Back in my undergrad days, there was a second year course titled "Groups, Rings and Modules". The lecturer (NSB) was notorious for his very dense and formal delivery, which left a lot of work to the reader. Another - much more popular - professor (IL) decided to offer an alternative version of the course during the same timeslots in a friendlier style. Despite NSB's absent-minded nature, it cannot have escaped his attention that ~10 people attended his lectures while the bulk of the year group queued up outside a different hall. Some friends and I attended NSB's lectures out of bloody-mindedness. We anticipated (correctly, it turns out) that if you managed to understand his notes, then you really understood the topic.
The reason I'm reminded of this is that one of the big theorems of the course is the simplicity of A5. I vaguely remember that NSB provided the icosahedron-based proof that's mentioned in the linked article, whereas IL provided a more intuitive and sensible proof.
One of the perks of being the official lecturer is that you set the exam questions. Imagine our delight when we opened the paper and saw the question: "By considering the rotations of an icosahedron, prove that the group A5 is simple"!
> In 1963, the Russian mathematician Vladimir Arnold gave an alternative
topological proof of the unsolvability of the quintic in a series of lectures to high school kids in Moscow.
...
> Arnold’s insight was to show that if there is a radical formula for the roots of a general polynomial, then the “dance of the roots” cannot be overly complex, in the sense that the image of the monodromy map must be a solvable subgroup of . But, for example, the 1-parameter Brioschifamily of quintics
() = 5 + 103 + 452 + 2, ∈ ℂ
has monodromy group 5 (see Figure 1), which is certainly not solvable since it is simple, as we will see by relating it to the icosahedron in the next section. Hence the unsolvability of the quintic.
...of course, well within the reach of most high school students.
I'll paraphrase a famous piece of public speaking advice "avoid talking down to your audience, they will catch up".
Also, who knows how the lecturer presented the information. Perhaps it was well composed, and in fact could be understood by anybody.
When fields highly specialize they seem to attract a certain personality of people who revel in their 'insider knowledge' and make the learning of it for others more difficult in order to inflate their standing in the eyes of their students, "How did you learn this stuff!?"
The reality of these fields, like mathematics, music theory, computer science, and the like is that the ideas themselves are so ridiculously simple that they could be understood by most anyone with a rudimentary understanding of the 5 basic arithmetic operators if presented coherently.
After all, “Coding is basically just ifs and for loops.”
I don't know if I agree with this. Advanced mathematics deals with layer upon layer of abstraction, and oftentimes there aren't significant shortcuts. A good friend of mine did her doctorate in algebraic geometry, and there were important open questions in her field that would take two years of study in order to be able to understand the problem statement.
Take the proof of Fermat's Last Theorem. In this case the problem is very easy to state (a^n + b^n = c^n has non-zero integer solutions only of n < 3). But the proof is very long (like over 100 pages at least?) and involves quite a lot of abstraction, well beyond integer arithmetic. Nobody's being pretentious here, it's just the nature of the field.
> When fields highly specialize they seem to attract a certain personality of people who revel in their 'insider knowledge' and make the learning of it for others more difficult in order to inflate their standing in the eyes of their students
This is one reason why Wikipedia is so very valuable. A well-done article gives you a foot in the door, even if by doing little more than decoding jargon.
The actual argument is a bit simpler than implied by the quote's use of the terms 'monodromy', 'solvable subgroup', etc. Here's an explanation of Arnold's proof that doesn't use those words at all, and while not precisely easy is much easier than you'd think from the quoted bit of the OP article: https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...
Not “most” high school students, but the more talented ones? Sure. Likely they were exposed to a lot of algebra before this lecture to bring them to a point where they could profitably follow along. The kind of kids who take AP Calc can learn abstract algebra (and, for that matter, I’d say that any kid who’s had Algebra II and properly taught geometry (with proofs) can learn abstract algebra).
The kinds of AP Calc kids you see must be super different from the ones I've seen then. I'm in a uni with a large amount of kids who've taken AP Calc and yet most of them struggled a lot with the first semester intro-to-proofs course. The second semester linear algebra course (it's a middle ground between a proof-based and a computational course) was even worse. I know many kids with 5's in AP Calc BC who resort to memorizing basic proofs (which is the sort of thing that helps with AP exams) instead of learning how to write one on their own, and some of the TA's have told me that the most common mistake on the midterm was incorrectly negating the statement "A is a subspace of B".
This is not to say that high schoolers can't do abstract algebra (or higher mathematics more generally). In senior year I attended a week-long camp (Arnold had a full semester) in my local uni where we proved the impossibility of squaring the circle, doubling the cube, etc using field extensions. And I was in the older side! Most of the kids there were 10th grades. Though Arnold's class was probably substantially harder than ours. I worked through V. B. Alekseev's book after the camp was over and the exercises were substantially harder than the ones we did at the camp. The material on Riemann surfaces was very hard to understand as well, much harder than the group theory part (I still don't understand Riemann surfaces lol).
In conclusion, AP Calc, and students' performance in it, is a terrible metric for assessing mathematical ability. Sorry for the long rant.
But it actually is! It's just written in an excessively jargon-laden manner. Here's an explanation that's less jargon-laden:
We can describe a polynomial by its coefficients, standard x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0 form. Actually, we're going to restrict a, b, c, d, and e so that they are all determined by a single parameter B (this is the Brioschi family of quintics). Remember that all values here are complex numbers, not real numbers.
Now what happens if we move around B a little bit (https://duetosymmetry.com/tool/polynomial-roots-toy/ is very helpful for seeing what I mean here)? Well, the roots will change a little bit. There's a continuous mapping going on. Now move around B in a large, closed path that includes a specific point in the interior. It turns out that the continuous motion of the roots will result in them swapping places--a permutation. This is the "monodromy map".
Now it turns out that the set of possible permutations has implications on whether or not you can solve it with polynomials. The paper here doesn't go into details, since it's written for a high-math audience (this is the "A_5 is not solvable since it is simple" bit).
If the terms: fundamental group, covering space etc are jargon then more so is the case for words like 'explanation', 'laden' and well, 'jargon' itself. They have precise definitions meant to capture mathematical concepts and ideas for which there can be no adequate words in natural language.
Composing radicals can only express certain types of permutation groups, and it's not too hard to see how it works. The quadratic formula term `-b +- sqrt(a^2 - 4ac)` is basically exploiting the permutation created by `sqrt(b)` which swaps between `+sqrt(b)` and `-sqrt(b)`. A cube root `cbrt(b)` rotates between three values, which are separated by roots of unity `e^(2pi/3 i)`. A fourth root swaps between four values, which are separated by the fourth roots of unity `(1, i, -1, -i)`. Etc.
So the basic "building block" is "rotate 2 points", "rotate 3 points", "rotate 4 points", etc. Even those doesn't express very much: a generic cubic has symmetry S_3, which is arbitrary permutations of three roots, and even that can't be expressed with just basic sums of these, because the only way to get three-fold symmetry is to use a single cube root.
But take a look at the cubic formula: https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_f.... Turns out if you nest them you can create more interesting symmetries. The cubic formula structure has the structure "sums of cube roots of square roots", which give six total values but really it makes three because of cancellation.
I interpret this to mean that as you try to make the larger permutation groups you're increasingly stressing the expressive powers of radicals. The quadratic did fine with two: the quadratic formula is basically "take the midpoint of the roots -b/2a and add the offset between the two `sqrt((b/2a)^2 - c)` with a plus or minus sign". The cubic formula gets messy because the roots don't have to be evenly spaced so you need a way to express that too. The quartic formula is insane (https://www.curtisbright.com/quartic/quartic-png.html) and involves sums of square roots of sums of third roots of square roots. So it is not too surprising that things fall apart soon after, apparently at the quintic.
The specific way that the quintic fails is that the symmetry group S_5 is not "solvable", which is hard to explain without a lot more space, but which basically means that it includes a list of operations (P, Q, ...) which when you replace them with their commutators (PQP^-1 Q^-1, ...) don't get any simpler. I guess that is what radicals don't do: commuting radicals always makes them simpler, and so at N=5 you hit a group they can't fake.
I agree that this is not within the reach of most high school students, and if I had to guess, I think that either these Moscow kids were very carefully selected to be in this group or most of them didn't follow it.
But I do think some very talented and motivated high school students could at least follow the gist of this argument! I've worked at a summer program for mathematically talented high schoolers, and while I didn't teach this particular class, one of the years I was there someone taught what I think is essentially the same proof this article is talking about, and they did get through it in five weeks.
The pictures of monodromy are called dessins de enfant because they are little pictures that are simple to understand. The hard part is why we care so much.
The reason I'm reminded of this is that one of the big theorems of the course is the simplicity of A5. I vaguely remember that NSB provided the icosahedron-based proof that's mentioned in the linked article, whereas IL provided a more intuitive and sensible proof.
One of the perks of being the official lecturer is that you set the exam questions. Imagine our delight when we opened the paper and saw the question: "By considering the rotations of an icosahedron, prove that the group A5 is simple"!