the physicalization of metamathematics

[Harvey Friedman]

Reponse to Stephen Wolfrm
https://cs.nyu.edu/pipermail/fom/2022-March/023177.html

I have always believed in the idea that a radically new approach to
fundamental matters can be given relatively brief simplified accounts
that readily create excitement starting from scratch, which is
sufficiently compelling for people to want to look seriously at the
much longer detailed accounts. I write this posting with that in mind.

On Tue, Mar 8, 2022 at 6:31 PM Stephen Wolfram <s.wolfram at wolfram.com> wrote:

" I just posted something I think may be of interest to FOM subscribers:
 https://writings.stephenwolfram.com/2022/03/the-physicalization-of-metamathematics-and-its-implications-for-the-foundations-of-mathematics

" It’s a (rather unexpected, at least to me) outgrowth of our recent
(and very active) Physics Project https://www.wolframphysics.org/ "

I would like to hear more about what specifically you have found that
is unexpected.

" Both metamathematics and physics are posited to emerge from
samplings by observers of the unique ruliad structure that corresponds
to the entangled limit of all possible computations."

As I see it,

i. "metamathematics" is a word that arose in a relatively early period
in the development of f.o.m. and whose precise meaning is open to
question.
ii. Generally speaking it refers to formal developments arising from
the formalization of mathematics - or at least from the formalization
of mathematical proofs.

Conventional wisdom is that the choice of these formalisms is driven
by careful consideration of (aspects of) the mostly informal thought
processes of mathematicians as they identify key mathematical
concepts, support them with "precise definitions" make "precise
claims" and give "rigorous proofs".

Conventional wisdom is that this process is almost entirely
conceptual, with little or no computational component, and is in the
"semantic" realm as in the conventional "semantic" versus "syntactic"
dichotomy.

It seems like you may be questioning this conventional wisdom and in
particular the conventional distinction between "semantic" and
"syntactic".

" The possibility of higher-level mathematics accessible to humans is
posited to be the analog for mathematical observers of the perception
of physical space for physical observers."

I would like to hear more about how you view "the perception of
physical space for physical observers". I'm now looking around my
office and looking from left to right and up and down and think I am
"perceiving physical space". And I think, with some struggle, we can
build a structure from that that is 3 dimensional and not 1,2 or 4
dimensional, and there is also a clear sense of time. I wonder what
you or others have done in the direction of starting with the naive
observations we make as persons sitting at a desk in their office (or
elsewhere) and building up the usual 3 dimensional space time
informally rigorously step by step incrementally.

On the informal mathematics side, we can also ask for a similar
development from first naive principles. I don't know of any standard
way of going about this, but perhaps the key idea is that

a. There are "different" things. In particular things separated by
space, and normally physically connected like an unbroken piece of
wood (a connection with the physical side of things above!).
b. These "different" things can be put together to form a new thing -
no longer physically connected.
c. These new things in b are called sets.
d. These sets have a fundamental equivalence relation, namely being
equinumerous.
e. In terms of what is and is not a member of a given such set, there
is basically only one fundamental notion, and that is of inclusion
(and containment).

Presumably, one can carefully and incrementally build up a
considerable amount of physics and a considerable amount of
mathematics through such origins. The challenge is to not make the
obvious, second nature to us, conceptual leaps in such a development.
And also to develop a clear notion of what is "cheating" when we try
to do this.

And I wonder how relevant the old Piaget work on the child's
conceptions of the mathematical and the physical are here.

At first blush, these two plans, one for the physical, and one for the
mathematical, appear to be substantially different. And I don't
readily see how to view this profitably as part of any space of
computational items. Above all, is there a profitable way of unifying
these two plans into a single plan?

" A physicalized analysis is given of the bulk limit of traditional
axiomatic approaches to the foundations of mathematics, together with
explicit empirical metamathematics of some examples of formalized
mathematics."

What does "bulk limit" mean here? Is there a suggestion here that
there is a legitimate approach to the foundations of mathematics that
is other than the traditional axiomatic approach through classical
f.om.? Or is the idea that the usual f.o.m. is either wrong headed or
is uniquely inevitable? Can you give a simple condensed empirical
finding that is significant?

" General physicalized laws of mathematics are discussed, associated
with concepts such as metamathematical motion, inevitable dualities,
proof topology and metamathematical singularities."

Can you give a perhaps simplified version of a "physicalized law of
mathematics"? Also perhaps simplified instances of the other four two
word phrases here?

" It is argued that mathematics as currently practiced can be viewed
as derived from the ruliad in a direct Platonic fashion analogous to
our experience of the physical world, and that axiomatic formulation,
while often convenient, does not capture the ultimate character of
mathematics."

You are suggesting a kind of foundation for mathematics that is not
axiomatic. Can you give a little simplified example that would
illustrate something clear about the nature of a non axiomatic
foundation for mathematics? The "ultimate character of mathematics" is
a rather open ended phrase, and the usual axiomatic f.o.m. is
extremely flexible and is continually able to incorporate more and
more features of mathematical practice.

" Among the implications of this view is that only certain collections
of axioms may be consistent with inevitable features of human
mathematical observers. A discussion is included of historical and
philosophical connections, as well as of foundational implications for
the future of mathematics."

I have had longstanding projects aimed at showing the usual systems of
f.o.m. are inevitably unique in certain senses, but have only
scratched the surface of this. What I have in mind is couched entirely
in the traditional f.o.m. But I think you have something very
different in mind which I would like to see a simplified hint of. Also
something about the "foundational implications for the future of
mathematics".

Harvey Friedman