History of Science and Mathematics Asked by PhD on September 2, 2021
I’ve been intrigued by the similarities between what Eudoxus’ Theory of Proportions and Dedekind cuts.
However, I wish to question this “perceived similarity” and would like to where the flaws are, in my understanding.
Eudoxus could see that similarity continues to exists irrespective of the “numerical magnitude” of the side. So side-stepping number theory, just casting this problem geometrically it’s relatively straightforward to come up with the idea of a proportion IMHO:
Now, Eudoxus tried to avert infinity by having definitions (per Euclid) like so:
Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
However, the definition of equality of proportions seems to apply an infinite process (i.e., Dedekind cut) since $m, n$ are expected to be integers/rationals to the best of my knowledge.
Questions:
I'm not sure that my answer will add anything to your understanding of the problems as expressed in your question, but it may be useful to remind us of the conventional reading of Eudoxus.
John Stillwell, in his text The Real Numbers, summarises the situation as follows:
The longest book of the Elements, Book X, is devoted to the classification of irrational quantities arising in geometry. And the most subtle book of the Elements, book V, is devoted to Eudoxus' "theory of proportions," which seeks to deal with irrational quantities by comparing them with the rationals.
In the next chapter we will see how the idea of comparing an irrational quantity with rationals was revived by Dedekind in the nineteenth century to provide a concept of real numbers making up the number line, thus providing an arithmetical foundation for geometry. The novel part of Dedekind's idea is its acceptance of infinite sets - and idea that the Greeks rejected.
Stillwell then goes on to introduce Dedekind cuts, closing the chapter by saying:
A slogan to sum up this chapter might be: the basic theory of $mathbb R$ equals Greek mathematics + Infinity (or even Euclid + Infinity). The ancient Greeks gave us integral and rational numbers and the principle of induction by which their properties may be proved. They also gave us infinite processes for approaching irrational numbers, though they did not dare to complete them or "take them to the limit". Thus Euclid, in his Elements, Book X, proposition 2, gave nontermination of the Euclidean algorithm as a criterion for irrationality. He gave no example at this point, but his proposition 5 of Book XIII immediately implies periodicity, and hence nontermination, of the Euclidean algorithm on the pair $frac{1+sqrt 5}{2}, 1$. This leads us to the continued fraction representation [$1; 1, 1, $...] but it would not have been accepted by the Greeks, since it implied "completing" a process that does not end. The Greeks accepted the "potential" infinity of a process, but not the "actual" infinity of its completion.
So while all of the necessary ingredients were available to the Greeks, Stillwell appears to be emphatic in his view that they never took the necessary final step.
Correct answer by Nick on September 2, 2021
A reliable source on early Greek mathematics is Fowler's Mathematics Of Plato's Academy, freely available. Unfortunately, there are too many inaccuracies propagated by textbooks and popular sources to address here. So let me start by brief statements.
Pythagoreans did not discover irrationality of $sqrt{2}$, they had no such notion. There is no such thing as "length" of a segment in Euclid. There was no "irrationality crisis", and there existed a theory of incommensurables, perfected by Theatetus, before Eudoxus got on the scene. The "infinite process" it exploited is called anthyphairesis. It is a geometric version of the Euclidean algorithm, and the modern arithmetic analog of it is expanding irrationals into continued fractions. Unfortunately, it becomes intractable beyond quadratic irrationals.
Eudoxus's invention of the ratio theory, and the double reductio technique that came with it (currently misnamed "method of exhaustion"), was indeed a breakthrough that advanced geometry by a mile. It is still underappreciated due to lack of direct sources, we only have Euclid's retelling and scattered secondary references. But it had little to do with the "crisis" of Pythagoreans discovering incommensurability.
So why so much inaccuracy? Because it would take too long to convey the context of Greek mathematics, with its strict separation of numbers, magnitudes and ratios, and no analog of Cartesian coordinates or even Vieta's assignment of numbers to geometric objects. So what is done instead is a "translation" into the closest things available today. This translation must collapse the Greek distinctions and inject quick moves they could never make or think of. The moves they did make then become inexplicable and drop out of the story. Fables about the "crisis" and Hippasus being thrown to the sharks are made up in their stead. Incommensurable ratios become "irrationals", and Eudoxus becomes Dedekind's thesis advisor.
There are indeed some similarities in proof moves and techniques in Eudoxus and Dedekind. But consider the differences too. Dedekind constructs real numbers as cuts, this kind of hypostatic abstraction is unheard of not only in antiquity but even in the early 19th century. Frege still struggled with its analog (the Caesar's problem), as do many modern students when they encounter quotient groups. Eudoxus, on the other hand, defines equality for ratios of pre-constructed magnitudes. With the standard tools (straightedge and compass) the magnitudes produced correspond to only a small fragment of algebraic numbers, let alone of real numbers.
Dedekind then proceeds to define arithmetical operations on cuts. Eudoxus, on the other hand, can only add/subtract magnitudes. There are already problems with multiplying them because it changes the type of magnitude involved: the product of lines is an area, but the product of areas already has no geometric interpretation. And even addition and subtraction are hard to transfer to ratios because it requires the Cartesian move of choosing a unit magnitude arbitrarily, something that runs counter to Greeks' invariance intuitions (not unlike those of modern differential geometers). So Eudoxus mainly uses only comparison of ratios by a process described in Euclid's definition 5 of Book V.
Dedekind proves continuity for his cuts. Eudoxus, on the other hand, assumes it (even admitting the "translation") by assuming that something like line segments, angles and areas are, in fact, magnitudes. One can see it in the Elements where Euclid wields the "definition" 5 of Book V as a postulate. And so on. This is not to diminish the Eudoxus's breakthrough, which was the root of a tall tree that eventually bloomed with Dedekind and Cantor. But it needed to grow a lot of other branches to get there.
Answered by Conifold on September 2, 2021
All work of Eudoxus on the theory of proportions is lost. So we cannot really say what he did. Our knowledge of the theory of proportions is based on Euclid (which is easily available online, but difficult to read. There are several excellent commentaries). Euclid himself does not credit Eudoxus (or anyone else). That the theory of proportions really belongs to Eudoxus is known only from the later ancient historians who lived hundreds years after Euclid.
(Imagine for comparison that all work of Newton is lost, and a modern historian tries to describe what he did based on secondary and tertiary sources). So speculations on what Eudoxus REALLY did are groundless. We can only discuss what is written in Euclid on the subject.
Answered by Alexandre Eremenko on September 2, 2021
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