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Topic: some amateurish opinions on CH
Replies: 57   Last Post: Apr 16, 2013 8:12 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: some amateurish opinions on CH
Posted: Apr 9, 2013 2:37 PM
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On 4/9/2013 11:27 AM, Dan wrote:
> On Apr 9, 8:34 am, fom <fomJ...@nyms.net> wrote:
>> On 4/9/2013 12:15 AM, William Elliot wrote:
>>


<snip>

>>
>>>> If I am permitted to be ambivalent about the role of model theory, I am
>>>> in agreement with your prochoice affiliation. Stop worrying about
>>>> models, and the axiom of determinacy becomes almost preferable.

>>
>>> What's that? The determination to needlessly multiply entities?
>>
>> The axiom of determinacy is inconsistent with the
>> axiom of choice. Under the axiom of determinacy,
>> every set of reals is Lebesgue measurable.
>>
>> It probably trades one set of needless entities for
>> another.

>
> The universe of ZF + AD appears to be, in all respects, 'smaller'
> than ZF+AC .
> So, it does eliminate a set of entities,but it doesn't seem to
> generate another .


What about the two little munchkins who choose
the numbers in the game-theoretic description?

:-)

> Anyway, may be a little off topic, I found this paper
> interesting ,even though I don't agree with everything it says :
> http://arxiv.org/abs/0905.1675
>


I have run across another paper by Nik Weaver.
He is a good writer, and, the paper has its
merits, although I just mostly scanned the
introductory remarks and then looked at
the bibliography.

It is odd. I have worked hard in my life
to understand the continuum hypothesis. It
caused me to read a great many things in which
I had no interest. It forced me to do my best
to understand things I did not believe.

Most of what I think of Weaver's position
goes back to what is discussed in that webcast
you posted in another thread:

http://www.youtube.com/watch?v=cG7MyZtGSB0

These arguments always revolve around who is "right"
and who is "wrong", what is "believable" and what
is "not believable", and how mathematics is "properly
conducted" and hot it is "not properly conducted".
It seems mostly to differentiate along the lines
discussed in those Gary Geck podcasts.

As for Nik Weaver's remarks, predicativism is
just a certain form of realism. To the modern
predicativist one should simply insist that a
little infinity is infinity and one should demand
that they return when they have a real predicativist
solution. They cannot do that because the nature
of their position is deny the need for philosophical
explanation so long as what they have presented is
"workable" in the sense of "it works".

There is a nice statement of the situation in the
discussion of Voltaire at

http://plato.stanford.edu/entries/voltaire/#NewEmpSci

"Newton pointed natural philosophy in a new direction.
He offered mathematical analysis anchored in inescapable
empirical fact as the new foundation for a rigorous account
of the cosmos. From this perspective, the great error
of both Aristotelian and the new mechanical natural
philosophy was its failure to adhere strictly enough to
empirical facts. [...]

"Critics such as Leibniz said no, since mathematical
description was not the same thing as philosophical
explanation, and Newton refused to offer an explanation
of how and why gravity operated the way that it did. The
Newtonians countered that phenomenal descriptions were
scientifically adequate so long as they were grounded in
empirical facts, and since no facts had yet been discerned
that explained what gravity is or how it works, no scientific
account of it was yet possible. They further insisted that
it was enough that gravity did operate the way that Newton
said it did, and that this was its own justification for
accepting his theory."

The modern version of "its own justification" consists
of various quotes, most notably from Wittgenstein,
about man's inability to speak of certain things.
Apparently, inanimate pieces of slightly impure silicon
are "smart". So men should "see no evil", "hear no
evil" and, most importantly, "speak no evil".

I say it that way because I fall on the side
of Leibniz and Goedel, although I strive for
the center. With regard to the continuum hypothesis,
Goedel probably found that center with the
constructible universe. Solovay quotes him with

"However, as far as, in particular, the continuum
hypothesis is concerned, there was a special
obstacle which really made it practically
impossible for the constructivists to discover
my consistency proof. It is the fact that the
ramified hierarchy, which had been invented
expressly for constructive purposes, has to
be used in an entirely non-constructive way."

Returning to the paper to which you provided
a link, Weaver is at least honest about one
particularly important item. He just does
not make the choice as stark as it needs to be.
The arguments over the use of infinity in the
calculus have led to two choices. Either,
one has the individuation of points for
the arithmetic operations involving representation
of the geometric continuum, or, one has the
arithmetic of synthetic differential geometry
with axioms admitting

x^2=0

without being able to conclude

x=0


Of course, there are other ways to frame the
same choice. But, there is a conceptual
efficiency with set theory that mathematicians
are unlikely to give up. The advent of
computers is creating the conditions where
the relationship between real analysis and
numerical methods is weakening. So, when
the need for set theory does seem to pass
away, it will do so because the working
conditions for a significant proportion
of mathematicians will have changed.
The argument will not have been won in the
bowels of foundational debates.

The situation is similar to one a professor
of mine once expressed: "We no longer
teach solid geometry. It is a shame because
it is such a beautiful subject."






Date Subject Author
4/7/13
Read some amateurish opinions on CH
fom
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
Bergholt Stuttley Johnson
4/7/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
Virgil
4/8/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/8/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/8/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/8/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/8/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/8/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/8/13
Read Re: some amateurish opinions by WM
Virgil
4/8/13
Read Re: some amateurish opinions on CH
Virgil
4/9/13
Read Re: some amateurish opinions on CH
apoorv
4/8/13
Read Re: some amateurish opinions on CH
Virgil
4/7/13
Read Re: some amateurish opinions on CH
Virgil
4/9/13
Read Re: some amateurish opinions on CH
Guest
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/10/13
Read Re: some amateurish opinions on CH
Guest
4/10/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/10/13
Read Re: some amateurish opinions on CH
fom
4/10/13
Read Re: some amateurish opinions on CH
JT
4/11/13
Read Re: some amateurish opinions on CH
apoorv
4/11/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/11/13
Read Re: some amateurish opinions on CH
apoorv
4/11/13
Read Re: some amateurish opinions on CH
fom
4/15/13
Read Re: some amateurish opinions on CH
apoorv
4/15/13
Read Re: some amateurish opinions on CH
fom
4/16/13
Read Re: some amateurish opinions on CH
Shmuel (Seymour J.) Metz
4/16/13
Read Re: some amateurish opinions on CH
fom
4/7/13
Read Re: some amateurish opinions on CH
Virgil
4/7/13
Read Re: some amateurish opinions on CH
William Elliot
4/7/13
Read Re: some amateurish opinions on CH
fom
4/7/13
Read Re: some amateurish opinions on CH
fom
4/8/13
Read Re: some amateurish opinions on CH
William Elliot
4/8/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
William Elliot
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
William Elliot
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/10/13
Read Re: some amateurish opinions on CH
fom
4/11/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/11/13
Read Re: some amateurish opinions on CH
fom
4/11/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/11/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
fom

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