Status Updates From Levels of Infinity: Selecte...

He also emphasises how the Lorentz group of special relativity is not a compact group and not orthonormal. It can function only as locally compact, due to the minus sign in the "light-time" dimension. This explains why kants synthetic intuition of space had to fail in relativity. This would require a globally compact group such as the orthonormal group where all dimensions have the same sign.

Reading through some of this work, Weyl seems to be one of the first to appreciate the depth of the difficulties for the foundations of mathematics that result after the failure of attempts to formalise mathematics under logic of Hilbert, Russell and others.

Argues that mathematical foundations are less certain than ever, between axiomatic and constructive approaches, indicating a deep connection between mathematics and human mind and nature still yet to be fully grasped. (Contrast this to say a Wittgenstein who would rush to embrace conventionalism at this point in the debate)

Very interesting first chapter, overviewing the uses and understanding of infinity, and how it has developed from ancient greeks like Zeno and the atomists and Anaxagoras to early modern approaches of Leibniz, and recent approaches of Cantor, Brouwer, Hilbert and others in their attempts to place the infinite. Infinity resists crystalisation in a closed structure and we must remain in an open relationship to reality.