Comment by kragen | Hacker News Reader

Comment by kragen | Hacker News Reader

Lucio Russo's The Forgotten Revolution points out somewhat indignantly that Dedekind was ("just") restating in modern number-theoretical language a definition of proportion or ratio that he found in Book 5 of Euclid.

The translation at https://mathcs.clarku.edu/~djoyce/elements/bookV/bookV.html says:

> Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

This is a bit hard to understand, but I think it does come out to the same thing: positing the existence of ratios of magnitudes that cannot be expressed as ratios of integers, but which divide the ratios of integers into those that are less than or greater than the ratio of magnitudes in question.

Further discussion at https://math.stackexchange.com/questions/499395/how-to-best-... seems to agree.

The real breakthrough with Dedekind cuts (as far as I can see) is not that you can define irrational numbers this way, it’s that when you do, calculus works. In other words, they obey all the axioms you would want and in particular they have the Archimedean property and possess a least upper bound.

You could always say that Dedekind was “just” restating real numbers in modern analytic language, but the fact is that no-one had managed to do that up till then, and it wasn’t obvious how it could be done. This is what sparked the crisis in analysis with some people saying that calculus was more like a religion that you just had to believe in rather than being able to prove.

By the way, there is an accessible and (in my opinion) brilliantly clear explanation of the construction of numbers on the “Another Roof” youtube channel.

“What is a number?” https://youtu.be/dKtsjQtigag?si=EPzNVgk47gXzNLra Explains what natural numbers are and why we should care about this set of definitions

And “defining every number ever” goes from there up to the complex numbers https://youtu.be/dKtsjQtigag?si=DI39LQtGvBsOU00y

Oh and the reason I picked on the Archimedean property specifically is that it had been this that both Leibnitz’”infinitesimals” and Newton’s “fluxions” had handwaved around. They both relied no some sort of non-Archimedean numbers for limits and derivatives to work.

The Archimedean property is that given two numbers x and y there exists an integer n such that nx>y. Infinitesimals and fluxions relied on the idea that they were so small this wouldn’t work and therefore they didn’t fit the definition of real numbers.

Is that not the same thing OP was talking about re: Eudoxos? Hard to tell because I'm not exactly sure what an "equimultiple" is, but comparing ratios is a constrained enough problem that I don't know what else it could be.

You're right, it is! I'm embarrassed to admit I stopped reading before that point.

I think the "Eudoxus reals" construction is worth thinking about too. It builds directly on integers instead of building on top of the rationals. https://ncatlab.org/nlab/show/Eudoxus+real+number

I wish history recorded how Eudoxus got to this. There was a paper speculating (iirc) that the Greeks started by thinking about continued fractions as a way to deal with incommensurables, and then realized you don't need that much machinery to define them. (You get continued fractions by running Euclid's algorithm on incommensurable input.)

Isn't this the same thing the linked post is saying?

Yes, oops! See above.

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