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.
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.
Pound for pound, mining and processing minerals for batteries has a much smaller environmental impact compared to extracting and processing fossil fuels for gas.
I'm not sure if I believe that, at least the "pound per pound" bit.
Extracting oil usually involves drilling a hole and getting a material you can mostly (I think about 80%) turn into useful products, though hydrofracking involves handling a lot of water and oil from some places in Saudi Arabia contains a lot of sulfur that has to be removed.
On the other hand, many minerals are found in concentrations of less than 10%, often much less
The real advantages of the minerals are: (i) a car consumes about its own weight in fuel every year so in its lifetime it consumes maybe 10x it's weight in fuel, (ii) the use (as opposed to production) of that fuel has environmentally unacceptable effects, (iii) the minerals ultimately will be incorporated in a "circular economy".
Note that the automobile industry is a lot more circular than many (say food packaging) in that your local junkyard sells whatever parts it can (I know a guy who just bought a used Ford truck with doors rusted out who just bought two doors from a junkyard) and will send what is left to get crushed when metal prices are high. If you smack your car up at 110,000 miles likely you will get some body panels from this source.
Battery recycling is not a big industry now but it will be. Mining will still be more important than recycling as long as the world is in transition to electric cars. I was quite amused to find that the techniques planned for battery recycling are very similar to both established and in development techniques for recycling spent nuclear fuel.
This issue of the review includes a very cool submission from a few weeks ago which did pretty well on the front page: https://alex.miller.garden/grid-world/
I don't know about incurious, maybe "extreme". A lot of the tools, techniques, etc. mentioned in the article aren't appropriate for ... well, most websites probably. But they do have their roles to play in large, complicated systems.
And, if you're interested in some historical context for this "type characters and jump to point" functionality, the Canon Cat: https://youtu.be/o_TlE_U_X3c
The 2:00 timestamp in this Veritassium video on expertise goes into exactly that phenomenon. We acquire expertise by recognizing the patterns in what we do. But once you take the patterns away, a lot of that "expertise" disappears too: https://youtu.be/5eW6Eagr9XA
Jazz piano vs classical is a perfect example of this: same instrument, two paradigms with totally different idioms.
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