On Oct 7, 2009, at 8:12 AM, John F. Sowa wrote: (01)
> Chris,
>
> CP> I can see the problem for types - which one does not escape
>> by saying "the same" could be replaced by "provably equivalent.
>>
>> However, I cannot see the problem for sets - and cannot see
>> where you explain this in your response (maybe I missed it).
>> Could you give us some idea what you mean.
>
> I have mentioned those issues in many different ways in many
> emails over the years. I'll summarize the points briefly. (02)
John, just to clarify. Your point is that real-world identity criteria
are complex and nontrivial to state, and even can be subject to
rational disagreement; and that identity criteria for sets do not face
up to these challenges. All this is true, but it does not address
Chris' point, above, which is simply that the identity criteria *for
sets* are indeed extremely simple and completely beyond controversy:
identity of sets simply means having the same members. There is no
such thing as an intensional set, or a non-extensional set, or a set
which changes with time, etc.. The use of set language burns away all
such complexities and doubts: indeed, that is what it was invented
for. So it is misleading and strictly incorrect to use phrases like
'intensional set' when talking about a collection defined in some way
that leaves some room for doubt or interpretation regarding exactly
what it is a collection of, or what exactly its members are. Such
collections may be important, but if their identity criteria are in
any way blurry or dependent upon some contextually provided source of
information, then they are simply *not sets*. So the problems you
mention, below, are not problems with identity between sets. (03)
> One of the problems with 20th century logic is that the great
> technical advances were made in the application of logic to the
> foundations of mathematics. As a mathematician, I consider that
> a perfectly fine use for logic.
>
> Unfortunately, there was comparatively little work in analyzing
> the application of logic to the real world. Carnap, for example,
> did make the attempt in his Logische Aufbau der Welt, but he
> abstracted away all the messy details. The later chapters of his
> book, which include all the interesting stuff, are pure hand waving.
>
> The reason why Peirce, Whitehead, and the later Wittgenstein are
> my favorite logicians is that they faced the messy details head on.
> Following is a quotation from Whitehead's last book _Modes of
> Thought_, which I highly recommend: (04)
+1 on that. It is very hard to find a clearer thinker than ANW. (05)
> ANW> In logical reasoning, which proceeds by use of the variable,
>> there are always two tacit presuppositions -- one is that the
>> definite symbols of composition can retain the same meaning as the
>> reasoning elaborates novel compositions. The other presupposition
>> is that this self-identity of each variable can be preserved when the
>> variable is replaced by some definite instance.... 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? (pp. 145-146)
>
> In mathematics, those sets and the variables that refer to them
> are always perfectly well defined. But whenever you talk about
> anything in the real world -- including people -- those things are
> constantly in flux. You can never be certain that two different
> references to what is supposed to be "the same thing" are really
> "exactly the same" for whatever abstraction is being considered. (06)
Well, never is a bit strong. But this needs to be spelled out
carefully, yes. (07)
Pat (08)
>
> John
>
>
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> (09)
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