The Planck length, denoted as "ℓP," is a fundamental constant in physics derived from Planck's constant, the speed of light, and the gravitational constant. It represents a tiny scale at which our current understanding of physics breaks down, and the effects of quantum gravity become significant.
In the context of your question, if mathematics were continuous rather than quantized, it implies a departure from the principles of quantum mechanics, which are fundamental to our understanding of the microscopic world. Quantum mechanics describes the behavior of matter and energy at small scales, where quantization plays a crucial role.
Einstein's theories, particularly the theory of general relativity, successfully describe gravity and the behavior of the universe on cosmological scales. However, they do not incorporate quantum mechanics. The reconciliation of general relativity with quantum mechanics is an ongoing challenge in theoretical physics, and a complete theory of quantum gravity is currently elusive.
At the Planck length, where quantum gravity effects become significant, it is expected that our current understanding of physics breaks down, including both Einstein's theory of general relativity and conventional quantum mechanics. It is in this regime that physicists believe a more comprehensive theory of quantum gravity is necessary to describe the fundamental nature of space, time, and gravity.
Therefore, it is not accurate to say that Einstein's theories would collapse at the Planck length solely due to the assumption of continuous mathematics. The breakdown of our current understanding of physics at the Planck scale is a consequence of the inherent limitations of our current theories in dealing with quantum gravity, which necessitates the development of a more complete and consistent theory.