Re: Cantor's circular "proof" that evens = integers

On 8 May 2007 01:11:57 -0700, "R. Srinivasan" <sradhakr@xxxxxxxxxx>
wrote:


Classically the position you take may not be tenable. Existence of
infinite sets will have to be "prior" to axioms. I doubt if your
colleagues in this NG will agree with your stand.

Of course, most of us _will_ agree.*) Claiming that "existence of
infinite sets will have to be 'prior' to axioms" seems to embrace a
"realistic" (->mathematical Platonism) point of view, imho. While
George formulated a "formalistic" (or "fictionalistic") position.
This means, (form this point of view) we can justly claim that
there is a an empty set in ZFC BECAUSE we can derive the statement

ExAy~y e x

in this theory, but NOT because there "really" is an empty set.

On the other hand, there ARE Platonists out there, especially
concerning set theory. (Famous example: Gödel.)


»On foundations we believe in the reality of mathematics,
but of course when philosophers attack us with their
paradoxes we rush to hide behind formalism and say
"Mathematics is just a combination of meaningless symbols,"
and then we bring out Chapters 1 and 2 on set theory.
Finally we are left in peace to go back to our mathematics
and do it as we have always done, with the feeling each
mathematician has that he is working with something real.
This sensation is probably an illusion, but is very convenient.
That is Bourbaki's attitude toward foundations.«

(Jean Dieudonné)


»The working mathematician is a Platonist on weekdays, a formalist
on weekends. On weekdays, when doing mathematics, he's a
Platonist, convinced he's dealing with an objective reality whose
properties he's trying to determine. On weekends, if challenged to
give a philosophical account of the reality, it's easiest to
pretend he doesn't believe it. He plays formalist, and pretends
mathematics is a meaningless game.«

(R. Hersh)


F.

_______________________________________

*) A crank wrote in sci.math:

"Integers are an illusion."

Virgil replied:

"All mathematics is equally illusional, but
those illusions build very real bridges."

:-)

--

E-mail: info<at>simple-line<dot>de
.

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