> On 8 May 2018, at 18:01, John Clark <[email protected]> wrote:
>
> On Mon, May 7, 2018 at 8:12 PM, Russell Standish <[email protected]
> <mailto:[email protected]>> wrote:
>
> >> I think you're confused about the difference between what a model says
> and what reality says. One model may say you can safely march across that
> bridge and another model might say the bridge will collapse, but it makes
> no difference which model you believe when you cross the bridge, it will
> either fall down or it won't.
>
> > Unfortunately you are using "model" in a different sense to how Bruno(or
> logicians generally) uses it. The real world bridge is a model.
> If the real world bridge is a model then give me a example of something that
> is not a model,
>
Logician use “model" like painter. Model = the real thing, like the naked woman
waiting to be painted. In that analogy, the theory is the painting.
Physicist use “model” like children. Model = toy model, like a small car to
play with.
Example in mathematics.
A theory is a interpretation set of formula. A model is a mathematical
structure. So here is a theory:
(Classical logic) +
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
A model of that theory is the structure (N, 0, +, *), where N is the set of
natural numbers, 0 denotes 0, and + and * is the usual addition and
multiplication.
Exercise: find a model of the theory above in which 0 + x is ≠ x, and where
there is a biggest natural number.
The theorem are true in all model, and a theory is consistent (does not prove A
and ~A, for any A) iff it has a model.
> if you can not then the word "model" has no meaning. Unlike the Continuum
> Hypothesis the Goldbach Conjecture is subject to the potential of
> experimental falsification, if logicians eventually proved that it is true,
>
We don’t prove that something is true. We just prove it. That plays some role.
No sound machine can ever prove that whatever she proves is true.
> that is to say they started with nothing but their axioms and derived it, but
> then the next day a computer found a huge even number that was NOT the sum of
> two prime numbers I think logicians would be very upset, or at least the
> competent ones would be. I think they would say that shows their present
> axioms must not be "sound" in the technical sense and need to be modified. I
> don't think they would say "the laws of physics that the computer runs on
> must be wrong and our model is right and every even number is the sum of two
> primes and thats that and I don't want to hear anymore about it"; but if I'm
> wrong and they did say that then I would no longer be interested in anything
> logicians said in the future because they would be jackasses. But I don’t
> think they’re jackasses because good logicians know there is a difference
> between proof and truth, physics will always tell you the truth but a proof
> is only as good as the axioms it is based on.
>
>
Aristotelian faith, like St-Thomas. You believe in what you see. Platonism are
skeptical on every cilamied truth, especially about what they see, because they
know they could be dreaming, or deluded, or that there is a systematic
experimental error, etc.
> > Your models would be called theories, and the real world bridge either
> satisfies it or not.
>
> You say the real world bridge is a theory,
I don’t think Russell says that. He said that what you call “model” is called
“theories” by logicians, and that logicians tackle the notion of reality with
the notion of model. Logicians studies the relation between theories (finite or
enumerable set of beliefs) and truth (a notion relative to interpretation, and
which is usually infinite-.
> so now we have theories about theories?
Yes of course. That is what mathematical logic is all about. A synonym of
“mathematical logic” is Metamathematics (used by Gödel, Kleene, etc.).
The whole point of Gödel’s discovery, is that a large part of the
metamathematics can be made in the mathematics. G formalises the logic of the
meta-arithmetic that Peano Arithmetic can prove. G* formalises the whole truth
of the standard model of arithmetic (what we learn at school). G* \ G
formalises what is true but cannot be prove in that meta-arithmetic (roughly
speaking, avoiding technical details).
> Give me an example of something that is not a theory, if you can not then
> like "model" the word "theory" has no meaning either.
Nothing are more distinct than theories and models. It is as much different
than a far away galaxy and Hubble telescope, or, the finger and the moon.
The theory (in logic) is a finite set of axioms, or a (recursively) enumerable
set of axioms.
The model of a theory is a reality which satisfy (in some precise sense which
would be long to describe here) the axioms and the theorems of the theory.
Think of the theory of group and an example of a “concrete”, like the group of
permutations of the set {a, b, c, d}, etc.
Mathematician works mainly with models. Formal theories are the object of study
of the logicians.
Bruno
>
> John K Clark
>
>
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to [email protected]
> <mailto:[email protected]>.
> To post to this group, send email to [email protected]
> <mailto:[email protected]>.
> Visit this group at https://groups.google.com/group/everything-list
> <https://groups.google.com/group/everything-list>.
> For more options, visit https://groups.google.com/d/optout
> <https://groups.google.com/d/optout>.
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
