What We've Learned from NKS Chapter 4: Systems Based on Numbers

In this episode of "What We've Learned from NKS", Stephen Wolfram is counting down to the 20th anniversary of A New Kind of Science with [another] chapter retrospective. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twitch channel of Stephen Wolfram here:   / stephen_wolfram  

Read all of NKS here: https://www.wolframscience.com/nks/

00:00 Stream Begins
3:20 SW begins speaking
8:21 Section 1: Notion of Numbers
9:40 Section 2: Elementary Arithmetic
25:30 Blog Post: After 100 Years, Can We Finally Crack Post's Problem of Tag?
34:15 Section 3: Recursive Sequences
45:50 Section 4: Sequence of Primes
52:49 Section 5: Mathematical Constants
1:08:07 Section 6: Mathematical Functions
1:15:20 Section 7: Iterated Maps and the Chaos Phenomenon
1:24:38 Section 8: Continuous Cellular Automata
1:32:32 Section 9: Partial Differential Equations
1:40:42 Section 10: Continuous Versus Discrete Systems
1:43:06 Q&A Begins
1:43:34 Those functions remind me a bit of signal theory in the discrete time domain. Doing laplace or fourier transforms on them could be interesting.
1:44:30 Do numbers have an intrinsic complexity? Are certain numbers more "common" than other numbers?
1:47:53 ​What is the empirical evidence in favor of the following hypothesis? Hypothesis: If the positive integer n occurs in nature, then the positive integer n+1 occurs in nature.
1:50:55 I found two Elementary CA's that computes 3n+1 sequences. The idea is that the parity-bit selects which rule to apply to the configuration. Any ideas what to do next?
1:52:16 Do the digits of pi necessarily exist? Could pi have any other value?
1:56:22 why did we humans invent primes such that it became to be one of the biggest mysteries in mathematics?
1:57:18 why did natural selection select for the prime number patterns being recognizable? Given examples of number recognition in birds bees and dragonflies t.b.c. Basically dragonfly brains appear to do operations we need quaternions to perform which suggests all numbers are real structures in the universe which evolution selects for when useful for survival.
2:02:08 Does this suggest that if two systems can be mapped to each other they have a representation that is common to both systems by "factoring" both systems? Then if two systems cannot be reduced to each other they do not share any common "factors"? Ie irreducibility is akin to being relatively "prime" in their representations in that they do not share any "factors" in common?
2:04:15 Are integers fundamental or not? Reducible or irreducible? The number 1 doesn't seem like it's made out of anything.
2:05:49 ​Is there computational equivalence between solving the Navier-Stokes equations and solving Einstein's field equations?
2:14:20 You said all sequences of digits appear equally often in 0.12345667891011121314... Why doesn't the Champernowne constant conform to Benford's law?
2:18:55 could you speak a little more on the slice of the zeta function being an approximation for any function?
2:21:07 so the einstein equations are kind of Godelian in some sense?
2:22:22 ​Have you read Inhomogenous and Anisotropic cosmology The No big crunch theorem they prove I suspect might constrain the Einstein F E if gneralize. basically the proof for 3 bounds restricts the EFE in such a way to eliminate solutions which reduce the spacelike volume hypersurface in flat or open geometry when inhomogeneous anisotropic sol
2:23:21 If you attached a quantum random number generator to your algorithm would this imply that the numbers that are more "common" depend on how "easy" it is to make a quantum random number generator? Then the regularities in the numbers we see are the deviations from pure quantum randomness?

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