# Re: Arithmetical Realism

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Le 31-août-06, à 22:20, 1Z a écrit :```
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>
>
> Bruno Marchal wrote:
>
>> Le 29-août-06, à 20:45, 1Z a écrit :
>>
>>
>>
>>> 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" ?
>
> "P is true" is not different to "P". That is not the difference I
> making.

I am glad to hear this.

>
> I'm making a difference between what "exists" means in mathematical
> sentences and what it means in empiricial sentences (and what it means
> in fictional contexts...)

Of course I do that difference too! Each hypostase has its own notion
of existence.
When I say that a number exists, it is in the usual sense of a realist
arithmetician.
But physical existence is a completely different things having a logic
of its own. The UDA shows that the logic of the physical propositions
should emerge from the logic of what will be true in all accessible
worlds. The world correspond to the relative consistent extension and
are eventually characterized by the discourse which remain invariant
through world-transition, themselves eventually given by the interview
of the lobian machine.
I am certainly not identifying many different notion of existence, on
the contrary. Recall perhaps that each hypostase (that is "notion of
person") defines some "canonical" Kripke "multiverses".
Perhaps search on "Kripke" in the archive, but I guess we will go back
to this at some point.

>
>
> The logical case for mathematical Platonism is based on the idea
> that mathematical statements are true, and make existence claims.
> That they are true is not disputed by the anti-Platonist, who
> must therefore claim that mathematical existence claims are somehow
> weaker than other existence claims -- perhaps merely metaphorical.
> That the the word "exists" means different things in different contexts
> is easily established.

>
>           <snip; ok but not completely relevant or premature>

>
> empirical
> truth. The anti-Platonists want sthe two kinds of truth to be
> different, but
> also needs them to be related so as to avoid the charge that one class
> of
> statement is not true at all. This can be achieved because empirical
> statements rest on non-contradiction in order to achive correspondence.
> If an empricial observation fails co correspond to a statemet, there
> but insufficient justification for truth in empircal statements, but
> a sufficient one for mathematical statements).

Even for math, non contradiction is not a sufficient criteria. This
follows immediately from the second incompleteness theorem. PA cannot
prove its own consistency (PA does not prove ~Bf). This means you will
not get a contradiction by adding to PA the formula stating that PA is
inconsistent (Bf). Sp PA + Bf, although quite insane in some sense, is
actually consistent, but mathematically unreasonable (but useful in
self-reference theory for getting a simple example of arithmetically
unsound but consistent machine).

>
>
>
>>> 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.
>
> But again this is just "mathematical existence". You need some
> reason to assert that mathematical existence is not a mere
> metaphor implying no real existence, as anti-Platonist
> mathematicians claim. I do not think that is given by computationalism.

When I say that there is an infinity of prime number, it is not a
metaphor.
I am not saying that prime numbers exists like electrons, only that the
"physical existence of electron" emerge in the stable dreams of the
lobian machines, and those dreams are reducible to relative and local
finite computations which, relatively to universal numbers (which exist
by CT), exist then, in the same sense than the prime number, that is
the interpretation of formula like "ExP(x,y)" in the standard model of
arithmetic (the one we learn at school).

>
>>>> 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.
>
> Occam does not support conclusions of impossibility. It could
> be a brute fact that the universe is more complicated than
> strcitly necessary.

With comp the universe is more complicated than necessary. Once lobian
machines appear relatively to themselves, complexity grows locally, in
an unbounded way.

>
>> 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).
>
> As is the way with these things, we anti-Platonists appeal
> to Occam as well (although not qua impossibilia).

Please I have never said that primary matter is impossible. Just that I
have no idea what it is, no idea what use can it have, nor any idea how
it could helps to explain quanta or qualia.
So I am happy that with comp it has necessarily no purpose, and we can
abandon "weak materialism", i.e. the doctrine of primary matter, like
the biologist have abandon the vital principle, or like the abandon of
ether by most physicist.
But with comp it is shown how to retrieve the appearance of it, by
taking into account the differences between the notions of n-person
(and of n-existence) the universal machine cannot avoid.

>
> All the facts about mathematical truth and methodology can be
> established
> without appeal to the actual existence of mathematical objects.

I am not *that* platonist.
But I can revert your sentence: All the facts about *physical* truth
and methodology can be established (and with comp: *have to* be
established) without appeal to the actual existence of physical object.

> In fact, the lack of such objects actually explains the
> objectivity and necessity of maths. Mathematical statements
> are necessarily true because there are no possible circumstances
> that make them false;

Not really .....

>  there are no possible circumstances that
> would make them false because they do not refer to anything
> external.

It depends what you mean by "external". You beg the question because
you talk like if we *knew* there is a "primary material reality" out
there.

> This is much simpler than the Platonist
> alternative that mathematical statements :
> 1) have referents
> which are
> 2) unchanging and eternal, unlike anything anyone has actuall seen
> and thereby
> 3) explain the necessity (invariance) of mathematical statements
> without
> 4) performing any other role -- they are not involved in
> mathematical proof.

I am not *that* platonist at all!!!!! (Neither Plato or Plotinus, ...)

>
>>> 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.
>
> So the specialness of Time depends on the specialness of nautral
> numbers, depends on the specialness of Robinson Arithemtic ?

Robinson Arithmetic is Turing-universal, and, unlike just any UTM,  RA
can easily be extended into Peano Arithmetic, which is not only turing
universal, but which, in some precise sense, know that she is turing
universal.
Mathematician does this all the time. They show that what they want to
prove about some mathematical object O does not depend on the choice of
representations used to represent O, so, after, they choose a special
representation in which O appears as something as simple as possible
for reasoning about it and then they can conclude by statements true on
O in all representations.

For the math you are far more platonist than I am. My position is just
that propositions like :"all non negative integers can be written as
the sum of four squares", [which was already known by Diophantus
(probably a contemporary of Plotinus), but proved much later by
Lagranges], are true independently of me or of any cognition abilities.
... If only because "cognition" has to be defined or isolate by number
relations, once the comp hyp is taken seriously enough.
And then physics will reappear as "relation between cognitions", higher
order "meta" relation between numbers ...

What I say is far more concrete than what you try to ascribe to me, I
think. I could say more: the standard comp-particles are probably
related to the irreducible presentation of permutation groups operating
on the roots of some (any?) universal diophantine polynomial(s), or
something like that. Why waves? I am still asking the lobian machine.
But thanks to Godel, Lob, Solovay ... it is possible to manage the
difference between quanta and qualia, intelligible matter and sensible
matter, sharable and unsharable truth, and many other n-person notions,
etc.
Don't forget I propose an empirically testable theory.

Bruno

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

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