[FOM] The deductive paradigm for mathematics

[Marc Alcobé]

Dear FOMers,

I'm highly interested in knowing your opinion about the following.

Matthew Foreman in his preface to the HST writes "The methodology for
settling the independent statements, such as the Continuum Hypothesis,
by looking for evidence is far from the usual deductive paradigm for
mathematics and goes against the rational grain of much philosophical
discussion of mathematics."

What I understand by "the usual deductive paradigm for mathematics" is
perfectly illustrated by Kenneth Kunen, who makes the following
assertions in his The Foundations of Mathematics (pp. 3 – 4): "in
mathematics, one writes down axioms and proves theorems from the
axioms. The justification for the axioms (why they are interesting, or
true in some sense, or worth studying) is part of the motivation, or
physics, or philosophy; not part of the mathematics. The mathematics
itself consists of logical deductions from the axioms."

Foreman also writes: "set theory finds itself at the confluence of the
foundations of mathematics, internal mathematical motivations and
philosophical speculation." Let me call your attention on the
"internal mathematical motivations" component.

Do you think the "deductive paradigm" view is shared by many mathematicians?

What kind of "internal mathematical motivations" do you think give
support to foundational work in mathematics?

Isn't it an oversimplification to say that the motivation for the
axioms of set theory is only philosophical (hence not mathematical)?

Isn't the need for a foundational framework at least as much
mathematical as philosophical? Does the fact that one has to face
problems and methods that lay outside the deductive paradigm change
the mathematical character of foundational research?

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.

Best,

Marc.