If there were a world in which algebra had been learned only through reading essays, without subskill-by-subskill practice, it would not be surprising if the world’s best algebra practitioners could be outperformed by an ordinary student who worked diligently through the exercises in a standard textbook.
This actually happened. The ancient Greeks weren’t very capable algebraists because they didn’t develop a symbol system that they could systematically manipulate according to prescribed rules. Their description of formal logical inferences were insane to read: “If the first and the second, or the third, implies the fourth, then the first or the fourth, implying the third....” The reason our word “algebra” comes from Arabic isn’t because the Muslims were better algebraists; it was because they used symbol systems (to avoid making icons of Mohammad) in order to encode the material they were reading in the Greek literature. The result was something reasonably close to our modern symbol-manipulation system, which made it possible to train in algebra.
So this isn’t just a theoretical example. Really, honestly, the first textbook (“al-jebr...” I don’t quite remember the title) absolutely trounced several hundred years of careful, intelligent Greek thought on the topic of numerical reasoning.
Edit: Please see this. There’s some question about the accuracy of my statement here.
This description is very plausible, but entirely wrong. It was almost completely the opposite of what you’re saying. The Muslim mathematicians used fewer symbols than the Greek tradition they inherited for almost the entire timeline of medieval Arabic/Islamic mathematics. The “first textbook” you’re referring too, Al-Khwarizmi’s Al-jabr wa’l muqabalah, the one ultimately responsible for the word “algebra”, did not use any symbols at all, and wrote everything out in words.
Greek mathematicians started to use something like symbols (abbreviated letters with fixed positional meaning) by the time of Diophantus around 3rd century CE. The Arab mathematicians did not adopt that when they translated the Greek texts, and for the first 500 years of their work, wrote everything out fully. Moreover, it is those texts devoid of any symbolic systems that were translated into Latin and used to help fuel the European tradition in 12th-13th centuries CE. Even though some Islamic mathematicians later did develop the beginnings of a symbolic notation, in 14th-15th centuries, this happened roughly in parallel with the Europeans inventing their own symbols, and did not influence the modern tradition that derives from those European symbols.
To be sure, Muslim mathematicians were much better algebraists than the Greeks. But that was because Greeks never quite reached the idea of decoupling numbers (and unknown quantities) from geometry and manipulating them as separate objects on their own. The Muslim mathematicians were able to do that (and as a result, much more), despite not having any symbolic system at their disposal.
Huh. This directly contradicts what I encountered. I’ll have to explore this a bit. I knew the Greeks had a problem with decoupling their idea of number from their concepts of geometric construction, but I was told that certainly in formal logic and I thought in numerical reasoning as well, their lack of symbol system machinery handicapped them. The Muslims, on the other hand, wouldn’t use pictures of the ideas to which they wanted to refer because of the ban on iconography, so they had to encode their concept of quantity differently, I thought that’s where symbol machinery came from.
So… I’ll have to look into this. Upvoted for offering a correction, although I don’t know yet if it’s actually correct. Thank you!
To a Greek mathematician, a number was fundamentally a measure of something geometric, like the length of a segment. The square of a number is not some abstract operation: it’s just the area of a particular figure, a square. An equation was a way of describing a geometrical problem. Equations were solved geometrically.
Here’s an example. Suppose you have unknown numbers x and y, and you know the difference between them and also their product. Can you find x and y? In algebraic terms, you manipulate some unknowns, express y in terms of x and substitute, arrive at a quadratic equation in x, and find the result. Greek mathematicians weren’t able to write it this way, in words or in symbols. That just wasn’t a way of looking at this problem, or a method of solving it, that they could recognize.
Here’s how they thought: you have two unknown lengths. You know by how much one is greater than the other, and you also have a square whose area is equal to the rectangle built on those lengths. Can you find these unknown lengths? Well, you can do it this way: take the difference between them, drawn as a line segment AB. Find its middle point C. Draw a line BQ perpendicular to AB at point B, of length equal to the side of the square you have. Now take the hypotenuse CQ, and add it to the original line AB, prolonging it to the point D. You have one of the unknown lengths in the segment CD.
This is straight out of Euclid. There’s also a proof that what I just described actually solves the problem; the proof is based on considering the rectangle built on the unknown lengths, cutting it into a few parts, reassembling them elsewhere, etc. That’s how Greek mathematicians solved equations. They didn’t have the mental image of x and y as these abstract entities that you can shuffle around in an equation (another abstract entity), multiply/divide by some numbers (more abstract entities) to simplify, and arrive at the algebraic result. To them, x and y were lengths you don’t know how to measure yet, and all the manipulations were inherently geometric.
Arab mathematicians changed that, and opened the way to looking at numbers, unknowns and equations “algebraically”, as separate abstract entities of their own which can be manipulated according to strict rules.
They measured time on the large scale accurately enough for astronomical purposes, and on the small scale to build something as amazing as the Antikythera mechanism. They probably didn’t measure their days into minutes and seconds the way we do, the everyday functioning of the society didn’t need and couldn’t use such precision.
This actually happened. The ancient Greeks weren’t very capable algebraists because they didn’t develop a symbol system that they could systematically manipulate according to prescribed rules. Their description of formal logical inferences were insane to read: “If the first and the second, or the third, implies the fourth, then the first or the fourth, implying the third....” The reason our word “algebra” comes from Arabic isn’t because the Muslims were better algebraists; it was because they used symbol systems (to avoid making icons of Mohammad) in order to encode the material they were reading in the Greek literature. The result was something reasonably close to our modern symbol-manipulation system, which made it possible to train in algebra.
So this isn’t just a theoretical example. Really, honestly, the first textbook (“al-jebr...” I don’t quite remember the title) absolutely trounced several hundred years of careful, intelligent Greek thought on the topic of numerical reasoning.
Edit: Please see this. There’s some question about the accuracy of my statement here.
This description is very plausible, but entirely wrong. It was almost completely the opposite of what you’re saying. The Muslim mathematicians used fewer symbols than the Greek tradition they inherited for almost the entire timeline of medieval Arabic/Islamic mathematics. The “first textbook” you’re referring too, Al-Khwarizmi’s Al-jabr wa’l muqabalah, the one ultimately responsible for the word “algebra”, did not use any symbols at all, and wrote everything out in words.
Greek mathematicians started to use something like symbols (abbreviated letters with fixed positional meaning) by the time of Diophantus around 3rd century CE. The Arab mathematicians did not adopt that when they translated the Greek texts, and for the first 500 years of their work, wrote everything out fully. Moreover, it is those texts devoid of any symbolic systems that were translated into Latin and used to help fuel the European tradition in 12th-13th centuries CE. Even though some Islamic mathematicians later did develop the beginnings of a symbolic notation, in 14th-15th centuries, this happened roughly in parallel with the Europeans inventing their own symbols, and did not influence the modern tradition that derives from those European symbols.
To be sure, Muslim mathematicians were much better algebraists than the Greeks. But that was because Greeks never quite reached the idea of decoupling numbers (and unknown quantities) from geometry and manipulating them as separate objects on their own. The Muslim mathematicians were able to do that (and as a result, much more), despite not having any symbolic system at their disposal.
Huh. This directly contradicts what I encountered. I’ll have to explore this a bit. I knew the Greeks had a problem with decoupling their idea of number from their concepts of geometric construction, but I was told that certainly in formal logic and I thought in numerical reasoning as well, their lack of symbol system machinery handicapped them. The Muslims, on the other hand, wouldn’t use pictures of the ideas to which they wanted to refer because of the ban on iconography, so they had to encode their concept of quantity differently, I thought that’s where symbol machinery came from.
So… I’ll have to look into this. Upvoted for offering a correction, although I don’t know yet if it’s actually correct. Thank you!
What do you mean by decoupling geometry and numbers/unknowns? Sounds interesting but I don’t understand.
To a Greek mathematician, a number was fundamentally a measure of something geometric, like the length of a segment. The square of a number is not some abstract operation: it’s just the area of a particular figure, a square. An equation was a way of describing a geometrical problem. Equations were solved geometrically.
Here’s an example. Suppose you have unknown numbers x and y, and you know the difference between them and also their product. Can you find x and y? In algebraic terms, you manipulate some unknowns, express y in terms of x and substitute, arrive at a quadratic equation in x, and find the result. Greek mathematicians weren’t able to write it this way, in words or in symbols. That just wasn’t a way of looking at this problem, or a method of solving it, that they could recognize.
Here’s how they thought: you have two unknown lengths. You know by how much one is greater than the other, and you also have a square whose area is equal to the rectangle built on those lengths. Can you find these unknown lengths? Well, you can do it this way: take the difference between them, drawn as a line segment AB. Find its middle point C. Draw a line BQ perpendicular to AB at point B, of length equal to the side of the square you have. Now take the hypotenuse CQ, and add it to the original line AB, prolonging it to the point D. You have one of the unknown lengths in the segment CD.
This is straight out of Euclid. There’s also a proof that what I just described actually solves the problem; the proof is based on considering the rectangle built on the unknown lengths, cutting it into a few parts, reassembling them elsewhere, etc. That’s how Greek mathematicians solved equations. They didn’t have the mental image of x and y as these abstract entities that you can shuffle around in an equation (another abstract entity), multiply/divide by some numbers (more abstract entities) to simplify, and arrive at the algebraic result. To them, x and y were lengths you don’t know how to measure yet, and all the manipulations were inherently geometric.
Arab mathematicians changed that, and opened the way to looking at numbers, unknowns and equations “algebraically”, as separate abstract entities of their own which can be manipulated according to strict rules.
Thank you. Makes much more sense now. The greeks failed to abstract number from length, so they failed to develop abstract mathematics.
How aware were they of measurements of time?
They measured time on the large scale accurately enough for astronomical purposes, and on the small scale to build something as amazing as the Antikythera mechanism. They probably didn’t measure their days into minutes and seconds the way we do, the everyday functioning of the society didn’t need and couldn’t use such precision.