Random thoughts on Gödel

Last year I blogged about fiction books I wanted to read that year. I only ended up reading three of them. I finished reading War and Peace, which was very long but very good, Stands a Shadow by Col Buchanan, and The Vindication of Man by John C. Wright. Point is, I ultimately didn’t read that much fiction, but read more non-fiction instead.

I’ll probably continue reading more non-fiction than fiction this year as well; there just seem to be more non-fiction books capturing my interest.

I recently finished reading Journey to the Edge of Reason : The Life of Kurt Gödel by Stephen Budiansky. It’s short, less than 300 pages, but provides a very good overview of his life and important mathematical contributions. I wouldn’t have minded if it went deeper into the math, but that’s something I can keep exploring on my own.

A good explanation of Gödel’s most famous contribution, his Incompleteness Theorem, can be found at the 15:16 mark of this YouTube video, though while the video makes the main idea understandable, it still glosses over the finer details of the proof that make it definitive.

To quote page 241 of the book:

Both of Incompleteness Theorems proved that no finite process of inference from axioms within a well-defined system can capture of all mathematics. But that, Gödel pointed out, leads to an interesting either-or choice: either the human mind can perceive evident axioms of mathematics that can never be reduced to a finite rule — which means the human mind “infinitely surpasses the powers of any finite machine” — or there exist problems that are not merely undecidable within a specific formal system, but that are “absolutely” undecidable.

Both choices point to a conclusion “decidedly opposed to materialistic philosophy,” Gödel observed. If the mind is not a machine, then the human spirit cannot be reduced to the mechanistic operation of the brain, with its finite collection of working parts consisting of neurons and their interconnections. If, however, the mind is nothing but a calculating machine, then it is subject to the limitations of the Incompleteness Theorem, which leads to the thorny fact that numbers possess at least some properties that are beyond the power of the human mind to establish: “So this alternative seems to imply that mathematical objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions, that is to say some form or other of Platonism or ‘realism’ as to the mathematical objects.”

Of course, none of this should be mind-blowing to anyone who already believes in God and has already rejected the notion of materialism, but I still find it thought-provoking. As to whether or not the human brain is nothing but a calculating machine, I don’t know. But even if it were, it would not be inconsistent with religious belief, as it would still point to metaphysical truths beyond itself. (This also states nothing about consciousness or the nature of Free Will. Is Free Will born from a necessary limitation of the Incompleteness Theorem?)

While Gödel was not open about whatever he believed regarding God (especially in the more atheistic-leaning circles in which he worked), he did write a letter revealing he certainly believed in an afterlife (from page 267-268):

You pose in your last letter the momentous question, whether I believe we shall meet in the hereafter. About that I can only say the following: If the world is constructed rationally and has a meaning, then that must be so. For what kind of a sense would there be in bringing forth a creature (man), who has such a broad field of possibilities of his own development and of relationships, and then not allow him to achieve 1/1000 of it. That would be approximately as if someone laid the foundations for a house with much effort and expenditure of money, then let everything go to ruin again. Does one have a reason to assume that the world is set up rationally? I believe so. For it is certainly not chaotic and arbitrary, but rather, as science shows, the greatest regularity and order reign in everything. . . . So, it follows directly that our earthly existence, since it in and of itself has at most a very dubious meaning, can only be a means to an end for another existence.

This of course to me echoes C.S. Lewis’s famous quote: “If I find in myself desires which nothing in this world can satisfy, the only logical explanation is that I was made for another world.”

That said, Gödel seems to not have been fond of organized religion. He also says, “… according to Catholic dogma omnibenevolent God created most human beings exclusively for the purpose of sending them to Hell for all eternity.” This is, of course, completely wrong; he obviously spent no time looking for an honest understanding of Catholic dogma.

Gödel was also plagued with mental disorders. He suffered from hypochondria, obsessive-compulsiveness, and harsh periods of maniacal fear and paranoia. While it may be tempting to regard these mental instabilities as an unfortunately side effect of his brilliant mathetical logician’s mind, as the contrast seems starkly ironic, I believe they are more likely unrelated. It would seem more likely to me that his mental disorders arose from emotional overreactions in his assessment of various perceptions. That is, assessing the meaning of a troubled stomach, for instance, has nothing to do with logic, so being a genius logician is perfectly compatible with overreacting to such a thing. He probably would have been able to received better psychological treatment had he been born decades later. Also, if he had been more open to the implications of the teachings of religion, he may have been less inclined to obsess over his social status and whether or not he “achieved” anything, another source of excessive worry and doubt for him.

Overall, it was a very interesting biography, and I would definitely like to explore more of his work and better understand his theorems, especially in how they apply to computer science and AI. Recommended to anyone with similar interests!