Editing The Road to Reality Study Notes (section)

===3.3 Real numbers in the physical world===
Penrose states that while the historical driving force for mathematical ideas were to find constructs that mimic the behavior of the physical world, it is normally not possible to examine the physical world in such precise detail to abstract clear cut mathematics from it. Instead, the mathematics provides its own ‘momentum’ within the subject itself whereby it may seem to diverge from what it had been set up to achieve, but ultimately leads us to a deeper meaning of the world.

The [https://en.wikipedia.org/wiki/Real_number real number] system is given as an example of this since no direct evidence from Nature exists to show that the physical notion of ‘distance’ extends from arbitrarily large scales to the indefinitely tiny.  Though, from cosmological scale to the quantum, to volumes and the metrics of spacetime, the ranges are increasing, and the fundamentals of calculus rely on the infinitesimal.

We may still ask whether the real number system is ‘correct’ for describing the physical universe or have we gotten lucky with extrapolation thus far.  Penrose describes quantum mechanics as not implying any discreteness to Nature, nevertheless this idea may persist and echoes a quote from Einstein’s last published work where he suggested ‘discretely based algebraic theory’ might be the way forward for future physics.  Penrose ends with stating that his idea of [https://en.wikipedia.org/wiki/Spin_network spin networks] are both discrete and foundational in quantum gravity, led to his [https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory], but real numbers are still fundamental in our understanding of the physical world.