Bruno Marchal wrote:

> > As usual, the truth of a mathematical existence-claim does not
> > prove Platonism.
>
> By Platonism, or better "arithmetical realism" I just mean the belief
> by many mathematician in the non constructive proof of "OR" statements.
>

Lest we go yet another round in the 'reification' debate, is it not
possible to reconcile what is being claimed here?

Bruno, I'm assuming that when you eschew 'Platonic existence' for AR,
you are thereby saying that your project is to formalise certain
arguments about the logical structure of possibility - and through
this, to put actuality to the test in certain empirical aspects.
Questions of how this may finally be reconciled with 'RITSIAR' (I hope
you recall what this means) are in abeyance. Nevertheless, some aspect
of this approach may ultimately be ascribed a status as 'foundational
existent' analogous to that of 'primary matter' in materialism.
Alternatively, such a hypothesis may be shown to be redundant or
incoherent.

Peter, as we've agreed, materialism is also metaphysics, and as a route
to 'ultimate reality' via a physics of observables, is vulnerable to
'reification'. Might it not be premature to finalise precisely what it
is that physical theory decribes that might actually be RITSIAR? You
may be tempted to respond, Johnsonianly, that it is precisely the world
that kicks back that is RITSIAR, but theoretical physics and COMP are
both in the business of modelling what is not so directly accessible.
This notwithstanding that we may believe one or other theory to be
further developed, more widely accepted, or better supported
empirically. Or is there some irreducible sense in which 'primary
matter' could be deemed to exist in a way that nothing else can?

David

> Le 20-oct.-06, à 17:04, 1Z a écrit :
>
> > As usual, the truth of a mathematical existence-claim does not
> > prove Platonism.
>
> By Platonism, or better "arithmetical realism" I just mean the belief
> by many mathematician in the non constructive proof of "OR" statements.
>
> Do you recall the proof I have given that there exists a couple of
> irrational numbers a and b such that a^b is rational? The proof was not
> constructive and did show only that such a number was in a two element
> set without saying which one. AR means we accept such form of
> reasoning. Formally it means I accept that the principle of excluded
> middle holds for the arithmetical propositions (that is those build in
> first order predicate calculus + the symbols =, 0, s, +, *).
> For example I believe that either every positive integer bigger than
> four can be expressed as the sum of two primes or there is a positive
> integer which is bigger than four and which cannot be written as the
> sum of two primes. This is exactly what I mean by being "platonist" or
> better "realist" about numbers and their relations. I put it explicitly
> in the hypotheses for avoiding sterile debates with
> ultra-constructivists or ultra-intuitionist. Note that I have no
> problem with moderate constructivism/intuitionism: non classical
> results can be recasted there through the use of the double negation.
> Without AR I am not sure CT makes any sense.
> 
> Bruno
> 
> 
> http://iridia.ulb.ac.be/~marchal/


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