[FOM] Dependence relations in model theory, 1

[Harvey Friedman]

Reply to Baldwin, Dependence relations in model theory, July 19, 2003, 
at 10:55 AM.

The reply will be in two parts. This is the first part.

Let me say at the outset that I very much appreciate the efforts of 
Baldwin in discussing contemporary model theory in such a strategic 
manner. Some of the discussion is readily understandable by the 
preponderance of FOM subscribers, and I'm sure that is greatly 
appreciated.

Part of the reason that I am so happy to see Baldwin making these basic 
survey postings is that it gives us an opportunity to perhaps see what 
is foundational about some of contemporary model theory, and perhaps 
help take some of it in more foundationally motivated directions.

Each of the four traditional branches of mathematical logic have their 
roots in the deep and fully foundational developments of obvious great 
general intellectual interest that have captured the imagination of 
scholars in a great variety of fields, and to some extent, the general 
intellectual community. I wrote about some evidence for this by 
referring to the list of 20 "great minds of the 20th century" compiled 
by Time Life books in 2000, in my posting

http://www.cs.nyu.edu/pipermail/fom/2001-February/004794.html

It was nice to see Simpson refer to this also in his posting Thu Jul 
17, 2003  5:34:38 PM.

In that Time Life list, fully 3 of 20 scholars listed were closely 
associated with foundations of mathematics. The clearest f.o.m. icons 
listed there are of course Kurt Godel and Alan Turing, rather than 
Ludwig Wittgenstein, who remains deeply controversial. In any case, a 
considerable contingent of philosophers back his presence on the list, 
but also a considerable contingent would back Bertrand Russell on this 
list instead (I'm a Russell fan).

It is interesting to note that Godel and Turing are also the only 
persons on this list of 20 who could reasonably be classified as 
mathematicians. Also, only two of them (Einstein and Fermi) can 
reasonably be classified as professional physicists.

Note how all four traditional branches of mathematical logic have their 
roots in Godel and Turing.

Model Theory. The completeness theorem, which asserts that a set of 
sentences in a countable first order language is true in all models if 
and only if it is provable in an explicitly given formalism associated 
with mathematical practice.

Recursion Theory. The analysis of discrete deterministic algorithms via 
abstract machines.

Set Theory. If the standard axioms for set theory are consistent, then 
they remain consistent if the axiom of choice (and the continuum 
hypothesis) is added .

Proof Theory. No consistent suitably axiomatized first order system is 
complete, or can even prove (any suitable formalization of) its own 
consistency.

There are a few other events in the 20th century in f.o.m. of perhaps a 
similar kind of great general intellectual interest than these events, 
but the list is not long. Also, the confidence that we have in the 
great importance of most of these events is made clearer by further 
associated developments, also of general intellectual interest, which 
serve to establish the robustness of the notions involved.

I could considerably elaborate on this spectacularly successful story 
of the foundations of mathematics in these general intellectual terms, 
but this is not the purpose of this posting.

When you reflect on such events, you see clearly how most people with 
substantial theoretical scientific instincts or philosophical instincts 
- and many people with strong general intellectual instincts - are 
moved. Surely for many such people, considerably more has to be said 
than these brief accounts above. Sometimes a major gap has to be filled 
in connection with many people's lack of familiarity with the general 
logical structure of mathematics. But this is generally not difficult 
to fill. Of course, sometimes one runs into professional mathematicians 
who are particularly uncomfortable with the very idea that there exists 
a mathematical subject of such great general intellectual interest and 
power (foundations of mathematics) that their mathematical knowledge 
and mathematical methods do not seem to illuminate. Such professionals 
sometimes have a tendency to dismiss f.o.m. as fruitless third rate 
mathematics that should not be pursued in a good mathematics 
department. Even these professionals can be turned around if one can 
establish an appropriate person to person interaction.

To give a startling indication of how unexpectedly striking f.o.m. is 
to a well known applied mathematician/historian, see

Mathematical Thought from Ancient to Modern Times, by Morris Kline, 
Oxford University Press, 1972, 1238 pages.

This book ends with Chapter 51, The Foundations of Mathematics. I quote 
the first paragraph:

"By far the most profound activity of twentieth-century mathematics has 
been the research on the foundations. The problems thrust upon the 
mathematicians, and others that they voluntarily assumed, concern not 
only the nature of mathematics but the validity of deductive 
mathematics."

Again, this is yet more evidence of the great general intellectual 
interest of foundations of mathematics, when one considers that this is 
written by an applied mathematician/historian who never worked in 
foundations of mathematics.

So in deference to this fantastic legacy that affects us all on the FOM 
list, that propels foundations of mathematics at its heights to levels 
of general intellectual interest that are only matched by the greatest 
of all intellectual achievements of the 20th century, we should not use 
the term "foundational" lightly. When using this term, we need to keep 
this legacy clearly in mind.

If one is to think of some development as "foundational", one should 
apply the standards that are appropriate in light of this legacy. This 
usually means that the particular recent advance that is regarded as 
"foundational" has at least the potential of some substantial general 
intellectual interest, but falls quite short of this standard when 
examined objectively keeping the legacy in mind. One generally has to 
follow up intensively on the advance, in order to bring out the 
"foundational" aspects. This generally requires a great deal of 
"foundational" imagination, and a clear sense of just what constitutes 
general intellectual interest.

Unfortunately, this is not what is normally done. Some advance is made 
that has some sort of "foundational" component, generally quite 
undeveloped and rough at the edges, and one goes on to further 
developments for much more ordinary - but perhaps valuable - purposes. 
One generally does not dwell on the truly foundational aspect, and use 
it to drive further developments.

This dwelling on the foundational aspect and using it to drive further 
developments is absolutely essential to create something of truly 
general intellectual interest. In particular, looking back at a series 
of mathematically rich developments, however mathematically 
interesting, and using the word "foundational" to describe the context, 
is not appropriate in light of our fantastic legacy.

It is obvious that model theory, and for that matter, all four of the 
traditional areas of mathematical logic, have substantially strayed 
away from their roots in foundations of mathematics. To a certain 
extent, I believe that this was appropriate and even necessary, in 
order to build up the mathematical technology needed for serious 
advances in the foundations of mathematics, of general intellectual 
interest.

I am optimistic about using this advanced mathematical technology in 
order to make such advances. Two approaches to f.o.m. are to

i) revisit various topics that have at least some semblance of a 
"foundational" nature and focus on that "foundational" aspect and 
rethink the topic in "foundational" terms in order to build the topics 
up into something truly foundational, of general intellectual interest.

Of course, my usual approach is to

ii) simply start with the most overarching foundational issues of 
obvious general intellectual interest, and develop things from there, 
using whatever mathematical tools are available, and trying to invent 
whatever mathematical tools are needed.

My keen interest in seeing such valuable postings as Baldwin's and 
others, is in connection with approach  i).

Harvey Friedman