Re: Cerberus and Quine
From: Paul Holbach (paulholbachSPAMBAN_at_freenet.de)
Date: 02/24/05
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Date: 23 Feb 2005 16:46:24 -0800
> examachine@gmail.com wrote:
> I agree with your explanations, Paul.
> We merely don't agree that
> realism satisfies questions at all these levels. It seems, in
> particular, on ontology.
I think that mathematical realism is especially viable with regard to
semantics and ontology.
Obviously, its crux are the epistemological problems.
> Let me point out my position briefly,
> so it becomes more apparent where
> I stand. I would like a metaphysics
> which is compatible with the modern
> view of the mind.
So would I.
> This view excludes any kind of dualism and Platonism.
> So, these are out of question for me.
As far as the mind-body problem is concerned I reject any kind of
substance dualism too.
(Instead I approve of Searle´s 'biological naturalism'.)
I generally adhere to metaphysical naturalism and atheism.
But I don´t think that makes it impossible for me to be a mathematical
realist in ontology, who believes in the existence of abstract
entities.
Even though abstract entities are non-sensorial, this doesn´t
necessarily mean that they are also supernatural, because I think that
if abstracta exist, they belong to natural reality (to Being) just like
any physical body.
(Guess why the natural numbers are so-called ...? ;-))
By the way, Gödel apparently was a substance dualist:
"Mind is separate from matter: it is a separate object."
[Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: MIT Press. (p. 192)]
And it cannot be denied that Gödel also held some fairly outlandish,
blatantly unscientific views such as:
"I don´t think the brain came in the Darwinian manner. In fact, it is
disprovable. Simple mechanisms can´t yield the brain. [...] Life force
is a primitive element of the universe and it obeys certain laws of
action. These laws are not simple, and they are not mechanical."
[Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: MIT Press. (p. 192f)]
> However, you can still be a
> realist on some matters, you don't have to go all solipsist or
> fictionalist (agnostic). What I say is that
> the Quine-Putnam argument
> is not true as it stands, but there is indeed
>some objective truth to
> mathematics which aliens would discover.
> The trouble is that, this
> objective truth has many intriguing forms
> that might deceive a simple
> metaphysics.
There are two basic facets of mathematical realism:
(1) Mathematical truths are objective:
"Define realism in truth-value to be the view that mathematical
statements have objective truth-values, independent of the minds,
languages, conventions, and so on of mathematics."
(2) Mathematical objectivity implies the existence of mathematical
objects:
"Define realism in ontology to be the view that at leat some
mathematical objects exist objectively, independent of the
mathematician."
[Quotations from S. Shapiro: "Thinking about mathematics: The
philosophy of mathematics" (OUP 2000), pp. 25+29]
Since you say above that "there is indeed some objective truth to
mathematics", you seem to some kind of truth-value realist but some
kind of anti-realist with regard to ontology.
The crucial question is whether mathematical objectivity is actually
possible without there being any mathematical objects.
Gödel answers: "NO WAY!"
"The real argument for objectivism is the following. We know many
general propositions about natural numbers to be true ('2 + 2 is 4',
'There are infinitely many prime numbers', etc.) and, for example, we
believe that Goldbach´s conjecture makes sense, must be either true or
false, without there being any room for arbitrary convention. Hence,
there must be objective facts about natural numbers. But these
objective facts must refer to objects that are different from physical
objects because, among other things, they are unchangeable in time."
[Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: MIT Press. (p. 211)]
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.
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.
> 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!)
> 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.
> 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.
> 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)]
Regards
PH
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