# Re: Arithmetical Realism

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Le 29-août-06, à 20:45, 1Z a écrit :```
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> The version of AR that is supported by comp
> only makes a commitment about  mind-independent *truth*. The idea
> that the mind-independent truth of mathematical propositions
> entails the mind-independent *existence* of mathematical objects is
> a very contentious and substantive claim.

You have not yet answered my question: what difference are you making
between "there exist a prime number in platonia" and "the truth of the
proposition asserting the *existence* of a prime number is independent
of me, you, and all contingencies" ?

> Where is it shown the UD exists ?

If you agree that the number 0, 1, 2, 3, 4, ... exist (or again, if you
prefer, that the truth of the propositions:

Ex(x = 0),
Ex(x = s(0)),
Ex(x = s(s(0))),
...

is independent of me), then it can proved that the UD exists. It can be
proved also that Peano Arithmetic (PA) can both define the UD and prove
that it exists.

>
>> Tell me also this, if you don't mind: are you able to doubt about the
>> existence of "primary matter"? I know it is your main fundamental
>> postulate. Could you imagine that you could be wrong?
>
> It is possible  that I am wrong. It is possible that I am right.
> But you are -- or were -- telling me matter is impossible.

Only when I use Occam. Without Occam I say only that the notion of
primary matter is necessarily useless i.e. without explanatory purposes
(even concerning just the belief in the physical proposition only) .
This is a non trivial consequence of the comp hyp. (cf UDA).

> But the negative integers exist (or "exist"), so it has
> an existing predecessor.

Yes. But the axiom Q1 "Ax ~(0 = s(x)" is not made wrong just because
you define the negative integer in Robinson Arithmetic. The "x" are
still for "natural number". The integer are new objects defined from
the natural number. All right? To take another example, you can define
in RA all partial recursive functions, but obviously they does not obey
to the Q axioms, they are just constructs, definable in RA.

Bruno

http://iridia.ulb.ac.be/~marchal/

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