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.
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.