[FOM] Falsify Platonism

W.Taylor@math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Sun Apr 25 06:48:45 EDT 2010

Quoting Richard Pollack <pollack at cims.nyu.edu>:

> I've always thought about
> Platonism as a primarily emotional view of many Mathematicians

This is perhaps a little simplistic!  I think there is *some* technical
concern, beyond mere demarcation of "comfort zones".

> expressed by the remark 'All Mathematicians claim to be formalists
> while in their heart they are all Platonists'.

A cute expression, which I have always liked for its allusions.
But yes, in their hearts they are realists, (almost all of them),
and merely adopt the Formalist camouflage as a means of warding off
pestiferous philosophers when they want to be doing real work.

> 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.

That is, indeed, Formalism, and is, IMHO, self-defeating.
"Deriving consequences of axioms" is ITSELF an objective, yes/no,
true/false, right/wrong sort of thing, which is surely the hallmark of
objectivity, that is, realism.   They have just moved their regard
for numbers over to that for finite strings, which is, effectively,
the same thing.  I have never seen any Formalists tackle this objection,
except perhaps some ultra-finitists who admit reality only to the first
many (they never say HOW many!) natural numbers.  Henle was a clear-cut
exponent of this idea in his Math Intelligencer Article of many years ago.

If we ignore these latter, there is really no difference between a Formalist
and a numerical realist like myself (and many others).

> Nevertheless, many of the same Mathematicians ... ...
> (feel) that the objects they study are real

It's largely irrelevant whether the objects are "real", which is only
a philosophical stance; the vital thing is that their properties are  

> and have properties among which they (hope) their postulates are "true"

Similar remarks apply.  The derived properties must be OBJECTIVE - checkable.

> and thus they are finding other "true" (and hopefully new) properties

This is the key, of course, as you are working around to.
Having "defined" (not technically OC) what we are talking about by axioms,
we get consequences *which we never expected*.
The consequences *force* themselves on us, whether we want them or not!
That is objectiveness.  That is realism - "Platonism".  Almost no subject
other than Math has this same definitive objectivity.  Physics, engineering
and computer science come closest, but do not quite attain the same level.
The humanities, of course, including philosophy, remain unutterably
subjective.  This is not a criticism (!) - subjectivity has its worth too,
but it is a completely different endeavor.

There is one technical matter to be emphasized.  I have been glibly declaring
that being part of a formal system in FOL(=) is tantamount to "existing".
This is a somewhat philosophical stance, but is borne out by noting the
completeness theorem of Godel, whereby consistency is found to be equivalent
to "objectively existing" - i.e. being a genuine objective abstract object.
This in turn is heavily dependent on the whole idea of model theory for FOL,
which is largely due to Tarski.

Godel is always given heaps of credit, but IMHO Tarski desrves even more for
making precise exactly what it means to be an abstract object in this way.
I think we may well call Tarski the father of modern mathematical Platonism!

>  Given this, I don't see that a self contradiction in PA
> would have any affect on the Mathematicians beliefs (feelings),

You are completely wrong in this, in almost every respect.

> 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.

See my parallel post for why this is altogether mistaken.

-- Booming Bill

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