[EMAIL PROTECTED] wrote: > > > On Aug 31, 6:21 am, Brent Meeker <[EMAIL PROTECTED]> wrote: >> 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 Meeker > > Mathematical concepts have more than one sense, is the point I think > Bruno was trying to make. For instance consider algebra - there's > *Categories* (which are the objectively existing platonic > mathematical forms themselves) So you say. > and then there's the *dynamic implementation* of these categories: > the *process* of algebraic operations (like counting). But processes > themselves (computations) are *not* equiavalent to the *descriptions* > of these processes. Sure. Counting sheep and goats and adding them up isn't equivalent to Peano's axioms. Who said otherwise? > The description itself is an algorithm written in symbols. Peano's axioms aren't an algorithm. Algorithms are computational procedures and aren't necessarily written in symbols. Writing the symbols might be an *instance* of an algorithmic process. As I type my computer is executing algorithms that are embodied in electronic processes. > > So three senses of math here: > > (1) The platonic forms (which are timeless and not in space and > time) > > (2) An actual implemenation of these forms in space-time (a > *process* or computation) > > and > > (3) The symbolic representation of (2) - an algorithm as written on > a peice of paper, described , drawn as diagram etc. > > > You can see that the *process of counting* (2) is not the same as the > description of counting (3). When you (Brent) engage in counting > your brain runs the algorithm. But a description of this process is > simply symbols written on a piece of paper. No, a description is Peano's axioms or some other axioms that describe the numbers and their relations. > > As to Godel, I agree with Bruno. The point is that there are > *perfectly meaningful* mathematical questions expressed in the > language of some formal system for which the answers can't be found > within that system. This shows that math is bigger (extends beyond) > any system as described by humans ; so math itself is objectively > real and can't be just descriptive. If math were just descriptive, > all meaningful math questions should be answerable within the human > described system. > > --- > > PS Hee hee. This is getting easier and easier for me. My old > opponents elsewhere are getting slower and slower. Or they're just getting tired of dealing with unsupported assertions. Brent Meeker > That's because they started from the 'bottom up' and are progressing > more and more slowly as they try to go to higher levels of > abstractions. (so they've run into a brick wall with the problem of > 'reflection'). I, on the other hand, started at the very highest > level of abstraction and my progress is getting faster and faster as > I move down the levels of abstraction LOL.. > > (Note: The PS was just a digression - nothing to do with this thread > or list). > > > > > > > --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
