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Topic: Matheology § 256
Replies: 4   Last Post: Apr 22, 2013 5:12 PM

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mueckenh@rz.fh-augsburg.de

Posts: 14,733
Registered: 1/29/05
Matheology § 256
Posted: Apr 20, 2013 5:30 PM
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Matheology § 256

In his dissertation of 1907, Brouwer had actually explained how he
could accept some of Cantor?s ideas, including his transfinite numbers
omega, omega+1, ? up to a certain point (as long as they are
denumerable and in a certain sense constructible {{i.e., given by a
finite formula or rule}}) but not the further concepts of ?a totality
of all such denumerable numbers.?[...]. And it? was not the set-
theoretic paradoxes that caused his reaction. As he remarked in 1923,
?an incorrect theory, even if it cannot be checked by any
contradiction that would refute it, is none the?less incorrect, just
as a criminal policy is none the less criminal even if it cannot be
checked by any? court that would curb it. [...] The point for the
intuitionists is that mathematics is a mental construction erected
freely by ?the mind. It is simply an illusion to conceive of
mathematics as dealing with independently ?existing objects, with an
objective reality somehow external to the mind. {{"God created man in
His own image? Rather man created God in his. [Georg Christoph
Lichtenberg, Göttingen]. If there are Gods, then only such that are
man-made. And if there are numbers, thenb only such that are man-made.
Mental constructions cannot exist without mind. - even if
matheologians are prepared to prove the contrary.}} But this is what?
modern mathematics does: the objects of the theory are conceived as
elements of a totality or set that is regarded as given, totally
independently of the thinking subject. This feature is deeply embedded
in the methods employed in mathematics, and (following Bernays, a key
collaborator of Hilbert) it is often called the ?Platonism? of modern
mathematics.

{{Nothing is so out of date as the utopies of yesterday: 1984, 2001,
transfinite set theory.}}

Meanwhile, the constructivists? treatment of mathematics ? exemplified
by intuitionism ? is based on careful consideration of the processes
by which numbers, etc., are defined or constructed. Each and every
thing that a mathematician can legitimately talk about must have been
explicitly constructed in a mental activity.

As time went by, Brouwer realized that it was better to avoid talking
of ?sets? at all, and he introduced new terminology (?species? and
?spreads?). [...]

As Brouwer?s reconstruction of mathematics developed in the 1920s, it
became more and more clear that intuitionistic analysis was extremely
subtle, complicated and foreign. Brouwer was not worried, for ?the
spheres of truth are less transparent than those of illusion,? as he
remarked in 1933.

[José Ferreirós: "Paradise Recovered? Some Thoughts on Mengenlehre and
Modernism", (2008)]

Regards, WM




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