[FOM] Weaver's error?

[Harvey Friedman]

This is a reply to 4/11/06 7:56 PM, "Nik Weaver" <nweaver at math.wustl.edu>.

Weaver stated *) and **). The Feferman/Schutte analysis asserts *) and
denies **).

We now know that Weaver claimed to have "refuted" the Feferman/Schutte
analysis merely by pointing out that

1. Weaver believes *).
2. Weaver believes that Feferman/Schutte don't argue properly for *).
3. Weaver believes **).
4. Feferman/Schutte deny **).
5. Weaver believes that Feferman/Schutte must give explicit reasons for
explicitly denying **). [NOTE: Weaver believes that Feferman/Schutte cannot
rely on the fact that there is no implication from *) to **). Weaver
believes that If they don't, then Feferman/Schutte has been REFUTED by
Weaver.]
6. Weaver is not convinced of any reasons that have yet been put forth for
explicitly denying **).

Since Weaver continually uses the word "refute", I had automatically assumed
that 

7. Weaver MUST be claiming that *) implies **), in some relevant sense of
implies.

for how else can he claim to have "refuted" Feferman/Schutte?

I pointed out that not only does *) not imply **) mathematically, but also
*) does not imply **) philosophically either. It would be both a
mathematical and philosophical mistake. I see no relevant sense of implies
here. This situation is completely familiar and very common throughout
f.o.m.  

So now that Weaver has explicitly disavowed any *implication* from *) to
**), Weaver no longer has any claim to having **refuted** Feferman/Schutte.

If Weaver had instead been claiming all along that

8. Weaver does not find the Feferman/Schutte analysis of predicativity
convincing.

then I would not have disagreed with him.

As you all know by now, I don't believe in any unique notion of
predicativity, and Feferman/Schutte has an analysis of one notion that they
make an excellent case for being a very interesting notion and arrive at a
very interesting stopping place. In light of that, I would have no quarrel
with 8.

It is essential that Weaver change his clear misusage of the word "refute"
so that the discussion of predicativity between Friedman and Weaver can now
focus on some matters of import. It hasn't yet, and the obstacle has been
Weaver's wholly inappropriate use of the word "refute".

Harvey Friedman