[FOM] Followup comments from Frank Quinn

[Timothy Y. Chow]

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