[FOM] R: Preprint: "The unification of Mathematics via Topos Theory"
Olivia Caramello
oc233 at hermes.cam.ac.uk
Sat Jul 10 18:55:39 EDT 2010
First, I should clarify that my approach is not at all intended to be the
'final and complete' solution to all the problems in the foundations of
Mathematics! Rather, the aim is to provide effective means for transferring
knowledge between distinct mathematical theories, so that problems
formulated in the language of a specific field of Mathematics can be
translated into different languages and therefore be tackled by using
different methods. For example, one might be able to prove the completeness
or consistency of a theory by means of arguments which are not logical in
nature but possibly combinatorial, algebraic or geometric (cfr. my
topos-theoretic interpretation of Fraisse's construction in Model Theory),
and similarly for other properties of theories. These methods unify
Mathematics to the extent that properties of different mathematical theories
which have a common 'semantical core' but a different 'linguistic
presentation' come to be seen as different *manifestations* of a *single*
property lying at the topos-theoretic level. So whatever happens at the
level of toposes has 'uniform' ramifications into Mathematics as a whole;
for example, the fact that the subtoposes of a given topos form a lattice L
implies that for any theory classified by that topos (and notice that there
are in general many different such theories), the quotients of that theory
(considered up to syntactic equivalence) form a lattice which is isomorphic
to L (cfr. section 8 of my paper). It is also worth to note that the
transfer of knowledge between two theories which are Morita-equivalent to
each other is not carried out - as it normally happens - by using the
explicit description of the equivalence, but rather by going through the
classifying topos, which acts as a 'bridge' connecting the two theories and
enabling us to transfer invariants across them (in fact, for transferring
'global' invariants of toposes one can well ignore the actual description of
the equivalence).
Overall, on the basis of my research experience, I can say that these
methodologies are extremely powerful, at least because they allow us to
generate a huge amount of surprising mathematical results in a
semi-automatic way; of course, most of the results generated in this way
could be rather 'weird' according to the usual mathematical standards
(although they might still be very difficult to prove by using more
traditional methods), but, as shown in the paper, with a careful choice of
Morita-equivalences and topos-theoretic invariants, one can easily get
interesting and deep mathematical results or recover classical theorems. As
far as it concerns the generality of these methods, since most of
Mathematics can be formulated in terms of geometric theories (cfr. section 2
of my paper) or in terms of categories, the approach described in the paper
is abstract enough to shed light on essentially every mathematical field.
I realize that this might sound unbelievable, but this machinery has really
the potential to 'automatically generate' results (a careful reading the
paper is necessary to fully understand the precise meaning of this claim);
on the other hand, the sensitivity and experience of an educated
mathematician are necessary in order to 'program the machine' to yield
results of current mathematical interest.
Olivia Caramello
-----Messaggio originale-----
Da: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] Per conto di
Brian Hart
Inviato: venerdì 9 luglio 2010 0.49
A: Foundations of Mathematics
Oggetto: Re: [FOM] Preprint: "The unification of Mathematics via Topos
Theory"
I agree that in this time of increasing fragmentation in the
foundations of mathematics that an approach emphasizing unity is very
welcome but is your approach open-ended and therefore powerful enough
to encompass any future foundational fragmentation? Since mathematics
is an open-ended discipline (as Gödel's incompleteness result reveals)
who knows where the future of foundational studies will take us and so
any overarching plan for final and complete unification seems to
likely be only temporary and thus illusory. Set theory could be
considered one of these foundational approaches as a generalization of
the concept of number but in time it was shown to be inadequate to
found the entire and vast mathematical edifice. Now category theory
largely supplants it as a more versatile foundational theory which is
compatible both with classical _and_ intuitionistic logics. Category
theory is a very capable and flexible tool, no doubt, but can it
really exhaust the inexhaustible?
On Thu, Jul 8, 2010 at 8:05 AM, Olivia Caramello <oc233 at hermes.cam.ac.uk>
wrote:
> The following paper, recently presented at the International Category
Theory
> Conference 2010, might be of interest to subscribers to this list:
>
> O. Caramello, "The unification of Mathematics via Topos Theory"
>
> Abstract:
> We present a set of principles and methodologies which may serve as
> foundations of a unifying theory of Mathematics. These principles are
based
> on a new view of Grothendieck toposes as unifying spaces being able to act
> as 'bridges' for transferring information, ideas and results between
> distinct mathematical theories.
>
> The paper is available from the Mathematics ArXiv at the address
> http://front.math.ucdavis.edu/1006.3930.
>
> Comments are welcome.
>
> Olivia Caramello
>
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> FOM at cs.nyu.edu
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