[FOM] Physics for Mathematics

[Harvey Friedman]

I was going to just post on 2, but I just saw 1 moments ago.

1. COMMENTS ON http://www.cs.nyu.edu/pipermail/fom/2016-September/020061.html
2. PHYSICS FOR MATHEMATICS

1. COMMENTS ON http://www.cs.nyu.edu/pipermail/fom/2016-September/020061.html

"PCTT+ states more or less: `every real number that we measure in
Nature is a computable real number' (a position also known as digital
physics)."

This is the kind of statement that I am dubious about in terms of its
being meaningful. However, I think the idea is to actually assign an
indirect meaning to this statement through the offered gedanken
experiment?

"At the same time we flip a coin an indefinite number of times, and we
interpret heads as 1 and tails as 0, creating a binary sequence which we
interpret as a binary expansion of a real number y between 0 and 1."

I acknowledged that this is using at least potential infinity in a
comparatively defensible way in the following sense. Maybe the coins
are being flipped pretty quickly - say really some pulse from nature -
and randomly, and where the time between flips is constant. So we are
really only relying on the idea that time is potentially infinite in
duration - and as I have said before, this is more reasonable than
infinite or arbitrary finite divisibility of space/time/spacetime. But
of course the coin flips do not actually finish in a finite amount of
time.

However,

"the odds that
at any given time y is seen to lie in one of the intervals of R_40 that we
have enumerated thus far, is less than 2^(-40)."

seems to be to rely on the idea that we can really "see" that y lies
in some extremely tiny interval(s), of length far less than 2^-40. But
to "see" this seems to require infinite divisibility of space - in the
sense of arbitrary finite divisibility of space.

Assuming that my objection is correct, I was thinking that there may
be a proof that all such attempts of this kind will fail.

But I thought of the following somewhat related possibility: maybe all
PROBABILITIES that occur in nature are recursive? i think that to make
sense of this, there is much less reliance on dubious notions of
infinity in the physical world. E.g., one could at least verify this
by constructing some experiments with an element of randomness, and
talk about indefinitely finitely many repeated trials and talk about
evidence that the data confirms that the "true probability" is some
given recursive real number predicted by theory. Or maybe the
recursive real number is surmised from a lot of data.

In a sense, this kind of thing already goes on in physics for the
number 1/2 and other simple numbers, with "true" randomness in quantum
mechanics. There "true randomness" is supposed to be meaningful. That
could set the standard for this kind of discussion.

2. PHYSICS FOR MATHEMATICS

In http://www.cs.nyu.edu/pipermail/fom/2016-August/020044.html  I
expressed severe doubts about the use of physics for the usual
infinite Church's Thesis. I surmised that physicists nearly
universally do not believe in the infinite divisibility of spacetime.

There have been five FOM responses to this to date:

http://www.cs.nyu.edu/pipermail/fom/2016-August/020046.html
http://www.cs.nyu.edu/pipermail/fom/2016-August/020047.html
http://www.cs.nyu.edu/pipermail/fom/2016-September/020053.html
http://www.cs.nyu.edu/pipermail/fom/2016-September/020054.html
http://www.cs.nyu.edu/pipermail/fom/2016-September/020059.html
http://www.cs.nyu.edu/pipermail/fom/2016-September/020061.html

I would say that taken as a whole, these five tend to back up my severe doubts.

But I'm not exactly sure what a survey on this among physicists would
show. It may depend of course on the branch of physics the respondents
work in.

A questionnaire might read something like this:

1. Time is infinitely divisible.
2. Space is infinite divisible.
3. Spacetime is infinitely divisible.

ANSWER: Certainly meaningful, probably meaningful, probably
meaningless, certainly meaningless.

If you answer probably meaningful,

ANSWER: Certainly yes, probably yes, probably no, certainly no.

On the other hand, I am optimistic about the use of physics in
connection with appropriate forms, yet to be formulated as far as I
can tell, of FINITE CHURCH'S THESIS.

JUST ADDED: And maybe a similar survey about "absolute randomness in
physics" and also maybe about "indefinite duration of time".

******************************

Here I want to take up a tangentially related matter.

CAN PHYSICS BE USED TO OBTAIN INTERESTING MATHEMATICAL INFORMATION
THAT WE CAN'T GET WITHOUT USING PHYSICS?
CAN PHYSICAL PROCESSES BE USED TO OBTAIN INTERESTING MATHEMATICAL
INFORMATION THAT WE CAN'T GET WITHOUT USING PHYSICS?
CAN PHYSICS BE USED TO OBTAIN INTERESTING MATHEMATICAL INFORMATION
THAT WE *KNOW* WE CANNOT GET WITHOUT USING PHYSICS?
CAN PHYSICAL PROCESSES BE USED TO OBTAIN INTERESTING MATHEMATICAL
INFORMATION THAT WE *KNOW* WE CANNOT GET WITHOUT USING PHYSICS?

The answer to the first is YES, with lots of examples. For whenever we
make serious use of a computer, we are using physics to argue that it
works properly. Even if that is not so clear, we are still using
physical processes.

It is clarifying to distinguish two cases of this:

1. Show that a particular ad hoc bit string(s) have an interesting property.
2. Show that all bit strings with a first interesting property also
have a second interesting property.
3. Show that an interesting finite set of bit strings has an
interesting property.

In connection with 2, the needed upper bound on the length of the bit
string is always OK as part of the "first interesting property".

It would be interesting to compile a list of the most striking
examples we can find where 1 or 2 or 3 have been achieved with
computers where it is inconceivable right now that we can do it
without computers, and dubious whether we will ever be able to do it
without computers. Of course, even more interesting would be to prove
that we will never be able to do it without computers, in ZFC.

(For some aspects of this discussion, of relevance is
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
83. Testing the Consistency of Mathematics, 7 pages, July 23, 2014.
Extended abstract.

referred to in the earlier
http://www.cs.nyu.edu/pipermail/fom/2014-July/018044.html

which needs to be updated in light of the recent Continuation Theory.)

To get started with this, let me throw out something now.

*Let n be an integer. The number of primes <= 2^n is even.*

QUESTION: For which n can your desktop prove or refute this statement
in a day with certainty? With high probability? And same for the most
high powered computers running for a year, networked?

It seems hard to imagine that for any but rather small n, this can
actually be proved or refuted by humans on their own.

OK, there might just be some sort of theory concerning primes below
power of 2. So we can modify this some.

*Let n be a specific randomly generated big integer. The number of
primes <= n is even.*

Even more far fetched is that humans will be able to prove or refute the above.

Harvey Friedman