Re: Humanistic mathematics: response to David Petry

Neil W Rickert wrote:
[snip]

                                             By gathering up all
the disparate branches of math into one big philosophical bundle,
and putting all math objects on the same ontological footing,
Cantorian set theory has rendered an incalculable service to math,
one which I'm sure mathematicians will never want to give up.


Quite right, and very well stated. 

Seconded (4wiw).

But still David has a valid point.  There IS another side to
the typical student repsonse, (well, to mine anyway, of long ago).
It is undeniably the case, that the Cantorian approach to the topic
can lead to having to swallow some very dodgy-looking methods,
typically very nonconstructive ones, and some very dodgy looking
results, often consequences of the Axiom of Choice, a bete noire
of mine as many folk here will know.



This is where I begin to disagree.  To use some computer science
terminology, I consider the non-constructive aspect to be a
feature, not a bug.

I'm thinking here of the applicability of mathematics to the
sciences. As we know from physics, various events might cause a
photon to be emitted. I cannot think of any a priori reason to
suppose that the path taken by that photon should be one that is
mathematically constructable.

We also can't guarantee that that path is present in some model of ZFC, can we? So is this really relevant? Isn't the relationship between mathematics and physics is a bit different? E.g. for quantum physics, the functional analysis of Hilbert spaces plus operators seems more than enough. What makes you think these theories are not constructive? (Or could be made constructive, without loss of power or utility for physics.)

And the consequence is that, if we 
restrict ourselves to purely constructive methods, then there is no
reason to believe that such a constrained mathematics could model
physics.

'Model' physics or 'do' physics?
Some constructivists are currently working on exactly that: showing that constructivism is more than enough to perform (the mathematics needed for) physics. And they're quite a long way, see e.g. the phd thesis of Bas Spitters, available online via his homepage.
(Constructive and intuitionistic integration theory and functional analysis. Ph. D. thesis, University of Nijmegen, 2002)

>
> I understand the appeal of constructive methods.  I normally prefer a
> constructive proof to a non-constructive one where that choice is
> available.  But, ultimately, much of the richness of mathematics
> comes from innovations that originated in the desire to model
> physics.  And I think it would be a mistake to constrain mathematics
> in such a way as to limit that source of ideas.

Who says that constructivism puts such a limit?
It just reduces the intuitions of physics a but further 'down to earth'.
Anyone's still free to use any ideas as a source of inspiration for models. You're just a bit more restrictive on what kind of models really make sense as truely rigorous models.

Or do i misunderstand you?

[snip]
> Geometry, of course, comes from idealizing measurement.  Ruler and
> compasses constructions are really an idealization of what can be
> done with the measuring rod.  And the mathematics we use in much of
> the sciences has its conceptual origins there.

BTW, euclidean geometry is very, very constructive. It's 1st order theory is even decidable. That the intuitions behind those axioms are traditionally called mathematics doesn't mean that we're eternally forced to call them so. They could be taken to be vague intuitions, only -inspiring- the axioms.

--
Cheers,
Herman Jurjus
.

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