# [hist-analytic] Methods of Proof: Re: Clarity Is Not Enough

steve bayne baynesrb at yahoo.com
Mon Feb 23 11:02:34 EST 2009

```Yes, I think talking in terms of domains of discourse
is the right way to go. Now on arithmetization of
syntax, one would think that the Godel numbers occur
in the meta-language. Making explicit the ontology
of arithmetization would, then, seem to require a
yet higher order language. What we are talking about
in the case of the formalization of arithmetic is
wff in the object language and these seem to contain
few "Godel numbers." Note that being a prime number
is part of how we assign the Godel numbers and so
there may an implicit "ontology" of arithmetic
involved. I better shut up. I haven't done much
logic in a lot of years.

Regards

Steve

--- On Mon, 2/23/09, Roger Bishop Jones <rbj at rbjones.com> wrote:

> From: Roger Bishop Jones <rbj at rbjones.com>
> Subject: Re: Methods of Proof: Re: Clarity Is Not Enough
> To: baynesrb at yahoo.com
> Cc: hist-analytic at simplelists.com
> Date: Monday, February 23, 2009, 10:56 AM
> On Monday 23 February 2009 15:27:26 steve bayne wrote:
> > I'm pretty sure I get what you are talking about
> w.r.t
> > the metalanguage stuff. But take this fragment:
> >
> > "represent the syntax of A in the ontology of
> B."
> >
> > Could you give an example of syntax being represented
> > "in the ontology"?
>
> Take arithmetisation.
>
> The arithmetisation of syntax consists in assigning to
> each syntactic entity a natural number so that talk
> (and partially vice-versa).
>
> The ontology of arithmetic is the natural numbers.
> Arithmetisation provides numerical representatives
> for syntactic entities, and hence represents syntax
> in the ontology of arithmetic.
>
> It would perhaps have been clearer to talk of
> "the domain of discourse of B", but that is of
> course just the set of things which exist so far
> as B is concerned, i.e. "the ontology of B".
> (is this an odd usage of "ontology"?
> I have thought about foundations "ontologically"
> for so long that I may have become accustomed to
> a way of thinking and hence writing which is not common)
>
> Roger

```