[FOM] extramathematical notions and the CH
joeshipman at aol.com
joeshipman at aol.com
Fri Feb 1 13:40:44 EST 2013
You misunderstand the meaning of "dimensionless". I am talking about quantities like the fine structure constant, or particle rest mass ratios, or probabilities and ratios of probabilities, which are completely independent of length scales and coordinate systems.
-- JS
-----Original Message-----
From: Steven Ericsson-Zenith <steven at iase.us>
To: tchow <tchow at alum.mit.edu>; Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Fri, Feb 1, 2013 12:52 pm
Subject: Re: [FOM] extramathematical notions and the CH
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.
Steven
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.
>
> Tim
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