[FOM] Solution (?) to Mathematical Certainty Problem
friedman at math.ohio-state.edu
Thu Jun 26 23:37:29 EDT 2003
Reply to Lindauer 5:05AM 6/26/03.
I am not sure that we have, to quote myself,
"agreed on these fundamental aspects of the setup, so that ther eis
any point in continuing the discussion."
Nevertheless, I will continue the disucssion a bit.
It appears that the thrust of what I am saying is that
**in order for us to achieve absolute certainty, or some particular
brand of certainty, about ALL of our rigorous mathematical knowledge,
we need only achieve absolute certainty, or some particular brand of
certainty, about a TINY PORTION of our rigorous mathematical
knowledge, together with very weak assumptions about physical
processes, far weaker than is contained in any substantial physical
theory. In fact, it is promising that all, or almost all, of the
physical component can be replaced by weak statistical reasoning.**
This seems to be a genuine reduction, and as such, is very striking.
Of course, in some sense, this idea has been lurking around for quite
some time, well before I have come to see this. But perhaps, it has
not been stated so explicitly, together with a program that one can,
in fact, make the relevant TINY PORTION more convincing that would be
a priori expected. I have not gotten into the serious matter of
analyzing how weak the rigorous physical process theory can be, nor
how weak the rigorous statistical reasoning can be.
In fact, this may a perfect place to do some things that have never
been done successfully before. Why? Because this context is very
specific, and the cards are stacked very much in our favor, since one
has only to "verify" just a little bit of stuff.
1. Write a generally applicable multitape Turing machine program
(CORE PROOF CHEKCER) and rigorously specify its behavior and verify
2. Analyze actual electronic hardware to run it on, and see how weak
the physical statements have to be in order to argue that we have the
appropriate kind of infallibility.
3. Do 2 rigorously, perhaps invoking various physical redundancy
procedures, as well as error correcting codes and the like, but in a
totally rigorous, verified setting.
4. Analyze all relevant statistical reasoning rigorously. Rigorous
foundations for statistical reasoning don't really exist up to the
standards we expect here in such a project. But in this narrowly
focused situation, they probably can be handled appropriately. This
might lay the groundwork for pathbreaking foundational ideas
concerning statistical reasoning.
But a skeptic would say: where are you going to get any absolute
certainty out of this? Once you admit statistical reasoning as part
of the mix, you have already admitted defeat, since you don't 100%
from such considerations.
I was waiting for this question.
It is considered very reasonable - if not very likely - that the
universe is COARSE. That the universe has been around for only
reasonably small amount of time, that there have been only a small
amount of states, a small amount of events, etcetera.
So under this viewpoint, if a probability is rigorously established,
in some appropriate sense, to be less than, say, 2^-1,000,000, then
that is not distinguishable in reality from 0. Perhaps in much the
same way that a distance like that in meters is not distinguishable
from 0, because of coarseness.
So once one admits coarseness, according to this point of view, in
order to achieve absolute certainty, we need only get down to
probabilities of failure at levels like 2^-1,000,000, or some such
But that is easily imaginable with very clever redundancies, error
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