[FOM] Frege's error
William Tait
williamtait at mac.com
Sat Jul 30 10:30:47 EDT 2005
On Jul 14, 2005, at 11:27 AM, Neil Tennant wrote:
> But Frege would have resisted the use of closed lambda terms
> in any explanation of what he might have meant by transforming the
> function x>y into the function x>x. For Frege, functions were
> inherently
> unsaturated.
Perhaps *that* was Frege's error---that he thought that this notion
of incomplete or unsaturated object was either needed or even could
serve as a foundation for analysis.
The fact that bound variables are in principle always eliminable
shows that it is not necessary (e.g. to understand the semantics of
propositions compositionally---which ultimately seems to be the
grounds Frege gives for their necessity or "priority", as he puts
it). The fact that the notion of saturating an incomplete object is
entirely parasitic on that of substituting a closed expression into
an open expression would seem to limit incomplete objects to those
expressible by open expressions---and so to be countable in number.
(Of course, in logic we consider languages with uncountably many
constants. But we draw on the fact that we can take the constants to
be real numbers or transfinite ordinals, or whatever. And this is not
available to Frege; since it is a foundation for the theory of real
numbers and other infinite systems that he wants to establish. So I
would conclude that it is not possible to found analysis on the
notion of incomplete object in Frege's sense.
I agree that I am changing the subject, Neil. On the one you
addressed, you are entirely right. But it is hot and humid in Chicago
today and I feel grumpy.
(I first sent this message on july 17. It was returned because some
of it was not plain text. The weather has recently improved and I am
not at all feeling grumpy. But maybe my message is still of some
interest.)
Regards,
Bill Tait
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