We know that voting rules experience all sorts of difficulties, e.g., as shown in this lecture, the problems are sufficiently bad that we should seriously worry whether the person elected is who the voters really want. The central issue is to understand why this is so and whether any rule has reliable outcomes. Because voting rules serve as a prototype for a wide class of aggregation rules, ranging from statistics, much of what is done in the social sciences, to even engineering multiscale processes, answers could be of wide value. In this lecture, Donald Saari will outline the mathematical structure of voting rules, and show how these structures completely explain all of those many troubling paradoxes.