[FOM] CH and mathematics

Bill Taylor W.Taylor at math.canterbury.ac.nz
Tue Jan 22 00:46:53 EST 2008

A compendium of replies:

> This was the Feferman question that was itself questioned:
>      (i) Is the Continuum Hypothesis a definite mathematical problem?
> What seemed puzzling to me was why Feferman's own succinct explanation
> was not taken into consideration as part of the ensuing discussion.

In fact, this paragraph of Feferman reflects my own thinking
virtually to the letter!   Three cheers for Feferman!

"Concerning (i), I came to the conclusion some years ago that CH is
an inherently vague problem (see, e.g., the article (2000) cited above).
This was based partly on the results from the metatheory of set theory
showing that CH is independent of all remotely plausible axioms extending
ZFC, including all large cardinal axioms that have been proposed so far.
In fact it is consistent with all such axioms (if consistent at all)
that the cardinal number of the continuum can be anything it ought to be
i.e. anything which is not excluded by Konig's theorem. The other basis
for my view is philosophical: I believe there is no independent platonic
reality that gives determinate meaning to the language of sets in general,
and to the supposed totality of arbitrary subsets of the natural numbers
in particular, and hence not to its cardinal number.

Spot on.  The man is clearly a deep and profound thinker; (after all,
he agrees with me...  ;-) )

I doubt his subsequent remarks about math millenium problems are very
relevant, though...  times change in other ways as well.

Feferman continues:

> I have been asked to explain what I mean by the statement of
> a problem being inherently vague. The idea is that, not only is it vague,
> but there is no reasonable way to sharpen the notion or notions which
> are essential to its formulation without violating what the notion is
> supposed to be about. For example, the notion of feasibly computable
> number is inherently vague in that sense.

This comment on feasible numbers is very apt!  It is also another
possible example to Tim's query about non-formalizable ideas.
Somewhat similar to my earlier comment about infinitesimal angles
within cusps.  Though I doubt CH will ever come to be seen in those lights.

CH is quite formalizable after all!  But similar remarks as above
could probably apply to other independent results, like Souslin's
conjecture and others similar.

Benjamin writes:

> 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.

This looks like a similar case as a historical remark, but philosophically
it is quite different.  After all, the other models of geometry are quite
standard and regular parts of math - they can be treated by similar
methods to Euclidean, and are the subject of various undergrad courses,
and so on.  To do anything similar to that with CH/~CH would require
setting up a great machinery of non-standard models etc, somewhat
similar to Robinson's approach to infinitesimals.  It needs a whole
different technical approach.   Logic, rather than math.
I see it as very different, anyway.

> Godel considered appealing to a notion of "fruitfulness",
> either within mathematics itself or within some, perhaps,
> future physical theory utilizing infinite cardinal numbers

This was a reasonable view for Godel, but history has overtaken it.
Physics has shown not the *slightest* sign of needing CH or anything
beyond continuum cardinality, and there is nothing on the horizon.

Regarding fruitfulness within math, I feel the proved conservativity
of CH for number theory is already a killer blow.

> 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

This is a very good point, and I like to see the proper use of "abductive" -
a word that gets too little action I feel!  :)   However, as I say,
history seems against it having any effect on CH.

Alex Blum writes:

> Is Feferman's question in effect an expression of doubt about
> the meaningfulness of CH? Would you say that the same would have been
> true of Fermat's last theorem would have been proved independent of
> the axioms of arithmetic?

These are in no way parallel.  FLT is arithmetical, so is subject
to the usual straightforward quantifier-extension in meaning as explicated
from Hilbert to Tarski.  No such remarks apply to set-theoretical
results, as Feferman was at pains to make clear in his quote, I feel.

And BTW, FLT is a bad example, being Pi-0-1.  If it was undecidable
it would automatically be true.  Twin-primes would have been better.

Bill Taylor.

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