I think you're missing the core component. We care __WHY__ the theorem is true. To be honest, the __IF__ part matters a lot less.
The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important.
I'll give an example from physics. (If you're unsatisfied with this one, pick another physics fact and I'll do my best. _Any_ will do.) Here's a "fact"[0]: The atoms with even number of electrons are more stable than those with an odd number. We knew this in the 1910's, and this is a fact that directly led to the Pauli Exclusion Principle, which led us to better understand chemical bonds. Asking why Pauli Exclusion happens furthers our understanding and leading us to a better understanding of the atomic model. It goes on and on like this.
It has always been about the why. The why is what leads us to new information. The why is what leads to generalization. The why is what leads to causality and predictive models. THe why is what makes the fact useful in the first place.
I do think the why that the Four Colour Theorem is true is captured my statement. The reason why it is true is because there exists some finite unavoidable and reducible set of configurations.
I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason. I suspect that if the size of this finite set were 2, instead of 633, and you could draw these unavoidable configuration on the chalk board, and easily illustrate that both of them are reducible, everyone would be saying "ah yes, the four colour theorem, such an elegant proof!"
Yet, whether the finite set were of size 2 or size 633, the fundamental insight would be identical: there exists some finite unavoidable and reducible set of configurations.
> I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason.
I think that is exactly correct, except for the "no good reason" part. There aren't many (any?) practical situations where the 4-colour theory's provability matters. So the major reason for studying it is coming up with a pattern that can be used in future work.
Having a pattern with a small set (single digit numbers) means that it can be stored in the human brain. 633 objects can't be. That limits the proof.
To me, the four-color theorem is a very interesting proof of concept, perhaps the most interesting mathematical proof of the past 50 years. Perhaps the "pattern that can be used in future work" is the idea of having computers enumerate a large number of cases, which they then solve in individually straightforward ways.
But, I can understand if pure mathematicians don't feel this way. This might be only really an intriguing and beautiful concept to someone who is interested in scaling up algorithms and AI.
> So the major reason for studying it is coming up with a pattern that can be used in future work.
Surely, reducing the infinite way in which polygons can be placed on a plane to a finite set, no matter how large, must involve some pattern useful for future work?
> The reason why it is true is because there exists some finite unavoidable and reducible set of configurations.
OK but respectfully that's just restating the problem in an alternative form. We don't get any insight from it. Why does there exist this limit? What is it about this problem that makes this particular structure happen?
Perhaps the key insight is that there is no concise explanation that underlies this particular structure. Many mathematical statements are true for no concise reason. If you want to discover if these things are true or not, perhaps you need a computer-assisted search, and that's the intellectual lesson to be drawn here.
I really doubt this. I mean mathematicians spent decades trying to answer if the number 2 exists. People spend a lot of time on what seems fairly mundane and frankly, the results are quite beneficial. What's incredible or mind blowing is really just about your perspective, it is really just about your choice to wonder more. https://www.youtube.com/shorts/lcQAWEqPmeg
The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important.
I'll give an example from physics. (If you're unsatisfied with this one, pick another physics fact and I'll do my best. _Any_ will do.) Here's a "fact"[0]: The atoms with even number of electrons are more stable than those with an odd number. We knew this in the 1910's, and this is a fact that directly led to the Pauli Exclusion Principle, which led us to better understand chemical bonds. Asking why Pauli Exclusion happens furthers our understanding and leading us to a better understanding of the atomic model. It goes on and on like this.
It has always been about the why. The why is what leads us to new information. The why is what leads to generalization. The why is what leads to causality and predictive models. THe why is what makes the fact useful in the first place.
[0] Quotes are because truth is very very hard to derive. https://hermiene.net/essays-trans/relativity_of_wrong.html