FOM: Re: SET vs. TOP

[Diskin]

Peter Aczel is absolutely right:  FOM is indeed like a drug, and  though I had
have no possibility  to trace the recent hot discussion "SET vs. TOP" in the
entire details, it's very tempting to add some remarks to clarify some of the
issues (at least for myslef, I apologize in advance if they repeat something
that was already said). 

Let me begin with formulating some known statement just to start. Let ZX be
the family of formal set theories of the kind "Z + something reasonable" (so
that X ranges thru F, FC, FC+CH etc). Similarly,  let  TX be the family of
formal theories of the kind "T + something reasonable"  where T is the set of
basic topos axioms and X ranges thru NNO, Boolean condition  etc. 
FACT. For any theory A from ZX there is some theory B from TX  in which A can
be interpreted. Conversely, any B from TX is interpretable in some A from ZX.
[]
So,  I don't see any big problem in relating  SET and TOP on the formal level
(the level of the pure FOM as such), if they are being thought of as ZX and TX
resp.  

However, mathematics is hopefully something more than a formal game, and
behind either of  the two formalisms some Natural-Set-World (NSW) is
presupposed as something independent of them (below I'll return to this
statement).  NSW may appear as either some standard universe of sets, or a
family of  standard universes, or a family of set-and-function universes, or
something else. What is important is that  NSW does exists and so the
following question is reasonable:  Which of the formalisms, SET or TOP, is a
more direct,  more immediate, more suitable for human reflections about NSW?
In the terms of the FACT above, something reasonable for SET might be foreign
for TOP and conversely, so that in this informal, actually humanitarian
context, interpretability stated in the FACT again becomes an open issue. 

Let me remind  one  very hot discussion  now belonging to the history of
physics. I mean the old discussion of what is an electron: whether it's a
particle or a wave. The answer found in Copenhagen, and now widely recognized,
is that the question itself is incorrect: electron is a too complex phenomenon
to be modeled (explained) by a single mathematical formalism. In function of
the observer's goals, an electron could be successfully considered as either a
particle,  or a wave (or something else, who knows?) but  does not amount to
the either of models.  

The same is for NSW: it is a too complex intellectual phenomenon to be
completely captured by SET or TOP separately. In some contexts SET is
preferable, in others - TOP is surely better so that only together, in some
integrated way, they give the most adequate mathematical notion of NSW at *the
present level of our knowledge*. 

The reference to "the present level" is essential. Indeeed, in the above
treatment NSW appears as something Platonic. However, the Cantian  view seems
to be more relevant: NSW is mainly (if not at all) our knowledge about it
rather than independent entity (though, of course, thinking it of as some
independent Platonic universe is a good working tip).  After invention of the
TOP-view on sets, and its successful applications in algebraic geometry, model
theory and  computer science, NSW has changed.  Maybe, this explains the
boiling temperature of the discussion "SET vs. TOP":  NSW  of SET and NSW of
TOP are essentially different simply because in mathematics a way of thinking
something essentially contributes to constituting this thing. (Such a
relativism has  recently gained some popularity even in physics where objects
of study are more  independent).

In the consideration above,  SET and TOP appear in a symmetric fashion. To my
mind, however,  their relation is asymmetric rather  than symmetric. Let me
describe one simple real life example where TOP is more preferable than SET,
it  is somewhat opposed to the Harvey Friedman's  example of cards on the
table.

What is the list of FOM-subscribers? It is a set varying  in time, and it
seems this can be well explained in either of the two paradigms.  But what
about elements  of this set? In fact, they are addresses: john at aol.com,
johann at use.net  etc.  Note however that  it may happen that john at aol.com  at
some moment  t1 and johann at use.net  at some moment t2  are the same person.
On the other hand,  john at aol.com  now and john at aol.com  a year later when he
studied and tried category theory  in his everyday work  should be considered
as  different FOM-subscribers. So, while in SET the notion of element  is
elementary and does not present any problem ,  in the real life we often deal
variable objects accessible only via also variable identification so that the
very notion of object identity becomes a highly non-trivial issue. Category
theory  offers a convenient apparatus for dealing with such dynamic
collections consisting of dynamic elements, and is gradually becoming the
basic machinery of software engineering  where managing such things is a must.

No doubts, all this complexity of object identification and dynamics can be
also described in the pure SET-terms. However, this would be a very bulky
specification blurring some essential aspects of the phenomenon. In contrast,
the TOP-based specification just throws light on these non-trivial questions
(I have checked this in my communication with real  database designers. Note
also that, in fact, database designers and software engineers  rediscovered
implicitly many of topos-theoretic concepts (in their own, terribly awkward,
terminology)).  

>From the view point of the example just described, the relation  between TOP
and SET is somewhat like the relation between the statistical and the
classical mechanics. Explaining dynamic objects in the SET-framework, and more
generally, explaining TOP via SET,  is somewhat like that reductionism   when
one says that statistic termodynamics  is reducible  to the classical
mechanics.  On a whole, SET-motivated objections against TOP somewhat remind
me old objections against quantum mechanics and relativity from the standpoint
of the classical  physics (at any rate, their temperatures are close  :-). 

My last remark is about standard models for TOP, the issue where it seems
there was  some mess.   In algebraic logic there is well known an essential
distinction between abstract algebraic models and concrete models arising from
sets. For example, for propositional calculi, this is the distinction between
abstract Boolean algebras (BA) and  BA of subsets, SetBA, or abstract Heyting
algebras, HA, and Heyting algberas arising from Kripke structures, SetHA, and
so on.  Similarly, for  first-order predicate  calculi,  one should
distinguish between abstract cylindric algebras, CA,  and CA  arising from
sets (Kripke frames for intuitionistic logics), SetCA.  Representation
theorems (like Stone's one) are always deep results showing when and how
abstract structures can be represented in concrete ones.  Now, topos
structures (in fact, algebras),  TA, is an algebraic version of HOL like CA is
algebraic versions of  FOL. Of course, concrete TA, SetTA,  do exist  - they
are  nothing but the presheaf toposes ( as always, concreteness  is
algebraically characterized by some subdirect-irreducibility-property
correspondingly formulated,  Lambek & Scott called such toposes models). Thus,
from the algebraic perspective TOP does possess standard models - these are
presheaf  toposes.

Thank you for your consideration,
Zinovy Diskin

Head of  Lab. for Database Design,       E-mail: diskin at fis.lv
Frame Inform Systems, Ltd.                            Diskin at aol.com
Latvian-America Joint Venture,             Phone (USA):  248 968-9511
Riga, Latvia