Le 06-sept.-06, à 21:23, Tom Caylor a écrit :

>
> Bruno Marchal wrote:
>> Le 16-août-06, à 18:36, Tom Caylor a écrit :
>>
>>> I noticed that you slipped in "infinity" ("infinite collection of
>>> computations") into your roadmap (even the short roadmap).  In the
>>> "technical" posts, if I remember right, you said that at some point 
>>> we
>>> were leaving the constructionist realm.  But are you really talking
>>> about infinity?  It is easy to slip into invoking infinity and get 
>>> away
>>> with it without being noticed.  I think this is because we are used 
>>> to
>>> it in mathematics.  In fact, I want to point out that David Nyman
>>> skipped over it, perhaps a case in point.  But then you brought it up
>>> again here with aleph_zero, and 2^aleph_zero, so it seems you are
>>> really serious about it.  I thought that infinities and singularities
>>> are things that physicists have dedicated their lives to trying to
>>> purge from the system (so far unsuccessfully ?) in order to approach 
>>> a
>>> "true" theory of everything.  Here you are invoking it from the 
>>> start.
>>> No wonder you talk about faith.
>>>
>>> Even in the realm of pure mathematics, there are those of course who
>>> think it is invalid to invoke infinity.  Not to try to complicate
>>> things, but I'm trying to make a point about how serious a matter 
>>> this
>>> is.  Have you heard about the feasible numbers of V. Sazanov, as
>>> discussed on the FOM (Foundations Of Mathematics) list?  Why couldn't
>>> we just have 2^N instantiations or computations, where N is a very
>>> large number?
>>
>>
>> I would say infinity is all what mathematics is about. Take any 
>> theorem
>> in arithmetic, like any number is the sum of four square, or there is
>> no pair of number having a ratio which squared gives two, etc.
>> And I am not talking about analysis, or the use of complex analysis in
>> number theory (cf zeta), or category theory (which relies on very high
>> infinite) without posing any conceptual problem (no more than
>> elsewhere).
>
> When you say infinity is what math is all about, I think this is the
> same thing as I mean when I say that invariance is what math is all
> about.  But in actuality we find only local invariance, because of our
> finiteness.  You have said a similar thing recently about comp.  But
> here you seem to be talking about induction, concluding something about
> *all* numbers.  Why is this needed in comp?  Is not your argument based
> on Robinson's Q without induction?


Robinson Arithmetic (Q or RA) is just the ontic theory. The 
epistemology is given by Q + the induction axioms, i.e. Peano 
Arithmetic.
This fix the things. The SK combinators (cf my older post on this 
subject) gives a more informative ontology, but in the long run none of 
the ontic theories play a special role. With regard to the TOE search 
they are equivalent. Now, RA is not interviewed. It defines the UD if 
you want (RA is turing equivalent). But RA cannot generalize enough. We 
need PA for having the machinery to extract physics from the ontic RA.





>
>> Even constructivist and intuitionist accept infinity, although
>> sometimes under the form of potential infinity (which is all we need
>> for G and G* and all third person point of view, but is not enough for
>> having mathematical semantics, and then the first person (by UDA) is
>> really linked to an actual infinity. But those, since axiomatic set
>> theory does no more pose any interpretative problem.
>> True, I heard about some ultrafinitist would would like to avoid
>> infinity, but until now, they do have conceptual problem (like the 
>> fact
>> that they need notion of fuzzy high numbers to avoid the fact that for
>> each number has a successor. Imo, this is just philosophical play
>> having no relation with both theory and practice in math.
>>
>>
>>> The UDA is not precise enough for me, maybe because I'm a
>>> mathematician?
>>> I'm waiting for the interview, via the roadmap.
>>
>> UDA is a problem for mathematicians, sometimes indeed. The reason is
>> that although it is a "proof", it is not a mathematical proof. And 
>> some
>> mathematician have a problem with non mathematical proof. But UDA *is*
>> the complete proof. I have already explain on this list (years ago)
>> that although informal, it is rigorous. The first version of it were
>> much more complex and technical, and it has taken years to suppress
>> eventually any non strictly needed difficulties.
>> I have even coined an expression "the 1004 fallacy" (alluding to Lewis
>> Carroll), to describe argument using unnecessary rigor or abnormally
>> precise term with respect to the reasoning.
>> So please, don't hesitate to tell me what is not precise enough for
>> you. Just recall UDA is not part of math. It is part of cognitive
>> science and physics, and computer science.
>> The lobian interview does not add one atom of rigor to the UDA, albeit
>> it adds constructive features so as to make possible an explicit
>> derivation of the "physical laws" (and more because it attached the
>> quanta to extended qualia). Now I extract only the logic of the 
>> certain
>> propositions and I show that it has already it has a strong quantum
>> perfume, enough to get an "arithmetical quantum logic, and then the
>> rest gives mathematical conjectures. (One has been recently solved by 
>> a
>> young mathematician).
>>
>> Bruno
>>
>>
>> http://iridia.ulb.ac.be/~marchal/
>
> What is the non-mathematical part of UDA?

If only the "yes doctor". UDA is applied math.


> The part that uses Church
> Thesis?

Also, but in a lesser measure. Church Thesis is at the intersection of 
math and philosophy. But is, nevertheless, 100% Popper-scientific by 
being clearly refutable.


>  When I hear "non-mathematical" I hear "non-rigor".


Gosh!   "Mathematical" can be non rigorous (cf Euler). Rigorous can be 
non mathematical (take any good text in any field, or just take the 
following non mathematical reasoning:
Socrate (a non math concept) is a human (also)
All human are mortal
Thus Socrate is mortal.



> Define
> rigor that is non-mathematical.

Clear definition, clear postulate, valid reasoning, etc.


> I guess if you do then you've been
> mathematical about it.  I don't understand.

Not necessarily. Or with "math" in a large sense: any applied math I 
would say.

OK?

Bruno

PS Must go now, I will comment Russell and Stathis asap.






http://iridia.ulb.ac.be/~marchal/


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