On 15 Jun 2015, at 17:15, Brian Tenneson wrote:
I had forgotten I wrote this a while back, from my FB feed "on this
day." Seems relevant.
Can truth ever be proven?
That has no meaning. Truth about what?
Here's something I wrote in a discussion I'm having.
Structure does not cause something to be non-fictional, nor does
lack of structure cause something to be fictional. A theorem in one
formal system might be false in another,
Formal system = machine = 3p-person. yes, they have all theior on
opinion, but this does not mean that some are not true or false.
a lot like how different video games have different rules. Even if
you "prove" something about all formal systems, that "proof" has
been carried out in a larger formal system;
Not necessarily. Formal systems can prove a lot about themselves,
including their own incompleteness conditionalised on their consistency.
so there is an inherent circularity,
I think the one you allude to is the one solved by the diagonals of
Kleene. Not the time to say much more, but I have explained this.
or more accurately, an inherent interdependency. It's like being in
a video game trying to prove that something is true of all video
games but that meta-game proof is being conducted in one of the
video games the proof is about. Thus, the concept of proof needs to
be anchored to something true but by this rationale, proof is merely
anchored to itself.
Anchoring proof on truth leads to the first person.
Therefore, perhaps proof of truth is an unattainable goal in math.
Proof of which truth. In logic, truth is defined by a model. "A" is
true if it is the case that A in the intended model.
Perhaps proof of truth is an unattainable goal anywhere.
?
Well, proof of Truth, with a big T is like the proof of the existence
of God: that does not exist, we have to start from something. But
since Gödel we know that "proof" means belief, and that it can't be
taken for granted.
If I were to say that both confirming and denying the statement
"there is no such thing as truth" implies that there is truth, I am
still formulating that theorem "there is truth" within yet another
formal system which, on the surface of things, gets us nowhere. It
is like inventing a two-player game with, from an outside point of
view, a bizarre set of rules, and claiming that checkmating someone
in that game amounts to producing not just truth but proof of truth.
The people outside our fishbowl looking in on us must be very
amused, just as are the people outside their fishbowl looking in on
them.
Ok, that is Truth with a big "T", and so it needs faith. It is
religion. OK.
Formal systems show us that our usual formal systems (the ones we
use to communicate, inform, and persuade in English for instance)
have the same relationship to truth that Earth does to the center of
the universe. No formal system is provably true and correct, though
there are formal systems that might conform to what we perceive.
Formal systems can only be proved relatively true compared to other
formal systems.
No, you can compared them to mathematical structure. That is why we
have a model theory. The situation is not that bad. We can use our
intuition of the finite to get the models. That is what we do when we
talk about limit, models, analysis, complex numbers, etc.
At least until that anchor is found.
That reduces math to a grand symphony. Grand symphonies aren't
inherently true or false and there is no hope in my mind of proving
the grand symphony that is math to be true. Another way to look at
is is a grand poem that makes up its own rules and even explicitly
acknowledges that fact.
The question of whether concepts referenced by the poem actually
exist is to open the door to many formal systems we might walk into
in order to answer the question. Moreover, it will be true in some
but not others that that concept exists. A really broad
interpretation of existence would be that something exists if it is
referenced by a grammatically-correct statement made in at least one
formal system.
I think that we can agree on elementary arithmetic, then we can agree
on the fact that almost all universal numbers will disagree on what
extends arithmetic.
Bruno
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