Re: Torkel Franzen on truth

MoeBlee wrote:
On Dec 17, 3:27 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
tc...@xxxxxxxxxxxxx wrote:
In article <xLw9j.7491$hQ3.4060@pd7urf3no>,
Nam D. Nguyen <namducngu...@xxxxxxx> wrote:

From what I gather, we don't call that "beliefs". We call it _interpretation

That we have a method for interpreting formal languages does not
contradict that one may believe certain things about mathematics.

which model basically is, and in which truths are true or false.

"in which truths are true or false". Are you sure you've eaten
breakfast this morning?

I usually have a glass of milk every morning. Some *interpret* that
as having breakfast; others would have opposite interpretation.
Actually my interpretation on that might vary from years to year.
What's about yours interpretation on this?


The problem
of this model-truth is over the same "structure" there could be opposite
interpretation.

No, that's completely wrong. Given a structure, there is only one
interpretation associated with that structure.

Given the structure "Nam's having a glass of milk every morning", how
many interpretations would one have for the sentence "Nam has breakfast
every morning"?


Religious truth on the other hand is supposed to *believed*
as true whether or not there is a model to reflect the truth.

I've never seen such a description of religious belief.

I'm sure there are descriptions that are very much *similar*!


That's why
belief doesn't have much of relevance in reasoning.

Belief may or may not have relevance in reasoning, but the confusions
you just posted don't lead to any conclusion on the matter.

Whose "confusions" are you talking about? I don't seem to have any here.


Then how do you become convinced that *anything* is true?
As I've explained above.

No you didn't.

That's one opinion of course.


Are you convinced, for example, that sqrt(2) is irrational? On what basis?

On the basis of model that "sqrt(2) is irrational" is true, of course.

Maybe you mean, on the basis that there is a model in which "sqrt(2)
is irrational" is true.

"Maybe"? My answer to Tim Chow's question is a straightforward short-one-liner
answer and you seemed to not understand?

And there is a model in which it is false also.

So far I don't see what your point here is!

What about operations on finite strings?

What about them?

Don't you believe, for
example, irrespective of any model, that the string "0011" is the same
as the string "0022 [with 1 substituted for 2]"?

In what context are you talking about "sameness", "substitute", etc...
Sorry your question is too vague in semantic and consequently is subject
to different interpretations.


On the basis of the proof?
No, not on the basis of proof: what is true or false is based strictly on model.
Syntactical provability is actually in a different (and independent) paradigm,
not withstanding Completeness.

But the proof starts with some axioms.
Of course.

On what basis do you become convinced of the correctness of the axioms?
What exactly does "correctness of the axioms" mean?

Or are you *not* convinced of the axioms?
The only senses for which we could talk about axioms are:

(a) They be independent from each other.
(b) They don't contradict each other.

No, there are lots of other properties of axioms. One, for example, is
that of a certain model being a model of the axioms.

Of course there are other properties: axioms' being finite formulas, etc...
All of these (and what you've mentioned) are utterly trivial not worth
being mentioned, it seems. So why are you mentioning here? Is it because
it has something to do with the purported "correctness of the axioms" that
is being discussed between me and the other poster?

(Besides, to be to be a property of axioms it has to apply to all axioms
in all circumstances, e.g. being finite formulas. Your "certain model being
a model of the axioms" is not applicable to all axioms!)


So, again, what does it mean to be "convinced of the axioms"?

But if you're not convinced of the axioms, then what good is a proof of
"sqrt(2) is irrational" from those axioms?
Proofs of course are good as a mechanism of assisting us in preventing
our reasoning from being inconsistent. Of course.

Except if the axioms are inconsistent. Actually, (first order) proof
doesn't ensure consistency but rather entailment.

Agree. Except that I only said "as a mechanism of assisting": I never said
anything about proofs guaranteeing/ensuring consistency. Of course not, in general.
But in some particular circumstances, proof would help consistent reasoning.


MoeBlee

.

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