I Who Rediscovered Archimedes' Lemma in Modern Mathematics?

[Gear300]

TL;DR Summary

How did it come to recognition?

Reading Stephen Hawking's "God Created the Integers", he includes all of Euclid Book V on Proportions. Then he includes Book IX on Numbers, and then Book X on Irrationals. Apparently an assumption is made in Book X, which Hawking points out in his scholia, that a certain multiple of a lesser magnitude exceeds a greater magnitude. Then later, Hawking includes Archimedes' work.

When studying the real number axioms, it is often taught that Dedekind's completeness axiom can just as well be replaced by the Archimedes' Lemma to yield the real measurables. Funny thing is I never looked into how recognition of this came about, which is why I'm asking here. "Who" discovered and posited Archimedes' work into the calculus?

 

[jedishrfu]

I think Archimedes was a part of Calculus from the start:

https://en.wikipedia.org/wiki/Archimedes

He used a method of exhaustion to determine the volume of sphere. This method led to the development of limits and infinitesimal quantities.

Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote that Archimedes "placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life",[28] though some scholars believe this may be a mischaracterization.[62][63][64]
Method of exhaustion
Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.

Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus.[6] Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the areas of figures and the value of π.

In Measurement of a Circle, he did this by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approx. 3.1429) and 310/71 (approx. 3.1408), consistent with its actual value of approximately 3.1416.[65] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle ( π r 2 {\textstyle \pi r^{2}} {\textstyle \pi r^{2}}).
Archimedean property

In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers.[66]

Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512) in Measurement of a Circle. The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."[67] It is possible that he used an iterative procedure to calculate these values.[68][69]

 

[Gear300]

Right, but I figured he had to have been "rediscovered" by more modern mathematicians, although Wikipedia does discuss this. I am more particularly interested how the lemma came to be accepted in modern context.

 

[fresh_42]

Summary: How did it come to recognition?

Reading Stephen Hawking's "God Created the Integers", he includes all of Euclid Book V on Proportions. Then he includes Book IX on Numbers, and then Book X on Irrationals. Apparently an assumption is made in Book X, which Hawking points out in his scholia, that a certain multiple of a lesser magnitude exceeds a greater magnitude. Then later, Hawking includes Archimedes' work.

When studying the real number axioms, it is often taught that Dedekind's completeness axiom can just as well be replaced by the Archimedes' Lemma to yield the real measurables. Funny thing is I never looked into how recognition of this came about, which is why I'm asking here. "Who" discovered and posited Archimedes' work into the calculus?

I have never seen that the definition of an Archimedean ordering can replace the Dedekind cuts. The latter simply assumes the former. We have this kind of ordering before we define a Dedekind cut. I have seen that Dedekind cuts can be replaced by equivalence classes of Cauchy sequences (Heine, 1872, also Cantor, 1872, reviewed version: Cantor 1883) also according to the Archimedean ordering. A different ordering gets a different completion. They all use the Archimedean axiom, they do not replace it.

The Archimedean ordering is probably as old as counting is. The question should be: When did we first discover that the Archimedean ordering is not the only possibility? And the answer to that question is: Hensel, 1908.

 

[Gear300]

I have never seen that the definition of an Archimedean ordering can replace the Dedekind cuts. The latter simply assumes the former. We have this kind of ordering before we define a Dedekind cut. I have seen that Dedekind cuts can be replaced by equivalence classes of Cauchy sequences (Heine, 1872, also Cantor, 1872, reviewed version: Cantor 1883) also according to the Archimedean ordering. A different ordering gets a different completion. They all use the Archimedean axiom, they do not replace it.

The Archimedean ordering is probably as old as counting is. The question should be: When did we first discover that the Archimedean ordering is not the only possibility? And the answer to that question is: Hensel, 1908.

Ah, Thanks. Lol, perusing wikipedia, I also found Otto Stulz. Thanks again.

 

[Office_Shredder]

Right, but I figured he had to have been "rediscovered" by more modern mathematicians, although Wikipedia does discuss this. I am more particularly interested how the lemma came to be accepted in modern context.

A decent chunk of greek mathematics just persisted continuously in Europe without ever having to be rediscovered.