Bruno Marchal wrote:
>
> Le 29-août-07, à 23:11, Brent Meeker a écrit :
>
>> Bruno Marchal wrote:
>>> Le 29-août-07, à 02:59, marc.geddes.domain.name.hidden a écrit :
>>>
>>>> I *don't* think that mathematical
>>>> properties are properties of our *descriptions* of the things. I
>>>> think they are properties *of the thing itself*.
>>>
>>> I agree with you. If you identify "mathematical theories" with
>>> "descriptions", then the study of the description themselves is
>>> metamathematics or mathematical logic, and that is just a tiny part of
>>> mathematics.
>> That seems to be a purely semantic argument. You could as well say
>> arithmetic is metacounting.
>
>
>
>
>
> ? I don't understand. Arithmetic is about number. Meta-arithmetic is
> about theories on numbers. That is very different.
Yes, I understand that. But ISTM the argument went sort of like this: I say arithmetic is a description of counting, abstracted from particular instances of counting. You say, no, description of arithmetic is meta-mathematics and that's only a small part of mathematics, therefore arithmetic can't be a description.
Do you see why I think your objection was a non-sequitur?
Brent aMeeker
>Only, Godel has been
> able to show that you can translate a part of meta-arithmetic into
> arithmetic, but that is not obvious (especially at Godel's time when
> the idea of "programming" did not exist). Obvious or not the
> disctinction between metamathematics and mathematics is rather crucial.
> It is as different as the difference between an observer and a reality.
>
>
>
>
>
>
>
>>> After Godel, even formalists are obliged to take that distinction into
>>> account. We know for sure, today, that arithmetical truth cannot be
>>> described by a complete theory, only tiny parts of it can, and this
>>> despite the fact that we can have a pretty good intuition of what
>>> arithmetical truth is.
>> But one would not expect completeness of descriptions.
>
>
>
> Why? After all complete theories exist (like the first order theory of
> real numbers for example). Incompleteness of ALL axiomatizable theories
> with respect to arithmetical truth has been an unexpected shock.
> Hilbert predicted the contrary.
>
>
>
>
>
>> So the incompleteness of mathematics should count against the
>> existence of mathematical Truth - as opposed to individual
>> propositions being true.
>
>
>
> I don't understand. Incompleteness of a theory is understandable only
> with respect to some interpretation or model, that is notion of truth.
> I do follow Godel on this question.
>
>
>
>
>
>
>> Doesn't it strike you as strange that arithmetic is defined by formal
>> procedures,
>
>
>
> Only a *theory on* arithmetic or number is defined by formal procedure
> (and does constitute an abstract machine).
>
>
>
>
>
>
>> but when those procedures show it to be incomplete, mathematicians
>> resort to intuition justify the existence of some whole? Theology
>> indeed!
>
>
> I don't understand. All mathematicians (except few minorities like
> ultrafinitists) accept the notion of arithmetical truth, which can be
> represented by the set of all true sentences of arithmetic (or to be
> even more specific, it can be represented by the set of godel numbers
> of the arithmetical sentences). But no theory at all can define
> constructively that set. That set is not recursively enumerable. No
> algorithm can generate it.
> A rich lobian machine, like a theorem prover for a theory of set like
> Zermelo-Fraenkel, can define that set, but still not generate it, and
> it can be proved that this remains true for all the effective extension
> (where an extension is effective when the extension is still an
> axiomatizable theory.
> So yes, arithmetical truth is a purely theological matter for a simple
> lobian machine like Peano Arithmetic, but is just simple usual math
> (despite non effectivity, but this you get once you accept classical
> logic) for a super-rich lobian machine like ZF.
>
> Although sometime you say correct thing in logic, I get the feeling
> that you miss something about incompleteness ... (to be frank). Are
> you aware that the set of true arithmetical sentences is a well defined
> set in (formal or informal) set theory, yet that it cannot be generated
> by any (axiomatizable) theory.
>
> (note: Axiomatizable theory = theory such that the theorems can be
> generated by a machine. You can take this as a definition, but if you
> know the usual definition of "axiomatizable theory", then this is a
> consequence by a theorem due to Craig).
>
> I have to go. I will say more to David tomorrow.
>
> Bruno
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
> >
>
>
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Received on Thu Aug 30 2007 - 14:21:33 PDT