[FOM] The deductive paradigm for mathematics
vladik at utep.edu
Fri Aug 6 18:27:42 EDT 2010
I think there may be a confusion between what is called mathematics and what mathematicians do. Mathematicians do explore different theories through their consequences, they do analyze motivations and argue about motivations, this is a well-recognized and well-respected part of our work.
Many good foundations papers have sections with motivations and semi-formal explorations, and these sections make the theorems more interesting. The question is how we classify this activity.
I think the point that Foreman and Kunen make is that, according to the viewpoint of mathematicians, these highly interesting and important activities are NOT called mathematics, they are classified as philosophy whatever.
This goes back to the point that I think I already made on this list: that there are two different meanings of the word "mathematics".
* according to mathematicians, mathematics is about proving theorems; informal arguments about mathematical objects and axioms are not mathematics
* according to many people outside math, any study of mathematical objects is mathematics; from this viewpoint, a heuristic algorithm for solving differential equations -- in which no theorem is proven - is mathematics.
Once we realize that there are two different terms, we will often see that besides the confusion caused by the difference between these two, there is actually very little disagreement.
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Marc Alcobé
I think that the deductive paradigm is too restrictive, assigning the
mathematician a unique role as a "theorem prover", and so explicitly
excluding the role as a "theory framer" every other kind of scientist
must play. Why shouldn't mathematicians explore different theories
through their consequences and choose those that best meet their
needs, just like any other kind of scientists do?
Thank you in advance.
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