[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
> (2) An actual implemenation of these forms in space-time (a
> *process* or computation)
> (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.
> 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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