Remembering Socrates

When asked whether the square root of 2 can be written as an exact 
fraction, i.e., as the ratio of two integers, many people will say 
they don't know.  This has always seemed interesting to me, because 
of course they DO know - which is to say, they are in possession of 
all the information and understanding necessary to know the answer.  
If sqrt(2)=M/N then 2 = (M/N)^2, and if they remember that numbers 
factor uniquely into primes, it's immediately obvious that 2N^2 = M^2
is impossible, because a square can't equal twice a square.  (The 
exponent of the prime 2 in M^2 is even, whereas the exponent of 2 
in 2N^2 is odd.)

Even without invoking unique factorization, Euclid described a simple
proof "from scratch" that anyone can easily follow.  If the square 
root of 2 equals M/N for some integers M,N, then 2=(M/N)^2, and we 
can assume that M,N are not both even, because if they were we could 
divide them both by 2 while preserving the ratio.  Thus, at most one 
of M,N is even.  Writing the equation in the form 2N^2 = M^2, we see 
that M^2 is even, and we also know that the square of an odd number 
is odd, so M itself must be even, and therefore N must be odd.  Now,
since M is even, there is an integer m such that M=2m, so we can
substitute this M into the equation 2N^2 = M^2 to give 2N^2 = 4m^2,
which implies N^2 = 2m^2.  But this shows that N^2, and therefore N,
must be even, contradicting the fact that there must be a solution
with M,N not both even.

It's interesting that nearly this very same example was used by
Socrates to illustrate the same point, i.e., how people know more
than they think they do.  In Plato's "Meno" he recounts a dialogue
between Socrates and a fellow Athenian on the subject of whether
virtue can be taught.  A recurring theme in Socrates' thought was
that all the knowledge we are capable of possessing is already 
within us, and the process of reasoning something out is really
just an act of recollection, i.e., remembering things we already
(in some sense) know.  

To illustrate this, Socrates questions an un-schooled servant boy 
about a simple geometrical proposition.  Socrates draws a square, 
and asks the boy how he would go about constructing a square twice 
as large.  Initially the boy says he doesn't know, but under further
questioning he thinks the answer is to make the edges twice as long
as the edges of the original square, making a figure like this:

                

But then Socrates asks him how much area this new square covers 
in relation to the original, and the boy correctly observes that
it has four time the area.  So Socrates re-iterates the question:
how would you construct a square with just twice (not four times)
the area of the original?  Obviously we need a square with half
the area of the one just constructed.  Socrates asks the boy if
we can cut each of the four squares in half by drawing a line
connecting opposite corners, and the boy answers Yes.  They draw
these lines ("clever men call this the diagonal") and arrive at 
the figure below:

                

They agree that the four diagonals describe a square, and its area
is twice the area of the original square.

 Socrates:  What do you think, Meno?  Has he, in his answers, 
            expressed any opinion that was not his own?
 Meno:      No, they were all his own.
 Socrates:  And yet, as we said a short time ago, he did not know?
 Meno:      That is true.
 Socrates:  So these opinions were in him, were they not?
 Meno:      Yes.
 Socrates:  So the man who does not know has within himself true
            opinions about the things that he does not know?
 Meno:      So it appears.
 Socrates:  These opinions have now just been stirred up like a 
            dream, but if he were repeatedly asked these same
            questions in various ways, you know that his knowledge
            about these things would be as accurate as anyone's.
 Meno:      It is likely.
 Socrates:  And he will know it without having been taught, but 
            only questioned, and find the knowledge within himself?
 Meno:      Yes.
 Socrates:  And is not finding knowledge within oneself recollection?

Socrates then goes on to speculate on when or how the boy had acquired
his true opinions about geometry, and suggests that it must not have
been during his present life (since Meno assures Socrates that the
boy has had no instruction in geometry).

Of course, we might observe that what clever men call the line 
connecting opposite corners ("diagonal") was not one of the boy's
own opinions.  He was taught this by Socrates, so one could argue
that the boy has in fact been taught something which he did not know,
and which he (presumably) could never "recollect" simply by examining
his opinions in isolation.  This highlights two different kinds
of knowledge, one that derives uniquely from first principles (the
common notions about geometrical shapes) and the other that is 
accidental and arbitrary (terminology). More fundamentally, one
could argue that people DO learn and acquire common notions about
spatial relations and proportions during their formative years,
as they organize their primitive sense perceptions.  On the other
hand, certain very basic aspects of our experience may be "hard-
wired" into the biology of our brains and sense organs.  This seems
to be a mode of transmission for information that Socrates doesn't
consider.

To his credit, Socrates concludes

 "I do not insist that my argument is right in all respects,
  but I would content... that we will be better men, braver
  and less idle, if we believe that one must search for the
  things one does not know..."

Incidentally, the very next section of the "Meno" dialogue contains
another geometrical example that is raised by Socrates to make a
point about whether knowledge is teachable.  Unfortunately the 
exact sense of his words is unclear, and the available translations
are all slanted toward one particular interpretation or another. 
Thomas Heath says that this example is

  "much more difficult [than the previous example], and it has 
   gathered round it a literature almost comparable in extent 
   to the volumes that have been written to explain the Geometrical
   Number of the Republic.  C. Blass, writing in 1861, knew 
   thirty different interpretations; and since then many more
   have appeared.  Of recent years, Benecke's interpretation
   seems to have enjoyed the most acceptance; nevertheless I
   think that it is not the right one...

Heath then goes on to give the interpretation that he thinks most
closely fits the text (based on ideas of S. H. Butcher and E.F.
August).  However, it seems to me that Heath's proposed interpretation
is not much more persuasive than any of the others for exactly what
Socrates (or Plato) had in mind.

The translation of Plato's text available from most sources today
is based on Heath's interpretation.  Here is how Heath thinks the 
passage should be read:

  If we are asked whether a specific area can be inscribed
  in the form of a triangle within a given circle, [we] might
  say... if that area is such that when one has applied it as 
  a rectangle to the given straight line in the circle it is 
  deficient by a figure similar to the very figure which is 
  applied, then [we have our answer].

This is not abundantly clear.  Heath gives a somewhat mundane
reconstruction based on ordinary rectangles and triangles on the
diameter of the circle, and his explanation is nominally plausible
(under the interpretation he provides).  However, Socrates' peculiar 
description has always reminded me of something else entirely.

Recall that Plato became a pupil and friend of Socrates in 407 BC,
and Socrates himself lived from 469 to 399 BC.  One of the most 
striking geometrical results of Greek mathematics was the quadrature
of the lune, accomplished by Hipppocrates around 440 BC.  This
would have been one of the most talked-about results during the
years when Socrates was beginning his teaching, because it was the
first time anyone was able to construct, by classical methods, the
area of a region with a curved boundary.

Moreover, it connects directly to the simple example of "doubling
the square" that is discussed earlier in Meno, as can be seen from
the drawing below:

              

The key to Hippocrates' argument is that the quadrant of the main
circle (consisting of the regions A and B) obviously has 1/4 the
area of the main circle.  Also, the smaller circle has a diameter
equal to 1/sqrt(2) of the larger circle, because it is what clever
men call the diagonal of a square whose sides are half of the main
circle's diameter.  Consequently we know that the smaller circle
has exactly half the area of the larger circle, which implies that
the smaller half-circle (the regions B and C) has exactly the same
area as the larger quarter-circle (the regions A and B).  Hence
we have A + B = B + C, and so A = C.  In other words, the area of
the "lune" (region C) equals the area of the inscribed triangle A.

In other words, the area of the lune is inscribed as a simple triangle
in the circle if, when we construct a circle on the edge of that 
triangle, the region that is excluded from the main circle is equal 
to the area of the inscribed triangle.  Also, the triangle is 
deficient (relative to the quadrant of the circle) by the very same
shape by which the smaller semi-circle exceeds the required area.  
Recall Heath's translation of Plato's account of Socrates' dialogue 

  ... a specific area can be inscribed in the form of a triangle
  within a given circle... if that area is such that when one has 
  applied it ... to the given straight line in the circle it is 
  deficient by a figure similar to the very figure which is APPLIED,
  then [we have our answer].

Here I've omitted the phrase "as a rectangle".  Without going back 
to the ancient Greek text, it's difficult to say how each term and 
phrase was intended, and words like "similar" vs "equivalent" are
often up to the translator to decide based on his understanding of
the context.  Heath's translation was naturally slanted toward 
his own guess as to what mathematical construction Socrates was 
describing, whereas other scholars have read and translated the 
same text differently.  It would be interesting to return to the
original Greek text, with quadrature of the lune in mind, to see 
if a translation based on this interpretation is feasible.