Re: Cerberus and Quine

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Date: 21 Feb 2005 15:33:14 -0800

JXStern wrote:
> Eray, I just crossed messages with you, so let me respond to this one
> just very briefly.
> >What's worse, Quine then rambles on about why we should make an
> >ontological commitment to such fictional things as "mathematical
> >objects", because they are part of the corpus of science. Should we
> >make an ontological commitment to every word in scientific talk?
>
> I too think that was Quine's big mistake, what Hacker calls Quine's
> apostasy to the linguistic turn. But from a common contemporary
point
> of view, the problem is that Quine didn't go far enough in this
> (mistaken) direction. I have small sympathy for this modern view. :)

Okay, as far as philosophy of mathematics goes, however, it becomes an
ultrarealist point of view, regarding in particular the Quine-Putnam
indispensability argument:
http://plato.stanford.edu/entries/mathphil-indis/

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>>From the rather remarkable but seemingly uncontroversial fact that
mathematics is indispensable to science, some philosophers have drawn
serious metaphysical conclusions. In particular, Quine (1976; 1980a;
1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that the
indispensability of mathematics to empirical science gives us good
reason to believe in the existence of mathematical entities. According
to this line of argument, reference to (or quantification over)
mathematical entities such as sets, numbers, functions and such is
indispensable to our best scientific theories, and so we ought to be
committed to the existence of these mathematical entities. To do
otherwise is to be guilty of what Putnam has called "intellectual
dishonesty" (Putnam 1979b, p. 347). Moreover, mathematical entities are
seen to be on an epistemic par with the other theoretical entities of
science, since belief in the existence of the former is justified by
the same evidence that confirms the theory as a whole (and hence belief
in the latter). This argument is known as the Quine-Putnam
indispensability argument for mathematical realism.

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As I commented on ai-philosophy, I think this is not a good argument.
Especially, it seems to have the form of medieval philosophy which
cannot distinguish the usefulness of a concept like "set" from the
physical existence of a "thought" or an "apple".

Why do I think it is medieval philosophy? Because I find this argument
to be dogmatic and circular. First, it does not explain how one can be
a positivist and at the same time believe in this mysticism. Second, it
is dogmatic, it says we know things exist when science uses them, how
do we know scientists know what they are talking about? How do we know
the fundamentals of mathematics have been "settled"? The problem is
this: you are trying to solve the entire ontology of mathematics by
giving mathematics an air of science. And you say this is determined by
language. Hmmm. Let's see, first you give "language" some mystical
metaphysical power, and then you give "science" a mystical authority on
truth, and then you say mathematics is part of science, so it must be
true, and if it is true, then its referents must truly exist BECAUSE
YOU JUST ASSERTED THAT IS HOW SENTENCES BECOME TRUE. That's rather
scholastic and simplistic.

The better alternative, I think, is to completely deconstruct the
mathematical corpus, and then see what each piece means. These holistic
accounts never worked, a good thing about holism is that it will deny
explanation to many difficult and interesting problems. What I
personally wish to find in mathematical talk is elements of common
sense reasoning, e.g. a machine learning and discovering abstractions,
and using them, making sense of the world around it.

> >Even in the corpus of physics, you can find many senses of
existence.
> >The existence of a wave function is nothing like the existence of an
> >observable. The existence of a singularity is not like the existence
of
> >an ordinary region in our universe. The existence of "sum of
histories"
> >is not like the existence of present. The existence of time is not
like
> >the existence of space! The existence of a possibility is not like
the
> >existence of a chair! The existence of an "orbit" is not like the
> >existence of a spacecraft. The existence of a photon may not be like
> >the existence of a physical law! I am sincerely hoping that, unlike
> >Quine, I won't have to explain every trivial thing on earth.
>
> Indeed. The better goal is to explain only what you say, and not
what
> there is. It is a separate question whether what you say is what
> there is, and one can get into great metaphysical questions about it,
> when all you really wanted was a sandwich. Quine respected all of
> these matters, though the pure logicist does not.

I do respect Quine's general motivation. However, I have a problem with
tying language, logic and metaphysics like that, it just doesn't work
well. You know that I have Russellian inclinations myself, it's such a
perfect world, unfortunately our world isn't like that. And perhaps we
just don't have the perfect lives that Russell and Quine lived, uh.

> >This is quite terrible, because he tries to come up with a slogan
and
> >he messes it up. The slogan wants to define the necessary conditions
> >for existence. But it then says, "Well I don't really define
anything,
> >it's up to you to define it." That's a great strategy, to recast the
> >question to the reader.
>
> Well yes, I think it is a great strategy, only it's not really a
> strategy, it's a definition of what it means to be a cognitive agent,
> and it's quite unavoidable, and I (in a somewhat revisionist reading,
> which you seem to share to some extent) give Quine credit for it.

This is a fine comment. Yes, it is perhaps something that'd work in
general for the cognitive agent, maybe it explores the cognitive sense
of existence, e.g. what existence means to somebody who has to do
real-world thinking. If I cannot speak descriptively of something, I
wonder how I could ever think it exists. I grant that, of course in the
wider sense of language which Quine seems to talk of, not just
languages like English or logic.

I do think that a linguistic approach to mathematics can be vastly
successful.

That is a better approach, in many ways, than Frege's definitions, I
think. However, I think some empiricism has to be thrown in for a good
mixture. And you have to remove the Platonism, I think, which will be
very difficult.

In trying to remove the Platonism inherent in our language, the
disciples of Wittgenstein went too far in declaring mind non-existent.
And in trying to apply the linguistic turn to language, the holists
declared a revival of Platonism. These are powerful weapons, but you
can always shoot yourself in the foot.

While Paul is obviously very right in saying that truth does not have
to conform to common sense, I think especially in caring about ordinary
language, you have to do ordinary philosophy of language, and respect
common sense, because in a strong sense, that is definitely what you
want to explain. So, maybe this point of view fits the above form of
"To be is to be the value of a variable". On the other hand, I don't
think that being counter-intuitive necessarily means that you're going
in the right direction. Some mathematicians seem to naively think that
if your formalization is counter-intuitive, then you must be working on
the true approach. What if mathematical truth is relative to which
axioms you accept as true? What if its truth is largely definitional?

The true philosophy of mathematics has to explain why and how axioms
are selected, regardless of this confusing metaphysical talk. It should
not merely consider itself with what mathematicians mean by existence
of sets, functions, numbers, etc. Some of the best mathematics was done
by people who believed in God, perhaps they (wrongly) saw a connection
between God and their math, this doesn't make their math bad. (There
are Islamic mathematicians who are mysticists obviously) Maybe it makes
it bad only when they manage to put their beliefs into axioms, and fool
the rest of the world in believing that it is some absolute
(mathematical) truth.

Regards,

--
Eray Ozkural

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