[FOM] BANNING impredicative mathematics

[Nik Weaver]

Harvey Friedman wrote:

> You state that impredicativity has no clear philosophical basis.
>
> You wish to draw a distinction between this view and the suggestion
> that impredicative mathematics should be banned.

Right!  I didn't think you were ever going to acknowledge that.
Wow, it almost looks like we can finally put this non-issue behind
us ... but ...

> I still regard you as promoting the idea that impredicativitive
> mathematics should be banned.

... maybe not. (sigh)

> However, perhaps you would prefer that I interpret this as
>
> *impredicative mathematics should be condemned*
>
> *impredicative mathematics is illegitimate mathematics*

What I'd really prefer is that you interpret it as "impredicative
mathematics has no clear philosophical basis".

> > As I've explained,
> > the predativist prohibition on circular definitions is a simple
> > consequence of a disbelief in a platonic universe of sets.
>
> PROHIBITION! Is that related to banning?

Good grief, what is this obsession you have with banning?

No, to say that one is "predicatively prohibited" from doing X
simply means that it is not predicatively valid to do X.  The
phrase is descriptive, not imperative.  (However, in this case
I will give you some points for misinterpreting me in a clever
way.)

> > I think true finitists have to use intuitionistic logic,
>
> HAVE TO? What does that mean? Related to banning? Certainly
> they don't have to.

Good gravy, what is this obsession you have with banning??

No, it means: I think that if you use classical logic in first
order arithmetic then you're not a true finitist.  (Not so
clever this time.)

> > and
> > I think that purported finitists who are willing to use
> > classical logic in first order arithmetic betray an implicit
> > acceptance of N as a well-defined structure.)
>
> BETRAY? This is just totally unconvincing. Certainly there is
> no betrayal.

Decidedly unclever.  (Mis)interpreting "betray" the way you want
to renders my sentence ungrammatical.

> You must have some special meaning for "foundational stance". What
> is a "stance"? Does this involve banning issues?

Um, this is getting a little creepy.

> We now agree that any talk of banning is silly.

(!)

Not sure how you got there, but okay, great, agreed.

Nik