[FOM] Book on model theory and the philosophy of mathematical practice
Harvey Friedman
hmflogic at gmail.com
Mon Aug 31 01:15:40 EDT 2015
On Sun, Aug 30, 2015 at 9:44 PM, John Baldwin <jbaldwin at uic.edu> wrote:
> Martin Davis posted a couple of days ago a message containing this sentence.
>
> Gödel showed us that the wild infinite could not really be separated from
> the tame mathematical world where most mathematiciansmay prefer to pitch
> their tents.
I think that this statement of Martin Davis is a bit misleading. I
would subscribe to the following only:
Goedel showed us that the wild infinite could not really be CRUDELY
separated from the tame mathematical world where most mathematicians
may prefer to pitch their tents.
>From Goedel's work, you only see a very CRUDE conclusion of the kind
Martin is talking about. And, in fact, mathematicians do not hesitate
to generally continue to subscribe to such a separation, because they
do not see Goedel's work to be sufficiently compelling in this vein.
Of course, some of us spend a lifetime endlessly engaged in moving the
bar from the crude to the compelling.
I should add that the kind of deep pathology of the wild infinite that
is well known to mathematicians (not other exotica) is easy to
separate from the tame.
> This is an excuse for me to publicize my book in progress. Much of it is
> dedicated to the proposition that modern model theory provides a systematic
> way to separate the wild from the tame.
This is of course generally extremely successful in separating the
tame from the wild (e.g., o-minimal, strongly minimal, and variants).
However, for the first time, this is looking directly vulnerable. See
http://www.cs.nyu.edu/pipermail/fom/2015-August/018920.html
Proposition 2. Proposition 1 of
http://www.cs.nyu.edu/pipermail/fom/2015-August/018907.html although
implicitly a tame statement, has the weakness that the set S cannot be
a tame object. Not that strong a weakness in this case, but
nevertheless the first one (prop 2) has no such weakness.
The attack that Prop 1 is too weird is going to be very difficult to
maintain as things evolve, especially since there are indications that
emulations/continuations have an immediate resonance in discussions
with a variety of people way away from math logic, and even outside
math.
> Claim 0.0.1. (1) Formalization of specifi c mathematical areas is a tool
> for studying mathematics itself as well as issues in the philosophy of math-
> ematics (e.g. axiomatization, purity, categoricity and completeness).
>
> (2) The systematic comparison of local formalizations of distinct areas is a
> tool
> for organizing and doing mathematics and the analysis of mathematical
> practice.
>
> (3) The choice of vocabulary and logic appropriate to the particular topic
> are
> central to the success of a formalization. The logic which has been most
> important for the study of mathematical practice is first order logic
>
> http://homepages.math.uic.edu/~jbaldwin/pub/philbook83015.pdf
>
It would be very useful for FOM readers such as myself, to see a
couple of basic examples of each of these Claims, so that we can see
these ideas (somewhat familiar) in action. That will probably spawn
some continuing useful discussion.
Harvey
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