I invite you to read this paper (Eudoxus' Axiom and Archimedes' Lemma) and share your thoughts

[MaxMath]

I bumped into this little paper, written by Johannes Hjelmslev, by chance and find it very interesting, but in the meantime very difficult to understand. I don't have any specific questions for now (perhaps I have too many but probably most are dumb questions—I just need to read it a little more). I will follow up with questions and thoughts later. If you have an interest in it, or you are knowledgeable in this space, your thoughts, insights, and answer to my questions (when they come) will all be appreciated.

Briefly and roughly, my aim is—beyond fully understanding the thoughts of the author—as suggested by him at the beginning of the paper, to penetrate the mists of history and get through into the thoughts of Archimedes and the Greeks alike, and grasp the essence, if this does not sound overly ambitious!

Of course, you are not obliged to read this or to answer my questions!

 

[MaxMath]

The tactic I always use when cracking something difficult is (let me know if you have a better tactic)---read carefully from the beginning to a point when it feels really hard; don't stop but keep pushing hard, to a point when reading further on becomes meaningless (because the combination of words stops making sense); restart again from the beginning, re-read what I have read--(hopefully) gaining new insights from there--plough to the point where I stopped last time; now due to the new insights gained, I'm likely to dig a little bit further, until reaching a new point just like last time; so on and so forth ...

Between rounds of attack, there will probably be peripheral readings, breaks, etc etc... Or I may never come back.

When this tactic eventually fails, what's left is usually a rabbit that is half dissected half intact, which I may pick up later, or may not.

 

[MaxMath]

At the centre of this paper are, apparently, the "Eudoxus' Axiom" and the "Archimedes Lemma". The aim of the author appears to clear a misinterpretation ("evident misunderstanding") among mathematicians taking the latter as equivalent to the former, hence using one to refer to the other or messing around by referring to "Archimedes' Axiom", etc.

According to the paper, these two are—

Eudoxus’ Axiom—Magnitudes are said to have a ratio if any one of them by being added a sufficient number of times to itself can be made to exceed the other: [math]a<b, a+a+...+a>b[/math]
Eudoxus’ Axiom is a part of Eudoxus’ theory of proportions.

Archimedes’ Lemma—… of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with [it and with] one another.

[In my first read, I was not sure what "itself" refers to—is it "the greater", "the less", or "such a magnitude"? According to section 5, it's clear that it refers to "such a magnitude", i.e. the magnitude by which the greater exceeds the less.]

It appears to be the author's view that not only they are not equivalent, Archimedes’ Lemma is more significant than Eudoxus’ Axiom, because the former is concerned with "the new field of magnitudes", such as the length of a curve, area of a surface, or volume of a solid.

I'm not yet able to fully appreciate this view, or exactly why such an extension is significant (of course, being able to deal with curves, areas and volumes is obviously one big step ahead of merely straight lines).

I would also be very interested in your interpretations of both the Axiom and the Lemma.

 

[MaxMath]

[This is related to p 2 of the pdf, shown as p 3 at the top-right corner.]

One particular paragraph (very short) in section 3 is baffling—

Quote
From the theory of proportions proper it is in Archimedes' investigations the main rule that is applied, stating that[math]a\gtreqqless b\;\text{yields respectively}\\a/c\gtreqqless b/c\;(\text{or}\;c/a\lesseqqgtr c/b).[/math]Unquote

The formula here is not the difficulty: it's the language itself. I have this question:

[Q1] Of, or in, what, is the statement (what is stated with the formulas) "the main rule"? Is it the main rule of Audoxus' theory of proportions, or anything else (what is it if the latter)?

It does not appear to be a part of what is quoted as Audoxus' theory of proportions in the paper, let alone a "main rule" (at least in the form of it)—

Quote
A. Eudoxus’ Axiom (Eucl. V, def. 4), which gives the conditions under which two magnitudes, a, b, can be said to have any ratio to each other:
Magnitudes are said to have a ratio if any one of them by being added a sufficient number of times to itself can be made to exceed the other: [math](a<b, a+a+...+a>b).[/math]Further
B. The definition of equal ratios a/b and c/d: if [imath]ma\gtreqqless nb[/imath], then respectively [imath]mc\gtreqqless nd[/imath] for each set of integers m, n.
C. The definition of unequal ratios: [imath]a/b > c/d[/imath], if there exists a set of integers m,n so that [imath]ma > nb[/imath] but [imath]mc\leqq nd[/imath].
Unquote

If it is, then it's probably what is not included in the paper, but is a part, and the "main rule", of Audoxus' theory of proportions.

 

[MaxMath]

[This is still related to p 2 of the pdf, shown as "p 3" at the top-right corner.]

What follows immediately, I also don't quite understand. It says—

Quote
But the proof of this proposition presupposes (Eucl. V, 8)—regard is here had only to the uppermost sign (a > b)—the existence of an integer number n so that[math]n(a-b)>c\text{.}[/math]Unquote

I happen to have a hard copy of Euclid's The Elements (but have not read it!), so maybe this can be figured out if I dive into it deep enough—it's a long proof.

Anyway, leave it as a question here—

[Q2] Why is it so?

The next question is about the following para (the last para on this page). This is linked to the central argument of the paper—Archimedes' Lemma is more significant than Audoxus' Axiom, because the former is concerned with a "new field of magnitudes".

It is true, more types (and apparently more sophisticated) fields are dealt with in Archimedes' Lemma. But the thing is, as is suggested by the Lemma, the magnitudes for comparison, or addition, in the Lemma still need to be of the same type, be they lengths of lines, or volumes of solids, or whatever else. In this sense, I really cannot see the greater significance of the Lemma—quite the contrary: the Axiom appears to be more general, which is one important measure of the value of a theory in mathematics.

So my question here is—

[Q3] How do you understand this paragraph (the last on "p 3") in relation to the central argument of this paper?