[FOM] Query for Roger Jones

Roger Bishop Jones rbj01 at rbjones.com
Sat May 13 04:12:38 EDT 2006

On Friday 12 May 2006 07:23, Bill Taylor wrote:
> Roger Jones, you write:
> > In this case the continuum hypothesis has a definite but
> > unknown truth value.
> Surely this attitude is almost a *definition* of Platonism in
> math?

I think its common to conflate two distinct positions.

The weaker is that the sentences of set theory, in the context of 
some semantics for set theory (of the kind we are discussing) 
have definite truth values.

The stronger is that questions about the existence of sets (not 
in the context of any particular language or semantics) have in 
some more absolute sense definite answers.

Some people not only confuse these two positions, but refuse to 
accept that there is any difference between them.

I think popular contemporary use of the term "Platonism" in 
respect of mathematics includes the stronger position (whether 
or not this is properly called Platonism seems controversial).

The distinction is related to Carnap's distinction between 
"internal" and "external" ontological questions.
It is also addressed by a paper of William Tait:

	"Beyond the Axioms: The Question of Objectivity in Mathematics"

republished in his recent collection "The Provenance of Pure 
Reason" (also available from his web site).

In this paper, I believe that Tait makes the distinction roughly 
as above, takes himself the weaker position, and argues that 
this is also the position taken by Cantor, Hilbert and possibly 
even, sometimes, Goedel.

Roger Jones

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