I've always wondered if it's possible to harness teen minds to solve significant math problems in high school, if you formulated them well and found the right scope. I think it's possible.
MIT's PRIMES program does exactly this - they give advanced high school students a mentor who picks out a problem, gets them up to speed on what is known, and then they work on the problem for a year and publish their results. It tends to work best with problems which have a computational aspect, so that the students can get some calculations done on the computer to get the ball rolling.
It is very possible!
Just this year these girls discovered a proof for the Pythagorean theorem using nothing but trigonometry, a feat considered impossible until they did it: https://youtu.be/VHeWndnHuQs
Unfortunately it seems their proof already had the Pythagorean Theorem embedded within its implicit assumptions - they define measure of an angle through rotation of a circle. They don't explicitly define circle, but from their diagram they hint at the "understood" definition, namely a set of points equidistance from a central point, while using Euclidean distance as the metric.
That's not true at all.
To understand why, read Euclid.
Geometry has made a bit of progress since Euclid's time. Its become a bit more rigorous.
Euclidean geometry is based on five axioms, and some other terms left undefined.
The fifth postulate - the parallel postulate - was considered so irksome that for hundreds of years, many attempted to prove it using the other four, but failed to do so, and almost drove some crazy. In the late 19th century it was shown you can generate perfectly valid geometries if you assume it to be false somehow - either no-parallel (spherical geometry) or infinite parallel (hyperbolic)
Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it. Like I could draw a "circle with a radius of 1" using taxicab distance, and it would look like a diamond shape.
Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
The (non-generalized) Pythagorean theorem is part of Euclidean geometry, so non-Euclidean geometry is irrelevant to this discusion.
> Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it.
You do it using an axiomatic compass, a device that copies length in a circular pattern but does not measure it. Lengths are measured using constructable line segments.
Are you implying that nearly all the hundreds of proofs of Pythagorean theorem, which do not use modern rigorous definitions, are not valid proofs?
> Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
So? The Pythagorean theorem is very easy to prove. There are hundreds of proofs created by amateurs. That doesn't make them "not proofs" simply because other proofs exist.
Strictly speaking, the postulates say nothing about compasses, or even straihhtedges/constructions. Also introducing lengths similarly, involves introducing number which is not a "pure" geometry concept. The third postulate just says that a "circle" exists defined by a point and a radii (which also, not a "pure" geometry construct since it involves a metric - i.e. number.
I would say yes, alot of the fundamental proofs while not striclty "incorrect" or false, are rather informal and contain some hidden axioms/circularities.
Tarski put geometry on a more secure footing using first-order logic.
Similar to how Calculus wasn't on a solid logical foundation until Riemann.
> a feat considered impossible until they did it
Hm? https://www.cut-the-knot.org/pythagoras/TrigProof.shtml
> J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009)
And Zimba's proof terminates in a finite number of steps.
Yes that's the education system. But I suspect you mean in some automated turk fashion.
Sorry, I meant while they were still in high school; I've edited my original comment to make that clear.