[HN Gopher] New Orleans Teenagers Found a New Proof of the Pytha...
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New Orleans Teenagers Found a New Proof of the Pythagorean Theorem
Author : malshe
Score : 138 points
Date : 2023-04-08 21:13 UTC (1 hours ago)
(HTM) web link (keith-mcnulty.medium.com)
(TXT) w3m dump (keith-mcnulty.medium.com)
| HL33tibCe7 wrote:
| > By all accounts, these two teenage math students are the exact
| opposite of the majority of the math establishment. They are
| female, they are African-American, and they come from an area
| which is not particularly renowned for producing high academic
| achievers. This is just an awesome turn of events and one which
| should inspire anyone -- no matter what their ethnic, gender or
| socio-demographic background -- that excellence in your chosen
| field of study is always attainable
|
| I think it's remiss not to point out here that these students
| attend a private, fee-paying Catholic all-girls academy.
|
| None of that detracts from the impressive achievement of
| discovering this elegant proof, of course.
| secondcoming wrote:
| I don't think Privilege matters in all cases, look at the
| background of Ada Lovelace.
| HL33tibCe7 wrote:
| It matters when you try and make the point that this case
| "should inspire anyone -- no matter what their ethnic, gender
| or socio-demographic background", as the article does
| JKCalhoun wrote:
| That's not the way I took the article. Rather I took it to
| mean something slightly different:
|
| Your ethnicity and/or gender should not be seen as a
| barrier to advanced mathematics.
|
| There is a nuanced difference there.
| pedrosorio wrote:
| 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.
| boomboomsubban wrote:
| ... The daughter of Lord Byron?
| pizza wrote:
| It would be really interesting to see if this could be applied to
| other analogous scenarios that are sometimes called Pythagorean
| theorems, in particular I'm thinking of the Pythagorean Theorem
| of Information Geometry- p* = argmin p of P (a
| set of possible distributions) of D_KL (p||q) (where q is eg your
| model's distribution)
| zem wrote:
| I'm impressed, I've been following the story in the news and
| various blogs and magazines, and this is probably the best write
| up I've seen of it.
| oceansea wrote:
| [flagged]
| johnfn wrote:
| The proof does indeed seem to demonstrate a^2 + b^2 = c^2 from
| first principles. If you have specific problems with
| assumptions the proof is making you should state them; as it is
| it's unclear what problem with the proof your post is
| responding to.
| civilized wrote:
| I have a PhD in math and I have no idea what you are talking
| about. This is a proof.
| joe_the_user wrote:
| I can't see how your claims relate to the article involved. The
| proof that's described uses a construction of infinitely many
| triangles and the formula for the sum of a geometric series, so
| it basically is a proof in it's form, whether it's correct or
| uses circular logic is another question but not it's merely a
| demonstration of properties (which is one way student proofs
| often fail but I don't see that here).
| silisili wrote:
| > This is the Louisiana education system in the USA, a state
| which is rather notorious for having sub-par education
|
| What does this have to do with anything? Exceptional people
| come from absolutely anywhere, and seek to learn on their own.
| It's as true in Louisiana as it is in Massachusetts.
|
| Ken Thompson was born and (at least partly, afaik) raised in
| Louisiana, for example. Are we all just dumb people using a
| dumb person's language and OS?
| KRAKRISMOTT wrote:
| Excellent points, I am not sure why you were flagged as dead.
| oceansea wrote:
| [flagged]
| beepbooptheory wrote:
| Perhaps this is a chance for some self-reflection on your
| part? This doesn't seem to be a mean pileon of people
| hating on you, and its not a particularly charged issue to
| begin with. You are simply not being clear at all in your
| argument and then otherwise using this whole thing as a
| platform to talk about your thoughts on the public
| education system in Louisiana, and most likely I think some
| broader thoughts of yours on how kids are just too damn
| coddled these days. (It should be noted that these students
| are not in the public education system anyway!)
|
| Maybe just try and reflect on whether you would take such
| an issue with this if the subjects of it didn't challenge
| your various implicit assumptions so much?
|
| Or at the very least, attempt to be clearer in your
| writing. and articulate your arguments. As a champion of
| life and knowledge, this should really be a priority for
| you.
| haskellandchill wrote:
| Because they are wrong? Read the article.
|
| > to make a proof of it would be to show why a^2 + b^2 = c^2
|
| That's what the article shows as a final step. And it
| outlines what assumptions are made and how the final step is
| demonstrated. I have no idea what OP is rambling about.
| kurthr wrote:
| Yeah, they made a symmetry argument, a series expansion
| argument, and a law of sines argument. None of those
| require the Pythagorean Theorem and are all separately
| proven. There may be a reason that it's wrong (I'm not a
| mathematician but the article's author is).
|
| So, they showed why the sum of squares is true for the
| hypotenuse of a triangle. It's even an exact proof rather
| than a limit argument. That I could follow the steps, and
| other mathematicians seem unaware of this combination of
| tools makes it seem relatively novel.
| oceansea wrote:
| [flagged]
| klyrs wrote:
| Speaking as a mathematician with a PhD, you're wrong
| about proofs. You're wrong about this site, which intends
| to promote a curious and celebratory attitude towards
| knowledge. Ironically, it is you who is displaying the
| hateful attitude that you bemoan. Please adjust your
| attitude, or see yourself out.
|
| And do re-read the article; the author is attempting to
| reconstruct their proof without seeing it. You're
| attacking the students without even seeing their work.
| Incomprehensible.
| roywiggins wrote:
| There's no actual requirement for a proof to give you a "why."
| The steps just all have to be be logically correct and not
| accidentally, eg, assume what is to be proven.
|
| Lots of proofs are basically unhelpful at giving you a "why,"
| that doesn't disqualify them from being a proof.
| oceansea wrote:
| I was using Stanford's definition. https://web.stanford.edu/c
| lass/archive/cs/cs103/cs103.1202/l...
| mathteddybear wrote:
| "A mathematical proof is an argument that demonstrates why
| a mathematical statement is true, following the rules of
| mathematics."
|
| yes, and?
| roywiggins wrote:
| That definition (as interpreted by you) would appear to, at
| the very least, preemptively disqualify any proof by
| exhaustion ("it's true because we brute forced all the
| possibilities and didn't find any counterexamples"), which
| is a perfectly valid proof method.
|
| https://en.m.wikipedia.org/wiki/Proof_by_exhaustion
|
| "All the steps are valid but it doesn't explain anything
| and I'm dissatisfied" is not an argument against a proof's
| validity. Sometimes the argument is just "A implies B
| implies C implies D implies E implies P, QED" and as long
| as you are convinced that each step is valid, you've got to
| take the proof itself as valid, even if the gestalt isn't
| satisfying.
| avindroth wrote:
| ...I took several classes with Keith (who teaches this
| class), and he would definitely consider this a proof.
| Also, you should look to the math department, not the CS
| department if you want to appeal to authority.
| yCombLinks wrote:
| That's a misc slideshow from a class, not a formal
| definition.
| oceansea wrote:
| [flagged]
| threeseed wrote:
| Pretty relevant when you're criticising the work of these
| teenagers because it doesn't meet your definition of a
| proof.
| SalmoShalazar wrote:
| It was a useful comment and your odd defensive posture is
| the problem here.
| threeseed wrote:
| That is not Stanford's official definition. That is the
| definition from one of their Computer Science lecturers.
|
| I can't speak to it's accuracy but I would imagine better
| sources exist.
| uberman wrote:
| Apparently there are some university math professors such as
| Lozano-Robledo who disagree with you.
|
| I can't say if their proof will hold up under real peer review
| but I hope it does.
| pdpi wrote:
| I can't see it failing to pass muster.
|
| It's more complicated than most proofs of Pythagoras's
| theorem I know, but it's still simple enough that several
| people have reconstructed it from just a diagram and some
| basic notes on how it should work. For so many people to have
| now independently reconstructed it without anybody finding
| issues is a clear indication that, even if the way they wrote
| it up has mistakes, the core ideas are solid.
| ivan_gammel wrote:
| Can you elaborate on what is a proof in your opinion? Having a
| degree in applied physics and mathematics from MIPT I'm
| struggling to understand your point. It does look like a proof
| of the theorem to me.
| oceansea wrote:
| https://web.stanford.edu/class/archive/cs/cs103/cs103.1202/l.
| .. I am using Stanford University's definition.
| klyrs wrote:
| You're pointing at a document without specifying what
| precisely in that document would indicate that the argument
| (which is presented only as an outline and extrapolation of
| the students' work) does not constitute a proof. If you're
| going to attempt to prove something, _do_ try to use at
| least a modicum of rigor.
| kurthr wrote:
| Ok, I have read the first 20 slides for a CS class at an
| institution from which I have a degree. I feel no closer to
| understanding why it is relevant.
|
| Can you specify which step in the "new proof" is wrong,
| unproven, or tautological (itself requires the Pythagorean
| theorem)?
|
| The simplest reason why it is explanatory as defined in
| your cited reference is that the Euclidean distance is
| defined by the hypotenuse of the triangle, that triangle
| (and similar triangles by construction) have ratios which
| are also similar, the distance along a side is the sum of
| those distances in an infinite series, the series
| expansions of sin and cosine are known along with the law
| of sines and are independently (from the Pythagorean)
| proven.
| [deleted]
| umanwizard wrote:
| In the first paragraph under "Deriving lengths of the sides of
| the smaller triangle", how do we know the length of the third
| side is 2a^2/b ?
| n3uromancer wrote:
| From cosb = a/c and cosb = third side/((2ac)/b)
| mathteddybear wrote:
| "This" in "This leads to the third side of this triangle to be
| 2a2/b." means "the same line of reasoning".
|
| it's not even trigonometry in that paragraph, just rudimentary
| proportions of similar triangles
| [deleted]
| Y_Y wrote:
| I've seen this construction before. It's very cool, but I don't
| think it's novel. In any case it's a cool article and it must be
| great for a couple of young students to present to the AMS.
| joe_the_user wrote:
| I have read about this proof for a bit and this is the first
| write-up that gives the slightest details. The phrase "using
| trigonometry" is confusing. What they do is assume functions sine
| and cosine exist, as normally defined, as ratios of triangle
| values, without assuming these have the various Pythagorean-
| theorem derived properties. They then construct an infinite
| series of nested triangles and use the formula for the sum of
| geometric series' to derive the length of the original triangle's
| hypotenuse. It certainly seems clever.
|
| I'm still confused what axioms they're effectively using relative
| to the usual Pythagorean theorem proofs - most of these use the
| formula for area of a right triangle and this seemingly doesn't.
| On the other hand, it seems an infinite construct would require
| things like the axiom of induction, which may be included in
| axiom of axiomatic geometry.
| scythe wrote:
| Induction isn't so bad. I'm not sure how they're getting away
| with the Law of Sines, though. The usual proof of LoS that I
| know is dependent on the existence of the circumcircle. But I
| don't know how to prove the existence of the circumcircle
| without dragging in a lot of geometry. Or you can use the area
| formula, which makes the proof similar to other arguments that
| use the area formula.
| dendrite9 wrote:
| 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 section is available as a pdf here:
| http://www.personal.psu.edu/mxl48/Welcome_files/Sample.pdf
| nolamark wrote:
| for completeness, here is a reference to the AMS presentation.
| https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa...
|
| [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 Pythagorean
| Proposition by Elisha Loomis) the author flatly states that
| "There are no trigonometric proofs, because all the fundamental
| formulae of trigonometry are themselves based upon the truth of
| the Pythagorean Theorem." But that isn't quite true: in our
| lecture we present a new proof of Pythagoras's Theorem which is
| based on a fundamental result in trigonometry--the Law of Sines--
| and we show that the proof is independent of the Pythagorean trig
| identity \sin^2x + \cos^2x = 1.
| malshe wrote:
| The actual title is "Here's How Two New Orleans Teenagers Found a
| New Proof of the Pythagorean Theorem" but I don't know why HN
| automatically converted it to this title.
| capableweb wrote:
| "Here's How" is usually used for clickbait stuff and adds
| nothing of substance to the title itself.
|
| New Orleans Teenagers Found a New Proof of the Pythagorean
| Theorem
|
| is both shorter and less editorial than
|
| Here's How Two New Orleans Teenagers Found a New Proof of the
| Pythagorean Theorem
| anigbrowl wrote:
| No. The short version titles an assertion, the longer version
| titles an exposition.
|
| HN automatically strips things like this on submission but
| you can edit the submission title after posting to put it
| back in when it's appropriate, as here.
| majormajor wrote:
| I get why that's the default behavior but "adds nothing of
| substance" is a huge generalization that fails in this case.
|
| An article with zero details about the proof could easily be
| titled "New Orleans Teenagers Found a New Proof of the
| Pythagorean Theorem" but couldn't accurately be titled
| "Here's How Two New Orleans Teenagers Found a New Proof of
| the Pythagorean Theorem." Whereas here it says it'll have
| more details and it does. (although TBH this is less a
| "here's how they found" and more a "here's what they found",
| if I'm being extra pendantic).
|
| The shorter title is less descriptive in this case.
| muststopmyths wrote:
| just taking out the "here's" and leaving "How New Orleans
| Teenagers Found a New Proof of the Pythagorean" would have
| preserved the meaning and also made it seem less
| clickbaity.
| varenc wrote:
| This is getting off topic, but IMHO the HN default behavior
| here is reasonable.
|
| There's certainly situations where removing the leading
| "Here's How" makes the title worse, but I think those
| instances are rare and in general this rule leads to better
| titles much more often than worse titles. Manual human
| review would of course be better, but dang only runs in
| O(n) time. Basically, it's not perfect but I think it does
| much more good than harm.
| satvikpendem wrote:
| HN strips clickbait titles. Same thing for listicles.
| johnfn wrote:
| HN has some sort of algorithm to rewrite post titles to be a
| bit less link-baity.
| bblpeter wrote:
| Sorry but that's not new
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