Chris, (01)
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. (02)
I have mentioned those issues in many different ways in many
emails over the years. I'll summarize the points briefly. (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. (04)
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. (05)
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: (06)
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) (07)
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. (08)
John (09)
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