[FOM] More Friedman/Baez

[Harvey Friedman]

More from https://plus.google.com/u/0/110536551627130071099/posts/6TiKLxjSCnu

Friedman:

So could you list two crucial issues from your point of view, with
similar specificity and prima facie philosophical coherence?"

Baez:

It's hard to know which two to pick, but okay.

1) Notions of sameness.  In a set it matters a lot whether two
elements are equal.  However, working mathematicians are almost never
interested in whether two groups are equal: what really matter is
specifying an isomorphism between them (or not, as the case may be).
When it comes to topological spaces an even more relaxed notion is
often more important: namely, specifying a homotopy equivalence.  (For
example, it's common in topology to say a circle is "the same" as an
annulus, meaning we have some homotopy equivalence in mind.)  These
various notions of sameness fit into a hierarchy that's become quite
well understood; I could write a book about this.  The crucial issue
is: how can we work with all these notions as easily and comfortably
as possible?   Various people have taken up this challenge, from
Quillen to Lurie to Voevodsky, and the results have transformed modern
mathematics.

2) Modularity.  Instead of a single overarching framework (e.g. set
theory), modern mathematicians prefer a network of frameworks, often
stripped down to the bare minimum necessary to accomplish specific
tasks.  Some of these frameworks are called "doctrines": examples
include the doctrine of operads, the doctrine of PROPs, the doctrine
of algebraic theories, the doctrine of finite limits theories, etc.
The doctrine of algebraic theories may be the easiest to assimilate,
since it's just an updated version of Garrett Birkhoff's work on
universal algebra.  Universal algebra lets us study mathematical
structures with n-ary operations obeying equational laws, like groups
or rings but not fields.  The crucial is: how can we best formalize
the network of frameworks so as to conveniently pass results from one
to another, and also to "hybridize" them in a number of ways?  This is
another lively area of research, both in pure math and computer
science. 

Friedman:

List of TIME Magazine's 100 most influential people of the 20th
century - Scientists and thinkers
•   Leo Baekeland
•   Tim Berners-Lee
•   Rachel Carson
•   Francis Crick and James Watson
•   Albert Einstein ("Person of the Century" #1)
•   Philo Farnsworth
•   Enrico Fermi
•   Alexander Fleming
•   Sigmund Freud
•   Robert Goddard
•   Kurt Gödel
•   Edwin Hubble
•   John Maynard Keynes
•   Louis Leakey, Mary Leakey and Richard Leakey
•   Jean Piaget
•   Jonas Salk
•   William Shockley
•   Alan Turing
•   Ludwig Wittgenstein

Interesting list?

Now what mathematicians made that list? It was constructed through a
survey of the "leading current scientists and thinkers across all
disciplines". Kurt Goedel and Alan Turing. Standard f.o.m. is the only
mathematics on this list! Gee, I wonder why. 

Friedman:

There are a lot of fundamental differences between your 1,2, which I
think are good typical examples of the abuse of the word "foundations
of mathematics" from its original use, that is fairly widespread.
These seem to be important strategic issues for internal abstract
mathematics, but of rather limited general intellectual interest.
There is a big difference between a strategic issue within mathematics
and a high traction foundational issue of great general intellectual
interest and anybody with any familiarity with very basic mathematical
thinking can relate to (and anybody with a fairly high IQ is not at
all familiar with very basic mathematical thinking).

There are several indirect ways to see the difference between your 1,2
and related matters, and my 1,2, which I will restate below in order
to make my points clearer.

A. What kind of people can readily understand and relate to clearly
stated specific yes/no issues being addressed?
B. What spectacular developments have led up to and surround the
issues being addressed that can be readily explained to what kind of
people?
C. What is the level of understanding and interest in the wider
general community of professional intellectuals?

With regard to A. I explained my 1,2 to my tax accountant, who
expressed interest in what I am doing in my "retirement". I explained
this leveraging off of his daily experience for decades with the
Federal and State Tax Code. This was a complete success, and he
thought that a huge variety of people like him would get it and be at
least casually interested in the issues, and find major developments
noteworthy.

I also explained my 1,2 to an elementary school teacher who teaches
6th grade math. Again, total absorption of the main points, and clear
interest.

I also tried the caregiver for an elderly relative. Again, there was a
high degree of understanding.

All of these people understood why anybody would care.

Regarding your 1,2 and A. Rather than relate this to someone like me
who is much more interested in philosophical and foundational issues
generally than he is in any kind of mathematics - who treats
mathematics as an important example where there are philosophical and
foundational issues about which there can be great revelations - let's
relate this to a colleague of mine with very broad interests and
reputation in PDE, math physics, asymptotics. He has no interest in
the abstract algebraic side of mathematics, and would give you a blank
stare if you said that what you are talking about "has transformed
modern mathematics". He knows that it hasn't even touched the
mathematics that he knows and works with, which is quite diverse.

Yet he immediately recognizes my 1,2 as of great general interest,
even though he is miles and miles away from math logic. But he readily
grasped and has a great deal of interest in serious foundational
issues in the sense that I am talking about.

With regard to B.  What obviously spectacular developments of great
general intellectual interest that can be described to anybody led to
your 1,2? In my 1,2, we have

i. Creation of predicate calculus (Frege), with its spectacular
completeness theorem (Goedel).
ii. Creation of epsilon/delta (Cauchy et al), set theory (Cantor)
iii. Creation of formalized set theory, ZFC, the accepted standard f.o.m.
iv. Spectacular Incompleteness of Goedel.

What have you got to compare with that?

With regard to C. Here I like to bring out the Time/Life 2000 book on
100 most influential people of the 20th century. Scientists thinkers
section of 20:

•   Leo Baekeland
•   Tim Berners-Lee
•   Rachel Carson
•   Francis Crick and James Watson
•   Albert Einstein ("Person of the Century" #1)
•   Philo Farnsworth
•   Enrico Fermi
•   Alexander Fleming
•   Sigmund Freud
•   Robert Goddard
•   Kurt Gödel
•   Edwin Hubble
•   John Maynard Keynes
•   Louis Leakey, Mary Leakey and Richard Leakey
•   Jean Piaget
•   Jonas Salk
•   William Shockley
•   Alan Turing
•   Ludwig Wittgenstein

Note that there are no mathematicians on this list -- except Goedel
and Turing, and them for standard f.o.m.! The list was constructed by
an elaborate feedback process from the current leading scientists and
thinkers, around 2000.

Here was my 1,2:

1. How, why, or should we believe that the various axioms/rules that
have substantial adherents, not only in standard f.o.m., are
appropriate, and in particular are free of contradiction? (Because of
interpretations, this issue has a crucial core that is independent of
choice of foundations).

2. Do the various axioms/rules that we use allow us to successfully
develop areas of CONCRETE mathematics that we find perfectly naturally
interesting, or do we need to expand these setups? For the purposes of
concrete mathematics, we have been adhering basically to a setup that
got engraved in stone around 1920. That is a very very long time for
something like this.

Here is a modified form that is aimed at the general intellectual
community - very short version.

1. By 1920, the "rule book" for proofs in mathematics was engraved in
stone and has stood the test of time. It comfortably allows
mathematicians great freedom in their construction of proofs. But does
it allow too much freedom? Do these rules allow us to obtain
contradictory results? What are the prospects for a complete or
partial collapse of mathematics because of any inherent
contradictions?

2. The rule book has not been updated since 1920, which is a very long
time for rule books. Have mathematicians run into proofs that they
would like to offer of valued mathematical results that are not
sanctioned by the rule book?  If so, should , and how should the rule
book be updated?

There have been and will be spectacular developments surrounding 1,2.
The clearest cases to date are of course Goedel's Completeness and
Incompleteness Theorems. What do you have to compare with the great
general intellectual interest of that?

Harvey Friedman