[FOM] Certainty in mathematical proofs, part 2

[Arnon Avron]

There were already some responses to the first part of my reply
to Davis, but I prefer to delay reacting to them (Except one)
after posting the other parts.

So  Martin Davis wrote:

> Harvey Friedman has emphasized the progression of 
> means of proof from the most basic to the highest 
> regions of the transfinite with a decision to 
> draw the line at at particular level a matter of 
> personal idiosyncrasy rather than the result of philosophical acumen.

The fact that there is a progression does not mean
that every choice of the line is reasonable. It
is also  impossible to draw the exact line between "old"
and "not old". This does not mean that one can reasonably
draw it at the age of 90. In the case of FOM the situation
is even clearer, because there is no continuity. There
are basically two major important gaps in the "progression".
The most basic one from philosophical and historical
point of view is between the discrete (e.g. N) and
the continuous (R, or P(N)):  in one guise or another,
justifying the use of (or even the talk about) the real numbers
is the major foundational problem since the time of 
the Greeks. The second major gap is in going further
from R, which is after all supported by a strong geometrical
intuition, to the full P(R) or full R->R (which do not have
such a support. I think that once P(R) and R->R
have been "accepted", there can be no rational objection to 
going a lot further, using the same principles). OBJECTIVELY, 
everything that assumes the existence without any restriction
of P(R) cannot be taken as absolutely certain, while everything
provable in PA *is* absolutely certain (I simply  find it 
hard to believe someone who denies the latter. I think
that s/he is fooling herself/himself). Now there is still 
a lot of room for research and debates in between - but 
not "everything goes".