[FOM] 461: Reflections on Vienna Meeting

Alex Simpson Alex.Simpson at ed.ac.uk
Thu Jun 16 16:02:51 EDT 2011

I did not intend to restart the discussion about the consistency of
PA, when I posted my comment about Angus' talk. But I do think that
Harvey's argument that Angus' overall position is untenable needs
a response.

> Incidentally, even "none of us know whether PA is consistent" is, I  
> believe, indefensible in the context of Angus' talk and known views  
> (i.e., adherence to standard mathematical practice). ...
> Modulo some very interesting issues in f.o.m., particularly reverse
>  math and strict reverse math, a consequence of this quote is
> "none of us know whether an infinite sequence of rationals from  
> [0,1] has an infinite 1/n style Cauchy subsequence"
> a claim that I am sure Angus would not subscribe to.

There is a perfectly rational explanation of this position.

Many people, including Angus in his talk, admire the mathematical
developments of set theory, although they concede that there
is the logical possibility that set theory might be inconsistent, in
which case perhaps little of the work would survive. I think
we would all recognise that such a position is not abnormal.

Similarly, one can accept more-down-to-earth mathematics,
such as the property Harvey states about infinite sequences, but still
with the caveat that there is an extremely remote possibility that
some low-level logical system will turn out to be inconsistent,
necessitating the rejection of some, and reorganisation of
other such core mathematics.

Ultimately, all mathematical "knowledge" is dependent upon some
degree of "belief". (And reverse mathematics plays an important role
in quantifying, modulo acceptance of the general framework,
the level of belief required.)

Thus it is perfectly consistent to have the view that: one
accepts and admires the great achievements of mainstream
mathematics; but one nevertheless allows that these
achievements might potentially need to be revised or even jettisoned
if it eventually turns out that inconsistent methods have been
used in their proof. Angus' remarks about the possible inconsistency
of PA included a sufficient degree of "shock" and "crisis" to show that
he holds inconsistency at this level to be extremely unlikely
and that he is aware that its mathematical ramifications would
be grave.

The above seems to me a wholly consistent interpretation of Angus' talk,
though of course I cannot say for certain that it correctly reflects his
own position. As Harvey says:

> In any case, Angus can speak for himself.

I entirely agree. And in the absence of any response from him, we
should take the video of his talk, in which he does speak for himself,
as the definitive source.


p.s. For the record, I reiterate that I am personally certain that
PA is consistent.

Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK
Email: Alex.Simpson at ed.ac.uk             Tel: +44 (0)131 650 5113
Web: http://homepages.inf.ed.ac.uk/als   Fax: +44 (0)131 651 1426

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