On Mar 20, 2007, at 12:22 PM, Chris Partridge wrote:
>> On Mar 19, 2007, at 5:25 PM, Chris Partridge wrote:
>>> ...
>>> A number of points:
>>> 3) If one has a Fregean notion of number, then as the set of all
>>> (actual and possible?) sets with 17 members would have an infinite
>>> number of members - working out the density would be difficult, if
>>> not impossible.
>>
>> This is a red herring. All that the Fregean (or, for that matter,
>> *any* set theoretic) representation of the numbers buys you is a
>> class of well-defined objects that collectively model the axioms of
>> Peano Arithmetic; they provide a convenient answer to the question,
>> "What are the numbers?". You don't actually have to "manipulate"
>> them in any sense that requires that they themselves be represented
>> in a computer, you just *do* arithmetic.
>>
>>> But not absolutely meaningless. However, I have already agreed
>>> numbers pose problems.
>>
>> Why? Arithmetic is undecidable, of course, but so is first-order
>> logic, so I take it that that is not the sort of problem you have in
>> mind. If anything, because the ontology of the numbers -- a.k.a.
>> Peano Arithmetic -- is universally agreed upon, they pose far fewer
>> problems than most domains.
>
> If you look further up the discussion, you will see that this
> comment was in
> the context of trying to deny the existence of abstract objects.
> Pat offered
> numbers as his favourite example of abstract objects. If you see no
> problem
> in making numbers (integers, rationals, reals, transfinite ...)
> concrete, I
> am interested. To repeat Pat's question, Where is the number one
> located? (01)
I haven't the foggiest idea what relevance the question of the
metaphysical status of natural numbers (or anything else, frankly)
has to do with ontological engineering. If we need to talk about
number in the development of a useful ontology, we include axioms for
them; end of story. Under what real world circumstances is anyone
*ever* going to agonize over the metaphysical implications of their
sums and products? Under what real world conditions would any
practical benefit accrue from even broaching the question of the
existence of abstract entities? A certain amount of philosophizing
is unavoidable in ontological engineering, but it seems to me that
this issue, while interesting and important in academic philosophy,
is badly misplaced here. (02)
Chris Menzel (03)
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