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This discussion seems like an apt place to drop Morris Kline's "Mathematics: The Loss of Certainty" (https://www.amazon.com/Mathematics-Loss-Certainty-Oxford-Pap...). As a non-mathematician whose only exposure with a lot of these ideas is through that book, I can't help but see these discussions as falling very much in line with the competing schools of thought described in the book.

The book is a non-academic tour of the history of mathematical foundations, and the way mathematicians struggled to rediscover "truth" or the purpose of their work when new crises were reached. For example, the book spends a good deal of time explaining the centrality of Euclidean geometry to people's worldviews, and the way that the discovery of non-Euclidean geometries shook people. Not just because they didn't assume other geometries could exist, but because people believed geometry to map onto Euclidean physical reality because it was God's way of revealing Himself to the world.

The other main crises that the book toured were the discovery of quaternions, Cantor's theories, and Godel's theorem.

Kline ends describing the arc over the last two hundred years of math as a splitting-off into four different schools: set theorists, intuitionists, formalists, and logicists. Each camp tried to reassert math on "solid ground". I hear echos of those debates in this thread, where some are asserting that there can possibly exist multiple foundations, which from my reading of the book is a very formalist idea (our rules of math are a formal system, and any internally consistent set of rules are just as valid as objects of study).

Not being a mathematician, I don't have a sense of where those schools played out to the current day. I'd be curious to hear if they're all still around in different forms, or whether some have more or less died out.

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