I get that mathematicians make up problems to be "interesting" and chewy to solve... but I keep coming back to one thing when I read through the article... WHY? The initial ruleset is just a human construction. It's a pattern that humans created. Why should it 'mean' anything mathematically? I know the value here is in the process, and there may be interesting other mathematical principles that are developed or evolved from the process of answering these three questions. But why this problem? Is it just mathematical curiosity that hooks certain brains, and makes problems like this interesting to a specific thinker? Or is there more here that I am missing...? (I'm thinking about a painter looking at another's painting... the brush strokes, the colors used, how those colors were mixed, the substrate that is being painted on, etc... is that analogous?)
"Math" to most people means "numbers." "Math" to mathematicians means "relationships." In a way, our understanding of higher math is hindered by our mastery of base 10 before we really deal with anything else. It's like learning to play with a capo. You'll get the fingerings that work, but you won't really get the relationship. This is why higher math rarely looks like "math" to normies; we can look at Fermat's last theorem and go "but that's just Pythagoras" and from the point of geometry, yeah. At least, Euclidian geometry at the scales we interact with it. But to get to how a chord works you need to know how waveforms constructively and deconstructively interfere. You don't need that if you're just making songs. If you want to make a new way to make sound? Suddenly you're dealing on a different level. Whenever math devolves to fractals or automata or stuff like that, you're dealing with the sound waves. They define what music is, not just how to play it. And by investigating the relationships that make up chords, you allow the creation of entirely new music, new ways to play music, new tools to make music with. Some people ban augmented 4ths. Some people figure out why augmented 4ths drive people crazy. This is along the lines of figuring out what the interaction of an augmented 4th looks like and where else we might see that interaction play out in ways we haven't thought about before.
There are lots of things that are "patterns that humans created" -- that doesn't necessarily make them uninteresting. There certainly ought to be space for personal curiosity in mathematics! Sometimes it's just fun to ask questions and solve problems. But also, cellular automata (and fractals) have a lot of relationship with dynamic "chaotic" systems which exist all over the place and are notoriously difficult to understand. This includes physical systems (which sometimes humans tend to be great at controlling, but explaining how to do it to a computer is devilishly tricky), quantum computing, and machine learning things.
The interestingness of a problem like this comes from its relatedness to other concepts in mathematics. The problems that are of interest are those whose solutions might contain concepts or truths that can be relevant to other fields of math. So a good math problem is not only something that is solvable and can lead to insightful truths about math, but also something that isn't too arbitrary or separated from the rest of math. Take the first question; it is about proving that the sequence of the central column isn't periodic. Whatever solution may come forth might also be relevant when proving or evaluating things related to prime numbers, which are equally a human construction based on a very simple rule, get equally complex equally fast, and are often looked at in terms of series of numbers. It may turn out to be a dead end, but it also may not, and the solution to this problem might just turn out to be transferrable to another set of problems.
That's what I expected, and what I was getting at with my attempt at recognizing that, in mathematics, the process of solving a problem can be even more valuable than the solution. Are they? I thought they were a basic element or fact of nature, that we just found an easy way to represent mathematically...? Like E=MC2... it is just a mathematical expression of a physical property. The Wolfram problem expressed in the article seems completely divorced from any practical, real world property. It's like he dropped six dice on a table, arranged them in some perceived order, and then made up some questions to ask about that order: Does it repeat? Is it infinite? What happens when you run the calculation a billion times? I'm mostly just marveling at how other people's brains work... not looking for an "answer", per se, just enjoying the window into other thought processes... "...The problems that are of interest are those whose solutions might contain concepts or truths that can be relevant to other fields of math...."
"...prime numbers, which are equally a human construction..."
Are numbers a fact of nature? You can count things, sure, but you'd never find just an idea of some number lying around. It's just as much of abstraction and tool as language or writing. The debate about maths (and, by extension, its study subjects like numbers) being invented or discovered is still ongoing. Philosophy aside, primes are just natural numbers that were observed to have some interesting property. Apart from being building blocks for numbers and them popping up everywhere in mathematics, there wasn't that much real-world use for them until the advent of modern cryptography (the 1970s or so). That's despite the fact people knew about and studied them since at least the times of Euclid. Fun and related to this thread story: classical electrodynamics, as formulated by Maxwell's equations, was compatible with special relativity good 40 years before special relativity was a thing. Even gave enough of a framework to derive Einstein's SR from them. It 'just' took someone pondering the right things to make it into its own thing.Are they? I thought they were a basic element or fact of nature, that we just found an easy way to represent mathematically...?
Like E=MC2... it is just a mathematical expression of a physical property.