[FOM] 600: Removing Deep Pathology 1

Nick Nielsen john.n.nielsen at gmail.com
Tue Aug 18 21:33:02 EDT 2015


On Sat, Aug 15, 2015 at 10:37:45PM -0400, Harvey Friedman wrote:

> TENTATIVE THESIS. The use of the surgical tools used in the PROJECT
> leaves all of the mathematics intact that is of sustained interest to
> the mathematics community.

Would it be worthwhile to make a distinction between the mathematics
that is of sustained interest to the mathematics community and the
mathematics that is of sustained interest to the philosophical
community (or, if you prefer, the philosophy of mathematics
community)?

For my part, I can't imagine anything more fascinating than the "deep
pathologies" investigated by Friedman and listed by Taylor.

Hendrick Boom wrote:

>I suspect there's a close relationship between pathology and use of
>nonconstructive methods.

>If you're not willing to go that far, consider for a start how much
>pathology depends on the axiom of choice.

Pathological, perhaps, but also very fruitful. The criterion of
methodological fruitfulness belongs to the same category of pragmatic
justifications as that of Harvey Friedman's pragmatic justification in
terms of utility (which I take to be implied by, "usefulness reflects
only the desirable part").

Best wishes,

Nick


On Sat, Aug 15, 2015 at 7:37 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
> 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
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