Le 08-juin-07, à 20:17, Tom Caylor a écrit :
> I should respond to your response. I'm in a busy pensive state
> lately, reading Theaetetus (as you suggested on the Incompleteness
> thread) along with Protagoras and some Aristotle (along with the dozen
> other books I'm always reading...) in the little time I have.
Take your time .... I will be extremely busy the next two weeks (exams
and then Siena Cie 2007).
>
> But you do make assumptions as part of the comp hypothesis, including
> assumptions about numbers.
I just assume the validity of the excluded middle principle on purely
arithmetical question. If you prefer I just assume that if you run a
machine then either that machine will stop or it will run forever.
I don't know anyone not believing in this, but I have to make it as an
assumption because I can not prove it from less, and I use it in the
proofs.
In the same spirit I assume 0 is not the successor of a positive
integer, etc. (I reason axiomatically).
Some technics can make this hypothesis weaker though.
> LRA looks to be about the particulars of arithmetic. PA, with
> induction, is trying to generalize to come up with some universal
> truths about arithmetic.
About numbers, ok.
> But LRA has access to only one particular truth at a time, with no
> awareness of generalities/universals.
Just few of them. OK.
>
>
>> LRA is, like PA, under the godelian limitation joug. Only, PA knows
>> it!
>> Lobian machine, like PA or ZF, are godel-limited, but they are aware
>> of
>> their limitation. OK?
>>
>
> So PA has this "awareness", by *definition*.
Not by definition at all! Showing this is the "difficult part" of
Godel's second incompleteness theorem. It is done for the first time in
some ugly way by Ackerman and Hilbert in their *Grundlagen*. It is the
work of Lob which has made possible to do it in a beautiful way, and
this has been a key step to the discovery of the modal logics G and G*,
which formalizes completely the propositional logic of self-reference.
> It is a useful *tool* in
> mathematics, but you are assuming it is a part of reality at the
> deepest level. This is part your Arithmetic Realism part of the comp
> hyp, is it not?
No. I consider PA as a clever being, a sort of *baby God* like any of
us could hope to be (with comp). PA is just a universal machine knowing
that she is universal (in a weak and precise sense). That is, PA is
what I call a lobian machine.
In some sense, PA "is" a turing machine having already the cognitive
abilities to begin being anxious about the length of its available
tape.
>
>>
>>> This is because even
>>> the statement 1+2=2+1 is a Plato-like statement. The Aristotle
>>> verification would be to take 1 object and then take 2 more objects
>>> and count the group as a whole. Then take 2 objects and then 1
>>> object
>>> and count the group as a whole. But, first of all, there are at
>>> least
>>> conceptually a (at least potentially) infinite number of objects you
>>> could use for this experiment, and you could do the experiment as an
>>> observer from an infinite number of angles/perspectives. Plus, a
>>> difference in perspective could make it so that you are taking the
>>> objects in a different order and so invalidate the experiment. I
>>> don't know what the implication is here other than there are very
>>> fundamental philosophical assumptions to deal with here. This is
>>> even
>>> without bringing multiplication into the picture. It seems, if you
>>> are going to base your reality on math, that these kinds of questions
>>> aren't unimportant because they remind me of the fundamental problems
>>> at the base of the quantum versus relativity.
>>
>> I cannot comment because it is a bit vague for me. Normally I can not
>> address physical question before getting the comp-physics.
>>
>> Bruno
>>
>
> The above does not require physical reality, but only concepts that we
> can think about looking inward (eyes closed view). But even though it
> is "only" conceptual, my point is that we are taking a "leap of faith"
> even when we talk about 1+1=2, classifying an infinite number of cases
> into one equivalence class.
Not at all. This could appears in engineering when you apply a theory,
not when you do math.
"1+1 = 2" means what you have learn in high school where the
mathematical structure written (N, 0, +, x) by algebraic minded people
has been introduced to you. "1+1 = 2" can be false in many mathematical
striucture. But then they admit other and different axiomatics.
Thanks to the *completeness theorem of Godel" what can be proved by PA
is true in ALL mathematical structure which verifies the axioms of PA.
But there is no machine which can prove all what is true in the
standard model (N, 0, +, x), which escapes all machine and all
theories.
>
> Perhaps at the core of this issue is whether things like "+" are
> prescriptive or descriptive. Is it possible that there are universes
> with mathematical "white rabbits" such that when you take 1 thing and
> 1 other thing ("physical" or not) and associate them in any way,
> including just thinking about them, then you don't necessarily get 2
> things (e.g. sometime you get 1 or 3 or 0)?
Yes of course. But they are not model of PA or LRA. They are model of
other theories.
Now there exist non standard model of PA, where for example a proof of
0 = 1 exist, but even in those model 0 is different from 1. It just
happens that in those models some weird object exists (different from
any usual number) playing the role of "a proof of 0=1".
Those who know the Lob formula knows that Bp does not necessarily
entails p, so that those structure are indeed possible.
Everywhere here I use model in the sense of the logicians. A model, I
recall, is a mathematical structure verifying proposition.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Jun 09 2007 - 09:41:34 PDT