[FOM] 461: Reflections on Vienna Meeting

[Harvey Friedman]

I remember distinctly during the question period asking specifically  
whether that was a "legitimate mathematical problem". I think I asked  
this without the hand held microphone, so that Angus heard the  
question, and responded with "yes" I believe, but my question would  
not have been picked up by the audio.

In any case, Angus can speak for himself.

As you know, I have written several times to Voevodsky and to  
MacIntyre, including open letters on the FOM to both of them.

Voevodsky responded, and we continued a lively correspondence, all of  
which I posted with his permission.

Angus never responded, despite the repeated attempts.

So let's let Angus speak for himself.

Until I hear otherwise from Angus, I am going to assume that my  
distinct recollection is correct.

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). I do recall that  
*upon hearing this*, I asked, without the microphone, whether Con(PA)  
is a legitimate mathematical problem, and getting the answer "yes".

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.

Angus, can you explain your position?

Harvey Friedman

On Jun 14, 2011, at 9:48 AM, Alex Simpson wrote:

Thanks to Anon Barfod for posting the link to Angus Macintyre's
April talk in Vienna:

  http://videolectures.net/godelfellowship2011_macintyre_fcm/

In his posting of 11th May, Harvey Friedman wrote of this talk:

> I asked very specifically whether
>
> The consistency of Peano Arithmetic
>
> is a legitimate mathematical problem in present day mathematical   
> culture.
>
> I asked this because the Fields Medalist Voevodsky has explicitly   
> considered this to be an open problem.
>
> To my shock, Angus declared, unequivocally, yes.

I was surprised when I first read this last month, because Harvey's
account did not correspond to my own memory of the talk, which I
attended in person. Memory being fallible, however, I was not
entirely confident of my position. But now that the talk is publicly
available, I wonder if Harvey could point us to the place at which
Angus makes his supposed unequivocal claim that the consistency of PA
is a "legitimate mathematical problem".

As far as I can see, the relevant exchange takes place around 32:40 and
goes:

  HF: [Voevodsky] is extremely confused about whether PA might be
      inconsistent

  AM: How is he confused? It *might* be inconsistent. I mean, it would
      be a shock. That would be a genuine crisis. I think the  
inconsistency
      of ZFC would not be. ... None of us know that PA is consistent.

I don't read Angus here as saying that he believes the consistency of PA
to be a "legitimate mathematical problem". For example, what Angus says
is wholly compatible with the view, expressed by William Tait in his
posting of 22nd May, that "the search for nontrivial consistency proofs
is off the board" (since, roughly, any consistency proof is at least as
doubtful as the system whose consistency it establishes). Also, given  
the
tenor of the rest of his talk, I imagine that Angus would very
much not view consistency investigations as the concern of legitimate
mathematics.

Perhaps I should add that my own view about the consistency of PA
is that it is as much a theorem of mathematics as other theorems of
mainstream mathematics (some of which are demonstrably interreducible
with it). I believe this is similar to Harvey's view.

Nevertheless, I think that the record should be set straight about
what Angus actually said in his talk. Or perhaps Harvey can clarify
where Angus made the claim he attributes to him.

Alex