Indeed. In fact, it is one of the most amusing aspect of the anglophone west (at least for the last few decades). Despite public perception (by public I mean those who have been to university since the 90s), Western historians of science and mathematics in general have never not acknowledged the previous works of the Persianate civilizations commensurate to their knowledge of them in their time. But somehow in the last few decades professional historians have had to waste time figuratively looking over their shoulders lest they be percieved as being Eurocentric. And, if they were to somehow find a way to show -- requiring whatever hermeneutical gymnastics -- that a prominent scientist was influenced (or even better, had stolen) from some other "cultures" than nothing better! (ex: Copernicus from the Maragha school as an example of interpretive gymnastics)
But, of course, this is one of the symptoms of the degeneration that now afflicts your particular civilization and is bringing about it's inevitable transformation to something else -- but better this than the fate of the Abassids or the Sung.
I'm going to get downvoted to oblivion for this. But it's still the truth: just wait until you try to get muslims to confirm what exactly about islam "safeguarded" science in the middle ages.
The answer is slavery, and patronage by very, very rich people (who outright owned the scientists, and these in turn kept libraries of the great scientific works of the past, as trophies for the sultan, with zero public access). Oh and the fact that they recreated the Roman habit of kidnapping slaves and then selling them, sometimes an enormous distance from where they were captured. That is how Hindu numerals spread.
One very famous example is the "Blue Mosque", the greatest piece of islamic architecture for over 500 years, the tallest building in the world for a very long time (only overshadowed by the Church it was copied from: the Aya Sofia) which is a copy of a Church building by a Jewish architect (who was a slave to the sultan). Yes, minarets are a Christian idea.
Perhaps this is the reason the Blue Mosque doesn't have one of the defining features of islamic architecture of mosques: it doesn't have a catwalk, a podium for selling slaves, which most ottoman mosques have.
Then, usually during periods of economic stress, muslims destroyed their science, usually for religious reasons. Of course, this happened in the Christian west too. In the west science (specifically the copying of books by the Catholic church, then giving public access to them. No public access existed in any caliphate) recovered faster than these religious attacks could destroy it. In islamic nations it didn't. Islam was more scientifically advanced in 800 than in 1800 (or 1900). Or, to put it another way: the more actual muslims a society had (in 800 that was almost none), the less science existed.
Why? The only disagreement with these claims comes from islamic supremacists. Even in islamic sources directly you can verify most of the claims (slavery, Blue Mosque - Aya Sofia + architect, slavery in mosques, barely any muslims in early muslim society ...)
Look up on Wikipedia, look up in history books. These are not small details.
On the contrary, what is presented by the OP is one of the many reasons that worship of science's heroes, unfashionable for decades, a whiggish pablum, is justified. If great results were birthed fully-formed -- a view I've frankly never heard anyone profess who has bothered to consider such things even briefly -- they would hardly be any heroes. Even little children who reflexively chomp on every superhero film aeroplaned towards their face understand this.
Let me take this opportunity to post one of the best texts on Galois Theory I have read -- and I had to go through quite a few while preparing for a class.
The subject is developed very naturally and every idea is beautifully motivated. It begins with a quick one chapter intro of Arnold's proof of Abel-Ruffini.
> It begins with a quick one chapter intro of Arnold's proof of Abel-Ruffini.
The key to understanding/motivating Galois theory is Abel-Ruffini, which is a corollary of Galois. And the simplest way to understand that is Arnold's topological proof, which i learned about from this video
Watching that video and rolling it around in my head completely demystified Galois theory for me, years after literally 2 semesters of algebra in undergrad. Everything about normal subgroups and commutators and splitting fields and blah blah blah immediately became tangible and obvious. It should be a crime not teach this proof first.
The coverage in Koch's book looks good too - lots of pictures - and funny enough it links to a different youtube video.
Edit: copy-pasting notes I took from the video after watching.
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The idea is to continuously perturb each of the coefficients of the polynomial along a loop (change each of them from their initial value such that they traverse a path that returns them to that initial value at the end of the path) and study what happens to the roots of the polynomial.
Note, once all coefficients have returned to their original values the entire set of roots also returns to itself, but each root does not necessarily returns to its original value. In general you get a permutation of the set of roots and so in this way we get a mapping between loops of the coefficients and permutations of the roots.
Also note, we can produce coefficient loops that map to any permutation of the roots by permutating the roots and “watching” the coefficients.
Hence, the way to prove Abel-Ruffini is to show that any expression involving the coefficients (ie formula for the roots in terms of the coefficients) returns to itself after the coefficients traverse their loops but the roots do not (and therefore the expression cannot capture all of the roots). For example, an immediate corollary of the construction of the mapping between loops of coefficients and roots is the fact that a general solution involving only -, +, ×, ÷ is not possible;
-, +, ×, ÷ are all single-valued and therefore no composition thereof could produce multiple roots.
There is indeed a deep connection between what is going on behind Arnold's proof and the classical Galois theory. But it needs quite a bit of sophistication to flesh out properly (not apparent in his famous lectures given to high school kids). There is a Galois theory for Riemann surfaces over algebraic functions where the coverings behave like fields do in the classical correspondence. If any one is interested, check out chapter 3 of Khovanskii's Galois Theory, Coverings and Riemann Surfaces.
i mean calling arnold's proof actually topological is probably a stretch (the "deep connection" you're talking about). it doesn't really use any topological facts about either the loops or the embedding space. continuity isn't really required i don't think? i should've said contiguity instead. it's just a very very nice model for the theory (in the sense of model theory) that lends itself to immediate visualization.
Is the idea of roots swapping places related to Riemann surfaces? A bit like the sqrt function being defined on two copies of the complex plane glued together.
You're responding to wrong comment. In the comment you meant to respond to I admitted I wasn't sure whether continuity was required to make the proof go through and you still haven't demonstrated that it is. Riemann surface here is just a fancy word for the loop itself and indeed the key part of the definition is connectedness not C.
You linked a math SE question with no responses (yes I read the comments), Arnold's book, and the same proof I've already read. So I'm not convinced.
If I'm being honest I really don't have a good recommendation for a text in intro abstract algebra. I learned it from Michael Artin's Algebra. Artin is a true master -- along with David Mumford, he was the main apostle for Grothendieck style AG in the US -- but his book was not very easy to learn from.
Atiyah is truly one of the giants of modern mathematics. I remember long ago I struggled through a reading course of his and Bott's Yang-Mills paper in graduate school. Like many great works of math it too had that paradoxical characteristic of transforming seemingly 'non-mathematics' into mathematics* by reversing the usual direction of application of one to the other, in this case, from physics to math. It would start a whole movement that'll produce much of modern geometries greatest hits like Donaldson's (his student) theorem in 4 manifolds to Witten's great papers.
* A reason I think modern LLM architecture as they currently stand with their underlying attention mechanisms will not produce interesting new mathematics. A few other ideas are going to be needed.
And why should I simply assume that "Education Economists"* really know the subject they purport to talk about? Because they are credentialed members of university departments with some label? Because a few of them won some Bank of Sweden award?
Just because a particular department or field of study exists in academia does not magically give them the imprimatur you think it does.
* Btw, I know for a fact that a few of them are not "education economists"
There's a suite of code-related tasks -- covering a diversity of areas, including dev ops, media manipulation etc., derived from issues I have faced over the years -- I perform for every new release. No model has solved the set of issues solved in one go but Claude still remains the best.
An example of the sort of problems in the suite:
> I have a special problematically encoded mp4 file with a subtle issue (something I ran into a couple of years ago while fixing a bug in a computer vision pipeline). In the question prompt I also pass the output of ffprobe and ask for the ffmpeg command that'll fix it. Only Claude has figured the real underlying issue out (after 4 interactions).
Are you aware of the book on [The Disc Embedding Theorem](https://academic.oup.com/book/43693) based on 12 lectures Freedman gave roughly a decade ago.
No, I am not an expert of 4-manifold theory and would not really understand most of the chapters. If this book fixes some of the literature issues in that field that is amazing! Does it finally resolve the issue of nobody understanding the construction of topological Casson handles?
Edit: I see from the MO comments "The fully topological version of the disc embedding theorem is beyond the scope of this book, since we will not discuss Quinn's proof of transversality."
But, of course, this is one of the symptoms of the degeneration that now afflicts your particular civilization and is bringing about it's inevitable transformation to something else -- but better this than the fate of the Abassids or the Sung.
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