On Aug 31, 6:21 am, Brent Meeker <meeke....domain.name.hidden> 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, marc.ged....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
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) 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. The description
itself is an algorithm written in symbols.
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.
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. 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).
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Received on Fri Aug 31 2007 - 02:13:15 PDT