> 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).
> > 

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