What are the being of numbers? Were the Greeks right in assuming that magnitude comes before number? Are numbers ratios of magnitudes? How can magnitude and number be seen as not immediately intertwined?

It's difficult for me to understand why the Greeks didn't view magnitude and number as intertwined, especially when they're not counting anything particular at all (e.g. just saying 3+9 = 12, without any reference to an object in the world).

Actually, a better way of putting it is I think I understand the what that is going on, but I don't understand the why. It seems like a self-imposed limitation.

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Book 5 in elements is magnitudes and Book 7 is number, so what you say about magnitudes before number is correct

I'm looking at the first propositions in books 5 and 6 and at definitions, I have to admit I had similar thoughts when starting book 5 and I did not do the whole book but only the beginning of the book, I have never read Klein and I think I should

I said 6 instead of 7

Jacob Klein was stupid, and so was Leo Strauss shilling his book.

You reminded me about a book I was reading. "A history of mathematics" by Luke Hodgkin.

Greeks knew about numbers, the way you can reason about them without any reference to an object in the world.

> The ordinary arithmetician, surely, operates with unequal units; his ‘two’ may be two armies or two cows or two anythings from the smallest thing in the world to the biggest; while the philosopher will have nothing to do with him, unless he consents to make every single instance of his unit equal to every other of its infinite number of instances. (Plato, Philebus, tr. in Fauvel and Gray 2.E.4, p. 75)

There's an example in Plato's Meno about doubling a square. Socrates calls a slave-boy, and the slave-boy tries numerical approach and fails, Socrates leads him to use geometrical approach and they find the solution.

Square's diagonal is irrational. If you want to represent it with a number, that's problematic.

> In fact, one reason why this particular problem has been chosen for the dialogue is perhaps—but here, as usual, we have to start attributing motives to the Greeks—because it shows the limitations of numbers, and the superior power of geometrical methods. The boy will never arrive at the right answer by guessing different numbers; but Socrates can draw a picture which solves it. The philosophers’ mathematics not only uses a more abstract idea of ‘number’, but when number becomes a problem, it can dispense with it.

Are you talking about the fact that Greeks didn't use algebraic approaches?

>Greeks knew about numbers, the way you can reason about them without any reference to an object in the world.

Isn’t there always an implicit reference though? What is the basis for it then? I’ve heard it explained as an “abstract unit” in genera vs the geometrical “physical unit” in particular. But that doesn’t really shed enough light on what’s going on here.

Can anyone make numbers like this, also what numbers are they, I say they are both one but different magnitudes or that a ratio can be found using Euclidean algorithm to arrive at a unit that measures both

Been one year since last thread on this book https://warosu.org/lit/thread/21887744 this sone kinda anniversary OP?

Not one of those posters did geometry, couldn't even conjure up a simple circle or line to accompany their thoughts

I didn’t get a good answer, so I revisited the book and the topic. I changed my mind about it being a contentless book, but it could have been far better explained.

Are people really incapable of drawing a line or circle in the sand while talking, this action is a very greek way to express thought

That cover looks like a lady's butt depicted with math.

For OP and anybody else who is interested in Klein's book, check out

>The New Yearbook for Phenomenology and Phenomenological, Volume XI (2011). There's a series of brilliant essays on Klein's book (still reading through, but Halper's and Romiti's stand out in particular) that can help shine light on the subject.

just learn category theory

not learning Neo-Kabbalah