The traditional formulation of the theorem of Pythagoras is
that the square on the hypotenuse of a right angled triangle is equal to the
sum of the squares on the other two sides. This is the Greek formulation, but
we know that some of the properties of Pythagorean triangles were known in
earlier cultures, such as in Babylonia.
The triangles of most interest were those constructed using
integral values, such as the 3,4,5 triangle. These values were commensurable,
and not irrational.
If we consider this in terms of the construction of the
version of the triangle constructed using the square values, we can see that it
is not a geometrical figure in two dimensions at all, but a line, built out of
the three squared values (9, 16, and 25), exactly 25 integral units in length,
with the three sides subsumed into one dimension.
This understanding of the geometrical arrangement perhaps
tells us something about the ancient perception of space, and why it exists.
There are several possible Pythagorean triangles which exist,
all of which exhibit the same properties, but with different combinations of
integral values.
There are 16 primitive Pythagorean triples with c ≤ 100:
(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)
(20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53)
(11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)
(13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)
Additionally these are the 29 primitive Pythagorean triples
with 100 < c ≤ 300:
(20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125)
(88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149)
(85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181)
(57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197)
(84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221)
(60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257)
(23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277)
(160, 231, 281)
All of these Pythagorean triples when
squared, represent mathematical and geometrical constructions which do not
exist in Euclidean space as geometries: they are lines existing in one
dimension.
We tend to look at the phenomenon of Pythagorean triangles
as just that: something which has practical usage, but no further implications
for the nature of the world in which we live, or our understanding of it. This
is not likely to have been the case in the ancient world, in which number was
venerated as something belonging to the divine and knowledge of the divine. The
triples meant something profound.
What might be the meaning of the existence of these
triangles and their properties?
In antiquity, it was obvious to anyone interested in number,
mathematics and geometry, that there were several aspects of the physical world
that involved irrationality, long before it was possible to provide logical
proof of such irrationality. One of these irrationalities was the relationship
between the diameter and the circumference of the circle. We know that
irrationality (understood as an absence of commensuration) was a major concern
in antiquity, since the existence of it seemed to undermine the idea that the
world was rational, and constructed by the divine on rational principles. In
other words, the existence of irrational things served to undermine the idea
that the world made sense, and that it was good.
What we understand as Pythagoreanism is actually a way of
approaching the world and reality on the basis of number, mathematics and
geometry. We have lost a grasp of this, particularly since the close of the
ancient world. Pythagorean ideas are not the creation of Pythagoras in the
sixth century B.C.E., but a range of ideas about the world, focussing
particularly on numbers and geometry, and the puzzles which the study of these
throws up (the Greek name for these puzzles is ‘aporia’). As such, these ideas
and puzzles belong to any culture which chooses to address the divine in terms
of how the universe is constructed. As already suggested, the Babylonians had a
sense of this, though they were also interested in the practical applications.
It is also the case that the inhabitants of Britain in the late Neolithic and
the early Bronze Age had such a sense.
The engineer Alexander Thom surveyed many of the megalithic
circles across Britain from the 1930s into the 1970s, and established that the circles
were constructed on the basis of a number of different Pythagorean triangles,
and that these circles were not in fact circular. The circumferences of these circles were
modified in order to make their lengths commensurate with the length of the
sides of the underlying triangles. These
modifications testify to the contemporary idea in ancient times that the
incommensurate nature of diameter and circumference shouldn’t be the case.
I’ve written elsewhere that Pythagoreanism, whether in the
sixth century or long before, was a transcendentalist view of the world.
Meaning that the world of physics and appearance in which we live, is not
reality itself, but simply a presentation of it. And the presentation of it is,
in a number of ways, crooked. So some aspects of physical reality are not
rational.
This does not mean that the ancient Pythagoreans were pitching
themselves against the workings of the divine, but rather that they were trying
to understand why what they saw, experienced and understood, was not rational.
The answer was that their place of refuge was not reality itself, but a false
representation of it.
In the physical world, they could therefore not expect rationality
to be woven all through it. Thom identified the obsessive concern of the
ancient Britons with whole numbers, and as a consequence (though this was not
understood at the time he was studying the megaliths), we know that they were
looking to a world beyond the puzzles and paradoxes, in which the relationships
of one thing to another were rational in nature.
The theorem of Pythagoras, however it was articulated in the
late Neolithic and the early Bronze Age, provided the answer to this. The
relationship between the sides of a 3, 4, 5 triangle is irrational in nature,
but by squaring the sides, the result is rational and commensurate. This would
have been understood to point to a world which transcended space, in that it
indicated a one-dimensional reality.
In that world, some things which are
incommensurate here, were commensurate. Which
they might have taken to indicate that, beyond that limited form of reality,
there was another reality with no dimensions at all, in which all irrational values
existed as commensurate with one another.
Plato echoed a range of Pythagorean ideas in his work, including
that reality itself exists in no particular place, has no form or shape or
colour. He also suggested that forms existed beyond geometrical figures
existing in space, and that these were to be accessed in the mind alone.
The Pythagoreans may have understood physical reality to
have been generated as the square root of mathematical values in a higher reality. The resulting
incommensuration would necessarily generate space. We could not possibly live
in a reality which embraced only one dimension, or even none at all. In which
case physical reality might have been understood by the ancient Pythagoreans as
a compromise of sorts, which made it possible for mankind to live.
