[FOM] Book appearing: Formalization without Foundationalism: Model Theory and the Philosophy of Mathematical Practice

[John Baldwin]

I thank Harvey for his thoughtful response.  I would like to clarify my
views on one of his points.
I want to think a bit about long proofs in ZFC but am in the santiago
airport and communication will be erratic.


Harvey wrote:
Obviously, applied model theorists did not have any incentive, nor did
they believe in, the matter of bridging the gap between Godel
phenomena and "real" mathematics. The assumption is simply that that
gap is very wide and totally necessary.

In reply, let me first emphasize the importance of the work of Godel,
Turing etc in clarifying the otion of
decidability.  On the one had there is  the active work in applied model
theory  on extending Hilbert 10th problem results
to other fields: e.g. Slapentokh
https://scholar.google.com/citations?user=jLnE3bMAAAAJ

On the other, one aspect of my book is the decidability and even finitary
consistency of parts of mathematics; some of this depend on work of Tarski
and Harvey.  In particular an extension of eucidean geometry that validates
the area and circumference formulas of circles, See

http://homepages.math.uic.edu/~jbaldwin/pub/axconIIfinbib.pdf

published

*Philosophia Mathematica, nkx031*

*John*

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607

On Tue, Jan 9, 2018 at 8:56 AM, Harvey Friedman <hmflogic at gmail.com> wrote:

> On Mon, Jan 8, 2018 at 11:48 PM, John Baldwin <jbaldwin at uic.edu> wrote:
> > I take this opportunity my about to be published book.
> >
> > Bound copies of the proofs of the above book will be available at the
> > Cambridge Press booth at the annual Joint Mathematics Meetings in San
> Diego.
> > The book will be available a bit later.
> >
> > Here is a lightly edited version of the book proposal.
> >
>
> John, thanks very much for putting out that substantive "edited form
> of the book proposal".
>
> I am briefly responding to what you wrote, and I expect to have a lot
> more to say after the book is available.
>
> A major part of the current ideology in applied model theory today is that
>
> *) the so called Goedel phenomena is a wrong turn and totally
> unnecessary turn in the history of mathematics, that needs to be, and
> is fairly easily, separated from the "good" or "productive" part of
> mathematics. This is done through "tameness", with one particularly
> clear embodiment being realized by 0-minimality, something with its
> origins in Grothendieck's "tame topology" and later firmed up and
> investigated deeply by applied model theorists.
>
> Obviously, applied model theorists did not have any incentive, nor did
> they believe in, the matter of bridging the gap between Godel
> phenomena and "real" mathematics. The assumption is simply that that
> gap is very wide and totally necessary.
>
> However, as the f.o.m. revolution of the 21st century continues, of
> course at a not quick enough pace for many of us, the gap is being
> removed.
>
> Also, closely related, the idea that there are appropriate independent
> autonomous foundations for separate parts of mathematics, where the
> foundational schemes do not interact deeply, is also being refuted.
>
> Of course, in this very early part of the 21st century, the known
> results are just not yet strong enough to make all of this completely
> obvious, as it likely will by the end of the 21st century, and without
> any doubt in my mind, in a few more centuries. That is the way things
> are clearly evolving.
>
> SOME EXPECTED FUTURE DEVELOPMENTS TOWARDS FIRMLY EMBEDDING THE GOEDEL
> PHENOMENA EVERYWHERE IN NORMAL MATHEMATICS
>
> 1. No less than Carl Ludwig Siegel (CLS) is famously created for
> establishing a decision procedure for the solvability of quadratic
> equations in the integers. (By the way, somebody questioned whether
> this has been done for the rationals. References?)
>
> 2. We know that the solvability of systems of quadratic equations in
> the integers is Goedelian. So CLS is banging his head potentially
> against the Goedelian. POSSIBLE FUTURE: Very few simultaneous
> quadratic equations in very few variables are already algorithmically
> unsolvable.
>
> 3. At the most EXTREME that I know of, there is the possibility that
> {n^2 + m^3: n,m in Z} is algorithmically unsolvable. If you find that
> unimaginable, fool with that expression a tiny bit, and then see what
> you think.
>
> 4. There is an integer 0 < r < 2^30 such that the solvability in
> integers of n^2 + m^3 = r (or use your favorite alternative very very
> simple expression, if you like, maybe with coeffieicnets also < 2^30 -
> is independent of the ZFC axioms. Provable from large cardinals, but
> not in ZFC.
>
> 5. It is possible to take, rather generally, simple combinatorial
> properties of long well orderings (too long for ZFC), which are
> sharper than well known such provable in the nonnegative integers,
> and, in a simple strategic uniform way, restate them in the rational
> numbers, so that the resulting statements can and can only be proved
> using large cardinals. As of December 2016, this is more or less
> essentially here.
>
> 6. 5 will be further pushed down into elementary number theoretic
> relations between finite sets of integers that will gradually be
> extremely simple and totally strategic. A probably significant but
> small step is in my recent series on 1-dimensional incompleteness.
>
> 7. These Goedelian combinatorial matters will be slowly moved into the
> realm of finitely presented groups. Then the longstanding connection
> with geometry/topology will be firmed up for this, to put Goedel
> inexorably into geometry/topology.
>
> 8. At a more abstract level, we already know that for consistent
> single sentences in predicate calculus, there is no maximal such under
> interpretability. But the natural proof, and in fact the only proof I
> am aware of, passes through the heart of Goedel. OK, sure, you want to
> call predicate calculus Goedelian and remove it as well (at least the
> extreme form of Baldwin). But what about finite systems of equations,
> with the tacit "there exists at least two objects" - same question
> about maximal interpretation power. This also passes through Goedel.
> And this can be made even more algebraic.
>
> 9. There will be an intelligible sentence in the usual (decidable)
> ordered field of real numbers with the following properties.
>
> i. It is true.
> ii. Any proof of it in ZFC is too large to fit in the existing worldwide
> web.
>
> To be continued, if people are interested!
>
> Harvey Friedman
>
>
>
> .
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> https://cs.nyu.edu/mailman/listinfo/fom
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20180110/6a81f565/attachment.html>