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Article 3559 of comp.ai.philosophy:
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>From: gross@befvax.uchicago.edu
Newsgroups: comp.ai.philosophy,sci.philosophy.meta
Subject: Re: Intelligence Testing
Message-ID: <1992Feb6.230645.27175@midway.uchicago.edu>
Date: 6 Feb 92 23:06:45 GMT
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In article <1992Feb6.214010.14199@aio.jsc.nasa.gov>, dlyndes@gothamcity.jsc.nasa.gov (David Lyndes) writes:
>In article <1992Feb5.204750.21898@midway.uchicago.edu>
>gross@befvax.uchicago.edu writes:
>
>|> The different philosophies of mathematics only arise when people start
>|> asking what mathematical statements mean.  The various theories are
>|> a result of their own theories of meaning.  
>|> 
>|> If we followed Wittgenstein's account of meaning about mathematics,
>|> we would avoid this whole discussion.
>|> 
>|> In a chess game, we ask whether a 'checkmate in 3 moves' exists.
>|> Somehow, there aren't any philosophical problems here.  We can talk
>|> about the existence of 'abstract objects' without being platonists
>|> or having any metaphysical views at all.
>
>I do not see how Wittgenstein's theories would allow us to avoid
>the whole problem.  Giving a name to a problem does not solve it.  All
>Wittgenstein did was to give the problem a name ("Language Game") and
>declared the problem solved.
No, that's not true.  

>But the problem remains: namely
>
>(1) platonism explains mathematical ontology and truth quite nicely but
>    fails to explain its utility and epistemology, while
>
I fail to see how platonism explains anything.  It is a metaphysical 
picture.  Without epistemology and utility, it explains nothing.  
What it says is that the relationship between the language of math
as a whole to the universe of mathematical objects is the same as
the relationship between a formal language and a particular model.

Models and semantics are something we do in mathematics, and an account
cannot be given for mathematics as a whole.  Platonism is just an 
application of the name-object theories of meaning to mathematics as 
a whole.  It explains nothing.

>(2) Wittgenstein, pragmatism, formalism, constructivism, etc explain
>    the epistemology and (some better than others) utility but fail to
>    explain its truth and ontology.

Wittgenstein's point is you can't separate the epistemology and utility
from its truth and ontology.  Actually the later Wittgenstein would 
take this position about almost any philosophical concept.  This is 
one of his main criticism of many philosophers.  (It also is related
to verificationist and other theory of meaning, which connect truth and
onotology with epistemology.) It's this distinction that leads to skepticism 
and all kinds of philosophical theories.  I could go through PI and show
how the examples he uses are based on the false picture.  
	examples		access to other people's emotions
				color words

Also, Wittgenstein rejects formalism for similar reasons that he rejects
of theories, so I'm not sure why you group them all together, unless
the grouping is one of non-platonist.

>As I recall, W's theory went something like this:
>
>(A) Logical terms do not denote.
>
>    Consider the problem often called the Bradley Regress:
>    Question: What makes a predication true?
>    Answer:   A relation (the copula) which holds between the subject and
>              predicate.
>    Question: Then what makes the copula relation hold, another relation?
>              And that relation another?  And another? ...
>
>    Wittgenstein's solution was that relations do not denote at all.  People
>    EXPRESS relations by relate-ING the names of the related objects.  We
>    can express the relation *is bigger than* by placing the words
>    "is bigger than" between the names of the smaller and larger objects.
>

This might make sense in the Tractatus, where there are no logical objects
and logical symbols do not have meaning in virtue of denoting logical 
objects.  This problem doesn't even make sense in the context of the
Wittgenstein's later work, and doesn't even appear there. 

Besides, in both the Tractatus and the Investigations, the 'Bradley Regress'
which Russell also deals with in Principles of Mathematics, is a pseudo
problem that does not require a solution at all.  The solution you 
present is more Frege's, somehow Frege does not arrive at this problem
at all. "Is bigger than" requires two arguments.  End of story. 

Denoting is only a problem if we think that is how terms have meaning.

>(B) Language meaning is irreducably tied to action.
Wittgenstein believes this.  But remember that speaking also is action.

>    "EXPRESS" and "relate-ING" are actions.  Hence W's "do not ask the
>    meaning, ask the use."  But beyond some hand waving toward "language
>    games" and "forms of life", there is nothing there to indicate that
>    he has anything strong enough to account for mathematical truth
>    beyond mere utility.  Nor has he shown that reference to mathematical
>    "objects" can be adequately accounted for by a theory of action.  He
>    has only given some examples of how it would work for some simple
>    mathematics.

>So, please say more about how "If we followed Wittgenstein's account
>of meaning about mathematics, we would avoid this whole discussion."

Wittgenstein has a whole book called Remarks of the Foundation of Mathematics,
wherein he discusses lots of issues in the philosophy of mathematics.
It's not just a question of handwaving from language games and forms of
life to develop a mathematical viewpoint. He has no theory of action.
To understand a mathematical sentence requires understanding mathematics.
That is how the 'form of life' things works.    

Wittgenstein is opposed to 'theories' of math, and doesn't give one.
If you attribute to him a theory, it will be a theory he himself does
not accept.  If you have a theory, no matter what it is, it's not
Wittgenstein's view.

My point was that theories in the philosophy of math are attempts to
give the meaning of mathematical statements, by saying what they are about.  
	Math is about			symbols
					mental entities
					proofs
					empirical objects
					a realm of platonic objects.
If we reject that we give the meaning of expression in terms of
these other things, then we don't have any more theories.   Once meaning
is separated from 'aboutness', we don't have these problems.

If we accept a prioirty of mathematical reasoning over mathematical
objects (which is what W means, in general, of meaning=use, i.e.
that use determines the content of concepts, not that the content of
concepts determines our knowledge of them), it make perfect sense
to talk about math as objective, and to talk about mathematical objects,
with out being a platonist.  We can say that 2+2=4 is about the 
numbers 2, 2 and 4, and that its true, necessary, and objective
without being platonist.  (Just like we can say that there are checkmates
with just a rook and a king, without making metaphysical claims). 

This relates questions of epistemlogy, ontology, and truth, and therefore
there is no gap to bridge by a theory.

Please see articles by WW Tait:  The Platonism of Mathematics, Proof and
Truth, J Phil.
Ricketts, Thomas.  Is Frege a Platonist?
(I can find the exact references if you want).

These view are not his but mine, and they are sort of rough sketches
of my views, but Tait is a teacher of mine, and I recommend the article.

Leon Gross
I'm not an actor, but I play on on TV.


