This paragraph puts to words something I’ve been mulling over for decades. Looking back to math class, or just looking at any discussion of math education, there’s always a discussion about students who just memorize and plug in the formulas, mentioned as the kinds of students who don’t understand what they’re doing. But that’s an incomplete explanation of what those students are missing that I believe the quoted paragraph completes.
It’s not about knowing how to apply a formula. It’s about understanding a formula as a property of the system you’re working with. It’s a great insight and one I’m probably going to mull over for a long while.
That kind of Gauss-level insight doesn't seem to be about properties so much as about understanding the relationships that define a domain - and being able to view problems in a kind of conceptual relationship space which can generate insights that massively simplify simple problems, and make harder problems tractable at all.
I remember school math was mostly rote learning, but some problems required that creative insight. Academic math seems to lean more heavily on creative insights.
Software dev is similar. You can always do things the hard way, but it takes a certain kind of insight to see time-saving shortcuts and simplifications that cut right to the heart of a problem. I suspect some people have a natural talent for that insight, while others are more likely to take the long way to a solution.
Gauss would have used a pre-(https://en.wikipedia.org/wiki/Karatsuba_algorithm) multiplication algorithm, with complexity at best O((log n) ^ 2).
Using an algorithm discovered in 2019 (cf. Harvey and van der Hoeven), it's possible to multiply in O(log n * log log n), if you have a lot of time. This is presumably optimal.
To given an example: the Pythagorean theorem, the parallelogram law, angles, transposes etc. are all properties/concepts a space has because it has an inner product.
Could you elaborate a bit? This sounds really interesting.
Pythagoras: <x,y> = 0 implies <x+y, x+y> = <x,x> + <y,y>.
Paralellogram: 2(<x,x> + <y,y>) = <x+y, x+y> - <x-y, x-y>
Angles: cos(angle from x to y) = <x,y> / sqrt(<x,x><y,y>).
Transpose: x^T is the dual vector defined by: x^T(y) = <x,y>.
the last one is especially subtle because most people think you can get the transpose by just flipping the vector, which is technically true but means you're basing your results on a possibly arbitrary choice of basis.
As an example of why knowing these relations is useful, imagine you want to measure angles on a map, then you might notice that there's no easily identifiable inner product, and indeed just measuring angles in an arbitrary projection will give different results depending on which projection you happen to be using. The trick is to go back to 3D space and measure the angles there (relying on the canonical inner product of 3D space) or to use a projection that preserves those angles like Mercator.
As another example conjugate gradient descent works better by using directions that are orthogonal through a more natural inner product rather than ones that just happen to be orthogonal because of your choice of basis.
Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward.
In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.
It was also called indirect proofs by textbook, and it’s when I first fell in love with math.
Note: Linear algebra killed said love of math for me ever so slightly, when the instructor chose a terrible textbook.
One could draw one on the surface of a sphere, for example, just as one can draw a triangle with three right angles (https://www.quora.com/How-can-I-draw-a-triangle-with-three-9...)
Seems it should be possible to draw the octagon that's shown in the article: draw a rectangle and flex the sides out a little bit.
Flexing the sides requires negative curvature or non-convexity. Triangles are easy to draw because they have fewer sides than a square, thus positive curvature makes it possible to create a 3-right triangle. Octagons are the opposite.
I've never had a great head for mathematics because it feels like most impossibilities arise from arbitrary constraints and definitions. Reading such proofs feels like:
1) I've created an arbitrary universe with made up rules
2) This condition does/doesn't break one of those rules I made up
For example, why on earth can't a straight line in a rectangle have a 180 degree angle at its center making that rectangle an octagon too? It's not clear why one system is epistemologically valid and the other is not. The proof of impossibility feels like a "no true scotsman"/"no true octagon" fallacy disguised by a bit of added notation.
If a stateful coin is permitted, the problem is indeed trivial; but if we start with the assumption that the problem isn't trivial, we can assume a "passive" coin more in line with the intuitions most people would have about coins.
Anyway, this is a bit outside the box; the point was that within a set of rules there was a problem that was impossible to solve.
The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle.
(The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))
(I know it's true, just saying it looks like you've offered an explanation in elementary terms, but I don't think one exists).
Why is a polygon with three consecutive collinear vertices not a polygon? Convention. Does anyone here have insight into why it's a good convention?
To me it sounds something like saying "9/12 is not a valid fraction because it can be simplified".
Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.
You can add various restrictions such as not allowing it to intersect itself or requiring that it is convex, but not allowing 3 successive vertices to be colinear doesn't really give you any useful properties so I'm not sure why you'd want that.