I had high hopes for this book. The premise is a good one: applying the ideas of metamathematics to philosophy and religion in a hopes of providing a new framework for considering these ideas. Sarah Voss’s project is, in effect, one of attempting to conceive a new metaphysics on a mathematical basis.
Unfortunately, it seems that not only does Voss not succeed in her effort, but she manages to fall into some of the traps that modern mathematical thought is meant to avoid, particularly, in one case near the end of the book, conceiving of her metaphysics as a set which contains itself. Bertrand Russell was the first to identify the problem with this in his definition of “perfect sets”: A perfect set is a set which does not contain itself. In this instance, we’re left with the question of whether P the set of all perfect sets is itself a perfect set. If it is a perfect set, it does not contain itself, but since P contains all perfect sets, it must contain itself and cannot be perfect. If P is not a perfect set, then it contains itself, but P is defined as the set of all perfect sets and thus is perfect.
It was just this sort of difficulty that led to Gödel’s examination of metamathematics and his famous incompleteness theorem, in short a set of mathematical symbols (with symbols defined broadly enough to allow all mathematical proofs to be expressed with these symbols) cannot prove all statements about those symbols, in short there are mathematical statements which cannot be proven false or true within the system of mathematical statements (it does not, however, provide any means of determining whether a mathematical statement is, in fact, among those unprovable statements, meaning that the efforts at proving such theorem’s as the Riemann Hypothesis could quite possibly be attempting to prove the unprovable).
Voss occasionally comes frustratingly close to seeing how this sort of metamathematical thought could be applied to philosophy or religion, but sadly she doesn’t seem to have sufficient mathematical sophistication herself to actually complete her project.
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