FOM: Godel, f.o.m.

[Harvey Friedman]

Reply to Silver Sun, 16 Jan 2000 09:21.

>    To me, this is an interesting viewpoint, but I'm wondering where you
>would locate the "essence" of Gödel's and Turing's achievements.   I'll get
>more specific below.

>    I thought perhaps you might locate the "essence" of G's and H's
>achievements somewhere in the field of logic, but I see from the above that
>that's out too.

I think you mean Godel and Turing. I would say that their most important
contributions were in what we now call f.o.m. In talking about this very
early period, it really makes little sense to distinguish between f.o.m.,
mathematical logic, and f.o.c.s. (foundations of computer science). Back
then, of course Godel's work would most commonly be called "logic", and
probably Turing's also.

>    Ok, I can now ask a more specific question:  Do you consider that the
>typical one-paragraph informal explanation of Gödel's (first) Theorem
>adequately captures the "essence" of his contribution?  Here's the schema:
>
>*    Let sentence G say 'I am not provable'.  Now, suppose G is true.
>     Etc. etc. etc.   What if G is false?  Etc.  etc. etc.

>    I personally do not like these informal expositions, and I'd be very
>surprised if you would think much of them.  Otherwise, I'm not sure what
>you're getting at when you speak of the essence of (in this case) Gödel's
>result.  Could you please explain more thoroughly what you take the essence
>of the first theorem to be (and, if you wish, the essence of Turing's
>achievements)?

When I write here about the essence of Godel's achievments, I try to put
myself into the context of that early period, taking into account some
relevant biographical material. I am not trying to be an historian.

My view of the essence of Godel's achievment(s) is naturally colored by
their impressive range - the completeness theorem, the two incompleteness
theorems, the relative consistency of the axiom of choice and the continuum
hypothesis, the Dialectica interpretation, the reduction of classical
arithmetic to intuitionistic arithmetic, the work on general definability
(OD and HOD), and perhaps to some extent the work on spacetime. (This is
not meant to be exhaustive, and in particular does not include his
contributions to the philosophy of mathematics, and some other important
topics such as modal logic, lengths of proofs, etcetera, where his work is
not as definitive.) See Godel's collected works.

The essence of his achievment can be stated in a very general way. Many
many times, Godel came up with definitive striking mathematical results
which are forever essential in the discussion of a variety of fundamental
issues in f.o.m. and p.o.m., and he did this in the context of there being
essentially little or nothing in the way of an organized mathematical field
to provide a mathematical context with feedback and background information.
The structure and scope of the results were entirely novel and
unprecedented. The very possibility of such decisive rigorous results was
beyond the imagination of all or almost all of his contemporaries.

This was not a matter of Godel being somewhat more clever and quick than
his contemporaries (although Kleene has written about the history of the
incompleteness theorems, suggesting that at least he was on the right
track). It was more a matter that he knew much better than others what was
of crucial importance, so that he could direct his mathematical efforts
most productively. This was a consequence of his great philosophical
acuity. In addition, he had the confidence that definitive mathematical
findings could be obtained on the crucial foundational issues, even if such
mathematical results had to be unlike any that had come before. He did not
have to work in any existing mathematical tradition. He founded his own
mathematical tradition.

There is nothing accidental about his achievments. There were realized by a
systematic approach to intellectual life, which proved to be incomparably
more powerful than those of his contemporaries. And he had the requisite
mathematical power to realize the fruits of his approach.

Had he not had this requisite mathematical power, the history of f.o.m. in
the 20th century would have to be written very differently. An interesting
question is: what is the relationship, if any, between the kind of
philosophical acuity and systematic approach to intellectual life
exemplified by Godel, and mathematical power? Are they positively or
negatively correlated, or not correlated at all?

By way of contrast, let me say what is not the essence of Godel's
achievments. The use of the beta function in his incompleteness theorems
for PA = Peano Arithmetic.

I mention this because a certain well known and active mathematical
logician is rather (in)famous for saying that

**the only thing interesting that Godel did was the beta function.**

This logician has since mellowed a bit and says that he was mainly being
provocative, and couldn't have meant it literally.

Actually, this beta function development was pretty clever and essential
for his results about PA formulated in the language of 0,S,+,x,=.

This is the most striking and well known example I know of the currently
popular practice among mathematical logicians and mathematicians of
stripping away the essence of advances in f.o.m. in favor of technical
points that are comparatively minor and pedestrian.