[FOM] Why Mathematics needs verbs

[Harvey Friedman]

Buckner has made a very large number of postings which at least imply 
some substantial relevance of natural language philosophy to the 
foundations of mathematics. In particular, substantial relevance of 
natural language philosophy to basic elementary set theoretic results 
such as Cantor's theorem.

Whereas I do not doubt the possibility that there are overlooked 
imaginative and productive connections, nothing in these postings 
suggest any to me.

What could they be? A principal one could be

***A new way to adequately formalize mathematics, which is strikingly 
different than the usual ways of formalizing mathematics. For 
example, one that is substantially weaker, demonstrably not 
supporting theorems like Cantor's, but nevertheless supporting very 
substantial portions of mathematics in a very natural way. And/or 
skirting the standard Godel phenomena.***

Another one could go in the other direction:

***The introduction of new primitives and/or modes of reasoning that 
go beyond the usual ones in f.o.m. E.g., the use of natural language 
primitives with obvious axioms that establish the consistency of 
large cardinals.***

Consideration of what mathematicians might mean when they talk about 
each other's claims does not count - at least on the face of it - as 
a substantial relevance of natural language philosophy to the 
foundations of mathematics.

I would expect Buckner to either

i) agree that he knows of no relevance of natural language philosophy 
to any substantive issues in the foundations of mathematics; or

ii) state explicitly what the substantive issue(s) in the foundations 
of mathematics is that he is concerned with, and just what the 
relevance of natural language philosophy is to it.

Then we can carefully analyze the issue(s) in f.o.m. and the 
relevance of philosophy of language to it.

Failing this, Buckner's postings then attain the character of 
presentations of various fragments of work in the philosophy of 
language, and it would be better to simply give the appropriate 
references to the literature from the philosophy of language.