March 9th, 2005

classic cylon

problem solving, infinity, the Kabbalah, and depression

Unfortunately, I'm still quite busy, so I don't have enough time to do this entry justice. I've been reading The Mystery of the Aleph by Amir D. Aczel. This is a book about how the mathematical concept of infinity was developed starting from its origins in ancient Judaism through today. Briefly, the Kabbalah was a secret society dedicated to the study of the Torah. The Kabbalists understood God as infinite in a very similar sense to how infinity had been previously described by the Greeks, such as Pythagoras. (Incidentally, when I showed my piano teacher this book, she showed me another book on the Kabbalah that listed Jesus as a Kabbalist, and described "walking on water" as part of the practice.)

The primary focus of the book is on the life and work of Georg Cantor, who proved that the integers and rational numbers were infinite sets that had the same cardinality (same number of elements), and that the real numbers, while also infinite, had a greater cardinality. However, he became depressed when he was unable to prove that the cardinality of power set (set of all subsets) of the integers was equal to the next highest cardinality. (This is known as the continuum hypothesis.) Quite possibly, his depression was worsened by his struggles with Kronecker, another mathematician who did not believe any of his theories and tried to discredit him.

Depression seems to be a common theme among those who studied concepts related to infinity. Several of the pioneers of algorithm theory (e.g. Gödel, Post) also seemed to suffer from some form of it. I don't think this is a coincidence. Actually, I have recognized, to some extent, my own depression in the difficulty of solving certain types of math problems, such as the puzzles and computability theory-related problems I've written about earlier. I think part of the problem is the paradoxical nature of such problems; I twist my mind into knots sometimes trying to solve them.
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