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Matheology § 256
Posted:
Apr 20, 2013 5:30 PM


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 manmade. And if there are numbers, thenb only such that are manmade. 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{{Nothing is so out of date as the utopies of yesterday: 1984, 2001, transfinite set theory.}}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



