929: Physical Infinity/randomness

Mon Mar 21 02:14:11 EDT 2022

NOTE: I put this among my numbered postings because of the segway into
Tangible Incompleteness that is addressed and the end and soon to be
elaborated on.

NOTE: This is a careful discussion of some things well known for
centuries with the hope of pushing them into some novel directions
where FOM tools will be useful.

******

There is no question that infinite sequences arise in Mathematical
Models of Physical Reality. This is already clear in the most basic
Newtonian physical theories and continues through all of the modern
mathematical models of physical reality.

However, infinities in actual physical reality are a prima facie very
dubious idea that would, to make coherent, require a radical kind of
justification that needs to be examined in great detail.

The simplest case of infinity in physical reality that some people
find attractive is the idea of a commonplace physical experiment being
reproduced infinitely many times.

This is taken to be a serious scientifically legitimate idea for many
people, who even go further and want to consider mathematical
properties of the associated infinite sequence of outcomes.

All of this goes against the grain of the compelling methodological
principle subscribed by the vast majority of scientists, even well
beyond mainstream physicists, that claims about physical reality must
be accompanied with reproducible incontrovertible scientific
experiment that are actually conducted in several credible scientific
centers.

But still there is the seductive siren of the basic infinite in
physical reality - especially infinitely many repeated trials of non
problematic or "non problematic" trials.

So what to do?

The answer could lie in the reformulation of infinitary concepts in
approximate finite form.

Now this move from the infinite to the large realistic finite
approximations of the infinite is a move that has been grossly
underdeveloped even in the context of mathematics.

FIRST IN THE REALM OF MATHEMATICS

Let n be a positive integer. Look at the space {0,1}^n sequences of
0's and 1's of length n. It is commonplace to speak of 0,1 as
representing "perfectly random coin tosses". We have no problem in
mathematics which this because we can fall back on just counting the
number of elements of {0,1}^n obeying certain properties. Or put
equivalently, the fraction of elements of {0,1}^n satisfying a certain
property.

EX 1. What fraction starts with 0? 1/2
EX 2. What fraction starts and ends with 0? 1/4
EX 3. What fraction is all 0's? Very very very very tiny. 1/2^n.

RESEARCH. Give a language for really "significantly interesting"
properties of elements of {0,1}^n, argue that these are really "good"
properties (maybe physically significant?), and calculate the number
of elements satisfying those properties, or approximations, and
compare different sizes. Do this for n = 1,2,3,4,5,6,7,8 say, and get
a feeling for when this gets really "complicated".

RESEARCH. Extend to {0,...,n}^m.

SECOND IN THE REALM OF PHYSICS

If n is not very big then we can easily generate lots of data about n
repeated trials of a physical experiment that is well designed. We can
even look at the results of say 10^6 repeated trials of a two outcome
experiment. I.e., the resulting length 10^6 length bit strings.

We can then consider the hypothesis that "the outcome of any of these
individual trials between repeated is random". Also "the total outcome
of these repeated trials is random". (alluding to notions of
independence of events).

Any direct attempt to define such notions appears to be wrought with
great difficulties.

PROBLEM. Prove that any proposed definition of this notion is doomed
in certain well defined senses.

Fortunately the mathematics of this situation does support very
reasonable approaches to this issue.

There seems to be a workable notion of "overwhelming chance". We know
mathematically that we can construct a nice readily testable property
of the outcome of 10^6 trials such that, mathematically,

1. If we have pure randomness (i.e., we use the count) then chances
that the property holds is "overwhelming". This statement is clear
mathematically, formulated in terms of counts.
2. If we have a "tiny" deviation from pure randomness of a "reasonable
kind" then the chance that the property holds is not overwhelming.
Again this is all mathematics once we pick "tiny" and "reasonable
kind".

Then we perform 2 many times and still get that the property holds.
Then with properly designed "tiny" and "reasonable kind" we can argue
with some serious level  of rigor that we have "practical physical
randomness".

"Overwhelming" can mean that it corresponds with current mathematical
physics concerning the probability that the Sun is not going
supernova, killing us all instantly, within 24 hours of this writing.

PROBLEM. Make especially good foundational sense of this to lay a
better foundation for physical randomness.

THIRD IN THE REALM OF LARGE CARDINALS

>From my work on Tangible Incompleteness, in the portion concerning
there being nondeterministic paths through certain basic computer
algorithms:

we can ask for short length paths, like length 10 or so.

Paths are sufficiently short (say 10) and the setup is sufficiently
basic that the search space is within the capabilities of a state of
the art desktop computer.

The desktop computer presumably finds such a nondeterministic path.

The existence of the path is predicted by the relevant large cardinal
hypothesis. Right now, SRP. Or even just Con(SRP). That prediction is
fully rigorous and maybe 25 pages of normal purely mathematical
manuscript.

However, there appears to be absolutely no way that any human being
can even find much more tiny nondeterministic paths in this and
related algorithms. It is obviously just simply far far far too
complicated.  And there seems to be no way for a human to use the fact
that we are only talking about short paths. So the human seems to be
forced, as is the case in a lot of other mathematics, to be working
out the general case of arbitrary finite length nondeterministic
paths. 10 is replaced by n as far as humans are concerned. The 10
appears to be of absolutely no help. Just like in many situations in
math (say 10 dimensions as hard as n dimensions).

SRP and Con(SRP) also show that there are nondeterministic paths of
every finite length. HOWEVER, we prove that the existence of non
deterministic paths of every finite length is outright equivalent
(over tiny base theory) to Con(SRP).

So here there is a real actual confirmation or "confirmation" of the
consistency of large cardinals - i.e., Con(SRP). The confirmation is
by the way of a certificate which is quite small.

What is the role of the physics in this situation? By the physics we
mean the correctness of the circuitry.

Physics used just to find the certificate.

So in this particular situation, there is no need to consider the
correctness of the physics.

##########################################

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 929th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-899 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

900: Ultra Convergence/2  10/3/21 12:35AM
901: Remarks on Reverse Mathematics/6  10/4/21 5:55AM
902: Mathematical L and OD/RM  10/7/21  5:13AM
903: Foundations of Large Cardinals/1  10/12/21 12:58AM
904: Foundations of Large Cardinals/2  10/13/21 3:17PM
905: Foundations of Large Cardinals/3  10/13/21 3:17PM
906: Foundations of Large Cardinals/4  10/13/21  3:17PM
907: Base Theory Proposals for Third Order RM/1  10/13/21 10:22PM
908: Base Theory Proposals for Third Order RM/2  10/17/21 3:15PM
909: Base Theory Proposals for Third Order RM/3  10/17/21 3:25PM
910: Base Theory Proposals for Third Order RM/4  10/17/21 3:36PM
911: Ultra Convergence/3  1017/21  4:33PM
912: Base Theory Proposals for Third Order RM/5  10/18/21 7:22PM
913: Base Theory Proposals for Third Order RM/6  10/18/21 7:23PM
914: Base Theory Proposals for Third Order RM/7  10/20/21 12:39PM
915: Base Theory Proposals for Third Order RM/8  10/20/21 7:48PM
916: Tangible Incompleteness and Clique Construction/1  12/8/21   7:25PM
917: Proof Theory of Arithmetic/1  12/8/21  7:43PM
918: Tangible Incompleteness and Clique Construction/1  12/11/21  10:15PM
919: Proof Theory of Arithmetic/2  12/11/21  10:17PM
920: Polynomials and PA  1/7/22  4:35PM
921: Polynomials and PA/2  1/9/22  6:57 PM
922: WQO Games  1/10/22 5:32AM
923: Polynomials and PA/3  1/11/22  10:30 PM
924: Polynomials and PA/4  1/13/22  2:02 AM
925: Polynomials and PA/5  2/1/22  9::04PM
926: Polynomials and PA/6  2/1/22 11:20AM
927: Order Invariant Games/1  03/04/22  9:11AM
928: Order Invariant Games/2  03/7/22  4:22AM

Harvey Friedman