161–170 of 287 posts
for completeness, here is a reference to the AMS presentation. https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa... [my post updated to include abstract] Abstract In the 2000 years since trigonometry was discovered it's always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythag…
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The sine function can be defined in terms of its own behavior, using its first-order differentiation and no reference to triangles. See this detailed article on sine. https://betterexplained.com/articles/intuitive-understanding... There’s section there titled Part 2: Understanding the definitions of sine.
You can define sine and cosine together using the functional equations S(X)C(Y)+C(X)S(Y)=S(X+Y) C(X)C(Y)−S(X)S(Y)=C(X+Y) The only solutions to this are the constant 0 functions and the sine-cosine pair.
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> > Your ethnicity and/or gender should not be seen as a barrier to advanced mathematics. > I would also take that interpretation if the author didn't explicitly mention "socio-demographic background". They should just leave that one out next time. Please help me understand. Are you saying that it’s okay to say that ethnicity and/or gender should not be seen as a barrier to advanced mathematics, so long as you don’t…
Read the thread I am responding to.
"...a few trigonometric proofs of Pythagoras have made the rounds since then. Claims in the media that Johnson and Jackson’s proof is the first trigonometric proof of Pythagoras are overblown..." I don't like how the author drops the bomb "this has been done before" and tries to compensate that with the "simple and lively" language of the proof, or the background of the girls. Why didn't author reference any of the p…
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 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.
I remember reading a book in high school and realizing there could be other ways to prove things that I had been taught only one way. One that particularly stood out later was using a rotating fishtank to prove the pythagorean theorum. A good friend of mine was so delighted by the example I gave him a copy of the book I found it in. https://press.princeton.edu/books/paperback/9780691154565/th... ) The relevant sectio…
The picture is drawn really poorly : The first composite angle : alpha+beta at the top of the A side, is a right angle and should be drawn as such (if comes from the hypothesis of the Pythagorean theorem that the two sides are orthogonal so alpha+beta+90° = 180°). Otherwise you can't understand why the ration A/C would be sin(2*alpha).
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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 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…
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 wit…
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?
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Why are being downvoted?, there have been crazy comment on this website, hyping to the max ChatGPT and how all the humans now are useless, I was just ignoring HN for the past weeks because it have been ridiculous
Probably because it has nothing to do with the article. It's just a rant about something entirely unrelated.
Assuming you have read the comment, it is totally relevant to the article with AI (in this case LLMs or GPTs like ChatGPT) still not being able to transparently prove unsolved mathematical problems or even generate such solutions with and without supervision, since even if it was supervised, it will still generate it incorrectly and as with its black-box nature, it cannot reason or explain transparently.
The fact the those teenagers were able to create and derive this proof without regurgitation and withstood the scrutiny of experienced mathematicians tells us that it requires creative thought with transparent reasoning in the field to go against the books written by experts that once said it was ‘impossible’ until proven otherwise.