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It seems pretty clear to me: - Trigonometric proofs of the Pythagorean theorem are rare, because many trigonometric identities depend on the Pythagorean theore

by svat 4y ago

It seems pretty clear to me: - Trigonometric proofs of the Pythagorean theorem are rare, because many trigonometric identities depend on the Pythagorean theorem, so it's easy to end up with an argument that is in fact circular. - Nevertheless, there have been a handful of trigonometric proofs in recent decades. - These two students have come up with a new trigonometric proof, and it is a nice one too (“could well be the most beautiful and simplest trigonometric proof we have seen to date”). That is the novelty/notability. Note that this is consistent with the talk abstract (quoted in https://news.ycombinator.com/item?id=35498423 https://news.ycombinator.com/item?id=35498423) — they don't claim to have the first trigonometric proof, just that “in our lecture we present a new proof […] based on a fundamental result in trigonometry—the Law of Sines—and we show that the proof is independent of the Pythagorean trig identity”. Pretty cool work IMO.

sedatk 4y ago
Your explanation is very clear, but I still think that the article is confusing about that. Note how the author sets the expecatations at the beginning of the article: "...the proof these young trailblazers have proposed might make a few established mathematicians eat their words. This is because their proof uses trigonometry." This wording implies to me that Jackson and Johnson are actually the first who came up with a trigonometric proof because otherwise how an nth in a series make a few established mathematicians eat their words? Why would they eat their words now? Why not after the first trigonometric proof? So, he raises the expectations then drops a "it's been done before" bomb, now what are we supposed to think about what's happened here? Is it what we read at the beginning? Or is it just the elegance of the proof that's cool? But, why does THIS one make the mathematicians eat their words then? The whole "yeah it's awesome, amazing actually, but hey, forget what I said earlier, it's been done before" kind of composition confuses me.
- galaxyLogic 4y ago
  I think any new proof of PT is pretty cool. it doesn't even have to be "elegant" or better than previous proofs. Just a new one. Will there at some point be a proof that all possible proofs of Pythagorean Theorem have been found. Or for any theorem? Is it possible to prove that all possible proofs of something have been found?
  - qez 4y ago
    I wonder, are there an infinite number of proofs of the theorem, each more complex than the last? Can I rephrase what they did, make it more convoluted, and call it a new proof?
    - FartyMcFarter 4y ago
      You can always insert irrelevant details into a proof, so it's easy to construct an infinite number of proofs from a single proof.
      - galaxyLogic 4y ago
        If they are irrelevant details then that should not count as a new proof, in my opinion. I guess it might be possible to have a "canonic form" of a proof so that if two proofs can be reduced to the same canonic proof then they are in fact the same proof.
- svat 4y ago
  I see your confusion. The thing is that “trigonometric proof” is not a well-defined mathematical object (even in this HN thread, you can see a lot of discussion about whether the trigonometry is really essential to the proof, ways to remove it, etc). So the “establishment” idea that “There are no trigonometric proofs” (Loomis, 1927) is not a mathematical conjecture, which can be demolished by the first counterexample (the first “trigonometric proof”), but a meta-mathematical statement or social convention, where minds change slowly. Note how he says “this point of view has been increasingly questioned in recent decades”: this is typical of the evolution of social consensus rather than of mathematical conjectures (which would quickly switch from false to true or vice-versa as soon as a proof or counterexample is found). The first few trigonometric proofs, being very complicated, might have just had a reaction (among the very few people who even care about this question) like “yeah ok, whatever, that's just too contrived, not very interested”, but when a proof like this comes along, being more beautiful and simpler, more people will change their minds — but even now it's not guaranteed, which is why the author says “might make a few established mathematicians eat their words”. Ultimately, all proofs are just pushing around of axioms and implications; there's no clear separation of whether a proof is “different” from another or whether it's “trigonometric”, but in this case the authors say their proof “is based on a fundamental result in trigonometry—the Law of Sines” and it seems pretty easy to believe that that is how they came up with the proof (so it seems fair to call it a trigonometric proof even if that can be got rid of). [PS: I just found that some scans of Loomis's book are online: https://personal.math.ubc.ca/~cass/Euclid/java/html/L.pdf https://personal.math.ubc.ca/~cass/Euclid/java/html/L.pdf (1927), https://files.eric.ed.gov/fulltext/ED037335.pdf https://files.eric.ed.gov/fulltext/ED037335.pdf / https://www.lapasserelle.com/documents/Pythagorean_Proposition_Elisha_S_Loomis.pdf https://www.lapasserelle.com/documents/Pythagorean_Propositi... (1940 second edition) — see the foreword where he says “Fifth, that no trigonometric proof is possible”, elaborated on p. 193(1e)/244(2e) in section called “No trigonometric proofs”.]
  - sedatk 4y ago
    Thanks, this has been very explanatory. It’s unusual that maths, the language known for precise definitions, has no unambiguous definition of trigonometry. It’s what it is though.

by svat 4y ago

sedatk 4y ago

Your explanation is very clear, but I still think that the article is confusing about that. Note how the author sets the expecatations at the beginning of the article: "...the proof these young trailblazers have proposed might make a few established mathematicians eat their words. This is because their proof uses trigonometry." This wording implies to me that Jackson and Johnson are actually the first who came up with a trigonometric proof because otherwise how an nth in a series make a few established mathematicians eat their words? Why would they eat their words now? Why not after the first trigonometric proof? So, he raises the expectations then drops a "it's been done before" bomb, now what are we supposed to think about what's happened here? Is it what we read at the beginning? Or is it just the elegance of the proof that's cool? But, why does THIS one make the mathematicians eat their words then? The whole "yeah it's awesome, amazing actually, but hey, forget what I said earlier, it's been done before" kind of composition confuses me.

galaxyLogic 4y ago
I think any new proof of PT is pretty cool. it doesn't even have to be "elegant" or better than previous proofs. Just a new one. Will there at some point be a proof that all possible proofs of Pythagorean Theorem have been found. Or for any theorem? Is it possible to prove that all possible proofs of something have been found?
- qez 4y ago
  I wonder, are there an infinite number of proofs of the theorem, each more complex than the last? Can I rephrase what they did, make it more convoluted, and call it a new proof?
  - FartyMcFarter 4y ago
    You can always insert irrelevant details into a proof, so it's easy to construct an infinite number of proofs from a single proof.
    - galaxyLogic 4y ago
      If they are irrelevant details then that should not count as a new proof, in my opinion. I guess it might be possible to have a "canonic form" of a proof so that if two proofs can be reduced to the same canonic proof then they are in fact the same proof.
svat 4y ago
I see your confusion. The thing is that “trigonometric proof” is not a well-defined mathematical object (even in this HN thread, you can see a lot of discussion about whether the trigonometry is really essential to the proof, ways to remove it, etc). So the “establishment” idea that “There are no trigonometric proofs” (Loomis, 1927) is not a mathematical conjecture, which can be demolished by the first counterexample (the first “trigonometric proof”), but a meta-mathematical statement or social convention, where minds change slowly. Note how he says “this point of view has been increasingly questioned in recent decades”: this is typical of the evolution of social consensus rather than of mathematical conjectures (which would quickly switch from false to true or vice-versa as soon as a proof or counterexample is found). The first few trigonometric proofs, being very complicated, might have just had a reaction (among the very few people who even care about this question) like “yeah ok, whatever, that's just too contrived, not very interested”, but when a proof like this comes along, being more beautiful and simpler, more people will change their minds — but even now it's not guaranteed, which is why the author says “might make a few established mathematicians eat their words”. Ultimately, all proofs are just pushing around of axioms and implications; there's no clear separation of whether a proof is “different” from another or whether it's “trigonometric”, but in this case the authors say their proof “is based on a fundamental result in trigonometry—the Law of Sines” and it seems pretty easy to believe that that is how they came up with the proof (so it seems fair to call it a trigonometric proof even if that can be got rid of). [PS: I just found that some scans of Loomis's book are online: https://personal.math.ubc.ca/~cass/Euclid/java/html/L.pdf https://personal.math.ubc.ca/~cass/Euclid/java/html/L.pdf (1927), https://files.eric.ed.gov/fulltext/ED037335.pdf https://files.eric.ed.gov/fulltext/ED037335.pdf / https://www.lapasserelle.com/documents/Pythagorean_Proposition_Elisha_S_Loomis.pdf https://www.lapasserelle.com/documents/Pythagorean_Propositi... (1940 second edition) — see the foreword where he says “Fifth, that no trigonometric proof is possible”, elaborated on p. 193(1e)/244(2e) in section called “No trigonometric proofs”.]
- sedatk 4y ago
  Thanks, this has been very explanatory. It’s unusual that maths, the language known for precise definitions, has no unambiguous definition of trigonometry. It’s what it is though.