Foundational Questions Institute: Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020) Support materials for the submission from Bruce Camber in April 2020. Last update: July 18, 2020.
|Background: An FQXi call for papers has forced us to focus on the raw power of mathematics to anticipate the structure of real realities. If there is mathematical cohesion, there is probably a real physical reality that it describes. Matching them up and learning where and how such a unit of mathematical cohesion fits within the larger frameworks is the challenge. More…
 Base-2 and Prime-Number Notations. We see the simple progression of numbers within our base-2 system. In light of it, how should we engage base-3? Wouldn’t that base-3 expansion necessarily be in sync with base-2? Might the expansion look more linear, going to Notation-201 in 67 jumps not just 3, 9, 27, 81? If base-5 is necessarily related by the base-2 foundation, might it require 40 jumps to get to Notation-200? Each prime number will require analysis. The mathematics and geometrics could vary from prime number notation to prime number notation. There may be different iterations. For example, would prime number 7 within the base-2 system move forward via Notations 14, 21, 28, 35, 42… and finally end up on Notation-196 in 28 steps?
Metaphorically speaking, perhaps we should think of these primes to be like an express train to its highest notation closest to 202, then a transfer on a local train, would then bring the effects into the present moment within Notation-202.
Again, let us ask, “What would be the effect of being necessarily tied to the base-2 platform?” Might it effect each base differently? A prime-number notation like 11 might be guided by its relation with base-2 to progress to Notation 198 (11×18) in 18 steps. Might there be a special equation of state at Notation 121 (11×11)? Prime number 13 might jump to Notation 195 in 15 steps. Would it also have some interactive qualities with the progression of 5? 17 jumps to 188 in 11, 19 jumps to 190 in 10, 23 to 184 in 8, and 29 to 174 in 6.
31 goes to 186 in six jumps. 37 goes to Notation 175 in five jumps. Just what is that passageway is anybody’s guess.
What happens with Notations-41, 43 and 47? How about 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, and 101? The nineteen primes thereafter — 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, and 199 — are themselves an especially unique place within the universe.
Each new notation initially replicates all prior notations, yet each evolves with its unique functionality.
Another key questions is about symmetry. Does each notation that is “completely filled” with planckspheres within its base-2 platform become fully symmetrical? Notation-202 which is being populated now as the current expansion, may well be literally filling up with planckspheres and is necessarily asymmetrical and directional.
Obviously, we are just being speculative, playing with ideas.