Bruno Marchal wrote: > > Le 29-août-07, à 23:11, Brent Meeker a écrit : > >> Bruno Marchal wrote: >>> Le 29-août-07, à 02:59, [EMAIL PROTECTED] 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/ > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---