Theory Reduction, Algebraic Number Theory, and the Complex Plane

How does mathematical knowledge grow? According to an influential formulation due to philosopher Ernest Nagel, when a scientific theory "reduces" another, the reduced theory is deductively subsumed under the reducing theory: thus for example chemistry is deduced from quantum mechanics, and molecular biology from chemistry. Recent critics, using examples from science, argue that Nagel's criteria for theory reduction are both too strict, and too weak. Prof. Grosholz reviews Nagel's model and its difficulties, and argues that theory reduction faces similar problems in mathematics. Certain proofs of Fermat's conjectures about whole number solutions of quadratic and cubic polynomials, by means of the alliance of number theory with complex analysis, lead not deductively but abductively (adding content) to the study of algebraic number fields, and class field theory. This extension of number theory is at once too strong and too weak to look like Nagelian theory reduction, which is precisely why it turns out to be so fruitful.