[FOM] extramathematical notions and the CH
steven at iase.us
Thu Jan 31 15:41:12 EST 2013
I agree with Tim. But I note that there appears to be a more basic epistemological question relating to the choice of metric and Joe's belief concerning the nature of relations.
Given any "experimentally measurable dimensionless physical quantity" a physicist is entitled to call this measure the unit length and compute other lengths with respect to it, thus all relative measures are subject to this choice of metric and accordingly it is these relative measures only that may or may not be a noncomputable real number.
On Jan 30, 2013, at 7:58 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
> Joe Shipman wrote:
>> It's possible we could use physics to gain access to noncomputable objects, if an experimentally measurable dimensionless physical quantity is a noncomputable real number; in that case ZFC would only decide finitely many bits of the number and if we could measure more than that we would have new mathematical knowledge coming from physics.
> As I've argued before, this makes no sense to me. In your scenario, we have a physical theory that predicts that such-and-such a physical quantity equals such-and-such a noncomputable real number. The theory predicts (let's say) that such-and-such a measurement will be 1 if ZFC is consistent and it will be 0 if ZFC is inconsistent. We make the measurement and it comes out to be 1.
> According to you, we now have "new mathematical knowledge"; we now "know" that ZFC is consistent. But do we really? How do you rule out the following possibility: ZFC is really inconsistent and it's the physical theory that is wrong? Physical theories by their very nature are supposed to be tentative hypotheses that can be disproved at any time by observation. They can at most give us some *evidence* that ZFC is consistent---e.g., our current physical theories about how computers work give us some such evidence whenever we program a computer to search for a contradiction in ZFC and let it run for a while.
> If you allow us to acquire "mathematical certainty" of physical theories, then I don't see why CH is immune. For example, I've just come up with a brilliant new physical theory, which predicts that if CH is true, then under the surface of the Moon is a layer of green cheese, and if CH is false, then there is no such layer. All we have to do to settle CH is to go to the Moon again and start digging. My brilliant physical theory will give us new mathematical knowledge of CH by physical means.
> The "physics boundary" is not between absolute and non-absolute, but between finite and infinite.
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