[FOM] 566: Bridge/Chess/Ultrafinitism

Thu Dec 25 10:46:55 EST 2014

I have placed a very polished extended abstract on my website which
greatly improves in many ways my #565
http://www.cs.nyu.edu/pipermail/fom/2014-December/018448.html

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87. Perfectly Mathematically Natural Concrete Incompleteness,  -
order theoretic relations. December 14, 2014, 25 pages. Extended
abstract.

I am now revisiting an idea from some time ago for finite
incompleteness, and expect to post later. It surrounds decomposing
positive integers into sums of positive integers. I have a new idea of
how to bring finite dags (directed acyclic graphs) into the approach,
which I am hoping will lead to a whole new generation of even more
natural finite Incompleteness. Recall that I use the ultimate phrase
Perfectly Mathematically Natural only for the implicitly Pi01
statements. The explicitly Pi01 forms in #87 are Natural but not
Perfectly Natural. I think this distinction is reasonably valid and
quite useful. Since ultimately Incompleteness is going to become
Perfectly Mathematically Natural even for explicitly Pi01, and
accepted mathematical subjects will emanate for them, this distinction
between Mathematically Natural and Perfectly Mathematically Natural
will lose its usefulness. Of course, if new criteria are placed on
what one wants for Incompleteness - e.g., that it is supposed to be
geometrically visible, or something like that - then one will have to
pass from the Artificial to the Natural to the Perfectly Natural all
over again.

Tim Chow posted the interesting
http://www.cs.nyu.edu/pipermail/fom/2014-December/018455.html  I
replied with http://www.cs.nyu.edu/pipermail/fom/2014-December/018460.html
offering the game of bridge as a good place to think about the issue
TIm Chow raised.

I.e., what is (or what are some of) the simplest entry points for
serious mathematics, which vividly illustrate the special status of
mathematical knowledge as opposed to scientific knowledge? And, is
there some deep associated f.o.m. associated with an analysis of such?

Bridge and Chess immediately come to mind as extremely interesting and
fruitful venues for this. And yes, I believe that there is deep
associated f.o.m.

There is considerable intellectually deep mathematics involved even in
rigorous formulations. Then there is considerable deep mathematics
involved in substantiation claims.

By "intellectually deep" here I mean deep from the general
intellectual point of view where one is starting essentially with no
math at all.

Consider what deep matters are needed in order to formulate

1. The most prevalent suit distribution in a 13 card bridge hand from
a normal 52 card deck is 4-4-3-2.

2. In Chess, either white has a winning strategy, or black has a
winning strategy, or white and black each have strategies assuring a
draw or win for them, respectively. Furthermore, these three cases are
mutually exclusive.

In both cases, the relevant structures, although finite, are not
accessible to us by nonmathematical means as they are simply too
large.

But a very large number of non mathematical people can reasonably
relate to 1,2, without being able to make the step to rigorous
formulations, let alone proofs.

To leverage off of the non mathematician's reasonable comfort zone
here, the above can profitably be approached from below. I.e., use n
card bridge hands from a normal 4m card deck. Also adapt chess to a k
x k board instead of an 8 x 8 board. Use various small n,m,k. Or look
at Tic-Tac-Toe http://en.wikipedia.org/wiki/Tic-tac-toe

What results is of course is a whole conceptual development deep
inside ultrafinitist mathematics, with practical computer algorithms,
and impractical computer algorithms, together with proofs that these
computer algorithms are correct.

I have written about this kind of topic before on FOM several years
ago, but I didn't develop this corner of f.o.m. systematically. This
is a deeply important corner of f.o.m.

In particular, the mathematics survives as we go from the roughly
directly observable (say a mate in 2 for a particular chess position
or arguably 2 card bridge hands), to the far too big to see but within
standard computer technology, to the explicitly finite inherent in
Bridge and Chess that is way outside standard computer technology, to
the general finite. The corresponding deep f.o.m. surrounds the nature
and general characterization of these more and more abstract venues.
Of course, far more issues with all kinds of deep f.o.m. arise as soon
as one makes the leap into the infinite.

************************************************************
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 566th 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 #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14 4:56PM
564: Perfectly Natural Review #3  11/24/14 1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM

Harvey Friedman