[FOM] The irrelevance of Friedman's polemics and results

[Arnon Avron]

On Thu, Feb 02, 2006 at 01:39:54PM -0800, John Steel wrote:
> 
> Arnon Avron wrote:
> ---------------------------
> why not choosing a good point
> which is *not* arbitrary (like predicative mathematics) and devote
> most of our efforts  to it? or at least wait until "core mathematicians"
> come themselves across problems in which they need stronger methods,
> -------------------------
> 
> 
> As I understand it, predicative mathematics is not such a "good point", as 
> the existence of least upper bounds for arbitrary sets of reals is not 
> predicatively justified. Not many would want to revamp the way we teach 
> undergraduate Analysis in a way that paid attention to predicativity. 
> Certainly not our revered "core mathematicians".  So "core mathematicians" 
> (whoever those people are-- the editors of Annals of Math?), who have 
> learned the naive lub principle, are forever in danger of using it in a 
> necessary way.

First, the phrase "core mathematicians" is not mine: it is used
extensively by Friedman. For the purpose of this message, it can mean:
"mathematicians who do not care about foundations when they are doing
their mathematics".

  Now in "our efforts" I meant not the "core mathematicians",
but mathematicians who are doing research in FOM. I cant even 
convince mathematicians to use \lambda notatoin for denoting 
functions and to be care about their use of variables (something
they really SHOULD do, no matter what their philosophical tendencies 
are), so how can I expect to convince them to do mathematics
predicatively? But in fact I *dont* think that they should do it
predicatively. Also Hilbert did not intend in his
program to convince "core mathematicians" to use only finitary 
methods. On the contrary: his goal was to allow them to safely use
their infinitistic methods with full justification. This is
how I see the main task  of the research in FOM. It is *our*
task to find out what is the degree of certainty of various 
pieces of mathematics. It is true that I identify "predicative mathematics"
with "absolutely certain mathematics", but I think I have 
made it clear that in many (most?) applications one does not need
absolute certainty. Moreover: it is very legitimate to do mathematics
which is not absolutely certain, as long as it is acknowledged 
that it is  not absolutely certain. Moreover: again it is 
our (the FOMers') tasks to
determine the degree of certainty of various mathematical 
theorems and methods. Thus I see the work on reverse mathematics 
as very interesting and important, because I see it as making a
big progress in this direction. 

  One side remark: in *my* version of predicativism every bounded *set* of 
reals does have an lub, since in this version the union of an "acceptable" set
is itself "acceptable". What is constrained is the notion of an
"acceptable" set to which one can apply this principle.

> Should we fight to reform our curriculuum, so as to encourage
> only "safe" predicative mathematics?

As I wrote above, currently the answer is "No" - at least until
(and if) a formulation of predicative mathematics will be developed
which is (almost) as convenient  to use as the conventional 
approach. If this happens then indeed it will be a good idea to
encourage doing things predicatively whenever this is possible.

> Pure mathematicians get out ahead of applications. They find natural
> structure, and later the less pure use it. They of course look for
> applications, so as to see the structure they have found is important.
> Mostly, these applications are to other parts of pure math. Much of what
> Avron writes seems to be a criticism of this procedure in itself.

If this is what you have understood from my messages, then I did a poor
job. In my defence I note that I was explicitly arguing with Friedman 
about what is important to *FOM*. Nowhere did I say anything about
what "core mathematicians" should or should not do. 


Arnon Avron