[FOM] 600: Removing Deep Pathology 1

[Hendrik Boom]

I suspect there's a close relationship between pathology and use of 
nonconstructive methods.

If you're not willing to go that far, consider for a start how much 
pathology depends on the axiom of choice.

On Sat, Aug 15, 2015 at 10:37:45PM -0400, Harvey Friedman wrote:
> There is an aspect of mathematics that I call
> 
> *deeply pathological*
...
...
> 
> In order to set the stage for further discussion, I offer one well
> known example of what I am talking about. Consider the equation
> 
> f(x+y) = f(x) + f(y)
> 
> where f;R into R. There is a great dichotomy: the solutions f(x) = cx
> are absolutely wonderful, whereas all others are deeply pathological.
> We will later take up just what available tools we have for
> systematically treating this kind of situation.

And theabsolutely wonderful ones are the constructive ones, right?

> 
> TENTATIVE THESIS. The use of the surgical tools used in the PROJECT
> leaves all of the mathematics in tact that is of sustained interest to
> the mathematics community. Revised developments in which the deep
> pathology has been cleansed, will be generally regarded as superior
> developments. The cleansing process INCREASES the totality of deep
> mathematical proofs Ideas from the old mathematics will resurface in
> the new mathematics.

There are constructive versions of much of classical mathematics.  
They tend to go into more  details than the classical mathematics they 
replace, for example to suggest methods that would be meaningful to, 
say, the numerical analyst (though possibly not at all optimum on 
digital computers).

> 
> CHALLENGE. It is commonplace in papers on functional analysis and
> operator theory to "motivate" the developments through their "use" in
> mathematical physics, engineering, and elsewhere. Establish that such
> usefulness reflects only the desirable part, and not any of the deep
> pathology. Or, alternatively, demonstrate that the deep pathology has
> a genuine "use".

I read a thesis on the use of constructivism in quantum mechanics a few 
decades ago.  It constructed wave functions whose solutions could not 
be computed reliably because of chaotic behaviour.  What it didn't 
seem to notice is that all these cases were concealed from 
observation by the uncertainty principle.

> In the next posting, I will present my first surgical tool, and its
> use on a dual space, and hopefully more.

That will be interesting.

-- hendrik