(Pardon possible embed problems, the links work . . . I am doing this under travel related constraints)
This one, gives a bit more of details on how Turing sharpened the theorem using the Turing machine, that led to the well known algorithm halting problem:
The issue of the intuitive imagining mind as opposed to an algorithmic machine, is discussed. Worth pondering.
At the same time, we must always bear in mind the famous Euler result:
ei*pi + 1 = 0
This speaks to astonishing unity in Mathematics, for in one expression, converging from wildly different directions, we find the five most important numbers, three of the most important operations, and work in entire fields of Mathematics.
So, here in the heart of mathematics, we at once confront our limitations, the limitations of logic, algorithm machines, and the issue of the rational, insightful, creative, intuitive power power of mind.
Then, stir in, the apparent pattern that starts with collecting nothing, and successively generating the natural numbers, then extending them via decimals to the continuum, using the rotational view of sqrt_-1 to give a plane, thus space, and the onward ijk vectors to give 3-d space, thence motion and the inertia- force- energy triad that gets us to physics.
So, now, let us further ponder, what is this that mathematics is telling us?
unreasonable effectiveness of mathematics” in helping us understand the physical world? END