[FOM] The deductive paradigm for mathematics

[Walt Read]

This touches on very important questions about math, with perhaps the
most important appearing in the line about mathematicians choosing
theories that "best meet their needs, just like any other kind of
scientists do". Most scientists believe in a reality that exists
independent of and prior to any theory of that reality and their need
is to describe that reality. What is the need of the mathematician?
Most math is a response to other math. It resembles art more than
science that way. If mathematicians want to be theory framers ala
scientists, they need to deal with "theory framer" issues. What is
this a theory of? How can I tell if it's a good theory of that? What
makes my new theory of that better than the previous theory? In
software engineering terms, the deductive paradigm is about
verification, not validation. Scientists are largely concerned about
validation, sometimes even to the point of sloughing over flaws in
verification. Validation depends on there being a way to check the
theory against the thing (external reality, in some sense) that it
purports to be a theory of.

What might mathematical theory framing, analogous to scientific theory
framing, look like? As a possible example, take PA to be a theory of
the natural numbers. The predictions of the theory would be the
theorems. How would we check whether the predictions are correct?
Closely examining the proof could only confirm that this is indeed a
prediction (theorem) of the theory. Or suppose I have a theory of sets
in which CH is provable. How would I test that theory? How would I
choose in any non-subjective way between that theory and one in which
not-CH is provable? More generally how would we test theories of
infinite Abelian groups or of Hilbert spaces? Or are these things in
fact defined by, and therefore essentially the same as, their
theories?

-Walt