On 03 Jun 2016, at 21:48, Brent Meeker wrote:
On 6/3/2016 11:26 AM, Bruno Marchal wrote:
I'd say a rational person is one who can give coherent reasons for
their beliefs.
Which makes PA into a rational person, even more so than most of
us. Humans still act like they believe that the only coherent
reason to believe in something is that the boss say so (and
everyone know that the boss is always right, especially when wrong!).
That is a coherent reason and a person giving it is rational. Being
rational doesn't mean being right or having all knowledge. "Because
the boss said so." may be a weak reason if the the boss is
expounding on international politics, but an excellent reason if
the boss said, "You'll be fired if you're late to work again."
OK, but to derive physics, we can limit ourselves to arithmetical
statement. It is irrational to believe such a statement just because
the boss or the teacher say so (in practice, it can be rational if the
goal is to have a good note, or to impress a mate, but again that is
not relevant).
But even PA, if asked about why she believes in x + 0 = x, might
say something like "obvious", or "Instinct",
Those are not coherent reasons. I find it telling that you use the
religious formulation "believes in" rather than just "believes".
Hmm, I am afraid it is just me mishandling english.
or "I have been told", or "I have many examples", or ....
Those are weak reasons.
All beliefs in physics are of that kind. We believe that F = GmM/r^2
only because we have many, but a finite number, examples.
I cannot prove to you that x + 0 = x. Well, I could prove it from the
K and S combinators axioms, but I will use implicitly my intuitive
number knowledge to choose a definition of numbers such that we
recover x + 0 = x from the combinators axioms. In fact all beliefs in
*any* theory is never 100% rational. But when we work in a theory, or
when we interview a machine, we start from things that we have few
doubts upon.
To be clear, I say that a belief by a machine M is rational when M can
makes its hypothesis clear (so I can define in the machine's language
"believed of p by M", say []p, and which is such that the machine
believability is closed form modus ponens, that is we have
[](p -> q) -> ([]p -> []q), and, in our case, that the machine can
prove (also) [](p -> q) -> ([]p -> []q). In fact, to make the proof
simple I use:
M believes p entails M believes []p (normality)
M believes []p entails M believes p (stability)
M believes [](p -> q) -> ([]p -> []q).
Löbian machines believes also []p -> [][]p (and eventually, it is a
theorem: []([]p -> p) -> []p. The reason is that by being a machine,
we get the diagonal closure and the fixed point properties from which
Gödel and Löb theorem follows (in the ideal case of sound machines).
To believe does not, indeed, mean to be sure that it is true. You can
use "assume" instead of beliefs. All laws are assumed, in the applied
science. In science, we *never* say that some statement are true. We
say that a statement is an axiom, or a theorem, or that is confirmed
experimentally. We just cannot prove anything from a finite number of
observations.
Bruno
Brent
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