God does not “need” evil to persist in order to show that He is good. God allows it to exist, as per His commitment to free will (for without which we cannot have true love), but your dialectic, ying-yang, Gnosticism just isn’t true. ]]>

Ever since Euclid (circa 300 BC), mathematicians have been trying to invent a formal mathematical system that is self-contained and does not rely on any preconceived assumptions. In Euclidian geometry, for example, Euclid’s mathematics relied on just 6 presupposed assumptions. It was believed that one day, these assumptions would be proved valid as part of an even more extensive system. Moreover, it was believed that one day a mathematical “theory of everything” would be formalized, one that didn’t rely on ANY assumptions. An assumption is something that you have to believe to be true, but cannot prove to be true.

Fast forward a couple thousand years to the 20th century, and the mathematical community believes they are very close to such a system, and are extremely eager to see it through. In 1910, the famed mathematician and atheist Bertrand Russell publishes a 3 volume book called Principia Mathematica. He believed that this rigorous presentation was, in fact, a quantum leap towards the mathematical theory-of-everything, which did not rely on any outside assumptions. The 3 volume set is so rigorous, we don’t get to “1 plus 1 equals 2″ until half-way through the second volume….

However, the German mathematician Kurt Godel not only proved Russell’s Mathematica to not be self-contained, he proved that ANY mathematical system MUST rely on outside assumptions, and a mathematical “theory-of-everything” is not possible. The math behind this is extensive, but relies on set theory. In short, Gödel proved that in any formal mathematical system, no matter how extensive, you could always formulate a mathematical statement essentially equivalent to saying “This statement is false.”. He proved that this form of self-reference (which proves that a system is not logically independent of outside assumptions) will always be evident. Using our Euclidian example, the minute you create a system that proves the assumptions (also called axioms) in Euclid’s system are valid, there exist other assumptions that you haven’t (and can’t) prove within that system. Doing this iteration again to yield those assumptions valid will still rely on more outside assumptions, and this is conclusion is inevitable no matter how many times you iterate the process.

Gödel’s theorems blew a hole wide open in mathematics, and the eager desire to have such a “theory of everything” was very conclusively squashed. An outworking of Gödel’s theorems can be seen in the infamous Continuum Hypothesis. The father of infinite set theory, another German mathematician by the name of Georg Cantor, postulated a certain hypothesis, but couldn’t prove it to be true. (Curiously, the hypothesis is predicated solely upon a problem that involves the idea of infinity). Kurt Gödel, a few decades after Cantor’s death, proved that you couldn’t disprove the Continuum Hypothesis. In other words, the Hypothesis was proven to not be false. This was an incomplete answer to the problem, because that kind of proof doesn’t mean the hypothesis is true, it just means it can’t be proven to be false. Well, this inconclusive proof did not sit well with mathematicians. For another 100 years, it remained number one on the list of important questions for mathematics. Finally, in 1964, another mathematician named Paul Cohen, solved the problem — but the solution blew an even wider hole in mathematics. Cohen’s solution was that the Continuum Hypothesis cannot be proven to be true. In other words, it proved that the answer is undeterminable. It proved that mathematics is not sufficient to solve everything in mathematics.

Bringing all this together, the only way mathematics is possible is if there exist an assumption, or set of assumptions, which must be true but cannot be proven to be true in any mathematical system. This source, whatever it is, must be a boundless and infinite entity. Furthermore, and piggy-backing on this conclusion, the Continuum Hypothesis proved as an example of how mathematics is not a sufficient system to prove everything within mathematics — the knowability of the Hypothesis exists outside our domain of reality.

If that’s not a mind-blowing evidence for God, I’m not sure what is.

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