FOM: Question: why is category theory viewed as relativism? wtait at
Wed Mar 11 12:26:54 EST 1998

Michael Thayer (mthayer at wrote (3/11)

> How
>does this differ from the discredited view that set theory does not offer a
>foundation for analysis, because there is no distinguished notion of natural
>number. (The tired old "is e a member of pi" objection).  I understand why
>relativist foundations is an oxymoron, and I don't wish to defend
>post-modernism or whatever on this list.   Just an answer to a simple

I would need first an answer to another simple question: What do you mean 
by saying that there is no distinguished notion of natural number? I know 
of lots of bad philosophy that seems to imply this, including the one you 
suggest (though 0 and 1 make better candidates for natural numbers than e 
and pi); but am astounded to find it accepted as a matter of course. 

Dedkind long ago answered the `tired old objection' in the case of the 
real numbers when Weber asked him why he did not simply identify the real 
numbers with the cuts of rationals. His answer, in essence, is that it is 
ungrammatical to do so---to ask, e.g. of two numbers whether one is a 
subset of the other or not. 

The `tired old objection' assumes that if the natural numbers are some 
one thing, then they must be sets or something else other than just the 
numbers; and then continues by pointing out that there is no 
distinguished such thing that they can be. Dedekind's answer is that they 
are simply the numbers. The proper answer to *what* they are is 
answered---as he answered it---by exhibiting the structure of the system 
of numbers. 
This seems like a very good answer, and I suggest that the tired old 
objection be put to rest.

Bill Tait

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