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. The law of sines is required. But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.
So afaict no circular logic is being used, but at the same time it doesn't seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold. I see it simply as a creative, unexpected proof.
The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, you can just taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.
Somehow the second gimmick (the infinite series construction instead of Einstein's elegant and simple construction) makes our monkey brains not notice the first gimmick.
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 necessarily be replaceable by the ratio of angles and lengths of some triangle. What am I missing?
Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible.
So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is imo the interesting part, and can stand on its own merits.
I don't think the "uses trigonometry" part is hype. They do use the definitions and law of sines, they just cleverly avoid the parts of trigonometry that depend on the Pythagorean theorem.
I address this in my OP. If you watch the video of the proof, you will see that the "law of sines" is 1 step away from the ratio definition of sin. You just drop one altitude, apply the definition again to the similar triangles, and re-arrange. It is almost content free as a result -- I see no reason using this in a proof would have special significance. For example, the standard proof using similar triangles (https://sumantmath.wordpress.com/2020/08/16/proof-of-pythago...) is implicitly using the law of sines.
The hype part is the implication that impossible trig barrier was shattered by their proof.
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.
4 dimensional triangles are the same as 3 dimensional lines. They don’t exist in the 4th dimension any more than they exist in any dimension >= 3. You would need a fourth side/point in the polygon in order for it to have any position in that dimension.
(It could be a triangle in dimensions 2-4 from our perspective but to the triangle it only has 3 dimensions any way you arrange it.)
Or you can bend a triangle in another dimension(s), but then it’s not a triangle by the commonly accepted definition. (E.g a 270° “triangle” on a sphere)
I wish I had a more modern summary of the papers mentioned in the linked paper
> Tannery, Fonctions d'une Variable, 1886, p. 147. Osgood, Lehrbueh
der Funktionentheorie, 1912, p. 582. Van Vleck and H'Doubler, Transactions Amer. Math. Society, vol. 17 (1916), p. 30
because we spent an entire semester at the university in one class working on these two.
> because we spent an entire semester at the university in one class working on these two.
Yeah, the Math Overflow answers are a bit sparse, and I think the Euler formulation (while clever) is a bit of a red herring and might be circular. Trying to slowly go through that 1917 paper, thanks for linking!
I'm not familiar with this result, but this comment is phrased in the language of Ordinary Differential Equations, so I'd look for a textbook on solving systems of ODEs and expect to find a technique that can prove that this is the unique solution (at least assuming differentiability of S(x) and C(x)).
I did not know that one! It's a more complex version of the well-known functional definition of the exponential function, i.e. the unique continuous function satisfying
The Pythagorean theorem is the most frequently re-proved thing in the history of math, with hundreds of published proofs and entire books dedicated to collecting them. The reason to come up new proofs these days is solely for the novelty, not because we have a need for a simpler proof. The fact that it's such a popular subject for proofs is why novel proofs are inherently interesting, regardless of their complexity.
> The reason to come up new proofs these days is solely for the novelty, not because we have a need for a simpler proof.
On the contrary, every new way to prove a known theorem has the potential to be applicable in other areas of the same or related fields, extending the mathematicians' toolset with new instruments. These new methods often serve as a seed for new discoveries.
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".
Hey, come to think of it, I'd be perfectly happy with it being the corner of a solid cube[*] of rock ploinked diagonally down into the sand; the whole idea of interior chambers and tunnels does kinda make it feel too fussy, not as clean and sleek as it could be...
For applied maths I agree with you. But when exploring novel and innovative approaches I disagree. There is so much we can learn from finding new ways to look at the world. The initial take may be complex or clever, but it can lead to a deeper understanding that allows for later simplification of entire subject areas.
'Aesthetic' appearance in math is important in helping drive mathematical innovation and help new human beings derive pleasure from that wonderful field.
Programming is kind of an applied mathematics where efficiency does matter because it's a tool, a means to an end.
Not to say that people can't find aesthetics in programming, nor that they shouldn't, rather in math at least the pleasure of discovering a new way of doing/proving something is the end in of itself.
It's pleasure for me to see another way to do or prove something; I can only imagine the feelings this teenager got from actually making a discovery.
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. The law of sines is required. But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.
So afaict no circular logic is being used, but at the same time it doesn't seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold. I see it simply as a creative, unexpected proof.
[1] https://www.youtube.com/watch?v=4xO8xqLyEbA