[FOM] 744: Philosophy of Incompleteness/6
Harvey Friedman
hmflogic at gmail.com
Tue Jan 10 00:15:57 EST 2017
FIRST, I want to clean up some typos in the previous 743: Philosophy
of Incompleteness/4. There is an omission of the word "I" in the
following long paragraph, which I hereby fix:
Now John and I AGREE that the mathematicians are DEAD WRONG on many
levels. The real difference between John and I is that whereas John
thinks that they are dead wrong because of the developments, post
Goedel, due to John and his close colleagues, through Projective
Hierarchy Incompleteness (being completed by large cardinal
hypotheses), I think they are dead wrong, post Goedel, entirely because
of me (with some early help in the 1960's,70's from Tony and Leo)
through Concrete Mathematical Incompleteness (being completed by large
cardinal hypotheses).
There are some awkward words in the next paragraph, which we hereby fix:
John clearly seems to accept that String Theory is a legitimate
subject in the sense of making claims that have a definite truth
value. Many physicists (maybe most for all I know) do not
accept that legitimacy in the absence of the established possibility
of experimental
confirmation and testing, which many (maybe most for all I know)
doubt. Also most mathematicians do not (at least readily) accept the legitimacy
of arbitrary coherently readable set theoretic statements (at least in
the sense of having a definite truth value).
LET ME ADD: Furthermore, I get the impression that most theoretical
physicists at least see the relevance or projected relevance of string
theory for major more established parts of physics, mathematicians do
not similarly accept the relevance or projected relevance of abstract
set theory for major more established parts of mathematics. So in this
sense, the mathematicians have not even been put in the position of
really having to pass any judgement except the judgement of silence
when it comes to abstract set theory. It is essentially a nonissue for
them until relevance of the kind that they accept as sufficiently
strong appears.
Natural statements that have been long since settled for Borel subsets
and functions in and between complete separable metric spaces, when
lifted to the projective hierarchy, and therefore covering many more
sets and functions, is not the kind of compelling relevance required
to make abstract set theory relevant in the sense required. This kind
of extra generality, where one is not getting one's hands on good
relevant examples of projective sets and functions that are not Borel,
is not going to energize mathematicians to confront abstract set
theory and the choice of axioms, evaluation of large cardinal
hypotheses, etcetera.
The prospects are entirely DIFFERENT for simple manipulations of
essential down to earth everyday objects common to almost ALL of
mathematics. But here do not UNDERESTIMATE just how perfect such
manipulations of essential objects must be, in terms of simplicity,
beauty, intelligibility, coherence, strategic conception, depth,
metaphorical content, richness of associated thematic programs, must
be in order to even get a hearing. MATHEMATICIANS are going to resist
tooth and nail any effort to upset the ZFC foundations that they have
come to KNOW and FORGET, because foundations has allegedly become such
a nonissue. It has become part of the DNA of mathematicians that
foundational considerations have long become pointless and irrelevant
long before they (the mathematicians) were born. They very much like
the idea that f.o.m. is a completely DEAD subject. Mathematicians are
going to have to be taken kicking and screaming by the back of the
neck and literally forced with the help of the general intellectual
community to be shamed and embarrassed into acting like f.o.m. is not
completely DEAD.
SITUATION TO BE ENVISIONED. The typical mathematically literate
INTELLECTUAL, but not professional mathematician, becomes at least as
or even more interested in some new AREA of concrete mathematics than
they are in the typical so called breakthrough result in the various
standard special areas of mathematics, AND that major results in that
AREA can only be established by going way beyond ZFC. And INTELLECTUAL
says to the mathematicians that "the AREA involves objects even
simpler and more basic than what you typically consider, and the
statements in the AREA are even clearer and more well motivated". So
the fact that YOU, mathematician, ignore this new AREA because you
didn't create it, and wish to avoid foundational issues at all costs,
is not convincing to the INTELLECTUAL. In fact, the INTELLECTUAL sees
an inevitable path to the new AREA that cannot be rationally resisted,
at least in the long run.
So we now have the ingredients for how we want to proceed in light of
the very poor prospects for our
A. An existing mathematical question that is widely known and of wide
interest, is shown to be neither provable nor refutable in ZFC.
But I promised to take care of one more typo from 743: Philosophy of
Incompleteness/4. I wrote a paragraph about my upcoming talks at U
Texas, in which I mentioned the situation with regard to mentioning
logic as a main research interest of faculty in Math and Philosophy.
The way I wrote it must have given the impression that I was invited
by the Philosophers. This is not the case at all. I was invited by no
less than Scott Aaoronson of their Computer Science department!
Once again, the main point is that in a not atypical place like UT
Austin, we have the following curious numbers:
MATH. 104 faculty listed. 0 list interest in logic. Logic is therefore
not on the 36 research areas listed on
https://www.ma.utexas.edu/research/interests.php
PHILOSOPHY. 39 faculty listed. 12 list interest in logic.
Philosophical Logic is listed as one of 9 major research areas listed,
and there is this page devoted to logic:
https://liberalarts.utexas.edu/philosophy/areas-of-study/Logic.php
COMPUTER SCIENCE. 73 faculty listed. 13 list interest in logic in the
category "formal methods". See
http://www.cs.utexas.edu/users/moore/atp/index.html
How typical in the USA is it that there is more interest in logic in
the Philosophy and Computer Science Departments than in the
Mathematics Department? What about in the rest of the World? Granted
the main focus of logic in Philosophy and Computer Science is not
really f.o.m.
In the next posting we will delve headlong into a critical examination
of where we are with regard to getting to the SITUATION TO BE
ENVISIONED above.
************************************************************************
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 744th 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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/2316 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 10/25/16 1:25PM
730: Consistency of Mathematics/2 11/17/16 9:50PM
731: Large Cardinals and Emulations/26 11/21/16 5:40PM
732: Large Cardinals and Emulations/27 11/28/16 1:31AM
733: Large Cardinals and Emulations/28 12/6/16 1AM
734: Large Cardinals and Emulations/29 12/8/16 2:53PM
735: Philosophical Geometry/13 12/19/16 4:24PM
736: Philosophical Geometry/14 12/20/16 12:43PM
737: Philosophical Geometry/15 12/22/16 3:24PM
738: Philosophical Geometry/16 12/27/16 6:54PM
739: Philosophical Geometry/17 1/2/17 11:50PM
740: Philosophy of Incompleteness/2 1/7/16 8:33AM
741: Philosophy of Incompleteness/3 1/7/16 1:18PM
742: Philosophy of Incompleteness/4 1/8/16 3:45AM
743: Philosophy of Incompleteness/5
Harvey Friedman
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