Publisher's Synopsis
There is a flaw in making set theory the foundation of mathematics: the definition of a relation is circular, because other relations are required to define it. The solution to this flaw is to make relations primitive, and define sets and set theory in terms of relations. It turns out that all of set theory can be built on relations, without any logical inconsistencies -- and by extension, all of mathematics, logic, and physics.This readjustment of foundations leads to some surprising results. It allows us to resolve many of the paradoxes that have plagued physicists and philosophers for centuries.To add even more clarity, we need to make a clear distinction between two worlds. The empirical world is everything that exists and is perceptible. And the noumenal world is everything that exists independent of the senses. Our everyday perceptions are of the empirical world; and theoretical science is the tool we use to describe the noumenal world. This distinction gives us further resolutions to a number of flaws in common-sense logic.We now have solutions to the problem of cosmic coincidences, the mind-body problem, the problem of induction, and the problem of being. And we have explanations of how theoretical science can predict empirical novelty, and why the noumenal world exists necessarily. And we also have a relational basis of truth, beauty, and goodness.Using symbolic logic, this book introduces formal proofs of these and other concepts -- remarkably, arriving at a proof of Leibniz's statement that we live in the best of all possible worlds.
