Re: Cerberus and Quine
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Date: 03/01/05
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Date: 1 Mar 2005 04:25:25 -0800
Paul Holbach wrote:
> So, Gödel´s argument runs as follows:
>
> - If mathematical truth is objective, then mathematical facts imply
the
> existence of mathematical objects.
> - Mathematical truth is objective.
> - Mathematical facts imply the existence of mathematical objects.
>
> Obviously, Gödel champions a correspondence theory of truth.
> An ontological anti-realist may reply that this kind of truth theory
is
> inappropriate as concerns mathematics and demand that it be replaced
> with some kind of coherence or consensus theory of truth.
I think the facts are more complicated than Godel thinks. Even his
first step of the argument does not follow. With that kind of reasoning
you can also prove God's existence, which he tried to do.
> Personally I doubt that the idea of mathematical objectivity can be
> saved if the idea of alethic correspondence is discarded altogether.
> In particular, I think that Tarski´s T-scheme can hardly be improved
> upon, it being a perfect logical encapsulation of the idea of alethic
> correspondence.
I didn't state my entire position, but I think it would suffice to say
that I believe in an objective mathematical science as far as it is an
accurate description of the logic of our universe, which seems to be
universal computation.
> > Anyway, the ontology I take is simple:
> > substance monism, and a
> > mechanical world. I deal in nothing more than that,
> > because I simply
> > don't believe in angels.
>
> Neither do I.
> The kind of naturalism I favour is monistic as well, insofar as I
> believe that abstracta do belong to the o n e, all-encompassing
> reality.
> Most naturalists appear to be strict physicalists, but I fail to see
> why one couldn´t be a monistic naturalist and embrace both physical
> and non-physical things.
> (I don´t mean to say that I´m prepared to affirm the existence of
> both natural and non-natural/supernatural things!)
Mainly because embracing non-physical things would flatly contradict
physicalism, such a view isn't considered a healthy position in phil.
of mind community. That *is* dualism.
> > I don't find this talk scientific, especially when one is
> > talking about unscientific concepts like transfinite ordinals.
>
> Mathematics is a science, and so the mathematically well-defined
> concept /transfinite ordinal/ is a scientific one. Of course, if only
> empirical science is genuine science, then you´re right.
Sure, I think so. I am really an empiricist who thinks logical
positivism and its later critics (Quine, etc.) went in the wrong
direction. Logical positivism went wrong by embracing these medieval
notions of analytic truth, that is theology. And Quine went wrong with
embracing holism, that is creationism.
Mathematics is partly a science, I think it is not the case that the
mathematically well-defined concept of a transfinite ordinal is a
scientific one (and furthermore you are not explaining why Cantor's
naive set theory has to be patched... Why should we take this or that
axiomatic set theory, or intuitionistic set theory to be "correct" at
all? Why isn't there a "single true" foundation? etc. etc.).
In particular, I can show how to construct alternative theories of
infinity that are just as consistent in their claims. Self-consistency,
which is satisfied only by endless tinkering by the way!, is no guide
to reality.
There are very deep problems with the admission of infinite entities,
and it seems that Brouwer did manage to identify some of these, but if
you follow general logical principles you can show more drastic
problems with the "unreality" of these constructs in a way better than
Chaitin did. In particular, there are "leaps of faith" that happen the
moment you go from finite to transfinite in *some* elementary subjects
that employ these concepts. I can also show how Cantor implicitly
assumed that some *arbitrarily* chosen principles do not change from
finite to transfinite, and some again *arbitrarily* chosen principles
do change. But that is, obviously, for another discussion.
I think the theories of transfinite have become more metaphysical than
scientific. They do not concern worlds like our own, they concern
worlds that may have been mentioned in Alice In Wonderland or the
Bible, but those are not scientific investigations by any degree.
Now of course, your hero Quine himself supported this nominalistic view
when it suited his publication calendar. :)
> > Therefore, I would like to first consider a positivistic
> > account that
> > is free of some convictions of Quine:
> > holism (non-explanation),
> > logicism (mixing knowledge with existence).
> > I think logic is just that,
> > a way to represent and process your knowledge.
> > But apparently predicate
> > logic is not the only way to do that, right?
>
> There are indeed several workable non-standard logics.
But the really painful truth of the matter is that there are
alternative foundations for mathematics....
> > Anyway, there are
> > reductions among many formalisms,
> > but this does not mean that one of
> > them is somehow special. It's just that whatever
> > formalism "works", the
> > human keeps. Sometimes, these formalisms
> > can be irrational. I don't
> > defend "irrationalism", but the situation is this:
> > logic does not even
> > tell us why its premises are true, right?
> > So, what does this tell you?
> > How do you know the premises are true?
> > By many layers of cognitive
> > processing, that are simply not rational
> > in the sense of logicism, or
> > even proper lambda calculus.
>
> Then you´ll probably deem the following statements by Gödel in some
> sense "irrational":
>
> "The axioms [of set theory] force themselves upon us as being true."
>
> "The mere pyschological fact of the existence of an intuition which
is
> sufficiently clear to produce the axioms of set theory and an open
> series of extensions of them suffices to give meaning to the question
> of truth or falsity of propositions like Cantor´s continuum
> hypothesis."
>
> [Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
> Gödel to philosophy/. Cambridge, MA: MIT Press. (pp. 209+242)]
These are worse than irrational I think.
The axioms of set theory force themselves upon us as being true.
Nothing could be further than truth. What happens is, we look at the
axioms one by one, and see if they match some of our intuitive ideas
about collections, when we sense that they do to some extent, and if we
are indoctrinated well enough to swallow the bits about infinity, then
we're ready to go. We pass these axioms from critical thinking,
experimenting as far as we can do in our minds, however these are still
just constructs. Just strings of letters with well-defined meanings.
Especially, such an absurd question like CH does not follow from the
psychology of the mathematical activity. Do we conclude from the
"obvious psychological facts" associated with what a priest feels when
he's praying that Jesus is the son of God? This is just the same kind
of reasoning, but centuries of Pythagorean indoctrination clouds
people's eyes. Like Hilbert, they do not wish to be driven from the
paradise that Cantor has created for them.
I think most mathematicians do not manage to see that it is not the
theory of infinite entities that provides any fundamental insight into
analysis, algebra or any significant mathematical subject. That is
simply an illusion.
What matters instead is finding ways to do abstract and general
reasoning, e.g. logic, and logic itself does not depend on the notion
of infinity.
In particular, what cannot be calculated is no reason.
Regards,
-- Eray Ozkural
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