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Learn how mathematics can be reimagined to create a more logical foundation for modern physics in this thought-provoking lecture. Explore the Principle of Mathematical Economy and discover why simpler mathematical frameworks might be more effective than current infinity-based systems that rely on real number arithmetic and infinite sets. Examine how rational mathematics and the Cayley transform, introduced in 1846, offer powerful alternatives to traditional angle-based trigonometry and circular functions. Delve into how this rational parametrization of the circle extends to higher-dimensional applications in quantum, relativistic, and particle physics. Understand the revolutionary implications of this purely algebraic rational approach for Lie groups and continuous groups like the orthogonal group O(n) and the Unitary group U(n), providing new perspectives for modern physicists to consider.
