Arithmetical Realism

From: Bruno Marchal <>
Date: Mon, 28 Aug 2006 15:18:29 +0200

Le 27-août-06, à 23:17, David Nyman wrote to Peter (1Z) :

>> 1Z: But you don't really address the existence question. You just
>> loosely
>> assume it is the
>> same thing as truth.
> Could I appeal to Bruno at this juncture to address this point
> directly?! At several places in our own dialogues, Bruno, you've
> implied that your 'number theology' was an 'as if' postulate, because
> (if I've understood) you are concerned to see how much can be explained
> by starting from this particular set of assumptions. I don't believe
> that you are claiming they are 'true' in an exclusive sense, rather
> that they are enlightening. Is this a correct interpretation of your
> position, or is there further nuance?

As a scientist, or if you prefer as a "willing to be a consistent
scientist" (you never know), I would say that *all* theories are third
person discourses which have to be taken with an "as if" proviso. Even
the grandmother physics (with propositions like 'objects fall', or
'the sun rise in the morning', etc.) is like that. Even our unconscious
theories that we have probably inherited from our ancestors are like

Of course, given a theory, we can harbor doubts about it, and we can
harbor those doubts differentially, that is more or less doubts on some
part of it.

"My" theory is a digital version of the very old mechanist theory,
saying that we are sort of natural machine. It is already explained by
Nagarjuna in the "Milinda's Questions" for example, or by Plato in some
place, and it has been developed by Descartes concerning animals (and
perhaps concerning humans too in some hidden way, if you take the
context of Descartes epoch).

I make that digital version more precise so as to be able to drive
precise conclusions. I have called in this list that more precise
version: COMP (but I called it digital mechanism in some places). Note
that what you call "number theology" belongs to the conclusion of comp
(I don't assume it).
The precise comp version is given by

a) the "yes doctor" act of faith YD
b) Church (Hypo) Thesis CT
c) Arithmetical Realism hypothesis AR

Now I can imagine "a)" to be false. In three ways actually: For example
I say yes to the doctor but the digital reconstitution of me remains
inanimate, or, I say yes to the doctor, and the digital reconstitution
is a zombie, or I say yes to the doctor, and the reconstitution is
alive but is not me. This I can logically conceived, and that would
make "a)" wrong.
Note that in the lobian interview we do not need anymore the "yes
doctor", except for giving a general sense to the *goal* of the

It is much harder for me to conceive that Church thesis could be false,
but this is due to more than many years of reflection on it. I am not
so much impress by the empirical evidence (all attempt to define
computable function lead to the same class of function, despite
completely different definitions and motivations), but I am infinitely
impressed by the closure for the diagonalization of the class of
partial computable functions. This is a quite convincing argument for
CT, as I try to explain periodically on this list. Still I can
"logically" doubt about CT. It is enough that someone comes up with a
function and a way to explain me how to compute and a proof that the
function cannot be programmed in Java (say), and CT would be refuted. I
would say that this is unlikely.

Now, it is still much more harder for me to doubt about AR. It is about
AR that I often say that I would have the feeling to lie to myself in
case I would pretend harboring doubt about it. AR just says that
elementary number theoretical statements (including existential one)
are true or false in a way which does not depend on me. Actually I am
even using a weaker version of AR, in the sense that for the ontic part
of the theory, I need only the independent truth of the formula with
the shape ExP(x), i.e. "it exist a number verifying the property P",
where P is an easily verifiable statement (like being prime, being odd,
etc.). I don't need universal (with the "for all" quantifier)
independent truth, only the simpler formula among the existential one.

(of course I don't believe at all in Peter Jones heavy form of magical

I have also never met someone doubting about AR, although I met
regularly people who pretend to doubt AR, but like Peter, they put in
it things which I don't put in it at all.

Somehow, to believe in NON-AR you have to believe in the possibility
that there is a proposition of arithmetic, stating the existence of a
number having some verifiable property, which truth value is capable of
changing according to the fact that you are alive or not. You need to
make a stronger and much weirder ontological commitment to get it.

Some people ask me: but if AR is so obvious, why do you postulate it?
I postulate it for reason of completeness, but also because I am aware
of contradicting 1500 years of implicit theological aristotelian
belief, and so I need to be quite explicit about what I assume.
Although very simple to believe, AR does play a key role, if not *the*
key role in the UDA proof.
AR eventually provides the whole comp ontology, although it has nothing
to do with any commitment with a substantial reality.

Hope this helps,


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Received on Mon Aug 28 2006 - 09:20:22 PDT

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