On Feb 11, 2009, at 10:41 AM, Chris Partridge wrote:
> PatH,
>
> Last time you raised the question of numbers, and I responded, Chris
> Menzel told me off for raising something NOT relevant. (01)
Really? I find that hard to believe. Given my own proclivities,
that'd be the pot calling the kettle black. :-) (02)
>>> 1. Are there certain things like e.g. numbers and possible worlds?
>>>
>>> 2. Are they abstract or concrete?
>>>
>>> Even the first may be controversial, but even if you agree that
>>> these things exist, whether they are considered abstract or not is
>>> a different question (and arguably more controversial).
>>
>> Whether they exist or not, I think you have to say that numbers
>> are not concrete. You can't weigh seven.
>
> As I think you must know there is a Fregean concept of number (where
> 2 is the set of all sets with two members). This Fregean proposal
> has modern supporters - e.g. Crispin Wright
>http://en.wikipedia.org/wiki/Crispin_Wright (03)
It is quite misleading to express the thesis in terms of sets, as it
is easy to show in modern Zermelo-Fraenkel set theory that there is no
set of all 2-element sets. Frege's account, famously, ended in
failure due to Russell's paradox. Modern reconstructions show that
arithmetic can in fact be reconstructed in second-order logic a la
Frege by the addition of "Hume's Principle": that the number of Fs =
the number of Gs iff there is a bijection between the Fs and the Gs.
The issues here are very subtle and extremely interesting. A masterly
account of both the history and the logic of Frege's account by Edward
Zalta can be found in the Stanford Encyclopedia of Philosophy: (04)
http://plato.stanford.edu/entries/frege-logic (05)
> There is debate about whether sets are abstract - David Lewis asks
> why we cannot say the set of cars in the car park is located in the
> car park - (06)
Are you sure Lewis asked that? (07)
> and if it is located, it cannot be abstract. Hence, there is an
> account of numbers that says they are not abstract. (08)
That, of course, is a complete non sequitur. Even ignoring the fact
that your proposed definition of "number" above doesn't work, all
Lewis's (?) question shows is that sets of *concrete* objects might be
considered concrete. But, on the "Fregean" definition of number you
propose, for it to follow that, say, the number 2 is concrete, you'd
need *all* 2-element sets are concrete, not just 2-element sets whose
elements are themselves concrete. (09)
-chris (010)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (011)
|