MAPPING DREAMS
PRO & CON
I am free to imagine a horse with or without wings.
RENÉ DESCARTES
Meditations V (1641)
A horse is a horse for everybody... Representation according to
nature is not itself nature; moreover, art is not nature.
PIET MONDRIAN
Natural Reality and Abstract Reality (19191920)
OLD REALITIES IN NEW DREAMS
Bosch, by means of paintings that interfused medieval forms in
Renaissance space, told what it felt like to live straddled
between the two worlds of the old and the new during this
revolution. Simultaneously, Bosch provided the older kind of
plastic, tactile image but placed it in the intense new visual
perspective. He gave at once the older medieval idea of unique,
discontinuous space, superimposed on the new idea of uniform,
connected space. This he did with earnest nightmare intensity.
Lewis Carroll took the nineteenth century into a dream world that
was as startling as that of Bosch, but built on reverse
principles. Alice in Wonderland offers as norm that continuous
time and space that had created consternation in the Renaissance.
Pervading this uniform Euclidean world of familiar spaceandtime, Carroll
drove a fantasia of discontinuous spaceandtime
that anticipated Kafka, Joyce, and Eliot. Carroll, the
mathematical contemporary of Clerk Maxwell, was quite avantgarde
enough to know about the nonEuclidean geometries coming into
vogue in his time. He gave the confident Victorians a playful
foretaste of Einsteinian timeandspace in Alice in Wonderland.
Bosch had provided his era a foretaste of the new continuous
timeandspace of uniform perspective. Bosch looked ahead to the
modern world with horror, as Shakespeare did in King Lear, and as
Pope did in The Dunciad. But Lewis Carroll greeted the electronic
age of spacetime with a cheer.
MARSHALL MCLUHAN
Understanding Media (1964)
Desargues' Theorem
The pairs of corresponding sides, AB and A'B', BC and B'C', and
AC and A'C' of two triangles perspective from a point meet,
respectively, in three points that lie on one straight line.
With specific reference to our figure the theorem says that if we
prolong sides AC and A'C', they will meet in a point P; sides AB
and A'B' prolonged will meet in a point Q; and sides BC and B'C'
prolonged will meet in a point R. And P, Q, and R will lie on a
straight line. The theorem holds whether the triangles lie in the
same or in different planes.
MORRIS KLINE
Mathematics in Western Culture (1953)
But I have resolved to quit only abstract Geometry, that is to
say the research of questions that serve only to exercise the
mind; this in order to have more leisure to cultivate another
sort of Geometry, whose purpose is to explain questions about
the phenomena of nature. If it pleases Desargues to consider what
I've written about salt, about snow, about the rainbow etc., he
knows well that all of my Physics is nothing other than Geometry.
RENÉ DESCARTES
Letter to Mersenne (27 July 1638)
I undertake to prove that God, in creating the universe and
regulating the order of the cosmos, had in view the five regular
bodies of geometry as known since the days of Pythagoras and
Plato, and that he has fixed according to those dimensions, the
number of the heavens, their proportions, and the relations of
their movements.
JOHANNES KEPLER
Mystery of the Cosmos (1596)
It is my impression that the very special serenity of the starry
sky is due to the geometrical relation of the stars to each other.
PIET MONDRIAN
Natural Reality and Abstract Reality (19191920)
By way of analogy we can now say that although it is not possible
to squeeze a fourdimensional body into a threedimensional space
without some parts sticking out, one can speak of the projections
of various fourdimensional figures in our space of only three
dimensions. But one must remember that just as the plane
projections of threedimensional bodies are twodimensional or
plane figures, so the projections of fourdimensional superbodies
in our ordinary space will be represented by spacefigures.
GEORGE GAMOW
One, Two, Three, ... Infinity (1947)
