[FOM] Thread and Two Open Letters

[Harvey Friedman]

1. THIS THREAD.
2. OPEN LETTERS TO VOEVODSKY, MACINTYRE.

1. THIS THREAD.

My http://www.cs.nyu.edu/pipermail/fom/2011-May/015391.html seems to  
have generated quite a substantial thread!

rom: <Andre.Rodin at ens.fr>
Date: May 16, 2011 6:24:25 PM EDT
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: [FOM] 461: Reflections on Vienna Meeting
Reply-To: Foundations of Mathematics <fom at cs.nyu.edu>

In my sense this is the most clear and sober and also original story  
about the
issue that I ever heard; it alone worths the whole series of meetings  
on the
same topic that I helped to organaze last Spring in Paris.
Andrei

TO THE extent that Andre was referring to my 461 from the subject  
header, I am very pleased. But perhaps Andre was referring to  
Professor V's DVD instead? What precisely is "the issue"?

As far as Professor V's DVD is concerned, I listened to some of it  
alongside a very knowledgeable applied model theorist (not from OSU),  
with a lot of knowledge of core mathematics. My colleague understood  
far more of this than I, but we were rather perplexed by it.

I am scheduled to be at a meeting in July where various kinds of  
logicians and philosophers and others will be present, and Professor V  
will have plenty of opportunity to explain what he is trying to do.

I made two points concerning the issue or "issue" of the consistency  
of Peano Arithmetic = PA. I'm not sure that people understood them, so  
I now elaborate.

1. There is a standard mathematical proof that PA is consistent. This  
proof does not seem easily distinguishable from many other proofs in  
abstract mathematics that are accepted without controversy.

In this vein, Chow's posting http://www.cs.nyu.edu/pipermail/fom/2011-May/015401.html 
  is very relevant, which purports to explain this apparent paradox.  
Of course, there is no substitute for Professor V (or, for that  
matter, Angus) from explaining what he has in mind.

2a. In some sense, there are plenty of totally accepted theorems of  
mathematics, T, which are at the level of ACA_0 and higher in the  
usual framework of RM = reverse mathematics. Thus, it is clear that  
any inconsistency in PA would give us a refutation of these theorems,  
T. So a consequence of the inconsistency of PA would force us to  
discard a whole host of actual totally accepted theorems of  
mathematics. That, I regard, is a significance consequence. It is not  
fatal, because mathematics could be expected to withstand a  
"cleansing" of all such theorems in favor of far more finitary theorems.

2b. However, item 2 has a subtle weaknesss. One only gets a refutation  
of T + RCA_0, and not a refutation of T. The point is that the base  
theory RCA_0 is needed.

2c. So this suggests that we need to strengthen RM to SRM = strict  
reverse mathematics, which operates with no base theory whatsoever.

2d. My initial work in SRM indicates that this can be carried out for  
a number of totally accepted theorems of mathematics, thereby showing,  
yet more strongly, that an inconsistency in PA would create  
refutations of a whole host of totally accepted theorems of mathematics.

The FOM email list has become the place of record for a discussion of  
issues in the foundations of mathematics.

There is no question that it is mostly dominated by what I call the  
traditional line of f.o.m. - but by no means exclusively.

I think it would be of great additional interest to have a greater  
representation of nontraditional lines of f.o.m. Even people who think  
that f.o.m. is a meaningless phrase - or a totally misguided one.

Accordingly, I am going to have a consistent policy of extending OPEN  
INVITATIONS to those with non traditional f.o.m. views.

I want to do this for three reasons:

i. Get a clearer idea of what views are out there, some of which may  
be responsible for the low esteem in which traditional f.o.m. is held  
in some quarters.
ii. Get these competing ideas into a moderated forum whereby these  
ideas can be tested in an open public forum populated with a variety  
of experts.
iii. Get experts such as myself involved in fine tuning our arguments.  
This inevitably results in major new conjectures, theorems, and  
research programs.

2. OPEN LETTERS TO VOEVODSKY, MACINTYRE.

Dear Professor Voevodsky,

I have become aware of your online videos at http://video.ias.edu/voevodsky-80th 
  and http://video.ias.edu/univalent/voevodsky. I was particularly  
struck by your discussion of the "possible inconsistency of Peano  
Arithmetic". This has created a lot of attention on the FOM email  
list. As a subscriber to that list, I would very much like you to send  
us an account of how you view the consistency of Peano Arithmetic. In  
particular, how you view the usual mathematical proof that Peano  
Arithmetic is consistent, and to what extent and in what sense is "the  
consistency of Peano Arithmetic" a genuine open problem in  
mathematics. It would also be of interest to hear your conception of  
what foundations of mathematics is, or should be, or could be, as it  
appears to be very different from traditional conceptions of the  
foundations of mathematics.

Respectfully yours,

Harvey M. Friedman


Dear Angus MacIntyre,

Greetings!

I have put up a detailed critique surrounding our presentations at the  
recent meeting in Vienna. See http://www.cs.nyu.edu/pipermail/fom/2011-May/015391.html 
  . This has attracted a lot of attention on the FOM email list. As a  
subscriber to that list, I would very much like you to send us your  
views on the relevant issues. In particular, your view as to the  
extend to which the consistency of Peano Arithmetic is a genuine open  
problem in mathematics - as you indicated in your talk. You may also  
recall that I challenged you to return to Vienna with me for a formal  
debate with strict rules of engagement. As you know, your views are at  
considerable odds in many respects with the traditional f.o.m. point  
of view. I am asking you to engage in a substantive debate on the FOM  
email list, which is carefully moderated by a dedicated moderator and  
FOM editorial board - in order that the relevant ideas being fully  
tested in depth, in an open and fair intellectual form.

Respectfully yours,

Harvey M. Friedman