Re: Successor Axiom: on what grounds TF?

examachine_at_gmail.com
Date: 02/08/05 

Date: 7 Feb 2005 16:59:49 -0800

Jeffrey Ketland wrote:
> <examachine@gmail.com> wrote in message
> > Jeffrey Ketland wrote:
> >> So yes, I'm perfectly happy with the notion.
> >>
> >> If you mean, "do I assume that mathematics is true"? Of course.
> >
> > True in what sense? What is this theory of truth about mathematics
that
> > you allude to, it completely escapes me.
>
> It was worked out in detail by Alfred Tarski. It is well known to
> mathematical logicians, for example. And anyone who works in
semantical
> theory.
> How exactly do you think that one proves that arithmetic truth is not

> arithmetically definable?

I'd read a few famous papers by Tarski and had enjoyed it. I think I
even took advanced classes in formal logic.

> > Do you mean correspondence theory of truth? Then, correspondence to
> > what?
>
> I mean the perfectly serious theory of truth worked out in detail by
Tarski
> (and the corresponding notion of objective truth which was assumed by
Goedel
> in his discoveries of the incompletenes results).
> Loosely speaking, it is a refinement of the correspondence view. It
says,
> roughly, that a statement is true just in case the things it refers
to are
> related as the statement says they are. Or---a slight modification of
how
> Aristotle put it in Metaphysics (Book 4)---a statement is true just
in case
> it says that such-and-such is the case, and such-and-such is the
case.
>
> You could try learning about this. See, e.g.,
>
> - W.V. Quine 1970: _Philosophy of Logic_, Chapter 3;
> - Susan Haack 1978: _Philosophy of Logics_, Chapter 7;
> - Richard Kirkham 1992: _Theories of Truth_, Chapters 5 & 6
>
> Tarski's 1944 paper is available here:
> <http://www.ditext.com/tarski/tarski.html>

Thank you, I'd read the paper. The third book reference looks
interesting. I know the correspondence view, but I cannot say I know
all theories of truth.

The problem is that mathematics and physics are quite different domains
of inquiry. While the first involves only thought experiments, the
second is empirical science. So, correspondence theory certainly works
for physics, but the situation is not as easy as it seems for
mathematics, because the referents are merely thoughts in some
situations, and not physical properties that you can be sure of.

For instance, cosmology tells us that (not certainly but with high
certainty) there is only a finite number of quanta in the world, etc.
So, actual infinity, say the size of Z, might not be a physical notion.
You are skipping that. Mathematics has always been metaphysical in
character.

> > What about the truth of the Continuum Hypothesis for instance?
>
> I have no idea (the realist view). Hugh Woodin seems to think it's
false.
> Perhaps we shall never know (the realist view).
> Similarly, I have no idea how many dinosaurs roamed the Earth's
surface on a
> particular day exactly 150 million years ago. On a realist view,
whether a
> statement is true or false is not related to whether we can *know*
that it
> is true or false.

And what ensures that mathematical realism is true?

There are people who might disagree with what you said about the
objective truth of CH.

> > Even saying that "spacetime" exists is idealism.
>
> No it isn't.
> Idealism (e.g., Kant) is the view that external physical spacetime
does
> *not* exist (Kant did not locate spacetime in noumenal reality, for
> example).
> Rather, according to Kant's idealism, spacetime is as a phenomenal
mental
> construct, and "cannot exist out of and apart from the mind"; and
there is
> no such thing as external physical spacetime. This is how Kant put
it:
>
> Time and space, with all the phenomena therein, are
> not in themselves things. They are nothing but representations,
> and cannot exist out of and apart from the mind. (Kant,
> Critique of Pure Reason 1787, Transcendental Dialectic, Section
VI)
>
> This is idealism.

Say, "realism" instead of "idealism".

> > Surely, many physical theories were only PARTIALLY true, such as
> > Newton's mechanics.
>
> Correct. This view is called realism. Strictly speaking, Newtonian
mechanics
> is *false*. But it's also a decent approximation to the truth, and
lives on
> in modern theories (such as special relativity) as a limiting case.
Again,
> this is the realist view.
> See any basic textbook on the philosophy of science, which will guide
you to
> serious research on the notion of truthlikeness, verisimiltude, etc.

But physical realism is so much different than mathematical realism.
You speak as if mathematics and physics are known to be identical.
While an amusing possibility for some time, and investigated by
metaphysicians, it remains merely a metaphysical argument like
solipsism which you are not fond of. That is not a scientific argument.

Regards, 

--
Eray Ozkural

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