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Gödel and unfathomable complexity

Gödel's Incompleteness Theorem had a profound impact on mathematics when it was first established in the 1930's and a profound impact on me when I learned about it. At the beginning of the 20th century a famous mathematician, Hilbert, proposed the construction of a formula or mechanistic process for deciding all mathematical questions. Gödel proved this was impossible.

Mathematics then and now is based on formal systems. These are mechanistic processes (in effect computer programs) for enumerating theorems. Gödel used Gödelization to show how formal systems could talk about or model themselves. He showed that any sufficiently powerful formal system could not prove that the system itself never generated a contradiction. To be sufficiently powerful, the system had to support the definition of a Universal Turing Machine.

One implication of this result is the Halting Problem. We can predict what a computer program will do at any time but we cannot predict if it will ever do something such as halt. Of course, if it does halt and we run it long enough, we will observe this. But, if it never halts, we can not know this from observation. There is no general way to determine when we will have waited long enough. For many programs we can decide the Halting Problem. Some programs have simple loops that continually repeat the same sequence in an obvious way. There are far more complex ways that a program can loop forever that we can understand. But we cannot do this in general for every possible computer program. There will always be programs that have some subtle way to loop or iterate that are beyond our current understanding. This suggests a creative aspect to mathematical truth. There will always be more interesting mathematics that we do not yet understand even if the human race, or at least the study of mathematics, is immortal.

Mathematical structure is devoid of any essential nature, but endowed with unfathomable complexity. Nowhere in the unbounded richness of mathematics and science is there anything that begins to touch on my own immediate experience. Science and mathematics can explain how aspects of experience are structured or related to each other, but the experience itself is completely beyond structural understanding.

This leads to the question: what is?


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