[FOM] Friedman/Lawvere/McLarty

[Harvey Friedman]

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

I will get to directly responding to the Lawvere quote within the Cody
time limit. But let me first say that I am not sure that the writers
took a serious look at that long list of versions of Goedel's
theorems. To appreciate why those are really (the best) forms of
Goedel's second, you have to be aware of Goedel's COMPLETENESS
Theorem. This shows immediately that consistency statements (which one
is avoiding having to formally state) provide interpretations.

I would be very pleasantly surprised if anyone on this list would have
a really good idea of how to prove any of those  statements that I
gave - let alone Lawvere. Yes, there is a diagonal argument buried
somewhere in the proofs.

But even if the proofs were trivial, - and I don't happen to know any
trivial proofs - that has almost nothing to do with the fact that it
gets Goedel a seat at the general intellectual interest table, which
the whole of the rest of mathematics (except Turing) does not as
readily - e.g., the Time 20 that I discussed earliere.

Also the remarks about the John Templeton Foundation are at best
superficial and misleading. At worst, the remarks are simply some
crazed manifestations of wildly incoherent leftist propaganda. 

DAVID ROBERTS

I will break silence to point out that André Joyal has an unpublished
proof of incompleteness using 'arithmetic universes', a particular
sort of category well-suited to this topic. This is not just a
diagonal argument but the whole shebang.

If you, +Harvey Friedman, wish to contact him privately to encourage a
public version of this proof, I would be grateful, since many people
have tried to get this to the top of his priority list; Joyal is very
busy making lots of other new mathematics and something from decades
back is not particularly on the front of his mind.

Thanks to +roux cody for hosting this discussion, for what it's worth.

FRIEDMAN

These are the cited Lawvere quotes (two of them):

"In Diagonal arguments and Cartesian closed categories we demystified
the incompleteness theorem of Godel and the truth-definition theory of
Tarski by showing that both are consequences of some very simple
algebra in the Cartesian-closed setting. It was always hard for many
to comprehend how Cantor’s mathematical theorem could be re-christened
as a “paradox” by Russell and how Godel’s theorem could be so often
declared to be the most significant result of the 20th century. There
was always the suspicion among scientists that such extra-mathematical
publicity movements concealed an agenda for re-establishing belief as
a substitute for science. Now, one hundred years after Godel’s birth,
the organized attempts to harness his great mathematical work to such
an agenda have become explicit."

"The controversial John Templeton Foundation, which attempts to inject
religion and pseudo-science into scientific practice, was the sponsor
of the international conference organized by the Kurt Godel Society in
honour of the celebration of Godel’s 100th birthday. This foundation
is also sponsoring a research fellowship programme organized by the
Kurt Godel Society"

Source: http://www.tac.mta.ca/tac/reprints/articles/15/tr15.pdf

I was a colleague of Bill's for five years at SUNY/AB. At first I was
puzzled how Bill was obviously not at all moved by what I considered
to be the obvious singularly high general intellectual interest of
Russell's Paradox, Godel's Incompleteness Theorems, and various
associated seminal events in standard f.o.m.

We had some interaction where we were led to his interest in hearing
about the usual logical setup - pre set theory - in standard f.o.m.
This is, of course, FOL=. I view the tire as hitting the road with the
existential quantifier.

I tried to explain the setup with regard to the existential
quantifier. To make a long story short, Bill found this utterly
incomprehensible. A meaningless bunch of nonsense.

At that point, it was clear that Bill mathematical thinking is so
completely different than mine and I think the preponderance of
mathematicians, that I quickly realized that standard f.o.m.
constructions, including FOL=, were a kind of mysterious, almost
meaningless, jumble of symbols.

>From that point of view, how could one possibly think that systems
like PA and ZFC have any special role in the history of ideas? They
are, for him, just some particular technical constructions, not
especially interesting, and really do not in any way shape or form
credibly model mathematical reasoning of any substantial kind. In
particular, for Bill, their consistency or inconsistency has nothing
to do with serious mathematical activity.

So since Goedel's first and second incompleteness theorems do not
deliver any kind of credible message about the fundamental nature of
mathematical reasoning - for Bill - one must turn to the actual
mathematics involved. Since at the essence of theorems, when you cut
out the fat - according to Bill's orientation - you are left with a
diagonalization argument at the core, which goes back to Cantor.

Now from the perspective of standard f.o.m., this "fat" is actually
the core of Goedel's results (and also Russell's paradox). The
"trivia" is the core diagonalization argument.

So how do you choose between Bill's perspective and the perspective of
standard f.o.m.?

I cannot expect to convince anyone who is profoundly allergic to the
existential quantifier, and hence all of the standard f.o.m. setups.

I can only turn to a wide ranging group of distinguished intellectuals
from various walks of intellectual life. Hence I continually talk of
general intellectual interest. I use the Time 20 as an indication of
the plausibility that standard f.o.m. is in a dominating position
relative to the whole of mathematics in terms of general intellectual
interest.

A tangential indicator is that the categorical perspective is always
last at the party. Yes, after Cohen, we know how to talk about models
of set theory with and without the continuum hypothesis, and also
models of set theory with and without the axiom of choice - and with
set theory as a special case of more general categorical
constructions.

But that is after Cohen's party.

Now Cohen left a great deal on the table. Susan's hypothesis, Lebesgue
measurability problems of various kinds, problems in uncountable
groups, Emil Borel's Conjecture (strong measure 0 implies countable),
and so forth and so on.

Even after categorical methods were introduced into independence
results, the categorical people were NEVER first at the party. ALWAYS
last.

Look at my long message above on the modern forms of Goedel's First
and Second Incompleteness Theorems. There are serious ideas there of a
nontrivial nature. Categorical people not even at the party.

And we are now moving to the independence from ZFC of mathematically
compelling statements that are sufficient concrete to be generally
considered to be matter-of-fact   ------  unlike the continuum
hypothesis. Where are the category theorists this time?

OK, you can say that this is not the focus of category theory. Agreed.
And I draw the conclusion that the focus of category theory is not
general purpose foundations for mathematics.  

FRIEDMAN

Now about the second Lawvere quote:

"The controversial John Templeton Foundation, which attempts to inject
religion and pseudo-science into scientific practice, was the sponsor
of the international conference organized by the Kurt Godel Society in
honour of the celebration of Godel’s 100th birthday. This foundation
is also sponsoring a research fellowship programme organized by the
Kurt Godel Society"

The JTF is a private foundation funded entirely with funds left from
the greatly successful investor/philanthropist John Templeton. One of
his granddaughters not runs the foundation, after his son died and ran
it for many years.

As is normal for private foundations, they must strictly adhere to any
instructions left by the donor - donor's intent. It turns out that the
donor was far from being a professional academic, and had his own
special and very detailed instructions for how this money is to be
spent.

One of the engraved in stone founding ideas is the promotion of
science and religion, both on their own, independently of one another,
and also in some form dependently. There is a lot of research support
there for pure science independently of any religious considerations
whatsoever, of pure religion independently of any scientific
considerations whatsoever, and also some research with complex
connections.

This seems to be exactly according to donor's intent.

Now normally, such a wildly successful private investor would simply
leave their money to their heirs, for names on buildings, for medical
research, for chairs at Universities, or whatever standard stuff.

But here you actually see money made available for purely no strings
attached scientific research, AND for other things.

As a scientist, Bill and I should be pleasantly surprised that ANY of
this private money is going to purely no strings scientific research
at all. It is highly unusual, as the donor in this case was in no way
shape or form any kind of scientist. We should all stop complaining. 

*Colin McLarty wrote:*

 I do know that Lawvere, very much like Angus Macintyre, believes
Godel's theorem gets a a great deal of shoddy popularization in mass
media.

FRIEDMAN

There is a reason that Goedel's Incompleteness Theorems "get a great
deal of shoddy popularization in mass media". So does Einstein's
theory of relativity, Crick and Watson's DNA theory, Darwin's
evolution, etcetera.

The reason is very simple. It has great general intellectual interest.

*Colin McLarty wrote:*

Actually I believe everyone likely to read a list like this one knows
that.  Lawvere, like Macintyre, also believes Godel's incompleteness
distracts attention from important logical work -- such as large parts
of current model theory that work precisely by avoiding the Godel
phenomenon.

FRIEDMAN

And there is a reason for this. What you call "logical work", the
model theory of tame structures, which as developed avoids "Goedel
phenomena", has a little bit of general intellectual interest, but not
at the overwhelming level of Goedel's Incompleteness Theorems.
Furthermore, the only real "logical" component in this model theory is
a rather simple use of FOL=, which is itself a mathematical structure
that was invented for standard f.o.m. purposes of great general
intellectual interest, going back to Frege.

Incidentally, what is emerging is that even within the most tame of
tame structures, (Q,<), the Goedel phenomena is rearing its head with
an uncontrollable viscous unremovable vengeance. What is emerging is
that there is absolutely no where in mathematics to hide from the
Goedel phenomena - even from extremely large doses of the Goedel
phenomena.

Distinguished scholars like Macintyre are going to be in a complete
state of denial, and will need to be confronted with public
declarations from mainstream math icons before being budged from that
state of denial. Very plausible but completely wrong ideas die very
very hard.

Harvey Friedman