Dear Bruno,

>> 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 

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.

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 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.

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).

> 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?

> 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!)

Best Regards,

Günther Greindl
Department of Philosophy of Science
University of Vienna


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