[FOM] 539: Absolute Perfect Naturalness

[Harvey Friedman]

Tim Chow wrote:

Friedman of course can speak for himself, but I interpreted him to mean
> that the "ordinary mathematician" who hopes that a single axiom (V = L)
> will eliminate all set-theoretical difficulties is perhaps being too
> optimistic, because "small large cardinal" hypotheses may still rear their
> head.
>


> But I agree that this shouldn't stop said ordinary mathematician from
> adopting V = L.  It will do nicely for now, and if some day Mahlo
> cardinals or whatnot intrude into ordinary mathematics, then we can cross
> that bridge when we get to it.


​Yes, I would like to speak for myself, and I don't want to wait to "cross
the bridge when we get to it."

Looking at
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
# 82, and ​
http://www.cs.nyu.edu/pipermail/fom/2014-September/018149.html

I claim that we already have examples which are

1. Perfectly mathematically natural in a robust sense that transcends
existing mathematics; i.e., transcends the particular path research
mathematics has taken at this time. Experiments with mathematicians,
especially including young budding mathematicians at the high school and
early college level, may be, I think, particularly revealing.
2. We have equivalence with consistency of massive extensions of ZFC via
"small large cardinals".
3. The examples are independent of even ZFC + V = L.
4. Moreover, no move like imposing V = L will remove this kind of
incompleteness.
5. Beyond 3,4, we can't even cut down the generality of the statements by
logically restricting the objects being quantified over, without changing
the significant mathematical purpose of the statement or making an ad hoc
move that does not preserve mathematics as we know it. I.e., we could move
to consider only finitely many rationals, excluding all whose numerators
and denominators are too large. HOWEVER,
6. For the maximal statements, we have an ALTERNATIVE defense available:
demand that all objects be algorithmically given; THEN the statements
become refutable. This move does some violence to mathematics as currently
practiced - but it is not entirely devastating. Moreover, it is a move that
may plausibly be made in the future, as so much of mathematics is preserved
in the algorithmic. NEVERTHELESS,
7. Other examples in #83 are explicitly Pi01, and they are at least within
the vicinity of this same kind of perfection. They are the finite forms of
the maximality examples which we know exist by the general procedure of
Goedel's completeness theorem that acts on the maximality examples. With
explicitly Pi01, no move like restricting to algorithmically given objects,
is available to us.

The examples are already in *ordinary* mathematics (although there are
different notions of "ordinary"), but not yet in *mainstream* or *core*
mathematics. For the record, here are some formulations, some of which I
have not previously given, each with their own advantages and
disadvantages. You can pick and choose. Experience shows that different
people have different favorites. I am starting to investigate reactions to
these statements with a variety of mathematicians, directly and indirectly.

PROPOSITION 1. Every order invariant subset of Q[-n,n]^2k has a maximal
square whose sections at (0,...,n-1) and (1,...,n) agree below 0.

PROPOSITION 2. Every order invariant subset of Q[-n,n]^2k has a maximal
square whose sections at (0,...,n-1), (1,...,n), (0,2,...,n) agree below 0.

PROPOSITION 3. Every order invariant subset of Q[-n,n]^2k has a maximal
square whose sections at the x in {0,...,n}^r< agree below 0.

PROPOSITION 4. Every purely universal property of subsets of Q[-n,n]^k has
a maximal solution whose sections at (0,...,n-1) and (1,...,n) agree below
0.

PROPOSITION 5. Every purely universal property of partial regressive
f:Q[-k,k]^k into Q[-k,k]^k has a maximal solution with f(0,..., k-1) ==
f(1,...,k).

There are many other variants, including 1,2,3 for roots and cliques.
HOWEVER, we don't yet know whether Proposition 1 is independent of ZFC. All
of the others are provably equivalent to Con(SRP).

Here is my prediction.

1. For the top senior mathematicians who are unusually reflective - and I
know some - they are pretty comfortable with the idea of "perfectly
mathematically natural in the absolute sense". Developing some sort of
necessary and, separately, sufficient criteria for "perfectly
mathematically natural in the absolute sense" is of particular importance
for f.o.m. and I am deeply committed to doing research on this.

2. Ideally, for such reflective top senior mathematicians, just ONE
"perfectly mathematically natural in the absolute sense" example should be
enough. I.e., enough to seriously join the issue of the fundamental nature
of mathematics and the underlying philosophical issues concerning choice of
axioms, etc.

3. WHETHER or not this is actually the case for such reflective top senior
mathematicians, remains to be seen. After all, I can theorize here about
how a top level reflective rational mind should think , but this does not
replace actual "facts on the ground".

4. I suspect that there may well be an evolutionary process, even with
many, most, or possibly all, of these top level reflective rational minds.
Intellectual revolutions sometimes sink in slowly over time.

5. Most even top senior mathematicians are not reflective at all, or not to
any substantial extent. Mathematics is not alone among noble professions
that attract few reflective minds. I envision at first the biggest growing
audience for this - before the examples really get much more diverse - a)
gifted young mathematicians at the high school and early undergraduate
level b) top senior reflective mathematicians. I am hoping and conjecturing
that the massive middle will be, slowly, over time, influenced by a) and
b). Of course, the whole process does greatly accelerate with the middle as
the examples become more diverse, and yet closer to what mathematicians
normally think about. But I think the ingredients are here to at least
start the process, at least after verification of the validity of the
mathematical claims via the appropriate manuscripts.

I do not subscribe to the analogy between Con(SRP) and RH, P not= NP.
Experience with FLT, 4 color problem, Kepler's conjecture, Goldbach's
conjecture, twin prime conjecture (largely solved), and so forth, shows
that there is every expectation that ignorance about RH, P not= NP is
merely a temporary condition, resolved as usual by clever and/or deep
intense (perhaps collaborative) normal mathematical activity (and maybe
with the help of computer technology). This is absolutely not the case, and
known not to be the case, with Con(SRP). It appears that Con(SRP) is not
going to be resolved without some sort of deep philosophical/foundational
reflection of a kind that we can't imagine, or have only a glimpse of.
Right now we only have metaphysical pronouncements of "facts" about the
"set theoretic universe", a completely foreign object to contemporary
mathematics. (Although there is use of computer technology for some kind of
"confirmation" which may enter into the picture - see
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#83, which we are about to extend further into large cardinals such as
HUGE).

Arguably at the moment, the best storyline in this foreign higher set
theory is probably "second order reflection". This however does not get up
to SRP, and so is insufficient - insufficient for the above even if it were
mathematically acceptable, which is dubious. LC's at the moment constitute
the introduction of a totally foreign kind of fantastic object to
mathematics, and such an introduction would constitute a major
revolutionary change in the mathematician's normal conceptual framework. Of
course, if nothing better comes along, and enough time elapses, then this
may become the default. However, at this stage we cannot really tell how
this is going to play out. E.g., there may develop a new hierarchy of
combinatorial principles that have their own maybe even arguably compelling
status independent of large cardinals, avoiding the acceptance of foreign
objects into mathematics. It's my job to get the process seriously started
- and I won't be around to see how it stabilizes.

****************************************
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 539th 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

Harvey Friedman
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