[FOM] CH and mathematics
Benjamin
somrh at yahoo.com
Sat Jan 19 21:41:41 EST 2008
Well, this is my first post so I should probably
introduce myself. My name is Benjamin. I noticed that
the reply features puts both the forum email address
and the individuals email address and I was wondering
if there is any convention on preference for this. In
any event...
--- Bill Taylor <W.Taylor at math.canterbury.ac.nz>
wrote:
> Now, on a Platonist/realist view of math, one of A
> or f must exist;
> the consistency of of the existence of either with
> ZFC is known,
> yet the consistency of the existence of both with
> ZFC is impossible.
>
> a: IMHO, this must destroy Thomas' view that
> "consistency is existence",
> at least for the mathematical realist. Do
> others agree?
>
> b: Also, the answer to which one exists seems to be
> closely involved with
> Alex's question. Is this so?
>
> Bill Taylor.
There is a sense, I think, that perhaps CH is like the
parallel postulate in geometry. Since it or its
negation can be added to ZFC then we can get two
consistent systems. Gauss' response to this, IIRC, was
to suggest that which geometry to "true" is an
empirical one. And that answer seems to be "it
depends". In some cases one geometry is favorable to
use and in others it seems favorable to use another.
This gives a more of a "contextualist" answer. It's
true in some cases but not in others.
On the other hand, there is a feel to it that CH must
either be true or false. If we can't give a
contextualist interpretation of it then it seems we'll
need some method for determining this. Godel
considered appealing to a notion of "fruitfulness",
either within mathematics itself or within some,
perhaps, future physical theory utilizing infinite
cardinal numbers which either needs CH to be true or
needs CH to be denied. That "fruitfulness" would then
decide whether we accept CH or not.
"...a probable decision about its truth is possible
also in another way, namely, inductively by studying
its "success." Success here means fruitfulness in
consequences [...] There might exist axioms so
abundant in their verifiable consequences, shedding so
much light upon a whole field, and yielding such
powerful methods for solving problems (and even
solving them constructively, as far as possible) that,
no matter whether or not they are intrinsically
necessary, they would have to be accepted at least in
the same sense as any well-established physical
theory."
-Godel, from "What is Cantor's continuum problem?"
As far as the consistency requirement is concerned
(and here I would prefer "coherence" over
"consistency" since one can have a set of propositions
with no relationship to one another that are
consistent) I think what one is going to get is
something contextual and you'll have to completely get
rid of a correspondence theory of truth (or something
akin to it) and replace it with some coherence theory
of truth and the truth would depend upon the context.
For example the Pythagorean theorem is true in
Euclidean geometry but false in hyperbolic.
Consistency as a requirement for existence seems quite
odd, in my opinion, but I think it's fairly common in
the formal treatment (I think Hilbert endorsed it to
an extent, Poincare, if I am not mistaken, may have as
well.) I'm personally more inclined towards Godel's
suggestion or something akin to it. We treat existence
of entities as "postulates" that give us explanatory
power in a similar way that the empirical sciences
postulate entities and postulates hypotheses that, if
granted true, explain phenomenon. This method is more
or less abductive and gives us a means to evaluate
propositions without proving them (which we need to
avoid regress problems anyway). If it turns out such
entities become unnecessary then we dispose of them
(take for example the ether which special relativity
disposed of).
Regards,
Benjamin
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