In the previous article (Found. Phys. Lett. 16:325–341, 2003), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, (...) due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

1. T. 1. Häfte, Allgemeine Einleitung in die platonische Philosophie.--1. T., 2. Hälfte. Die Früdialoge -- 2. T., 1. Hälfte. Die Dialogue der Übergangszeit Gorgias, Meno, Euthydem, Menexenus Cratylus -- 2. T., 2. Hälfte. Die Dialogue der literarischen Meisterschaft Phädo, Symposium, Staat und Phädrus --3. T., 1. Hälfte. Die Spätdialoge Theätet, Parmenides, Sophist und Politicus -- 3. T. 2. Hälfte. Die Spätdialoge Philebus, Timäus, Critias und Gesetze .. 4. T. Register.

Throughout this book, Gooch tells and retells the stories of Socrates and Jesus as he examines perennial human issues: why would anyone willingly die? To what do these two martyrlike deaths bear witness? What are the limits of words in explanation and defense? Why was Jesus silent during his trial? Why did Socrates' most powerful apologia fail? What words, if any, work in prayer? Do words work against the fear of death? Out of this philosophical and religious questioning, Reflections on (...) Jesus and Socrates throws new light on these two compelling figures and on the continuing meanings of their stories for us today. (shrink)