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

[hendrik@topoi.pooq.com]

On Thu, Feb 02, 2006 at 10:55:44AM +0200, Arnon Avron wrote:

> In principle,
> a scientific theory should be taken as a whole, and there is no 
> reason to demand one component of it (the mathematical one) to be
> more reliable and certain than other parts. So it is quite reasonable
> to employ in science mathematical methods that are not certain in
> the pure, ideal mathematical sense (including even differentials,
> diverging seies, and other means that are actually used).

Practicing physicist seem to act as if every set of real numbers is 
measurable, for example.  This lets them off the escalator of ascending 
set-theoretic axioms rather early, as they never get around to accepting 
the axiom of choice.

In fact, some physists seem to act as if every total function from R->R 
is continuous!  Does this make them unwitting constructivists?

> A case in point is 
> the well-ordering principle that Cantor wanted at a certain point 
> to accept as an axiom, but had no ground for doing so. The status 
> of this principle changes when it was inferred from a much more
> intuitive principle (that is obviously valid for anyone who is 
> a platonist): AC.

Intuitive for the platonist, maybe, but not for the physicists I 
mentioned.

>  
>   Another important observation: as you write yourself in this
> paragraph, the road you take is a never-ending road. According
> to your way of thought, mathematicians will need accepting
> *ever* stronger methods.

The trouble is, the path of *ever* stronger methods is not unique.

-- hendrik