Dear Günther,
Le 13-sept.-07, à 21:37, Günther Greindl a écrit :
>
>>> The problem is: in math what follows from the axioms is true per
>>> definition (that is what following from the axioms mean).
>>
>> Not at all. If you were true, no inconsistent theory in math would
>> appear.
>
> You are right, my above sentence was too simple.
>
> New try:
>
> All sentences that follow from axioms which do not lead to a
> contradiction - and therefore an inconsistency - we call true in this
> system. Better "valid" than "true", so new refinement:
>
> All sentences that follow from axioms which do not lead to a
> contradiction - and therefore an inconsistency - we call valid in this
> system.
I am more or less ok. Strictly speaking "validity" is for a deduction,
and you can deduce, in a valid way, a contradiction (for example from
some other contradiction), like when Bertrand Russel derives in some
valid way, the identity of himself with the pope from 0 = 1.
>
> We have now freed the word true for a different use (see below).
>
>> "Axioms" are just provisory statements on which we agree. For
>> simple filed like number theory, it happens that nobody doubts them,
>> and in that case I am willing to say I do believe them true, but I am
>> a
>> few bit less sure for ZF set theory, and quite skeptical for a theory
>> like NF (Quine's new foundation).
>
> Axioms and inferential rules can be formed arbitrarily. If they are
> consistent, they may be interesting.
As they may be uninteresting ! Like PA + (PA is inconsistent). That is
a consistent theories which is not very interesting. Of course the
existence of such unreasonable but consistent arithmetical theories
*is* interesting!
>
> If the axioms and the inferential rules are chosen in a way that an
> isomorphic mapping with the physical world is possible, we call them
> true.
(I could call them illusory and I think they are asking for deeper
explanations; as a platonist (say) the observable world is just the
border of what we cannot observe, but we can learn to infer it from the
observations, or still better, learn to deduce it from a deeper theory,
or by lobian machine's introspection (à-la UDA, or its purely
arithmetical version, I refer you to my url for more)).
> (I am somewhat unhappy with the word true here - I am trying to
> adopt your choice of words here; I would never use "true" in describing
> mappings of formal systems into reality: I would only call the mappings
> consistent/usefull with preservation of inferential validity.
That's an excellent move. (Actually that move is the main one well
captured by the category theory approach of logic).
>
> True should IMHO be reserved to propositions made about reality
> (propositions which relate formal systems to reality, for instance, but
> distinct from the formal system).
Yes, but eventually "true about any reality concerning me" is not even
definable by a lobian machine, and that is why, again with Plato, the
notion of truth, once encompassing enough to concern *you*, becomes
something unameable by *you*. It is the God of Plato: Truth. But lobian
machine can still deduce the complete theology of simpler lobian
machine (and understand she has to lift those theology only by *hope*
in self """""soundness""""""""). Many "quotes" to insist that a lobian
machine cannot define its own soundness (its own relation with truth or
its "intended model").
>
>
>> Well, you can doubt the axioms indeed, but this could lead
>> to long and useless debate. It is better, imo, to try to make the
>> postulate (axioms) sufficiently precise so that we can infer some
>> absurdity (internal or empirical).
>
> I agree. I also think it is interesting to develop this idea precisely.
> But do you think that discussing the assumptions is really useless?
No. But making it precise and searching consequences helps to avoid
misunderstanding. The comp hyp is really a religious belief: it *is* a
belief in the fact that you can be reincarnated through a digital
reconstitution of yourself relatively to some hopefully stable set of
computational histories (on which you can only bet). So the question is
not "is comp true"? The question is really: "do you accept your
daughter marries a computationalist".
And my point is only that IF comp is true then the mind body problem is
reduced into a derivation of physics (the eventually stable physical
beliefs) from ... addition and multiplication (and there is a gift: it
gives the quanta and the qualia, thanks to the G/G* separation
discovered by Godel, Lob, Solovay: see my url for more on this, Russell
says some words on it in his book)
>
>> I don't think so. Teaching in science, for adult, is (I mean ideal
>> teaching *should* be) an invitation to deduction in hypothetical
>> context, inductive inference and then the art of observation and
>> verification.
>
> Ok, that is of course correct - but you have to at least convince the
> people that it is worthwile to _reason_ correctly :-)
> (not all people seem to share this opinion, even at university!)
Yes indeed! But then how is it possible to convince someone who does
not reason correctly, of the advantage of reasoning correctly?
Answer: by letting him learn the consequences of reasoning incorrectly,
if he can still learn after!
Problem: about fundamental questions, this can take millennia, and more
...
I have to go, good week-end Günther and list (I will reply to John and
Russell later).
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Sep 14 2007 - 10:25:07 PDT