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: >>
>> >>>> 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:
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
"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
without being able to conclude
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."