[FOM] Frank Quinn article in January Notices
Vaughan Pratt
pratt at cs.stanford.edu
Thu Dec 29 04:24:27 EST 2011
Walt, I accept your premise that maths and physics should not be
conflated, but not your rationale.
Please correct me if I've misunderstood, but your basis for the
difference seems to be in terms of accuracy: maths is accurate, physics
less so.
Suppose for the sake of argument you put a black ball in a black urn and
a white ball in a white urn, and I do the opposite, putting the black
ball in the white urn and the white in the black. Surely this is
physics, but where is the inaccuracy?
The difference as I see it is that whereas physics is obliged to
describe the universe we live in (or minor variants thereof when
contemplating counterfactuals), mathematics recognizes no such
limitations and feels free to construct any and all imaginable
universes, about which it then proceeds to reason. If mathematics wants
to talk about urns mod 3, where a black ball placed in the 4th urn
magically winds up in the 1st one instead, it is entirely free to do so.
If this sort of insanity were to happen in physics it would lead in due
course to a richly deserved Nobel Prize.
The idea that physics is inaccurate while mathematics is accurate is
from the late 19th century, when such notable physicists as Lord Kelvin
were floating the idea that the future of physics was the perpetual
refinement of precision of its fundamental constants.
How does this bear on excluded middle, the unequivocal truth of "P or
not-P"? An algebraist might analyze this question of its truth in terms
of Boolean vs. Heyting algebras. But before I got into logic and
algebra I was by training a physicist. From that perspective it seems
to me that the question boils down to the definition of "not."
Take the property of being a black swan. What is the logical negation
of that property? Well, that's easy enough, it's the property of being
either not black or not a swan. But what are the options for not-black
and not-swan? Red and white are presumably not black but what about
grey? And is there a class of not-swans? Does it include letters to Santa?
The idealization of "not" as an involution can be understood in the
appropriate mathematics worlds, namely any Boolean algebra. A key
difference with the physical world, it seems to me, is that "not" is not
the symmetric involution that Boolean algebra makes it out to be.
Vaughan Pratt
On 12/28/2011 3:21 PM, Walt Read wrote:
> On Tue, Dec 27, 2011 at 1:55 PM, Monroe Eskew<meskew at math.uci.edu> wrote:
>> confusion can result from muddying the distinction between statements P and statements about P such as "We reject P," "We
>> believe P," "P is provable," etc.
>
>> Monroe
>
> This is particularly useful a distinction because I think a lot of
> confusion is coming from conflating math and physics. Insofar as
> scientific statements P are necessarily falsifiable, it makes little
> sense to speak of them as true or false. We might speak of them as
> approximations, as we do with classical physics, or as compatible with
> observations, as we do with QM or GTR (but maybe not both together),
> but the closest we can come to "true" is "makes pretty accurate
> predictions as far as we can tell so far". Whatever the facts of any
> actual reality "out there", we only have contingent models and the
> choice of EM or not is the modeler's option, subject to that accurate
> prediction thing. Even the - unfortunate - attempt to use
> probabilistic language doesn't help here. If P is highly probable and
> almost certainly implies Q, what can we say about the likelihood of Q?
> Almost nothing. Reasoning in physics or probability is a different
> beast - at best we have belief, not knowledge - although talking about
> (modeling) that reasoning might be of a more conventional logical
> nature.
>
> The situation is different in math. While some will argue that
> mathematical statements are little different from physics statements,
> it's also possible and common to claim that mathematical objects have
> an objective existence and that we somehow have direct accurate
> knowledge of that reality. In that case it makes sense to say that a
> statement is true and not just "true enough".
>
> -Walt
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