[FOM] Fwd: The deductive paradigm for mathematics

[Marc Alcobé]

2010/8/8 Vladimir Sazonov <vladimir.sazonov at yahoo.com>:
 >> Does the fact that one has to face
 >> problems and methods that lay outside the deductive paradigm change
 >> the mathematical character of foundational research?
 >
 > The last question is not very clear to me. But, anyway...

I just mean that under the view of the deductive paradigm given by
Kunen's quotation (I do not mean this is representative of his own
view, which I do not know, but only make use of his words to exemplify
what I understand is such view; similarly, I do not know if Foreman
has this kind of view in mind when he talks about the "deductive
paradigm", this is a supposition of mine), problems like the continuum
problem or methods having to do with exploring different theories and
deciding which best fits one's needs lay outside of it, and this
should not justify classifying all foundational research as
philosophical.

 > I essentially agree with your, as I consider, rhetorical questions

You are right, so do I consider them. I wrote these rethorical
questions as a motivation for the only non-rethorical one. It looks
like I exceeded on my emphasis to say: let us admit, as Foreman
proposes, there are "internal mathematical motivations" for
foundational work. The question is, then, which do you think are they?

I think this is a natural question to ask. It does not involve as much
answering to "what is mathematics" as to "what is mathematics for
mathematicians", something the deductive paradigm (as described) is
unable to do (under the given proposal).

I wish to remark my question is not vacuosly stated. See, for example,
Kanamori's "The Mathematical Development of Set Theory from Cantor to
Cohen" (possibly it is not casuality Foreman's views are near to his).

Best,

Marc.