Le 03-avr.-07, à 21:08, Brent Meeker a écrit :

>
> Bruno Marchal wrote:
>> Hi Tim
>>
>> Le 03-avr.-07, à 12:03, Tim Boykett wrote (in part):
>>
>>>    One of the recurring ideas here is that of "mathematicalism" - an
>>> idea
>>> that I understand to be that we perceive things as physical that have
>>> a certain
>>> mathematical structure. One of the "everything" ideas that results is
>>> that
>>> only certain of the all-possible universes have the right stuff to be
>>> perceivable, the right mathematical structure. We are in one such
>>> universe,
>>> and there are others.
>>
>> We can come back on this if you are really interested, but shortly:
>> once we assume the computationalist hypothesis (in the cognitive
>> science/theology), then the picture you give is most probably wrong.
>> Physics keeps a better role in the sense that physics emerges from the
>> "whole of arithmetic/mathematic". If you want, the physical world is
>> not a special mathematical world as seen from inside, but the physical
>> world somehow is the sum of all possible mathematical world where you
>> are.
>
> That brings up an issue which has troubled me.  Why arithmetic?  
> Mathematical physics commonly uses continua.  Most speculate that this 
> is an approximation to a more discrete structure at the Planck scale - 
> but I don't believe there has ever been any rigorous proof that this 
> kind of approximation can work.


I think that, assuming comp, the existence of third person actual 
infinities is undecidable for us. But infinities can be shown to be 
unavoidable from the first person perspective. Again we have to be 
aware that the same third person "truth" will appear different from the 
internal person points of view.  Note also that comp assume classical 
(non intuitionnist) arithmetic, and thus some actual number theoretical 
infinities.



>
> If we are to suppose that arithmetic "exists" because statements like 
> "2+2=4" are true independent of the physical world, then it seems that 
> calculus and analysis and geometry and topology should also "exist".


The situation is similar to the "Skolem paradox", the fact that that 
there are countable" model" of the Zermelo Fraenkel Set Theory. From 
inside ZF there are uncountable object, from outside, those object are 
countable. From inside Peano Arithmetic's mind there is indeed a sense 
for saying that analysis and geometry exists, and even necessarily 
exist. But we don't have to postulate them as ontic independent third 
person realities.



>
> I initially thought the idea of using arithmetic as the foundational 
> ur-stuff was attractive because I assumed that infinities could be 
> avoided,


Certainly not. At step 7 of the UDA, you should realize that infinities 
and continua are unavoidable. Indeed the measure we are searching on 
the OMs bears on a continuum of infinite computation+oracles. The 
Universal dovetailer does generate, from inside, all the real numbers.



> i.e. allowing only "potential infinities" as in intuitionist 
> mathematics.


Comp, as I define it, relies heavily on the excluded middle principle, 
that is, classical non constructive mathematics. Intuitionism is a pure 
first person view of math avoiding any bet on any external third person 
view. It corresponds indeed to the first person "soul-like" hypostasis.
To be sure, intuitionist arithmetic is quite similar to classical 
arithmetic, so that there is a non trivial intuitionistic form of comp, 
and it is a matter of technical simplicity for not using an 
intuitionist framework at the start.



> But it appears that diagonalization arguments are essential to Bruno's 
> program and those require realized infinities.


Yes that true, but as I said, many consequences of comp can be arrived 
at by restricting ourselves to effective diagonalizations, like the one 
I have presented with the growing computable functions and the 
constructive transfinite ordinals. But the very substance of comp is 
classical. You can make it intuitionist by using Godel -Glivenko double 
negation translation of classical arithmetic into intuitionistic 
arithmetic.

Each time we use the word "exist" we should make clear from which point 
of view we are talking.


Bruno


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


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