# [FOM] Frank Quinn article in January Notices

Tue Dec 27 20:01:22 EST 2011

```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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