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

On Wed, 09 May 2007 01:47:33 -0400, herbzet <herbzet@xxxxxxxxx>
wrote:


To be sure, we have comments such as Aatu Koskensilta's

" ... According to any ordinary mathematical
understanding the naturals and the reals are fixed objects, which must
thus be reflected in any formalization of our ordinary mathematical
reasoning. The idea that a formal theory could "vindicate" or "justify"
mathematical reasoning is wrong-headed; rather, we must judge the formal
theories on basis of how well they capture the relevant aspects of our
mathematical knowledge, conceptions, reasoning and so on ..."

That's certainly true.

»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)


although it must be added that Aatu rejects this as expressing any
sort of "realism", or indeed as any sort of philosophical position:

"Talking about our mathematical knowledge, conceptions,
reasoning, and the extent some of their features are captured in this or
that formal theory is not in any obvious sense related to any questions
about realism or anti-realism ..."

Hmmm...


"that the naturals are a 'fixed object' is not intended as any
philosophical or metaphysical claim ..."

Right, it is (usually) not _intended_ as an metaphysical claim by
the typical mathematician. (So it certainly has the form of one...
;-)


This all does seem to suggest that there is something '"prior" to
axioms' ...

Sure, mathematical thinking predates the axiomatic approach, that's
certainly right. BUT that does not justify RS' original claim:

"existence of infinite sets will have to be
'prior' to axioms."

That's nonsense. Sure, CANTOR used the notion of /set/ in his
/Transfinite Mengenlehre/ without an axiomatic foundation. But that
does not mean that (infinite) sets have to exist to do that.


[...] my personal opinion is that, assuming consistency of the
axioms of ZFC, that infinite set is a logically possible entity,
whatever its status as an actually existent entity [...].

Right. Completely agree with you.


F.

--

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

Relevant Pages

  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... | formalization, and Peano drew from centuries' worth of inspiration ... |> of mathematics they regard as meaningful without resorting ... meaning the terms being used literally. ... | model) or finding an axiomatization which picks out your private ...
    (comp.theory)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... | formalization, and Peano drew from centuries' worth of inspiration ... |> of mathematics they regard as meaningful without resorting ... meaning the terms being used literally. ... | model) or finding an axiomatization which picks out your private ...
    (sci.math)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... | formalization, and Peano drew from centuries' worth of inspiration ... |> of mathematics they regard as meaningful without resorting ... meaning the terms being used literally. ... | model) or finding an axiomatization which picks out your private ...
    (sci.logic)
  • Re: Cantorian pseudomathematics
    ... What I mean is set theory was not born just by the great thought of Cantor. ... of mathematics behind, which help greatly, and which even where necessary for the theory to be imagined. ... formalization, at a level up, for meta-theory. ... scientists and engineers doesn't have to be used by engineers. ...
    (sci.math)
  • Re: Name the thesis: "Formal sentences capture informal ones"
    ... > logicism, namely that informal mathematics reduces to informal logic, I ... directions from a foundational point of view, ... about the passage by formalization from a body of mathematics M to ...
    (sci.logic)
Quantcast