On Oct 8, 2009, at 9:05 AM, John F. Sowa wrote: (01)
> Pat,
>
> I agree that all sets are mathematical.
>
> JFS>> For mathematical sets...
>
> PH> There is no such thing as a 'mathematical' set.
>> Sets are sets, whatever they are sets of. Set theory
>> is ontologically neutral.
>
> Yes. I was using a convenient, but possibly confusing feature
> of English, which allows an adjective that modifies a noun to
> be used with the relationship left unspecified.
>
> The longer phrase I had in mind was
>
> "For sets composed of mathematically defined entities"
>
> as contrasted with
>
> "For sets composed of physical entities"
>
> JFS>> ... the "simple" criterion usually involves a proof of
>>> equivalence: for anything other than small finite sets, it
>>> is necessary to prove that the two specifications determine
>>> exactly the same elements.
>
> PH> That is needed in order to SHOW that two sets are identical, yes.
>
> Yes indeed. And for any sets composed of physical entities, it is
> necessary to SHOW that a variable S that represents a set at one
> instant of time can be assumed to represent "the same" set at
> a different time (02)
No, that never needs to be shown. In fact, as stated it doesn't even
make sense. (03)
> :
>
> 1. The set theoretical operations are ontologically neutral.
>
> 2. That neutrality lulls people into thinking that the precisely
> defined operations on sets composed of mathematical entities
> are just as reliable when carried out on sets composed of
> physical entities. (04)
They are as reliable, no lulling involved.
>
> 3. But as Whitehead pointed out, "The baby in the cradle and the
> grown man in middle age are in some senses identical and in
> other senses diverse. Is the train of argument in its conclusions
> substantiated by the identity or vitiated by the diversity?" (05)
So, when using set talk, one needs to specify what things are the
things in the set. Possibilities here are (1) babies in cradles, as
distinct from grown men (2) grown men, as distinct from babies (3)
people, considered to be the same person whether a baby or a grown
man. These are different kinds of thing. But sets of them are all just
sets. (06)
>
> 4. Therefore, any use of set theory for representing physical
> entities
> must recognize that a set at one instant of time is not "exactly
> the same" set of supposedly "exactly the same" things at any
> other instant. (07)
Sets have no temporal dimension. There is no such thing as a 'set at
one instant of time'. To paraphrase Gertrude Stein, a set is a set is
a set. What one does have to do is to specify, before even talking of
sets of things, whether or not *the things themselves* have temporal
dimensions, or how their identity criteria are affected by the passage
of time. But a set, once specified, is not a dynamic entity. That
would be something like a function from times to sets, or the like:
itself a thing that can be modeled using sets, of course, but then it
would be a set of <time, set> pairs. (08)
>
> 5. We can often say that they are "approximately" the same for
> one particular application, but the differences might be
> critical for a different application.
>
> Mathematicians never think about such issues in abstract set theory,
> but any application of set theory to the physical world must address
> them. (09)
You have to be clear what it is that you are talking about. But I see
this as something of an advantage: the use of set talk requires one to
be unusually rigorous about saying what one is talking about, exactly.
Its not enough to just say, a set of tomatoes, without being clear
whether one means tomatoes at one moment in time, or tomatoes
considered as continuants or as 4D histories or genera or species
or ... But one can have sets of any of these in any combination. (010)
Pat (011)
>
> John
>
>
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> (012)
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