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Re: Endorsement of Wolfgang Mueckenheim from a serious mathematician
Posted:
Feb 1, 2013 3:04 AM
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On 31 Jan., 23:35, david petry <david_lawrence_pe...@yahoo.com> wrote:
> It is eminently reasonable to believe that the purpose of mathematics is to provide a rigorous and practically useful conceptual framework for reasoning quantitatively about real world phenomena.
Just Cantor's aim. He devised set theory for application to reality.
In a letter to Hilbert he wrote about his plan of a paper on set
theory and its applications: The third part contains the applications
of set theory to the natural sciences: physics, chemistry, mineralogy,
botany, zoology, anthropology, biology, physiology, medicine etc. It
is what Englishmen call natural philosophy. In addition we have the so
called humanities, which, in my opinion, have to be called natural
sciences too, because also the mind belongs to nature. (Original
German text in Matheology § 039)
The objects that exist in a conceptual framework are concepts; the
universe of mathematical objects is a collection of concepts.
Concepts can be encoded in language. Languages are countable. The
Cantorian claim that "uncountable" infinite collections exist is
tantamount to the claim that mathematics should assert the existence
of things that are not within the mathematical conceptual framework.
That is a truly extraordinary and even bizarre claim that requires
truly extraordinary evidence, and such evidence is lacking, and
prominent and well respected mathematicians have pointed out that such
evidence is lacking.
This is due to a reversal of implication. Every finite definition
implies an infinite decimal, but not vice versa. Numbers
_are_definitions. 1/4 or pi or 0.333... or 2.718281828andnotsoon are
abbreviations of definitions. These definitions yield infinite
sequences of digits, some of which eventually get constant, others do
not. In any case we can never in mathematics, i.e., in the
mathematical discourse, obtain a number from an infinite decimal
because we never see the end and, hence, never can be sure what will
follow. All we can use in mathematics is the finite definition of a
number. Therefore undefinable numbers are not part of mathematics.
In Cantor's argument, these facts are inversed. There is an infinite
list of infinite sequences. Neither of them exists in mathematics
without a finite definition. But with a finite definition, every
diagonal is defined too. Cantor's argument only shows that there is no
completed "actual" infinity, i.e., there is no complete list. But if
we assume completed infinity, then Cantor's argument shows that the
_definable_ reals are uncountable, because every definable Cantor-list
yields a definable diagonal.
Regards, WM
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