feferman shows in his article, "Why a little bit goes a long way.
Logical foundations of scientifically applicable mathematics" on the
basis of a number of case studies that the mathematics currently
required for scientific applications can all be carried out in an
axiomatic system whose basic justification does not require the actual
infinite.
http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT#416,62,Folie 62
"The actual infinite is not required for the mathematics of the
physical world" [S. Feferman: "In the light of logic", Oxford Univ.
Press (1998) p. 30]
http://books.google.de/books?id=AadVrcnschMC&pg
Though Gödel has been identified as the leading defender of set-
theoretical platonism, surprisingly even he at one point regarded it
as unacceptable.
In his concluding chapters, Feferman uses tools from the special
part of logic called proof theory to explain how the vast part if not
all of scientifically applicable mathematics can be justified on the
basis of purely arithmetical principles. At least to that extent, the
question raised in two of the essays of the volume, "Is Cantor
Necessary?," is answered with a resounding "no."
[S. Feferman, loc. cit, Description from the jacket flap]
http://math.stanford.edu/~feferman/book98.html
Regards, WM
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