On 4/20/2013 4:30 PM, WM wrote: > > > 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) 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,
With all due respect to Brouwer, it is easy to whine about Cantor after the hard work is done. Newton never gave any defense or explanations of the calculus. Leibniz, at least explained himself. The notion of infinity permeates mathematics because of the principle of identity of indiscernibles. The Newtonian system is just as suspect of the same assumptions.
"All existential propositions, though true, are not necessary, for they cannot be proved unless an infinity of propositions is used, i.e., unless an analysis is carried to infinity. That is, they can be proved only from the complete concept of an individual, which involves infinite existents. Thus, if I say, "Peter denies", understanding this of a certain time, then there is presupposed also the nature of that time, which also involves all that exists at that time. If I say "Peter denies" indefinitely, abstracting from time, then for this to be true -- whether he has denied, or is about to deny -- it must nevertheless be proved from the concept of Peter. But the concept of Peter is complete, and so involves infinite things; so one can never arrive at a perfect proof, but one always approaches it more and more, so that the difference is less than any given difference."
> just > as a criminal policy is none the less criminal even if it cannot be > checked by any? court that would curb it. [...]
This is why Boole has to plea for his reviewers to set aside the received paradigm. This is why Cantor has to make a defense for mathematical freedom.
> The point for the > intuitionists is that mathematics is a mental construction erected > freely by the mind.
So Brouwer steals Cantor's argument.
> It is simply an illusion to conceive of > mathematics as dealing with independently existing objects, with an > objective reality somehow external to the mind.
Brouwer should have read Kant more carefully. Kant's a priori had never been intended as some sort of solipsism. It had been an argument in defense of objective knowledge against Humean scepticism.
The knowledge base of mathematics is an objective reality external to the mind. How an individual interprets the statements of mathematics is subjective. That is the benefit of formalist axiomatization.
> 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.
Actually, the statements are a formal syntax. The objects are given in model theory. This is arguably not a theory of truth in a philosophical sense.
> 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. >
Yes. Except that the formal interpretation of denotation is consistent with Russellian description theory. So, it is Platonism with uncertain denotation. Hence, it is "abstract".
> 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. >
WM pays no attention to the paragraph above, of course.
> 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?). [...] >
It is almost incomprehensible. But, it could be easily learned if WM actually used Brouwer's mathematics instead of using Brouwer to convince other's that his unconvincing views are substantive.
> 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. >
Yes. It became useful after Heyting put it into a formal calculus and intuitionistic theories had been characterized according to formalist methodology.
Furthermore, the logic within a calculus is different from the meta-logic outside the calculus.