[FOM] Frege's error

[William Tait]

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