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\centerline{MONT 107Q -- Thinking About Mathematics}
\centerline{Greek Thinking about Commensurability, Concepts of Magnitude and Number,}
\centerline{from Plato (ca. 424 - 348 BCE)
and Euclid, {\it Elements} (ca. 300 BCE)\footnote{Quotations
taken from the online version of the Heiberg edition of Euclid with English translation by Richard Fitzpatrick}}
\centerline{February 15, 2017}
\vskip 10pt
\noindent
{\it Definitions from Book V}
\bigskip
\begin{enumerate}
\item A magnitude is a part of a(nother) magnitude, the
lesser of the greater, when it measures the greater.\footnote{In other words, $\alpha$ is said to be 
a part of $\beta$ if $\beta = m \alpha$ (for some integer $m$, comment added by JL)} 
\item And the greater (magnitude is) a multiple of the
lesser when it is measured by the lesser.
\item A ratio is a certain type of condition with respect to
size of two magnitudes of the same kind.
\item (Those) magnitudes are said to have a ratio with respect to one 
another which, being multiplied, are capable
of exceeding one another.\footnote{In other words $\alpha$ has a ratio with respect to $\beta$ if $m\alpha > \beta$ and $n\beta > \alpha$ (JL: for some integers $m,n$)}
\item Magnitudes are said to be in the same ratio, the first
 to the second, and the third to the fourth, when equal
multiples of the first and the third either both exceed, are
 both equal to, or are both less than, equal multiples of the
second and the fourth, respectively, being taken in corresponding order, according to any kind of multiplication
whatever.\footnote{In other words, $\alpha : \beta :: \gamma : \delta$ if and only if $m\alpha > n\beta$
whenver $m\gamma > n\delta$, and $m\alpha = n\beta$ whenever $m\gamma = n\delta$, and 
$m\alpha < n\beta$ whenver $m\gamma < n\delta$.  This definition is the kernel of Eudoxus' theory
of proportion, and is valid even if $\alpha, \beta$, etc. are irrational.}
\item And let magnitudes having the same ratio be called
proportional.
\end{enumerate}
\vskip 10pt
\noindent
{\it Definitions 2,3 from Book VII}
\vskip 10pt
\begin{enumerate}
\item[2.] A number is a multitude composed of units.
\item[3.] A number is a part of a number, the lesser of the greater, when it measures the greater.
\end{enumerate}
\vskip 10pt
\noindent
{\it Definition 1 from Book X}
\bigskip
\begin{enumerate}
\item Those magnitudes measured by the same measure
are said (to be) commensurable, but (those) of which no
(magnitude) admits to be a common measure (are said
to be) incommensurable.\footnote{By JL:  In Proposition 5 of Book X, Euclid shows that if two magnitudes 
$\alpha, \beta$ are commensurable then $\alpha : \beta :: m : n$ for some integers $m,n$.  Then
Proposition 6 shows that the implication goes the other way too, so this definition is
actually equivalent to saying $\alpha : \beta :: m : n$ for some integers $m,n$.}
\end{enumerate}
\vfill\eject
\noindent
{\bf Proposition}.  Let it be proposed to show that the diagonal of a square is incommensurable with its side.\footnote{by JL: This proposition
and the following proof were included in Euclid, {\it Elements}, Book X, as Proposition 117 in
some versions.  However, modern scholarship has identified it as a later interpolation from a 
commentary on Aristotle's {\it Prior Analytics}, where Aristotle discusses the logical
structure of the proof -- a {\it proof by contradiction}.  Many modern editions of Euclid do not include it for that reason.  The standard edition of the Greek
by J.L.Heiberg includes it in an appendix to the main text, but this was not
translated by Fitzpatrick.  Non-literal, reorganized
and slightly modernized translation from the Greek by JL.}
\vskip 10pt
\noindent
{\it Proof:}  Let the square be $ABCD$
whose diagonal is $AC$.  I say that the diagonal $AC$ is incommensurable with 
$AB$, its side.  

\bigskip
Suppose that these magnitudes were commensurable.  I say their ratio is then
the same as $a : b$ be where $a,b$ are integers,
and let this ratio have the smallest numbers among all with the same ratio
 [JL: that is, in our language, the fraction $a/b$ is in ``lowest terms''].  
Then $a > b$ and necessarily $a > 1$ because $AC > AB$.    
It is clear the square on $AC$ is twice the square on $AB$ in area [JL: by
the form of the Pythagorean Theorem from {\it Elements}, Book I, Proposition 47]
and hence
$a^2 = 2b^2$.  Therefore $a^2$ is even and $a$ is even, since if a number is
odd, so is its square.  Since $a : b$ is
in lowest terms, the numbers $a$ and $b$ cannot have any common factors
and it follows that $b$ must be odd.  Put $a = 2c$.  Therefore $4c^2 = 2b^2$ and
hence $b^2 = 2c^2$, so $b$ and hence $b$ must be even.  But $b$ was also odd which is impossible.
Therefore the diagonal of a square is incommensurable with its side--what was 
to be proved.  
\vskip 20pt
\noindent
A Related Passage from Plato (ca. 424 - 348 BCE) {\it Theaetetus}\footnote{Plato, The Collected Dialogues, Hamilton and Cairns, eds., pp. 852-853}
\vskip 10pt
\noindent
...
\vskip 10pt 
Theaetetus:  Theodorus here was proving to us something about square roots, 
namely, that the sides [or roots] of squares representing three square feet and five square feet
are not commensurable in length with the line representing one foot, and he went on in this way,
taking all the separate cases up to the root of seventeen square feet.  There for some reason
he stopped. The idea occurred to us, seeing that these square roots were evidently infinite
in number, to try to arrive at a single collective term by which we could designate
all these roots.
\vskip 10pt
Socrates:  And did you find one?
\vskip 10pt
Theaetetus:  I think so, but I should like your opinion.
\vskip 10pt
Socrates: Go on.
\vskip 10pt
Theaetetus:  We divided number in general into two classes.  Any number which is the product
of a number multiplied by itself we likened to a square figure, and we called such a 
number `square' or `equilateral.'
\vskip 10pt
Socrates:  Well done!
\vskip 10pt
Theaetetus:  Any intermediate number, such as three or five, or any number that cannot
be obtained by multiplying a number by itself, but has one factor either greater or less
than the other, so that the sides containing the corresponding figure are always unequal, 
we likened to the oblong figure, and we called it an oblong number.
\vskip 10pt
Socrates:  Excellent.  And what next?
\vskip 10pt
Theaetetus:  All the lines which form the four equal sides of the plane figure representing
the equilateral number we defined as {\it length}, while those which form the sides of
squares equal in area to the oblongs we called {\it roots} [surds], as not being 
commensurable with the others in length, but only in the plane areas to which their
squares are equal.  And there is another distinction of the same sort in solids.
\vskip 10pt 
...
\vskip 10pt
\noindent
Comment by JL:  The general theme of this Platonic dialogue is the question, ``what is knowledge?''
The point made by Theaetetus here seems to be that simply having {\it names for things} can be 
knowledge of a sort.  But
as is true in many of the Platonic dialogues, Socrates is eventually going to get him to admit
that neither this, nor any of the other possible definitions of knowledge that they consider is satisfactory -- 
knowledge is not just having a name for something; 
it is not just \emph{perception}, it is not \emph{true judgment}; and it's not even \emph{true judgment with 
an argument in support of the judgment}.  There 
is no definitive answer to the main question at the end of the conversation(!)
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