FOM: PenMaddy's set-reductionism,Re: Friedman,fom,& Sheep's.Shop.

Robert Tragesser RTragesser at
Sat Feb 21 09:22:11 EST 1998

Pen Maddy wrote:
[[[One can think that all mathematical objects are ultimately (modelled as)
sets without thinking that all effective mathematical methods are set
theoretic methods.  Compare:  one can think that everything studied in the
sciences is ultimately physical without thinking that the only effective
scientific methods are those of physics.  Who would expect botanists or
psychologists to use the same methods as physicists?]]]

        Fair enough,  I guess,  given the weakness of my formulation
of my worry.
        But here is a stronger formulation:

[1] I'd like to see an exact formulation,  and solid justification of,
the claim that "all mathematical objects are ultimately (modelled on)
sets".  Undoubtedly some "one's" can think that,  and far better "one's"
than I have.  But how well how have they thought it?   With what rigor?
With what disciplined sense of final cause [see below]?  Are
the reasons empirical [i.e."we can in fact...for all known..."]?  Or 
something much stronger [e.g., something  a priori -- and definitely not
understanding "a priori" in the Kantian sense,  as for example 
Bill Tait would hope it was not understood in the Kantian sense]?
        [Is the request for "exactness" unreasonable of me? I see the work
of Ken Manders (rightly?) as pointing in a direction in which something 
like mathematical exactitude might be forthcoming. _Surely_ in any case
a scientific attitude -- I dislike this expression but it has been
used on FOM in a menacing way so I might as well draw on its
powers here -- requires a systematic _set-independent_
of those quite extensive parts of mathematics which practically owe little
or nothing to set theory and then determine in what momentous senses
they are modelled in SET -- uncovering necessary and sufficient conditions
for a piece of mathematics to be _significantly_ so modelled.] 

[2] For the sake of articulation of deeper worries,  I here { = in [2]}
grant that all mathematical objects can be ultimately modelled as sets.
        [2a] This brings me back to the question [in the original posting]
of the ends or aims of set theory based f.o.m.?  [The wrangling between 
the SETS and the TOPS have left me unclear about what the aims and results
are supposed to be, and so what is substantially gained by all mathematical
objects being ultimately modelled as sets.]

        [2b] I here reiterate the observation that
                (i) "one" does not of course directly model mathematical 
objects in SET.  Rather,  one models them through a conception.  But for
broad reaches of mathematics,  the resonant conceptions have emerged from
(typically) deep theorems proven without up-front reliance on nontrivial
set theory.
                (ii) In that case,  one wonders what the mathematical or
conceptual gain is of the modelling in SET?  One wonders if what is
__fundamental__ in mathematics is not SET,  but what gets us the goods in
first and second place (those deep,  conception engendering SET-free

[3] Pen Maddy's (by now old and long blown away hat?)
attitude is striking.  Maybe it is worth mentioning that within physics
itself there are serious and controversial problems about what is
"fundamental".  An
especially seminal piece (whose importance Sam Schweber pointed out to me)
the solid state physicist's (or as he instructively says, "the N-body
P.W.Anderson's essay "More is Different" in _Science_ 4 Aug 72, Vol.177, 
no.4047, pp.393-96),  arguing against elementary particle physics being 
fundamental physics.
        Anderson ends with that old story of the conversation between the
novelists Fitzgerald and Hemingway:

Fitzgerald: The rich are different from us.
Hemingway: Yes,  they have more money.

        I think that Hemingway is wrong in the same way that
(if I may be allowed the poet's _chiasma_) Pen Maddy is,  only less so.

POSTSCRIPT ON PLATONISM: By a terrible,  if not plain silly,  irony,
set-theoretic reductionism is rather widely identified with "Platonism".
It should rather be identified with Plato's murdered and most serious
rival,  the Democritian!  To be a genuine Platonist is to fully and
powerfully honor those conceptual differences the set-theoretical
reductionist [all too glibly [by my lights] effaces]. 

Robert Tragesser
Professor in the History and Philosophy of Science
Conecticut College 

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