FOM: Platonism

[Olivier Souan]

Reuben Hersh wrote :

'One more thing.  I don't find Platonism repulsive, disgusting,
upsetting,
'sickening or anything like that.  I find it inconsistent with an
'empiricist or (dare I say it?  materialist!) or skeptical or
'critical philosophical stance.  It is theology.  That's fine for
'those who like theology.  But I don't think people should pretend
'it's not inconsistent with empiricism or materialism or skepticism
'or any critical stance.  If I may quote myself, I think Platonism
'is in part a leftover from the long tradition of philosophies tying
'math and religion together.  Math may indeed  be a thought in the
'mind of God (although I don't think so.).  But if you don't base your 
'philosophy of math on religion, Platonism is a relic.  It's
'like the smile on the face of Lewis Carroll's Cheshire cat.
'(The holy cat vanished, yet the smile remains.)

Here we are!! I am sure that 95% of the issues of FOM and philosophy of
mathematics (and perhaps the very methods of the common mathematician)
are led by extramathematical content, like ontology or philosophy or
whatever. The point is that those remain hidden and private. Gödel wrote
one day in a letter that his mathematical investigations were led by his
philosophical convictions. And he never explained much himself about the
Gödel theorem (that would have avoided much misinterpretations though),
and one has to read Hao Wang's books to find out some precious quotes
where he delivers a laconic explanation (From mathematics from
philosophy, p.8). I'd like you all to be more explicit about your
philosophy and ontology
    But has platonism to be linked to religion? To a philosophy of some
kind certainly, but not necessarily to a religion and a theology. What
is theology? It is a coherent system of propositions about God and its
relationship with man based on the Scriptures. In that case, it is not a
philosophy, were everything has to be proved and argumented as far as
possible, independently of any Revelation. Platonism is a philosophy.
Platonism is not the residue, the "relic" of Christian philosophy, since
platonism was prior to Christianism. Medieval thinkers were in some
parts inspired by Plato, but much more by Aristotle. William of Occam
invented nominalism, which is the exact opposite of platonism. 
    Of prior concern in mathematics is the existence of transfinite or
infinite entities. Has any medieval thinker claimed the existence of
such entities? I do not know any. Listen to the prominent Christian
thinker Thomas of Aquino, the foremost theologian of Catholic Chuch :
"Deus non potest facere aliquid infinitum simpliciter" : God cannot
create infinite beings (Summa Theologia I, qu. 7 art. 1. 71), "Unde non
est possibile esse aliquam multidudinem actu infinitam" (ST I, qu.7 art
4. 96) : God cannot create actually infinite multiplicities. What could
be farther from Platonism? That's why I think M. Feferman is ambiguous
when he writes that platonism is "a medieval philosophy of
mathematics"("Infinity in mathematics : is Cantor necessary?" in
Infinity in science, p .207). It is the opposite of medieval philosophy.
    Can your personal dislike of some religion of the past - which I
respect - be relevant in the field of the philosophy of mathematics? Can
you be led to reject strong ontologies, the Continuum Hypothesis,
transfinite cardinals and the like only because you think they have some
very remote connection to religion?
    Platonism is the exact opposite of dogmatism. Dogmatism asserts
truthes without justifying them and refuses everything that is not its
very discourse, while platonism not only tries to check everything as
far as possible but also warns us that mathematical truths can be
surprising and unexpected. Think of Cantor writing to Dedekind : "I see
it but I can't believe it". Mathematical truths bear the same feature
than common reality : they can be unexpected and resist to our will not
to believe them.
    Best Regards


Olivier
Souan                                                                                            
Student of Philosophy, and
Logic                                                       
Université Paris IV et Paris I
Sorbonne                                            
Tél: [011 33 from the US/ 010 33 from the UK] +                     
           [0*]1.46.33.92.34   (*0 to be added if calling from France)
Areas of interest : Phenomenology, Greek philosophy,
Logic and Set theory, Mathematics and its foundations