[FOM] Falsify Platonism

[Brian Hart]

It seems some foundational views are easier to falsify than others.
Platonism may not be falsifiable.  Perhaps it may only be accepted as
a matter of belief, like religious faith -- inspiring the
mathematician or religious follower to continue their practice
zealously.  Even a great mind like Gödel's was lured by the promise of
philosophy.  I think he was interested in more profound truths than
simply trivial technical results in logic or mathematics which is why
he largely lost interest in logic after his most important results
were found.  Some even consider him to be a philosopher even though he
is considered to be one of the greatest logicians because he was so
engaged with philosophical issues throughout his life.  His
intellectual journey largely was in part guided by investigating the
philosophical ramifications of his incompleteness theorems.  Perhaps
even he thought he could somehow "prove" the truth of Platonism via a
meta-mathematical or philosophical argument and many may interepret
his later years as just an elaborate detour into the interminable
argumentation of philosophy but I don't think all was for nought even
though he struggled mightily to come to grips with his own ontological
and epistemological views which is one of the reasons why he published
so few philosophical works.  (Another reason might be his (likely
unrealistic in the philosophical arena) penchant for exact correctness
which could be much more easily obtained in a discipline like
mathematics where a proof is either right or wrong.)  Philosophy
teases the discursive intellect into thinking it can acquire
cumulative philosophical knowledge with a confusing array of smoke and
mirror tricks.  Just when one thinks one has a "proof" in one's hand
one looks there only to find one's hand empty.

As the analytical philosopher he informally was he thought he could
utilize the tools provided him by mathematics and logic to attempt to
precisely formalize and once and for all settle some mathematically
philosophical issues.  I think Gödelian scholars will agree that the
results were pretty much mixed, but his exploration of philosophy left
him (and us) with much food for thought including investigations
pertaining to the nature of the mind, the ontology of mathematical
objects, the existence and identifiability of absolutely unsolvable
problems, etc.  Do the Gödelian scholars on this list believe that
Gödel would've taken up philosophy with the same vigor and dedication
that he did had he had another chance at life or was this a
significant misttep of a great man which could've been more
well-suited to tackling the more technically-oriented problems of
logic and mathematics?

On Fri, Apr 23, 2010 at 11:23 PM, Richard Pollack <pollack at cims.nyu.edu> wrote:
> I'm a bit puzzled by this conversation. I've always thought about
> Platonism as a primarily emotional view of many Mathematicians
> expressed by the remark 'All Mathematicians claim to be formalists
> while in their heart they are all Platonists'. I've always understood
> this as an expression of the belief (held by many Mathematicians)
> that Mathematics has no content but is purely formal and consists
> only in deriving (proving) consequences of postulates. Nevertheless,
> many of the same Mathematicians (I count myself among these)  believe
> (feel) that the objects they study are real and have properties among
> which they believe (hope) their postulates are "true"  and thus they
> are finding other "true" (and hopefully new) properties of these
> objects. Given this, I don't see that a self contradiction in PA
> would have any affect on the Mathematicians beliefs (feelings), The
> Mathematician would give up his belief that PA is "true" (let alone
> "characterizes") of the whole numbers.  At best the Mathematician
> will give up his belief that PA is true for the whole numbers and may
> seek other postulates for them.
>
> Ricky Pollack
> Professor Emeritus Mathematics
> Courant Institute of Mathematical Sciences/NYU
> Office: Room 522, Warren Weaver Hall (Courant
>         Institute).
> Phone (office): 212-998-3167
> Phone (mobile): 646-734-0138
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>          New York, NY 10012, U.S.A.
> Email: pollack at cims,<http://nyu.edu>nyu.edu
> homepage url:
>    <http://math.nyu.edu/faculty/pollack/index.html>http://math.nyu.edu/faculty/pollack/index.html
>
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