Some Stories About Math And Science

I find certain stories of achievements in mathematics and science intriguing. In some of those I select, much that came before was overthrown. At any rate, these are stories about creations of the human mind that are tough to wrap your head around. I only claim to understand the last story.

Fermat's last theorem lacked a proof for three and a half centuries. When he first saw the theorem as a school boy, Andrew Wiles decided he was going to be a mathematican when he grew up and prove it. And he did.

I have written about the classification of finite simple groups before.

The twentieth century saw some amazing results in logic, set theory, and model theory. Gödel's incompleteness theorem, computability, the axiom of choice, the (generalized) continuum hypothesis, and the Löwenheim-Skolem theorem are very puzzling topics. Perhaps the question of the truth of the continuum hypothesis is, after last year, closer to being solved, whatever that might mean. As I understand it, both the assertion and denial of the continuum hypothesis are consistent with the axioms of Zermelo Fraenkel set theory. So its resolution would take agreement on additional axioms. Apparently, David Asperó and Ralf Schindler showed last year that one such proposed axiom implied another. I doubt I will ever understand this. I suppose perplexity at how maths mean goes back to, at least, the invention of non-Euclidean geometry.

In physics, quantum mechanics and the theory of relativity provide amazement. Their very existence is a surprise. Newtonian mechanics seemed to be the most empirically well-confirmed theory in all of science. Then, in the first couple of decades of the twentieth century, Newton was shown to be incorrect in his basic picture of the universe. At least, this is something like how Karl Popper saw it. Relativity has the surprising implication that time travel is possible in a rotating universe. Gödel showed this when he wanted to provide something for a festschrift for his friend Albert Einstein. I gather Bell's theorem shows that quantum mechanics and a limitation imposed by general relativity cannot both be right. I gather that Bell has been experimentally verified by astronomers looking at radiation passing through gravitational lenses formed from intermediate galaxies.

Political economy provides at least one story like the above. I refer to Sraffa's disproof of marginalism half a century ago.

References

• David Asperó and Ralf Schindler. 2021. MM⁺ implies (*).
• J. S. Bell. 1964. On the Einstein Podolsky Rosen paradox. Physics 1(3): 195-200.
• Stephen Budiansky. 2021. Journey to the Edge of Reason: The Life of Kurt Gödel W. W. Norton.
• Paul J. Cohen. 1963. The independence of the continuum hypothesis IProceedings of the U.S. National Academy of Sciences 50(6):1143-1148.
• Paul Cohen. 1964. The independence of the continuum hypothesis IIProceedings of the U.S. National Academy of Sciences 51(1):105-110.
• Torkel Franzen. 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Peters.
• Kurt Gödel. 1936. On formally undecidable propositions of Principia Mathematica and related systems I. Monatsheft für Mathematik und Physik 38:173-198.
• Kurt Gödel. 1938. Consistency-proof for the generalized continuum-hypothesis. Proceedings of the U.S. National Academy of Sciences 25: 220-224.
• Kurt Gödel. 1940. The consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory. Annals of Mathematic Studies 3.
• Kurt Gödel. 1949. An example of a new type of cosmological solutions of Einstein's field equations of gravitation. Review of Modern Physics 21: 447-450.
• Joel David Hamkins. 2011. The set-theoretic multiverse
• Morris Kline. 1982. Mathematics: The Loss of Certainty. Oxford University Press.
• Calvin Leung et al. 2018. Astronomical random numbers for quantum foundations experiments
• Edwin E. Moise. 1963. Elementary Geometry from an Advanced Standpoint. Addison-Wesley.
• Piero Sraffa. 1960. Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge University Press.
• Robert A. Wilson. 2009. The Finite Simple Groups. Springer.