Mathematical commentary on YBC 7289
There are a number of remarkable facts
about the tablet, which is one of the very oldest
mathematical diagrams extant. Given our vast ignorance
about the era, speculation is inevitable.
It is the last point that has attracted most attention, but to
one like myself, who has spent a great deal of time
and effort thinking about graphical proofs
of mathematical results, it is the second last point that
is the most striking. I believe that we are looking here
at the very origins of mathematical reasoning.
This claim doesn't seem to be commonly accepted
by mathematicians (see, for example, Jim Carlson's
remarks in his notes on ancient mathematics),
and historians might not appreciate it,
but I don't see how else to interpret
the figure itself. It amounts to a dissection
of the square on the hypotenuse of an isosceles right
triangle into pieces which can be reassembled to
make up the two squares on the sides, and I can't see why the
figure is exactly what it is if it weren't understood to demonstrate this.
To make this idea slightly more plausible, consider
the extrapolation of the figure on the tablet to the figure
on the left below:
The Babylonians, unlike the early Greeks much later on,
interpreted ratios of lengths as numbers.
- They possessed a place-value numbering
system. It involved a principal
base of 60, but the numbers from
0 to 59 were expressed in base 10.
It is not apparent in the picture,
but they didn't yet (i.e. in 1600 B.C.) possess a zero.
Their representation system was a true floating point system:
"30", for example, could be either 30 or 1/2.
- They weren't just finding a good
(very good) approximation to the ratio of
a diagonal to a side of a square.
They knew that
the ratio of the diagonal of a square to a side was a number
whose square was 2.
They possessed an algorithm for finding approximations to the square root
Incidentally, the figure on the right, also closely
related to our figure, is itself close to one of the figures found
in the equally old tablet B.M. 15285.
The second figure is also, remarkably enough, one of the diagrams that
accompany Socrates' dialogue with the Greek
boy in Plato's Meno, in which Socrates illustrates
his claim that all knowledge is somehow innate to
the human mind by drawing out of the boy
a proof of Pythagoras' theorem for isosceles right triangles.
So in effect, what we are seeing here is one of
the very first examples of human reasoning - but
visual reasoning, not verbal. Who are we
to say how mathematical reasoning evolved?
Or that for many purposes visual reasoning is not as valid as verbal?