Non-foundationalist foundational thinking

[Harvey Friedman]

Tim Chow wrote a lengthy FOM posting "Non-foundationalist foundational
thinking". The posting comes in three parts.
https://mail.google.com/mail/u/0/#search/tchow%40alum.mit.edu/WhctKKXXJxWwpDfGTWlvBndlqCmSmVnHWFrjDScmGgXzbcpqhCvcvXfKVzlDKbkgFqbgJcG

PART 1/Chow

Tim's notion of "foundational thinking" is "the type of thinking that
mathematicians engage in when they work on the "foundations of X" where X
is some specific subfield of mathematics". This is *NOT* a good example of
what I have in mind when I refer to "foundational thinking". This is
because a large component of that Tim is referring to involves issues of
exposition and simplification and effectiveness in presenting proofs, and
various in house issues mathematicians deal with when they are involved in
the practical presentation of mathematics to make the exposition clearer
and more readily understandable.

Tim puts that aside, and focus on his

(*) What is really needed to prove Theorem T?

This is more or less what I wanted to address when I founded RM and SRM
going way back to the late 1960's and culminating with the founding RM
papers in ICM and JSL abstracts. RM was not really quite what I was aiming
for - it really was SRM which at that time was entirely unwieldy and in
retrospect highly premature. I tamed SRM into RM with those founding
papers. An in depth account of the history of RM and SRM can be seen at
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
 #117

At that time and actually all the time up thru now, I am a working
foundational thinker who is most interested in creating illuminating, rich,
and productive areas of research that advance our knowledge and
understanding of fundamental ideas and issues of general intellectual
interest.

Tim's (*) is, or is close to, an example of what I was and still am looking
for. Of course, it was obvious at the time that nothing even remotely like
SRM or RM is going to fully deal with (*). For example, what is really
needed to prove T includes mathematical insights of various kinds, some of
which are non foundational in any sense of the word. E.g., see Polya's How
to Solve It.

It was obvious back then, even to me, that it was much more promising to
deal with a limited notion of "need" here in order to inspire the creation
or discovery of SRM and RM. The alternative is to simply stare at (*) and
not found a remarkable new subfield of mathematics with a seemingly
compound growth rate of interest and rich discoveries.

So I was and even still am (despite my age) fully aware of such obvious
points as

"But there is more to the story than
reverse mathematics, at least as the subject is usually thought of.  When
I say "what is really needed" I mean to refer to what practitioners
intuitively feel is essential, and not just to strict logical implication."

Now I challenge Tim to create a significant new subfield of mathematics out
of his obvious point. If Tim can set up a new subfield dealing with this
obvious point across mathematics, then I might be the first to want to
Simpsonize this by jumping on it with abandon.

"reverse mathematics is not the entire story"

I wouldn't exactly call this a revelation. RM is far from the entire story
even with regard to its focus on strict logical structure that Tim is
thinking of as "limiting". It uses logically structured base theories which
SRM seeks to eliminate. I am gauging that the most productive effort in
this rough area surrounding Tim's (*) above is the proper development and
launching of SRM. On a related point, RM is far from the entire story in
that it isn't really dealing with many mathematical objects lying outside
the scope of the language of Z2, in a fully satisfactory way. I wrote a
number of FOM postings concerning this.

PART 2/Chow

Tim writes

"I'm going to assume that if you want to reduce all
of mathematics systematically to indecomposable concepts, and analyze
questions of consistency and logical strength, then set theory is the best
way to go.  I make this assumption because I'm about to say some things
that might superficially sound anti-set-theoretical, and I want to make it
clear that what I'm about to say is consistent with a belief that set
theory is the best choice for carrying out a foundationalist project."

Tim emphasizes that the above does NOT imply any of the following:

-All mathematical objects - manifolds, finite simple groups, differential
equations, categories, etc. --- are "really" sets.

-The foundations of X, where X is a specific subject, are always best
formulated in set-theoretical terms.

--Set theory is always the best tool for foundational thinking.

Again these are basically obvious. And once again, if Tim can do anything
with such banalities, in terms of creating new subjects, then I would be
greatly interested.

In particular, I have never seen an account of what mathematical objects
"really are" that is even remotely convincing. The claim of set theory here
is that it is a remarkably philosophically coherent account of a certain
kind of mathematical object (sets) that can *simulate* the whole range of
mathematical objects. Furthermore, that this simulation is rather powerful
smooth and uniform.

This situation is reasonably similar to the situation in theoretical
computer science. The rough analogy goes like this: Turing machine model of
computation presents algorithms in a coherent irreducible fashion. We can
simulate all algorithms in terms of Turing machines - allowing step by step
building up with TMs calling previous TMs in an hierarchical development.
And of course, any serious computer scientist is going to say that their
algorithms are not expressed that way, and not thought of that way at all.

So trying to pull something productive out of Chow's posting, we have the
projects

a) Formally Expressing Mathematics as It is Thought of by Mathematicians

b) Formally Expressing algorithms as They are thought of my Programmers

The closest subject I know to the first of these is SRM = strict reverse
mathematics.

As for the second, there are various high level languages, but I think that
the prevailing view is that there is no really good way of uniformly
treating all important algorithmic ideas in one subject.

Incidentally, I think that SRM will succeed with a) and something new will
succeed with b).

At the moment, the only technical tools relevant to the Dramatic
Foundational Revelations that we have have arisen from the usual f.o.m.
Because set theoretic foundations is such a popular setup with incredibly
strong simulation power, it has spurred the development of powerful tools
to support various Dramatic Incompleteness and non Computability, as well
as the discovery of candidates for new mathematical axioms of Revelatory
effectiveness.

One question I have been asking is whether type theory or category theory
can provide any satisfactory form of Goedel's Second Theorem. With Goedel's
First Theorem, there has been no problem. But several of the leaders in
Category theory and Type Theory today say they have no insight as to how to
look at Goedel's Second Theorem categorically or type theoretically. I
thought they could do something with my new form of Goedel's Second Theorem
in
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/
#78 The no interpretation form of G2.

PART 3/Chow

True I am a serious advocate of gii = general intellectual interest, and in
fact I won't invest serious time on any mathematics - or any inellectual
activity for that matter - without knowing in advance what the gii will be
if I succeed.

Foundational thinking has an overwhelming gii anywhere, including way
outside mathematics. Foundational thinking is needed and is very effective
when it is mastered in diverse areas.

However, foundational thinking has been most highly developed and effective
in mathematics and computer science. This is a relative statement.

For example, foundational thinking in the physical sciences is
comparatively impoverished. But I maintain that that is not because it
has to be ineffective. It is because it seems to be much harder to make it
effective.

Foundational thinking in the physical sciences is, for example, far far
less developed than was foundational thinking in mathematics before Frege
with his (admittedly too messy) predicate logic.

Look, I have been involved to some extent with science writers for decades
writing about my work and others. Almost every one of them has told me that
the Dramatic Revelations I emphasize with them are of singular general
intellectual interest compared to the usual things that they write about in
mathematics.

Now, respecting Tim's distinction between foundational and foundationalist,
still the foundationalist way of going about foundational thinking is, in
my view, when properly implemented (very difficult!), singularly effective
everywhere.

In fact, I would like to think that the foundationalist method was the
method I used to go from an unnoticeable high school piano player nobody
would even recommend go to Music School, to an obviously professional level
classical and popular player who can hold my own now (significantly
improved from my YouTube recordings a few years back) on a very good day
with the world's leading players. I am considered self taught (by
professional standards) and this seems to not be known to have been done in
classical piano. So I seems like I did this by some special method - a
method that I think is foundationalist at its core.

Harvey Friedman
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