7
$\begingroup$

This is something I've been wondering about. When I think of "ratios" $x/y$ and $z/w$ as being "equal", with $x$, $y$, $z$, and $w$ being real numbers, this means the results of dividing the real numbers $x$ by $y$ and $z$ by $w$ are equal. Or that $xw = yz$, from manipulation of the fractions. Intuitively, we may say this means the "scale factor" going from $y$ to $x$ is the same as that going from $w$ to $z$, or that $x$ has as many "units" of size $y$ (allowing for non-integral numbers of units) as $z$ has of size $w$.

Yet, Euclid (~300 BCE) did not have real-number arithmetic to work with. Instead he had various kinds of "magnitudes", like line segments and shapes with areas and other things that had a kind of "size" to them. So he had to do something else, and this I don't get. If we have "magnitudes" $x$, $y$, $z$, and $w$, which for modern purposes could be taken as nonnegative real numbers, then we say $x/y = z/w$ iff for every pair of nonzero natural numbers $m$ and $n$, $mx < ny \rightarrow mz < nw$, $mx = ny \rightarrow mz = nw$, and $mx > ny \rightarrow mz > nw$. But how does one intuitively grasp this definition? How does it relate to our modern one? On Wikipedia, it says also "There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers." How exactly does this relate to Dedekind cuts? (this last bit is why I also file this under real analysis)

$\endgroup$
1
$\begingroup$

Take my answer as a grain of salt because my understanding might not be entirely correct:

Intuitively, the way I see Euclid's definition is you're trying to "squeeze" in rational numbers between $x/y$ and $z/w$ (or I suppose, the other way around) and if this fails, then they are equal. A Dedekind cut, partitions the rationals by creating two non-empty sets of rational numbers $A$ and $B$ with the condition that $A$ has no greatest element and all its elements are less than those of $B$. A real number $r$ is then defined to be a cut. Notice that Euclid's definition also cuts the rationals if we combine the equality condition with the greater than condition. So he has also essentially defined a cut. I think the difference between the two is that Euclid probably only had algebraic numbers in mind and not transcendental numbers, but I could be wrong..

$\endgroup$
  • $\begingroup$ Well, I think the old Greeks knew of $\pi$, which is transcendental, though they wouldn't have known of that property or the algebraic/transcendental division. The big division was between rational vs. irrational numbers, or what Euclid would've called commensurable and incommensurable ratios (e.g. a triangle (in area) to its corresponding parallelogram is an example of the former, while the side of a square to its diagonal an example of the latter). The definition of comparison was to handle comparison of such incommensurable ratios. $\endgroup$ – The_Sympathizer Sep 20 '13 at 11:44
  • $\begingroup$ @mike4ty4 This might help: books.google.co.uk/… $\endgroup$ – user70962 Sep 20 '13 at 16:46
  • $\begingroup$ Interesting. Thanks. $\endgroup$ – The_Sympathizer Sep 21 '13 at 1:29
  • $\begingroup$ @mike4ty4 No problem. Might be interesting to read Dedekinds original paper too if you haven't already: gutenberg.org/files/21016/21016-pdf.pdf Its pretty awesome being able to read these papers I must say. Thanks for the question. I've learned a few things because of it. $\endgroup$ – user70962 Sep 21 '13 at 16:03
  • $\begingroup$ Hmm. I also notice there seems to be a logical gap in Euclid: for the definition of equal ratios to work, it need be that "rational" magnitudes are dense in the space of magnitudes, however Euclid does not prove this anywhere. This follows from the Archimedean property, which he hints at in his Definition V.4 (but doesn't state explicitly as an axiom or postulate even though he really should have), but requires proof nonetheless. $\endgroup$ – The_Sympathizer Sep 21 '13 at 22:52
0
$\begingroup$

With the suggestions provided by 12F8916 and some more thinking and reading about this in the following time, I think I have managed to come up with a way to answer my own question on this one.

Euclid considers a system of geometric magnitudes, things like lengths, areas, etc. Given a pair of magnitudes of the same type, $x$ and $y$, we form their ratio $\frac{x}{y}$, which we can think of in modern terms as a member of some set of comparable elements ordered by some linear order $\leq$. As Euclid has ratios between whole numbers, we can consider that the rationals are a subset of this -- though, more specifically, only the positive rationals, since Euclid had neither negative numbers nor zero. The following then shows how Euclid's notions relate to our modern concepts:

So we have a set of ratios, call it $\mathcal{R}$. Then $\frac{x}{y} \in \mathcal{R}$. For any $\frac{m}{n} \in \mathbb{Q}^{+}$, we also have $\frac{m}{n} \in \mathcal{R}$. Now Euclid says that two ratios $\frac{x}{y}$ and $\frac{z}{w}$ in $\mathcal{R}$ are equal iff, for every pair of positive integers $m$ and $n$,

$$mx < ny \rightarrow mz < nw$$ $$mx = ny \rightarrow mz = nw$$ $$mx > ny \rightarrow mz > nw$$

. We can rearrange these with cross-division: if $ab <=> cd$, then $\frac{a}{c} <=> \frac{d}{b}$. So we get

$$\frac{x}{y} < \frac{n}{m} \rightarrow \frac{z}{w} < \frac{n}{m}$$ $$\frac{x}{y} = \frac{n}{m} \rightarrow \frac{z}{w} = \frac{n}{m}$$ $$\frac{x}{y} > \frac{n}{m} \rightarrow \frac{z}{w} > \frac{n}{m}$$.

The first equation means every positive rational greater than $\frac{x}{y}$ is greater than $\frac{z}{w}$, the third every positive rational less than $\frac{x}{y}$ is less than $\frac{z}{w}$, and the second that every positive rational equal to $\frac{x}{y}$ is equal to $\frac{z}{w}$. So we have that

The set of all positive rationals greater than $\frac{x}{y}$ is contained in the set of all rationals greater than $\frac{z}{w}$. The set of all positive rationals equal to $\frac{x}{y}$ is contained in the set of all rationals equal to $\frac{z}{w}$. The set of all positive rationals less than $\frac{x}{y}$ is contained in the set of all positive rationals less than $\frac{z}{w}$.

Due to the unstated, but presumably understood, symmetry of equality, we have that the converse subset relations hold as well. So actually this says those subsets are equal.

But the "set of all positive rationals greater than some ratio", unioned with those equal to it, is just the upper half of a Dedekind section $(L, R)$ of $\mathcal{R}$. Likewise, the "set of all positive rationals less than some ratio" is just the lower half. So we can interpret this as saying:

Two ratios are equal if the Dedekind sections they define in the positive rationals are equal.

Note that for this definition to make sense, we need for the ratio which makes the cut to be unique. This in turns requires that the positive rationals be dense in $\mathcal{R}$, which in turn requires the Archimedean Axiom, which is probably one of the reasons Euclid said that magnitudes only "have a ratio" if they satisfy this axiom ("when multiplied, exceed one another"), although of course Euclid would not have thought of it in terms of "density" of one set in another. Euclid also apparently implicitly assumes that all magnitudes, or at least straight-line magnitudes, i.e. lengths, satisfy this axiom, since he never qualifies any of the propositions which depend on it.

Note that this brings you half-way to the real number plane used in modern geometry. In particular, if you take the Archimedean Axiom as valid, you can have no more points in the plane than in the real number plane. To get all the way, that is, that there are no fewer points, you need a sort of converse of the above: the above contains an implied notion that for every ratio, there corresponds a cut. The converse is, "for every cut, there corresponds a ratio". This, of course, is what we call "Dedekind Completeness" now. However, Euclid probably wouldn't have liked this one, because it requires completed infinite sets to specify the ratio, and it seems the ancient Greek mathematicians did not like the idea of an actual, completed infinite (which probably also would aid in the acceptance of the Archimedean Axiom, which says "no" to actually infinitely-long line segments). In terms of modern logic, you can define equality of ratios with only a quantifier over individual naturals (without zero), whereas to define completeness, you need a quantifier over infinite sets of naturals. It would seem, then, that the Greek geometric plane would probably have been more like what we would call the "constructible plane" containing only those points constructible by compass and straightedge, or perhaps at a very great best the algebraic plane, containing only those points whose coordinates can be given by algebraic numbers.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.