# Re: Fw: Something for Platonists]

From: Joao Leao <jleao.domain.name.hidden>
Date: Mon, 16 Jun 2003 16:41:06 -0400

Jesse Mazer wrote:

> >From: "Hal Finney" <hal.domain.name.hidden>
> >To: everything-list.domain.name.hidden, lasermazer.domain.name.hidden.com
> >Subject: Re: Fw: Something for Platonists]
> >Date: Mon, 16 Jun 2003 10:46:56 -0700
> >
> >Jesse Mazer writes:
> > > Yes, a Platonist can feel as certain of the statement "the axioms of
> >Peano
> > > arithmetic will never lead to a contradiction" as he is of 1+1=2, based
> >on
> > > the model he has of what the axioms mean in terms of arithmetic. It's
> >hard
> > > to see how non-Platonist could justify the same conviction, though,
> >given
> > > Godel's results. Since many mathematicians probably would be willing to
> >bet
> > > anything that the statement was true, this suggests a lot of them are at
> > > least closet Platonists.
> >
> >What is the status of the possibility that a given formal system such as
> >the one for arithmetic is inconsistent? Godel's theorem only shows that
> >if consistent, it is incomplete, right? Are there any proofs that formal
> >systems specifying arithmetic are consistent (and hence incomplete)?
> >
> >Hal Finney
>
> Godel showed that if it's complete, a theorem about its consistency is not
> provably true or false within the formal system itself. We can feel certain
> that it *is* consistent nevertheless, by using a model that assigns meaning
> to the axioms in terms of our mental picture of arithmetic. For example,
> with the symbols for multiplication and equals interpreted the way we
> normally do in arithmetic, you can see that x*y=y*x must always be true by
> thinking in terms of a matrix with x columns and y rows and another with y
> columns and x rows, and seeing that one can be rotated to become the other.
> In the book "Godel's Proof", Douglas Hofstadter gives a simple example of
> using a model to prove a formal system's consistency:
>
> "Suppose the following set of postulates concerning two classes K and L,
> whose special nature is left undetermined except as "implicitly" defined by
> the postulates:
>
> 1. Any two members of K are contained in just one member of L.
> 2. No member of K is contained in more than two members of L.
> 3. The members of K are not all contained in a single member of L.
> 4. Any two members of L contain just one member of K.
> 5. No member of L contains more than two members of K.
>
> >From this small set we can derive, by using customary rules of inference, a
> number of theorems. For example, it can be shown that K contains just three
> members. But is the set consistent, so that mutually contradictory theorems
> can never be derived from it? The question can be answered readily with the
> help of the following model:
>
> Let K be the class of points consisting of the vertices of a triangle, and L
> the class of lines made up of its sides; and let us understand ?a member of
> K is contained in a member of L? to mean that a point which is a vertex lies
> on a line which is a side. Each of the five abstract postulates is then
> converted into a true statement. For instance, the first postulate asserts
> that any two points which are vertices of the triangle lie on just one line
> which is a side. In this way the set of postulates is proved to be
> consistent."
>
> As I think Bruno Marchal mentioned in a recent post, mathematicians use the
> word "model" differently than physicists or other scientists. But again, I'm
> not sure if model theory even makes sense if you drop all "Platonic"
> assumptions about math.

You are quite right! The answer is: it doesn't. Model Theory, in which Tarsky
built a workable notion of truth is as subject to Godel Incompleteness as any
other system of of axioms beyond a certain size. Basically the only
mathematical
models that do not suffer from this problem are isomorphic to binary boolean
algebra of classes (though Set Theory suffers from its own problems).

If you want to have an idea of what kind of back-flips people have to do
to avoid Platonism in the foundations of math and logic check this paper,
(silly as it is):

http://philsci-archive.pitt.edu/archive/00001166/

-Joao Leao

>
> Jesse Mazer
>
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Received on Mon Jun 16 2003 - 16:47:31 PDT

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