The assertion is that it is impossible to prove the Pythagorean theorem using trigonometry and they proved that this assertion is false by developing such a proof. This assertion is stated because trigonometry uses principles that are based on the Pythagorean Theorem, so you would be committing the fallacy of begging the question by using it to prove the Pythagorean Theorem.
So what exactly was the substance of their proof? No news reports give any technical details. I have been able to find only an abstract of the paper they presented, also no help.
I don’t doubt that these could be some brilliant students but frankly I am skeptical that a couple of high school students have developed a math proof that has eluded mathematicians for 2000 years. After all, there have been some pretty smart mathematicians in that time.
So far, the result has not yet been peer reviewed, apparently. Mathematicians, especially the ones who want to see kids learn and develop, have encouraged them to submit to peer review.
So, basically, this is all self-claimed for the moment.
It’s not impossible at all that they’ve come up with a heretofore unknown proof. Eluding mathematicians is not necessarily evidence against. The concept behind the Pythagorean Theorem itself is thousands of years older than Pythagoreans (though they did build it up solidly from principles) and there have been several proofs constructed over the years (including one by US President James A Garfield). And it’s not like there are hundreds of mathematicians desperately searching for such a proof (it’s already been proved, after all, so another one would be sort of cute but not earth shattering).
That said, the kids are clearly bright and worked hard. They certainly should be congratulated and encouraged in their efforts but they should also be encouraged to submit for peer review, because that’s how math works - your work needs to stand up to rigorous analysis from other people. There’s also a pedagogical benefit to going through that process, no matter the result.
For what it’s worth, something similar happened with Fermat’s Last Theorem. Andrew Wiles presented at a conference, and then his proof had to go through review (where they actually did find a bit of work that needed to be done, which was readily done).
The citation says they used the Law of Sines. Which is suggestive. That law is provable without other trig or Pythagoras. So it may well be. But there are a lot of caveats. Just what we consider to be “trigonometry” in the steps of the proof is the big question. Most of the usual trig laws are provable using Pythagoras. (Double angle and so on.)
Sometimes finding a proof is made vastly easier if you know it can be done. The accepted wisdom that it cannot be done for some apparently reasonable but in reality not watertight reason just stops people bothering.
However, it isn’t as if there are not already a huge number of existing proofs of Pythagoras.
The Law of Sines isn’t really quite trigonometry. It is in some ways a coincidence that the proof is called what it is. It does involve ratios of lengths of right angle triangles, but if you didn’t name them the sine of the angle and used some other nomenclature, the law would still be fine.
It would not surprise me that their proof is yet another proof of Pythagoras that does not otherwise trigonometry, but can be cast in terms of trig ratios in the manner in which it derives from the Law of Sines. In order to use other trig laws, you would need to show how they can themselves be proved without Pythagoras. That is the stumbling block of proving Pythagoras with trig. You can’t use a law unless it can be proven outwith Pythagoras’ Theorem.
The cite suggests that show a new proof that sin^2\theta + cos^2\theta = 1 is provided. That is of course just a restatement of Pythagoras. It is again, not really trigonometry in and of itself.
I’m pretty sure there are entire books listing different ways to solve this theorem.
Aha. This article from The Guardian cites the book and gives an explanation. Of sorts.
Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “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, the abstract counters, “that isn’t quite true”. The pair asserts: “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 sin2x+cos2x=1.” In short, they could prove the theorem using trigonometry and without resorting to circular reasoning.
Maybe @Hari_Seldon could give us a better isea what’s going on.
The article says “relies on an aspect of trigonometry which does not derive from Pythagoras’s theorem in the first place”
The second pic of the presentation is clearly showing part of the Law of Sines. As I noted above, this particular law can be proven without recourse to Pythagoras. But it does contain right angle triangles and a ratio that is the sine of an angle.
I begin to feel more sure that the proof depends upon the Law of Sines being classed as “trigonometry” because well, it uses the word “sine”. This sits on an odd dividing line. The term trigonometry isn’t as well defined as we might like here.
So, if we defined trigonometry as including any aspect of geometry where you calculate a sine, maybe. But if we define trigonometry as geometry that makes use of the gamut of laws derived from Pythagoras’ Theorem, well clearly that is going to involve circular reasoning.
So if they did come up with a new proof for the Theorem, it’s a good story but not revolutionary, but on the other hand, it could be a blockbuster? I’m not dismissing it, I’m just not a math guy and want to make sure I understand the general “wow” of this story depending on what we find out about their work.
It’s not as though this is some famous unsolved problem that mathematicians have been trying, unsuccessfully, to solve for hundreds of years, although some of the reporting about it, including the sensationalist Daily Mail article that @Petek linked to, make it sound that way.
The Guardian article that @Exapno_Mapcase linked to includes a link to a scanned PDF of the book The Pythagorean Proposition, a 20th century book (which math writer William Dunham has called “slightly kooky”) that attempts to catalogue all known proofs of the Pythagorean Theorem. The book’s foreward, on p.9-10 of the PDF, is worth reading for context. There the book’s author makes some claims about what types of proofs of the Pythagorean Theorem are possible, among them “that no trigonometric proof is possible.”
I don’t know whether others have checked or commented on his claims about what types of proofs are possible—it’s not a topic that I’ve encountered. At any rate, it’s not the kind of thing that mathematicians have been widely discussing for 2000 years. (To clarify, mathematicians have been widely discussing the Pythagorean theorem itself, and how to prove it, for 2000 years; just not, as far as I know, the possibility of a trigonometric proof.)
The only thing I can add is that the law of sines is independent of any appeal to Pythagoreas. It depends only on knowing that an angle lying on a circle is 1/2 the subtended arc and, in particular that an angle whose arc is a semicircle is right. The law of cosines, however, is not independent on Pythagoreas; it is in fact a generalization of it. Now I don’t have the faintest idea how to prove it using the law of sines, but I do not imagine that mathematicians have spent 2000 years trying to. That doesn’t mean it isn’t interesting.
Since I am here, I would like to point out that what @Great_Antibob said about Wiles readily filling in the gap isn’t really true. It took a year and was actually done by Wiles’s student Richard Taylor who doesn’t get enough credit.
So how is it still a theorem when it’s been used to calculate area for centuries? Maybe it doesn’t strictly fit the definition of a proof, but it’s not wrong.
I thought a year was readily enough but fair enough.
Agreed on Richard Taylor, though. He really does not get enough credit.
I’m not sure what the question is here.
The Pythagorean Theorem has been proven true for at least a couple thousand years now and gainfully used for thousands more.
Maybe the concept of a ‘theorem’?
A theorem is not a scientific theory, if that’s the confusion.
A theorem is a mathematical statement that has been proven true (given whatever starting assumptions you have - in this case, the 5 postulates that make up Euclidean geometry plus some definitions and rules of deductive reasoning and so on).
So, once you have a proof of such a statement, you have your theorem.
This is opposed to science, where there is very rarely (never?) really absolute proof but once you have a preponderance of evidence, you accept a theory as true.
By contrast, there are things we do not yet know to be true. Some of them we strongly believe are true (and have yet to find contradictions for) but have yet to be absolutely proven true. For example, the Goldbach Conjecture, which can be summarized as “every even number larger than 2 can be expressed as the sum of 2 prime numbers” .
ETA: on review, unless I’ve also been whooshed as well
By the way, this means that Fermat’s Last Theorem was not, strictly speaking, a theorem until it was proved by Wiles et al in the 1990s. Before that it would have been more accurate to refer to it as a conjecture or a hypothesis.
The reason it was known as “Fermat’s Last Theorem” for all those years is (I assume) that Fermat claimed to have a proof (that was “too big to fit in the margin” of the book he made the note in).
While we don’t know for sure, it’s reasonably certain that Fermat did not actually have a valid proof. This is because:
Fermat’s claim to have a proof was in a “note to self,” not something he published or shared with anyone else, so he wouldn’t have felt a need to back it up or retract it if he later realized his “proof” was invalid.
In the hundreds of years since, despite lots of trying, no one has been able to come up with a valid proof that Fermat could have found (i.e. using the mathematics available in Fermat’s day).
We have a pretty good idea what proof Fermat thought he had, but that “proof” is flawed.
Googling around finds me this page, which includes a video explaining a possible explanation of the proof:
The mathematician who made the video looked at some of the pictures from their presentation and worked backwards. So it might not be exactly the same proof, but it’s compelling. The proof does use summation of an infinite series. But if you can handle that, the rest is relatively easy to follow.
Also, we have Fermat’s own proof of the case of n=4 and it’s very possible he had one for n=3.
He very likely realized himself that his own general “proof” was flawed. He never wrote of this proof or offered it as a challenge to others (which he often did for theorems he had proven) later in his life.
It’s not like notes in margins are intended for general audiences, and his margin notes were published posthumously only because of his son.
I’m also not really a math person, but I am a software person with some light interest in math.
My sense would be that the primary advantage to having an alternate proof of Pythagoras mostly benefits the world of mathematics by:
Introducing an alternative way to teach Pythagoreas Theorem that might be more easily understand by some. (But, probably not, given how many are out there already and pretty visual.)
Helping to get some mathy kids into the spotlight, so they’ll be able to get scholarships to university and continue on in the math world.
Giving trigonometry an external proof might help to make it less axiomatically reliant. (But, probably not. It’s probably been proven through non-trigonometric/non-circular proofs already so it doesn’t need an external proof.)
#2 is probably the main one.
That said, in the world of computers, some calculations are easier to perform than others. I could imagine that their solution would lead to new trigonometric formulas that are easier to implement in software or in silicon than the old ones. If there is some big outcome from the discovery, that’s where I’d expect it to land.
Specifically, he came up with a technique that he later used to prove (and publish) the case for n=3 and n=4 (i.e., there are no positive integers a, b, c for which a^3 + b^3 = c^3, nor for a^4 + b^4 + c^4). He could very easily have thought that this technique could be generalized for all n. It turns out that it can’t be, but it was an easy mistake for someone to make, especially if they’re just doodling around in the margins and not writing it out rigorously.
And of course, the fact that he did publish proofs for the n=3 and n=4 case, but not for the general case, pretty conclusively establishes that, whatever else happened, he didn’t have a full proof of the general case, or he would have published that, too.
As for the proof in question in this thread, it’s really difficult to pin down what it means to prove something “using” such-and-such. It’s like the old line “That and five bucks will get you a cup of coffee”: There are already oodles of proofs known for the Pythagorean Theorem, and you could always just take one of those proofs and include an extra useless given in it that states some trignometric fact.
To slightly paraphrase Feynman, “Mathematics is like sex. It might give some practical results, but that’s not why mathematicians do it.”. To professional mathematicians, the purpose is irrelevant: It’s all about the game. What complicated and beautiful things can you construct, using a simple set of rules?