Re: Torkel Franzen on truth

tchow@xxxxxxxxxxxxx wrote:
In article <xLw9j.7491$hQ3.4060@pd7urf3no>,
Nam D. Nguyen <namducnguyen@xxxxxxx> wrote:
Unfortunately mathematical reasoning isn't religion where "beliefs"
would be much relevant.

You don't think beliefs are relevant in mathematical reasoning?

From what I gather, we don't call that "beliefs". We call it _interpretation_
which model basically is, and in which truths are true or false. The problem
of this model-truth is over the same "structure" there could be opposite
interpretation. Religious truth on the other hand is supposed to *believed*
as true whether or not there is a model to reflect the truth. That's why
belief doesn't have much of relevance in reasoning.

Then how do you become convinced that *anything* is true?

As I've explained above.

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.

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.

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.
.

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