[FOM] Frank Quinn article in January Notices

[Michael Blackmon]

Dear David, 

(Beforehand, let me apologize for jumping in from the shadows like so
many internet boogie men.)

It seems to me the issue Monroe has taken with your example is precisely
the one outlined in the article originally referenced; Namely that the
business of mathematics is certainty and 'truth.' 

While it might be the case that we study systems in which the principle
of excluded middle fails--as is the case with statistical hypothesis
testing after interpretation--we do so from a removed perspective.
Removed in the sense that we are concerned with the validity of
particular atomic statements (e.g. "1 - P(X < \theta) < 0.95") for which
the absolute (with respect to the system we are working) truth or
falsity can in principle be determined (as pointed out.) 

A running theme in Quinns' article is that: beyond the Facts a
Mathematician can establish about a system, there is little else he is
concerned with. In this sense, the example of statistical testing is not
adequate to refute this claim. This is because the interpretation of the
results of the test are not within the domain of mathematics proper.
This is precisely the point Quinn makes when he initially talks about
Hilberts axiomatization of geometry. The careful distinction being
ignored is the one made between semantics and syntax. 

>From a syntactic perspective the failure of excluded middle is easy to
spot and rather trivial: f(f(x)) != x (for some f a function of one
variable, and variable x). However, in order for any such example to
have weight in this context you must give the symbols f and x meaning,
which (strictly speaking) steps outside of the system being studied.

It is in this way the presence of the statement \exists x (f(f(x)) !=
x), and the failure of excluded middle are distinct entities which
cannot be identified within the confines of a fixed system.

This is also why your topos example fails here.

On Wed, 2011-12-28 at 11:31 +1030, David Roberts wrote:
> Dear Monroe,
> 
> On Dec 28, 2011 8:55 AM, "Monroe Eskew" <meskew at math.uci.edu> wrote:
> >
> > Dear David,
> >
> > Of course probability and statistics can be developed from a
> classical foundation. 
> 
> Of course it can and I don't dispute that, but I maintain that the
> identity not not A = A does not hold in the context of hypothesis
> testing. 
> 
> > I don't think your example is one of the excluded middle failing,
> just one where 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.  
> 
> I would say that statistics doesn't concern itself with statements of
> belief as modal logic does. And outside of the theorems of statistics,
> which can be seen as statements of analysis, measure theory and so on,
> statistics in practice doesn't have a concept of proof that a pure
> mathematician would recognise. Using the word 'reject' was perhaps a
> poor choice, it is jargon. Essentially it means 'not A is true' where
> A is a statement that is essentially of the form X=0.
> 
> Perhaps I took Quinn's article as meaning something different - I read
> it as saying that the internal logic, as it were, of mathematical
> sciences such as physics and statistics is not necessarily classical,
> not that the mathematics they use is intuitionistic. If I may venture
> an analogy, even in a purely classical foundation, the internal logic
> of a topos is not classical.
> 
> Best,
> 
> David Roberts
> 
> > Best,
> > Monroe
> >
> >
> >
> > On Dec 26, 2011, at 6:30 PM, David Roberts
> <david.roberts at adelaide.edu.au> wrote:
> >
> >> Dear Monroe,
> >>
> >> I thought I should point out that whenever statistical reasoning is
> involved in exact sciences (and some inexact ones), one inherently
> cannot assume excluded middle. Hypothesis testing - in its simplest
> form asking whether a measurement yields a null result - is full of
> phrases like 'fail to reject the null hypothesis at x level of
> uncertainty', which is definitely *not* the same as accepting the null
> hypothesis. This is one area where beginning students of statistics
> trip up all the time, mostly because the are expecting, implicitly, EM
> to hold.
> >>
> >> Witness two recent examples, namely faster-than-light neutrinos and
> the not-quite-discovery of the Higgs particle. Everything is stated in
> statistical terms, levels of significance and so on. And when they
> analyze things like the so-called look elsewhere effect in the latter,
> one can see vestiges of intuitionistic reasoning.
> >>
> >> David Roberts
> >>
> >>
> 
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