Pulling this topic out of the hellthread, tl;dr the arabs barely did anything mathematical
The history of 'arabic numerals' has lots of revisionist history surrounding it because it originates from the northern indian descendants of the aryan invasion and passed through the middle east and egypt at a time when both places were ethnically more aryan than arab. It's not fun going too deep into this topic because you start sounding like scalar (pbuh) very quickly, but regardless calling them "arabic" numerals is a liberal misnomer.
The history of calculus is more recent and better attested. There are two main branches of calculus, differential and integral. The differential calculus measures rates of change by comparing infinitesimal differences, this branch was not developed at all by the ancients because it was built out of the analytic geometry* of Descartes as well as the symbolic equations of medieval mathematicians (prior mathematics was written out in paragraphs, no symbols at all).
The integral calculus measures areas by summing up infinite amounts of infinitesimal areas. The greeks were able to calculate some areas by their "method of exhaustion" but this was limited to certain geometric shapes like conic section, and relies more on a version of aristotle's principle of non-contradiction than on directly manipulating infinities**. Prior to the 17th century this was just called quadrature, it was simply using shapes to make measurements because geometry derives from the greek word for measurement. It wasn't until Newton, Liebniz, and their contemporaries discovered the fundamental theorem of calculus that links the differential and integral together as inverse operations (this is completely unintelligible to prior mathematicians, you can't give them any credit for this).
what did the arabs do exactly? they translated greek, egyptian (greek), indo-aryan, and mesopotamian mathematics and preserved copies of these translations through the collapse of the roman empire. Mathematics didn't advance until these works were translated back into European languages and worked on by Europeans.
*Newton himself was eccentric and privately made his calculations using "infinite series" which is a related but still separate construction, and then he rewrote his results in terms of greek synthetic geometry when publishing his work, this was both to express respect for the ancients and to hide his methods from contemporaries
**Specifically the greeks would argue that the area of a shape could not be less than a certain value, and it could not be more than that value, so it must therefore be exactly equal to that value. It did not technically make use of infinity so it's inaccurate to call this calculus***
***the word calculus itself comes from the romans and just meant "method to calculate", usually referring to counting things with numbers. Our modern definition of calculus is an abbreviation of "differential/integral calculus" which are methods to calculate certain equations.
RT: https://poa.st/objects/975a6d9d-71fd-43cd-8830-8b094709bc97
The history of 'arabic numerals' has lots of revisionist history surrounding it because it originates from the northern indian descendants of the aryan invasion and passed through the middle east and egypt at a time when both places were ethnically more aryan than arab. It's not fun going too deep into this topic because you start sounding like scalar (pbuh) very quickly, but regardless calling them "arabic" numerals is a liberal misnomer.
The history of calculus is more recent and better attested. There are two main branches of calculus, differential and integral. The differential calculus measures rates of change by comparing infinitesimal differences, this branch was not developed at all by the ancients because it was built out of the analytic geometry* of Descartes as well as the symbolic equations of medieval mathematicians (prior mathematics was written out in paragraphs, no symbols at all).
The integral calculus measures areas by summing up infinite amounts of infinitesimal areas. The greeks were able to calculate some areas by their "method of exhaustion" but this was limited to certain geometric shapes like conic section, and relies more on a version of aristotle's principle of non-contradiction than on directly manipulating infinities**. Prior to the 17th century this was just called quadrature, it was simply using shapes to make measurements because geometry derives from the greek word for measurement. It wasn't until Newton, Liebniz, and their contemporaries discovered the fundamental theorem of calculus that links the differential and integral together as inverse operations (this is completely unintelligible to prior mathematicians, you can't give them any credit for this).
what did the arabs do exactly? they translated greek, egyptian (greek), indo-aryan, and mesopotamian mathematics and preserved copies of these translations through the collapse of the roman empire. Mathematics didn't advance until these works were translated back into European languages and worked on by Europeans.
*Newton himself was eccentric and privately made his calculations using "infinite series" which is a related but still separate construction, and then he rewrote his results in terms of greek synthetic geometry when publishing his work, this was both to express respect for the ancients and to hide his methods from contemporaries
**Specifically the greeks would argue that the area of a shape could not be less than a certain value, and it could not be more than that value, so it must therefore be exactly equal to that value. It did not technically make use of infinity so it's inaccurate to call this calculus***
***the word calculus itself comes from the romans and just meant "method to calculate", usually referring to counting things with numbers. Our modern definition of calculus is an abbreviation of "differential/integral calculus" which are methods to calculate certain equations.
RT: https://poa.st/objects/975a6d9d-71fd-43cd-8830-8b094709bc97
@KitlerIs6 @sun_eater @dushman @meso @vriska @KonataWagner @RahowasaurusRex1979 @udongle @special-boy The Arabs invented the numeral system we had ...
Here's an example of the difference between arab mathematics and European mathematics. This is an excerpt from Al Kwahrizmi's "algebra" (the book that coined the term), the paragraph is what mathematics looked like before we invented equations, wordswordswords very hard to follow. The footnote is the exact same mathematical idea just written in a symbolic equation.
Because of everybody's experience with schooling we think of mathematics as synonymous with equations, but that is a comparatively recent European invention.
Because of everybody's experience with schooling we think of mathematics as synonymous with equations, but that is a comparatively recent European invention.
>I cropped the wrong part of the page
The top paragrapgh corresponds to the equation in the foot note. The fact that you can't read a paragraph at a glance but you can quickly read an equation is why it's become the universal way of doing mathematics.
The top paragrapgh corresponds to the equation in the foot note. The fact that you can't read a paragraph at a glance but you can quickly read an equation is why it's become the universal way of doing mathematics.
CatSoc 
@udongle @confederatehobo @Groomschild God, I can't imagine doing maths like that. For maybe small things, yes, but to tackle many problems in this way would be hell.
To be historically fair the greeks wrote like this too, it took until Descartes for it to start being readable and it's not until Euler that things read like they would today.
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@udongle @confederatehobo @CatLord @Groomschild This is also how Newton worked, since he hated Descartes for some odd reason. Which is probably why Liebniz had the better notation.
@ThatWouldBeTelling @confederatehobo @udongle @CatLord @Groomschild I’ve read that. It’s a pretty decent read , even though it’s not completely rigorous the things it smudges, like transfinite induction, are not really important to understanding how hyperreals work. I think it might be hard as a first real (hyperreal?) analysis book though, because it’s fairly concise.
Keisler's book is a much better introduction, very graphical and articulate. A lot of the proofs that would normally need limits and inequalities can be done with simple algebra and it's very intuitive (the inverse function rule is is just division because dy/dx is a fraction with hyperreals). And it even introduces limits in terms of hyperreals and then teaches standard epsilon-delta proofs so you're not left confused by standard analysis.
@udongle @confederatehobo @CatLord @Groomschild @mandlebro H. Jerome Keisler’s books are the downloadable ones I recommended in my previous posting.
Based on other’s recommendations, didn’t know about all this when I started it in the late 1970s with Thomas, a journey unfortunately cut short.
Sound even better than I though when I learned about them almost a couple of decades ago.
>when I started it in the late 1970s
Wow You don't see a lot of old timers around here. The internet has been a boon for learning this stuff, any obscure topic has multiple pdfs floating around and at least one of them is great.
Wow You don't see a lot of old timers around here. The internet has been a boon for learning this stuff, any obscure topic has multiple pdfs floating around and at least one of them is great.
@mandlebro @udongle @confederatehobo @CatLord @Groomschild Have also read this Newton-Leibniz feud and/or English chauvinism resulted in English work getting retarded for some time. Newton’s dot notation might work for a genius like him, but Leibniz’s dx, “a little bit of x” was vastly superior and eventually won.
BTW, if you tackle calculus, might want to check out infinitesimal methods vs. limits, the latter got a thorough foundation quite some time ago, last century the former got the same, there’s an old standard style textbook you can find on-line, and … OK, now Dover has republished the short MIT Press hardcover book that’s a good overview and orientation on the basics: https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mathematics/dp/0486428869/