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New Orleans teenagers found a new proof of the Pythagorean Theorem

keith-mcnulty.medium.com

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Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#231

Post by svat »

sedatk wrote: post #164
svat wrote: post #159

Earlier quoted context omitted.

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…

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 (1927), https://files.eric.ed.gov/fulltext/ED037335.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”.]

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#232

Post by foldr »

boomboomsubban wrote: post #55
secondcoming wrote: post #17

Earlier quoted context omitted.

I don’t think Privilege matters in all cases, look at the background of Ada Lovelace.

... The daughter of Lord Byron?

Their point is that Ada Lovelace is still a salient example of a woman who made significant discoveries in computer science, even though she was also privileged in other ways. Indeed, in general, it's often otherwise-privileged members of disadvantaged groups who are the first to make advances.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#233

Post by deathlight »

fnordpiglet wrote: post #130
jonahx wrote: post #99

Earlier quoted context omitted.

It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…

Is there more to trigonometry? I’m not a abstract math person, so forgive the ignorance, but my understanding was all trigonometric functions derive from ratios of angles and lengths of triangles so in the end each occurrence of a trigonometric function can be replaced by the corresponding ratios in some triangle. There are other ways to construct things, such as power series representations, etc, but even these must…

Sine and cosine can take as their input any real number including negatives and including very large positive numbers. Their outputs can also be negative numbers between negative 1 and 1 if they have real inputs. None of this necessarily makes any sense if you're considering a purely geometric naive interpretation in terms only of ratios of lengths. You have to introduce concepts like modulo the angle in a circle and analytic coordinate system for it all to square with normal naive intuition.

In fact the sign and cosine can take as their inputs any and produce as their output any complex number. You have to come up with some very interesting triangles to make this makes sense. I'm sure it might be doable but they would potentially be four dimensional triangles and I haven't explored that concept very deeply.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#234

Post by Tao3300 »

cabalamat wrote: post #209
jonahx wrote: post #99

Earlier quoted context omitted.

It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…

> It's a very clever proof. Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.

The mistake is seeing this as a constructed thing. Math is already there, we only uncover it. If they revealed a previously unseen chamber in the Great Pyramid or something, you wouldn't say "aw nuts, that overcomplicates our existing knowledge of the structure".

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#235

Post by Tao3300 »

hypertexthero wrote: post #94
dendrite9 wrote: post #49

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…

This brings to mind the visual solution to calculating triangle area in James Somers post “I should have loved biology”: > In his “Mathematician’s Lament,” Paul Lockhart describes how school cheapens mathematics by robbing us of the questions. We’re not just asked, hey, how much of the triangle takes up the box? > That’s a puzzle we might delight in. (If you drop a vertical from the top of the triangle, you end up wi…

I hated math for most of my childhood. I tested into an advanced track and had to be sequestered into lower level courses in the next higher grade do to a lack of effort.

Then when I hit college and had Discrete and Calculus, I found out I loved it and wound up minoring in math. Though my arithmetic is still slow and my trig has major gaps in it due to school math just sucking in general.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#236

Post by yafbum »

I am puzzled by the article's spin on this, which really centers on the fact that this original proof was authored by two teenage African-American girls from the South, as if the interesting thing here was not so much the proof itself than the idea that there are gifted mathematicians from underrepresented backgrounds and skin colors. In my experience, pretty much in any place and social stratus you might visit, there are bright kids who love math. The challenge is more about what happens next in their career — can these kids get affordable higher education, and a career track that values their gift? My data point of one is a friend of mine, who was an extremely bright student of physics, but had to drop out of college early because he couldn't afford it and needed to start making money. That kind of thing could explain skewed representation in science more than lack of talented high schoolers...

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#237

Post by TheRealPomax »

"OK, so here’s how I think it goes" what do you mean "you think"? It was presented to the American Mathematical Society, there is no guessing here, you show what they showed. Why laud their background and then immediately dismiss their finding by not showing their proof? And then you can't even be bothered to draw a right angled triangle as a right angled angle, something literally any drawing program with straight lines will let you do? Forget you, buddy.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#238

Post by matthewdgreen »

yafbum wrote: post #236

I am puzzled by the article's spin on this, which really centers on the fact that this original proof was authored by two teenage African-American girls from the South , as if the interesting thing here was not so much the proof itself than the idea that there are gifted mathematicians from underrepresented backgrounds and skin colors. In my experience, pretty much in any place and social stratus you might visit, the…

I think it's excellent that you, a commenter on HN, understand that important contributions can come from all parts of society. I really wish that understanding was shared more broadly across this whole country. Unfortunately I've been alive long enough to see that it really is not.

I suspect that more people felt as you do, we wouldn't have so many barriers to opportunity for kids. Moreover, the existence of those barriers wouldn't be so disproportionately correlated to race and place-of-birth.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#239

Post by steve76 »

lern_too_spel wrote: post #113
pelorat wrote: post #111

Earlier quoted context omitted.

I think it's just fucked up these things needs to be mentioned at all. We're all fucking humans. Great job to these persons* * = a human being regarded as an individual

That's what centuries of oppression does. It makes this rare. Hopefully, elevating these achievements in the community can make a difference. Ignoring that this problem exists certainly won't. If you don't care about broadening the pool of mathematicians, simply ignore the part of the article about that subject.

[dead]

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#240

Post by boomboomsubban »

foldr wrote: post #232
boomboomsubban wrote: post #55

Earlier quoted context omitted.

... The daughter of Lord Byron?

Their point is that Ada Lovelace is still a salient example of a woman who made significant discoveries in computer science, even though she was also privileged in other ways. Indeed, in general, it's often otherwise-privileged members of disadvantaged groups who are the first to make advances.

The equivalent in Lovelace's case would be saying she's a woman from an area where most women are illiterate and work sixteen hour days in death trap factories. It's true, but ignoring that she was part of the nobility adds a disingenuous element to the statement.

And saying that privilege doesn't matter is nonsense, without it she's unlikely to have had the opportunity to do what she did.

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