[FOM] Followup comments from Frank Quinn
Timothy Y. Chow
tchow at alum.mit.edu
Fri Dec 30 10:43:38 EST 2011
Frank Quinn responded promptly to my email, and with his permission, I am
forwarding his reply (see below). He said that he did not want to receive
a lot more email about this topic, so please respect that. (On a related
note, FOM readers will no doubt disagree with many of Quinn's statements
about truth, knowledge, meaning, etc. I would recommend resisting the
temptation to flood FOM with rebuttals. Try to keep the discussion
productive.)
In light of Quinn's clarifications, I definitely have to back off from my
conjectured interpretation of his use of the term "excluded middle
reasoning." Still, I personally feel that one of the main distinguishing
characteristics of modern mathematics is the idea that you cannot claim to
"know" something unless you can prove that the negation leads to a
contradiction, and that most of Quinn's article makes a lot of sense if we
read "excluded middle reasoning" this way, even if that was not his
intent.
One idea in Quinn's article that I found very intriguing was the
suggestion that the revolution in foundations a century or so ago had a
democratizing effect. It has certainly been my experience that in the
humanities, and even the sciences to a lesser extent, it is much harder
than it is in mathematics for an outsider to get a breakthrough idea
accepted. Not that the mathematical world doesn't also suffer from
elitism and the tyranny of fashion to some extent, but there does seem to
be a qualitative difference here. I wonder if any other FOM readers also
feel that the mathematical world was less democratic (in this sense) and
more like other fields, prior to the revolution in foundations?
Tim
---------- Forwarded message ----------
Date: Thu, 29 Dec 2011 17:15:52 -0500
Subject: Re: Article in January Notices
On Dec 29, 2011, at 10:48 AM, Timothy Y. Chow wrote:
> Professor Quinn,
>
> I very much enjoyed your article in the January Notices. I mentioned on
> the Foundations of Mathematics mailing list and it has generated some
> interesting discussion. ... See in particular Monroe Eskew's objections
> to your use of the term "excluded middle reasoning" and my attempt to
> defend your usage.
Hi,
Nice to hear from you. I can respond now but probably won't be able to do
much in the future.
It may help to know that I have no interest in developing a philosophy of
mathematics. I am developing a *description* of mathematics, as it is
*done*, and this is full of surprises and much richer. There are things in
this world not dreamt of in philosophy, so to speak.
To begin the response, it seems to me that the word "true" has a lot of
philosophical baggage, most of which is irrelevant and some of which is
counterproductive to mathematical practice. The phrase "mathematics needed
a mathematical meaning for `true' " suggests redefining the term, but the
futility of that is illustrated by the discussion on the list. In the
long version I abandon `true' for `completely reliable' in the hope that
it will evoke less static.
> By "excluded-middle reasoning," I think Quinn does *not* mean
>
> the principle that if we know that "not P" is false, then we know
> that "P" is true
>
> but rather
>
> the principle that we cannot know that "P" is true unless we can
> prove that "not P" cannot be true.
Alas, the first version *is* what I meant. Your version needs
clarification, but in general usage `true', `known', or whatever, requires
both of these: it means *exactly* that `not P' is false. See Contexts,
below.
> Similarly, instead of defining mathematical *truth* as being that which
> is "impossible to contradict," Quinn probably meant to say that
> mathematical *knowledge* is that which is impossible to contradict,
> i.e., that we cannot claim to "know" some mathematical assertion until
> we have proved that it cannot be otherwise.
In careful current usage `true' actually means `known to be true'. For
practical purposes `true but not known to be true' is worthless and
invites abuse. As in "See, I told you twenty years ago this was true, and
you were wrong to doubt me even though I did not have a proof." Another
reason the word `true' is problematic.
Contexts:
There is always a context for mathematics and in standard practice, unless
specified otherwise, it is ZFC set theory. It is well-known that there are
many mutually-exclusive consistent extensions of this context. Note that
a disproof of `not P' in ZFC remains a disproof in any extension, so
implies that P cannot be contradicted in any extension. Similarly, a
direct proof of P in ZFC remains a proof in any extension, and implies
that `not P' cannot be correct in any extension. Current usage is that
`true' means "known to be impossible to contradict in *any extension* of
ZFC". In practice this is done either by proving P or disproving `not P'.
(or proving `not (not P)', etc.).
Notice that "impossible to contradict *within* ZFC " would include
"provable in some consistent extension", even if the assertion could also
be disproved in some other extension. Standard practice requires making
this explicit: eg. "assuming the continuum hypothesis, it is true that
...". I think this is done to ensure upward compatibility. If you look
back a few decades you will see that ZF (without C) was the standard
context and use or denial of the axiom of choice was explicitly
acknowledged. If `not C' was assumed, we know the material has to be
reconsidered before it can be used in the current standard context.
Someday something else might become part of the standard context, though
there don't seem to be any candidates at the moment.
Physical world:
in the physical world "excluded middle" reasoning can be used, but wisdom
and experience are needed about how far it can be trusted. In mathematics
you must follow certain explicit rules of argumentation very carefully,
but in that case excluded-middle is _always_ trustworthy. Experience and
wisdom help develop lawful arguments, but you can squeeze the method
itself as hard as you like. And we do that, far beyond limits of wisdom.
We accept very dense, delicate 100-page arguments that end with "so the
assumption on page 1 that there is a counterexample is false, and the
theorem is proved". We don't enjoy them because checking that the
arguments are lawful is painful, but we accept them even if the conclusion
is outrageously counterintuitive. (These are actually the best kind!)
Statistics:
Things like "this is true with 90% confidence" are inconsistent with
excluded middle, but that is because the statement is sloppy and
mathematically incorrect. The mathematically precise version would be
something like: "if the data has the type of distribution you assume, and
if you make a huge number of random selections of N points from the
distribution, then as huge goes to infinity the proportion of the
selections with characteristics within the stated tolerance will be
greater than 0.9." In real-life circumstances this has no logical force at
all. People may interpret it as having some sort of moral force but,
obviously, they should not expect excluded-middle argumentation to survive
this move.
Best regards, Frank
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