Can physics be rationalized? The Cayley Transform is key! | Sociology and Physics | N J Wildberger

We discuss the important question of how to use a better mathematics to help steer modern physics in good directions. The Principle of Mathematical Economy suggests that a simpler, more logical mathematics ought to be a more useful foundation than our current "infinity based" axiomatic framework, relying on a fictitious " real number arithmetic" and corresponding dreaming with "infinite sets" and "infinite procedures/functions". A prime example is the question of dispensing with the unfortunate reliance on "angles" and "circular trig functions" that pervades much or modern theoretical physics. By adopting rational mathematics, with rational trigonometry and a rational parametrization of the circle, we have a powerful alternative which is purely in the realm of finite computation, i.e. what our computers can actually do. And then the key to extending this to the higher dimensional situations that modern quantum, relativistic and particle physics needs is a remarkable mapping introduced by Arther Cayley in 1846: the Cayley transform. We show how this is a direct generalization of the classical rational (angle-free) parametrization of the unit circle, and how it upends our understanding of Lie groups, or continuous groups, like the orthogonal group O(n) over the rationals, or the Unitary group U(n) over the rational complex numbers. The power of the purely algebraic rational approach is strongly evident here, and should give modern physicists lots to mull over. The Playlist "Classical to Quantum" is found at Wild Egg Maths (Members only): https://www.youtube.com/watch?v=HzyK2NW15QU&list=PLzdiPTrEWyz4CVqTYS1fwInPOZxBxDaQf