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The Four Color Theorem is a great example! I think this story is often misrepresented as one where mathematicians didn't believe the computer-aided proof. Thurs

by nicf 2y ago

The Four Color Theorem is a great example! I think this story is often misrepresented as one where mathematicians didn't believe the computer-aided proof. Thurston gets the story right: I think basically everyone in the field took it as resolving the truth of the Four Color Theorem --- although I don't think this was really in serious doubt --- but in an incredibly unsatisfying way. They wanted to know what underlying pattern in planar graphs forces them all to be 4-colorable, and "well, we reduced the question to these tens of thousands of possible counterexamples and they all turned out to be 4-colorable" leaves a lot to be desired as an answer to that question. (This is especially true because the Five Color Theorem does have a very beautiful proof. I reach at a math enrichment program for high schoolers on weekends, and the result was simple enough that we could get all the way through it in class.)

rssoconnor 2y ago
Is the proof of the Four Colour Theorem really that unsatisfying? The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. To verify this computational fact one uses a (very) glorified pocket calculator.
- godelski 2y ago
  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. [0] Quotes are because truth is very very hard to derive. https://hermiene.net/essays-trans/relativity_of_wrong.html https://hermiene.net/essays-trans/relativity_of_wrong.html
  - rssoconnor 2y ago
    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.
    - gsf_emergency_2 2y ago
      If it were size 2, we could more easily make sure that the answer is definitely mind-blowing. Have programmers given up on wanting their mind blown by unbelievable simplicity?
    - godelski 2y ago
      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 https://www.youtube.com/shorts/lcQAWEqPmeg
    - roenxi 2y ago
      > 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.
- gsf_emergency_2 2y ago
  Fwiw fairly recently posted to HN was progress towards a more satisfying (read, likely mind-blowing) proof: https://blog.tanyakhovanova.com/2024/11/foams-made-out-of-felt/ https://blog.tanyakhovanova.com/2024/11/foams-made-out-of-fe... This is also nice because only pre-1600 tech involved
- eru 2y ago
  > The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. You just summarised (nearly) everything a mathematician can get out of that computerised proof. That's unsatisfying. It doesn't give you any insight into any other areas of math, nor does it suggest interesting corollaries, nor does it tell you which pre-condition of the statement does what work. That's rather underwhelming. That's less than you can get out of the 100th proof of Pythagoras.

by nicf 2y ago

rssoconnor 2y ago

Is the proof of the Four Colour Theorem really that unsatisfying? The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. To verify this computational fact one uses a (very) glorified pocket calculator.

godelski 2y ago
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. [0] Quotes are because truth is very very hard to derive. https://hermiene.net/essays-trans/relativity_of_wrong.html https://hermiene.net/essays-trans/relativity_of_wrong.html
- rssoconnor 2y ago
  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.
  - gsf_emergency_2 2y ago
    If it were size 2, we could more easily make sure that the answer is definitely mind-blowing. Have programmers given up on wanting their mind blown by unbelievable simplicity?
  - godelski 2y ago
    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 https://www.youtube.com/shorts/lcQAWEqPmeg
  - roenxi 2y ago
    > 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.
gsf_emergency_2 2y ago
Fwiw fairly recently posted to HN was progress towards a more satisfying (read, likely mind-blowing) proof: https://blog.tanyakhovanova.com/2024/11/foams-made-out-of-felt/ https://blog.tanyakhovanova.com/2024/11/foams-made-out-of-fe... This is also nice because only pre-1600 tech involved
eru 2y ago
> The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. You just summarised (nearly) everything a mathematician can get out of that computerised proof. That's unsatisfying. It doesn't give you any insight into any other areas of math, nor does it suggest interesting corollaries, nor does it tell you which pre-condition of the statement does what work. That's rather underwhelming. That's less than you can get out of the 100th proof of Pythagoras.