Algebraic number theory and the complex plane

I review Nagel’s model of theory reduction, with its projection of a collapse of two discourses into one, and then Jeremy Butterfield’s critique of it. He argues that in physics, definitional extension is both too strong and two weak to conform to Nagel’s strict criteria, and so blocks the collapse. I then argue that the same holds true for attempts at theory reduction in mathematics, by tracking the expositions of alleged reductions in a logic textbook by H. B. Enderton. I also review the proofs of a few of Fermat’s conjectures by means of the alliance of number theory with complex analysis, and show that these truly ampliative and explanatory proofs open up the study of the rational numbers to the study of algebraic number fields, an extension of number theory that is once too strong and too weak to look like Nagelian theory reduction, which is precisely why it turns out to be so fruitful.