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.
- Pythagoreans discovered irrationality of $\sqrt(2)$ - now the notion of a "number" was threatened and the (Greek) world wondered how could the laws/proofs of "similarity of triangles" be applied correctly (poetic exaggeration)
- A length of $\sqrt(2)$ clearly exists geometrically, but not number theoretically as a diagonal of a unit square. (The latter implying that $\sqrt(2)$ is only attainable as a limit)
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:
- Represent any pair of magnitudes as two straight lines $a$ and $b$.
- Take another pair of magnitudes represented by two straight lines $c$ and $d$
- If $na <=> mb \space\forall (m,n) $ and the lines $c$ and $d$ move similarly (i.e., $nc <=> md \space\forall (m,n)$) then there is definitely a "relationship" between the two magnitudes. If they move the other way (i.e., increasing one decreases the other or vice versa) then we have an inverse relationship
- Given the above, we can continue with similarity as if there wasn't any problems and voila! we're back, crisis averted, number theory stalled for 2000 years!
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.
- It doesn't seem that Eudoxus was really partitioning anything - he was looking at the problem to subvert the irrational crisis and just put forth the notion of proportionality geometrically vs. that of considering individual numbers. Is this correct?
- Was he (or his contemporaries) aware of the "hidden" infinite process in computing $m,n$ since "number theory" was absent - all they were doing was extending "lines" and seeing if another related pair moved "similarly". Did they really encounter this infinite process "explicitly" or is this a view from our modern understanding of real numbers, infinity and limits?
- What may I be missing?