[FOM] CH and mathematics

Staffan Angere Staffan.Angere at fil.lu.se
Sun Jan 20 11:44:32 EST 2008

I agree with some earlier posts that the meaningful answers are "yes", "no" and "neither". However, the interesting part is of course how these are to be elaborated.

As a non-platonist with structuralist-constructivist leanings, I would still want to pick the "yes" answer to this one. We must differentiate between "Is CH a definite mathematical question?" and "Is CH a definite mathematical problem?". For the first, I would be tempted to answer no, and for the second, as mentioned, yes.

The point seems to me to be that something can be a problem even if it does not admit of a unique correct answer. In this light, we can see CH as an invitation to construct (or describe) interesting abstract structures where it holds or not hold. Gödel's constructible universe is certainly one of these (a kind of "solution", if you will), and probably also Cohen's falsifying structures.

In this way, one could compare CH with, for instance, "the problem of what 0/0 is". Both, rather than being questions for a unique answer, can be seen as invitations to construct structures with certain properties, and both may be called "definite", since it is quite clear what these properties should be. Finally, from a constructivist viewpoint, both fall squarely within mathematics, seen as the art of constructing and describing abstract structures.

Staffan Angere

> In his recent entry in Philosophy of Mathematics:5 Questions, 
> Solomon 
> Feferman asks: "Is the Continuum Hypothesis a definite mathematical 
> problem?"
> Can someone elaborate on what the choices are or could have been?
>Alex Blum

More information about the FOM mailing list