[FOM] 600: Removing Deep Pathology 1

[Harvey Friedman]

There is an aspect of mathematics that I call

*deeply pathological*

I used the word deeply because there are much weaker forms of
pathology, and "arguable pathology", about which there is considerable
disagreement. But I believe that there is a clear threshold into the
"deeply pathological" that is familiar to almost all mathematicians.
This initial posting on this topic is to set the stage for a hopefully
fully productive and lively discussion. This initial posting is not
the place for systematic detailed discussion. That is planned for
later.

Although practically all mathematicians will instantly recognize what
I am talking about, let me state a PROJECT, TENTATIVE THESIS, and
CHALLENGE.

In order to set the stage for further discussion, I offer one well
known example of what I am talking about. Consider the equation

f(x+y) = f(x) + f(y)

where f;R into R. There is a great dichotomy: the solutions f(x) = cx
are absolutely wonderful, whereas all others are deeply pathological.
We will later take up just what available tools we have for
systematically treating this kind of situation.

PROJECT. Identify and systematically remove all involvement with
deeply pathological objects in mathematical theorems or conjectures,
while retaining the desirable objects. There are many naturally
defined structures which have some (many) elements that are themselves
deeply pathological, and the presence of these deeply pathological
objects often create complications that are not generally related to
the purposes of the naturally defined structure . Develop general
surgical tools for removal of the deeply pathological elements, while
retaining the rest. Replace the study of the entire bloated structure,
no matter how naturally defined, with its non deeply pathological part
- i.e., its desirable part. Establish strong dichotomy theorems. A
very typical large supply of naturally defined structures are the
duals of naturally defined Banach spaces. If the naturally defined
Banach space is separable, then there are not going to be any deeply
pathology elements of the dual, and so the dual does not need to be
cleansed. However, if the naturally defined Banach space is non
separable, then there are generally going to be deeply pathological
elements of its dual, and it is highly desirable to cleanse the dual.
Deep pathology may occur in other contexts where the deeply
pathological objects have not been placed into naturally defined
structures. In this case, the surgical tools are used to show that
there are no reasonable objects obeying the conditions under
investigation, or that the reasonable objects obeying the conditions
under investigation have a reasonable classification.

WARNING. I have spent almost 50 years struggling to come up with
examples of mathematically natural statements about desirable objects
which cannot be proved without using certain  which are at least
extremely unfamiliar to the mathematics community. These certain
objects (large cardinals) can be argued to at least exhibit deep
pathology. The kind of deep pathology that they exhibit is NOT the
kind that deep pathology that I am referring to in the PROJECT, which
is, on the other hand, a kind highly familiar to mathematicians.
Furthermore, in the PROJECT I am focusing on deep pathology involved
in the statements of theorems and conjectures, and not in the proofs.
Nevertheless, a rather important related issue is whether there are,
or to what extent there are, really convincing examples where we have
a statement that has been cleansed of deep pathology, yet its best
proof still involves the mathematically familiar deep pathology under
discussion.

TENTATIVE THESIS. The use of the surgical tools used in the PROJECT
leaves all of the mathematics in tact that is of sustained interest to
the mathematics community. Revised developments in which the deep
pathology has been cleansed, will be generally regarded as superior
developments. The cleansing process INCREASES the totality of deep
mathematical proofs Ideas from the old mathematics will resurface in
the new mathematics.

CHALLENGE. It is commonplace in papers on functional analysis and
operator theory to "motivate" the developments through their "use" in
mathematical physics, engineering, and elsewhere. Establish that such
usefulness reflects only the desirable part, and not any of the deep
pathology. Or, alternatively, demonstrate that the deep pathology has
a genuine "use".

In the next posting, I will present my first surgical tool, and its
use on a dual space, and hopefully more.

************************************************************
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 599th 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
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory  1/27/15  10:39 AM
577: New Pi01 Incompleteness  3/7/15  2:54PM
578: Provably Falsifiable Propositions  3/7/15  2:54PM
579: Impossible Counting  5/26/15  8:58PM
580: Goedel's Second Revisited  5/29/15  5:52 AM
581: Impossible Counting/more  6/2/15  5:55AM
582: Link+Continuation Theory  1  6/21/15  5:38PM
583: Continuation Theory 2  6/23/15  12:01PM
584: Finite Continuation Theory 3   6/26/15  7:51PM
585: Finite Continuation Theory 4  6/29/15  11:23PM
586: Finite Continuation Theory 5  6/20/15  1:32PM
587: Finite Continuation Theory 6  7/1/15  11:39PM
588: Finite Continuation Theory 7  7/2/15  2:44PM
589: Finite Continuation Theory 8  7/4/15  6:51PM
590: Finite Continuation Theory 9  7/6/15  5:20PM
591: Finite Continuation Theory 10  7/12/15  3:38PM
592: Finite Continuation Theory 11/perfect?  7/29/15  4:30PM
593: Finite Continuation Theory 12/perfect?  8/23/15  9:47PM
594: Finite Continuation Theory 13/perfect?  8/4/15  1:44PM
595: Finite Continuation Theory 14/perfect?  8/5/15  8:23PM
596: Finite Continuation Theory 15/perfect?  8/8/15 12:35AM
597: Finite Continuation Theory 16/perfect?  8/10/15  10:22PM
598: Finite Axiomatizations  8/10/15  5:05AM
599: Invariant Sequential Choice  8/15/15  4:22PM

Harvey Friedman