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DEGREES OF FREEDOM
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1800s. However, the philosopher Immanuel Kant (1724—1804) considered
some of the spiritual aspects of a fourth dimension:
A science of all these possible kinds of space would undoubtedly be
the highest enterprise which a finite understanding could undertake
in the field of geometry. ... If it is possible that there could be
regions with other dimensions, it is very likely that a God had
somewhere brought them into being. Such higher spaces would not
belong to our world, but form separate worlds.
Euclid (c. 300 B.C.), a prominent mathematician of Greco-Roman antiquity,
understood that a point has no dimension at all. A line has one dimension:
length. A plane had two dimensions. A solid had three dimensions. But
there he stopped—believing nothing could have four dimensions. The Greek
philosopher Aristotle (384—322 B.C.) echoed these beliefs in On Heaven:
The line has magnitude in one way, the plane in two ways, and the
solid in three ways, and beyond these there is no other magnitude
because the three are all.
Aristotle used the argument of perpendiculars to prove the impossibility of a
fourth dimension. First he drew three mutually perpendicular lines, such as
you might see in the corner of a cube. He then put forth the challenge to his
colleagues to draw a fourth line perpendicular to the first three. Since there was
no way to make four mutually perpendicular lines, he reasoned that the fourth
dimension is impossible.
It seems that the idea of a fourth dimension sometimes made philosophers
and mathematicians a little nervous. John Wallis (1616—1703)—the most
famous English mathematician before Isaac Newton and best known for his
contributions to calculus's origin—called the fourth dimension a "monster in
nature, less possible than a Chimera or Centaure." He wrote, "Length,
Breadth, and Thickness, take up the whole of Space. Nor can fansie imagine
how there should be a Fourth Local Dimension beyond these three."
Similarly, throughout history, mathematicians have called novel geometrical
ideas "pathological" or "monstrous." Physicist Freeman Dyson recognized this
for fractals, intricate structures that today have revolutionized mathematics
and physics but in the past were treated with trepidation: