[FOM] Frank Quinn article in January Notices

[Walt Read]

Vaughn:

I think I added to the confusion by conflating physics and statistics.
The point wasn't that physics is inaccurate, at least not in any sense
statistics speaks to, as in "refinement of precision of its
fundamental constants". It's rather that physics has very different
assumptions and goals. The physicist assumes a reality "out there" and
tries to understand that reality by building models, the criterion for
understanding being prediction. In the terms you suggest, he knows
it's *this* world but otherwise not what world it is. It's seriously
underspecified. The mathematician, by choosing a world or by faith in
intuitions about this world or maybe just by faith in a reliable
logical ability we all share (ignore the grumblings of that
evolutionary biologist), has certain knowledge about fundamental
patterns, so much so that we feel confident making infinite
statements. For example, there really are infinitely many primes. In
regard to excluded middle, we can discuss whether that was always true
or only became true when proved. But we generally agree that it's true
and will remain true forever. By contrast, the physicist has to be a
strict finitist. Rather than hypothetical balls and urns, about which
we can assume complete knowledge in the analysis, they have things
like electrons or neutrinos. And the point isn't so much that
electrons might not behave that way - that's a modeling issue - but
that even if the analysis does go through today they might learn
something about electrons tomorrow that completely changes the
conditions. So we get to fuss about next-level issues, like
consistency and completeness and excluded middle, while they have to
worry about just getting it right in a sometimes rapidly changing
environment.

-Walt

On Thu, Dec 29, 2011 at 1:24 AM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> 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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