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Chapter 3: No-Boundary Territory
ultimate metaphysical secret, if we dare state it so simply, is that there are
no boundaries in the universe. Boundaries are illusions, products not of
reality but of the way we map and edit reality. And while it is fine to map out
the territory, it is fatal to confuse the two. It's not just that there are no
boundaries between the opposites. In a much wider sense, there are no dividing
boundaries between any things or events anywhere in the cosmos. And nowhere is
the reality of no-boundary seen more clearly than in modern physics, which is
all the more remarkable considering that classical physics—associated with
such names as Kepler, Galileo, and Newton—was one of the true heirs of Adam
the mapmaker and boundary drawer.
Adam finally passed on, he left humankind his legacy of mapmaking and boundary
drawing. And since every boundary carries with it political and technological
power, Adam's bounding, classifying, and naming of nature marked the first
beginnings of technological power and control over nature. As a matter of fact,
Hebrew tradition has it that the fruit of the Tree of Knowledge actually
harbored knowledge not of good and evil but of the useful and the useless—that
is, technological knowledge. But if every boundary carries technological and
political power, it also carries alienation, fragmentation, and
conflict—because when you establish a boundary so as to gain control over
something, at the same time you separate and alienate yourself from that which
you attempt to control. Hence the Fall of Adam into fragmentation, known as
the boundaries Adam drew were very simple kinds of boundaries. They merely
classified, and were useful only in description, definition, naming, and so on.
And Adam didn't even make full use of these classifying boundaries. He had
hardly gotten around to naming vegetables and fruits when he fumbled the ball
and got kicked out of the game.
later, the descendants of Adam finally worked up enough nerve to start fooling
around with boundaries again, and more subtle and abstract boundaries at that.
In Greece men of brilliant intellectual powers appeared—that is, great
mapmakers and boundary drawers. Aristotle, for instance, classified nearly
every process and thing in nature with such precision and persuasion that it
would take centuries for Europeans just to question the validity of his
no matter how precise and complex your classifications, you can't do very
much—scientifically at least—with that type of boundary line except describe
and define. You have only a qualitative science, a classifying science.
However, once you have laid down your initial boundaries, so that the world
appears as a complex of separate things and events, you can then proceed to
much more subtle and abstract types of boundaries. And the Greeks, like
Pythagoras, did just that.
what Pythagoras discovered, looking over all the various classes of things and
events, from horses to oranges to stars, was that he could perform a brilliant
trick on all these different objects. He could, in fact, count them.
naming seemed magic, counting seemed divine, because while names could
magically represent things, numbers could transcend them. For instance, one
orange plus one orange equals two oranges, but so does one apple plus one apple
equal two apples. The number two refers impartially to any and all groups of
two things, and so somehow must transcend them.
abstract numbers, humans succeeded in freeing their minds from concrete things.
To some extent this was possible through the first type of boundary, through
naming, classifying, and noting differences. But numbers increased this power
dramatically. For, in a sense, counting was actually a totally new type of
boundary. It was a boundary on a boundary, a meta-boundary, and it worked like
the first type of boundary, we draw a dividing line between different things
and then recognize them as constituting a group or class, which we then name
frogs, cheeses, mountains, or whatnot. This is the first or basic type of
boundary. Once we have drawn our first boundaries, we can then draw a second
type of boundary on the first type and then count the things in our classes. If
the first boundary gives a class of things, the second boundary gives a class
of classes of things. So, for example, the number seven refers equally to all
the groups or classes of things which have seven members. Seven can refer to
seven grapes, seven days, seven dwarfs, and so on. The number seven, in other
words, is a group of all the groups which have seven members. It is therefore a
class of classes, a boundary on a boundary. Thus with numbers, humans
constructed a new type of boundary, a more abstract and generalized boundary, a
meta-boundary. And since boundaries carry political and technological power,
humans had thereby increased their ability to control the natural world.
these new and more powerful boundaries brought with them the potential not only
for a more developed technology, but also a more pervasive alienation and
fragmentation. The Greeks succeeded in introducing, through this new
meta-boundary of number, a subtle conflict, a subtle dualism, which has
fastened onto Europeans as a vampire battens on its prey. For abstract numbers,
this new meta-boundary, so transcended the concrete world that humans
discovered they were now living in two worlds—the concrete vs. the abstract,
the ideal vs. the real, the universal vs. the particular. Over the next two
thousand years this dualism would change its form a dozen times, but rarely be
uprooted or harmonized. It became a battle of the rational vs. the romantic,
ideas vs. experience, intellect vs. instinct, law vs. chaos, mind vs. matter.
Those distinctions were all based on appropriate and real lines, but the lines
usually degenerated into boundaries and battles.
new meta-boundary—that of number, counting, measuring, and the like—was not
really put to use by natural scientists for centuries, until the time of
Galileo and Kepler, around the year 1600. For the intervening period between
the Greeks and the first classical physicists was occupied by a new force on
the European scene—the Church. And the Church would have none of that
measuring or scientifically numbering-up of nature. The Church, through the
influence of Thomas Aquinas, was closely allied with the logic of Aristotle,
and Aristotle's logic, for all its brilliance, was predominantly one of
classifying. Aristotle was a biologist, and carried on the classifying begun by
Adam. He never really got the full swing of Pythagorean number and measurement.
And so neither did the Church.
by the seventeenth century, the Church was in decline, and humans began looking
carefully at the forms and processes of the natural world. And it was at this
time that the genius of Galileo and Kepler entered the drama. The revolutionary
thing these physicists accomplished was simply to measure, and measurement is
just a very sophisticated form of counting. So where Adam and Aristotle drew
boundaries, Kepler and Galileo drew meta-boundaries.
the seventeenth-century scientists didn't just resurrect the metaboundary of
number and measurement and then sophisticate it. They went one step further and
introduced (or rather, perfected) an entirely new boundary of their own.
Incredible as it seems, they came up with a boundary on the meta-boundary. They
invented the meta-meta-boundary, better known as algebra.
simply, the first boundary produces a class. The meta-boundary produces a class
of classes, called number. The third or meta-metaboundary produces a class of
classes of classes, called the variable. The variable is best known as that
which is represented in formulas as x, y, or z. And the variable works like
this: just as a number can represent any thing, a variable can represent any
number. Just as five can refer to any five things, x can refer to any number
over a given range.
using algebra, the early scientists could proceed not only to number and
measure the elements, but also to search out abstract relations between those
measurements, which could be expressed in theories, laws, and principles. And
these laws seemed, in some sense, to "govern" or "control"
all the things and events marked off with the very first type of boundaries.
The early scientists produced laws by the dozens: "For every action there
is an equal and opposite reaction." "Force is equal to the mass times
the acceleration of the enforced body." "The amount of work done on a
body equals the force times the distance."
new type of boundary, the meta-meta-boundary, brought new knowledge and, of
course, explosive new technological and political power. Europe was rocked with
an intellectual revolution the likes of which humankind had never seen. Just
imagine: Adam could name the planets; Pythagoras could count them; but Newton
could tell you how much they weighed.
then: this entire process of formulating scientific laws was based on three
general types of boundaries, each building on its predecessor and each being
more abstract and generalized. First, you draw a classifying boundary, so as to
recognize different things and events. Second, you search among your classified
elements for ones that can be measured. This meta-boundary allows you to shift
quality to quantity, classes to classes of classes, elements to measurements.
Third, you search for relationships between your numbers and measurements of
the second step until you can invent an algebraic formula embracing them all.
This meta-meta-boundary converts measurements to conclusions, numbers to
principles. Each step, each new boundary, brings you a more generalized
knowledge, and hence more power.
knowledge, power, and control over nature was, however, bought at a price,
for, as always, a boundary is a double-edged sword, and the fruits it slices
from nature are necessarily bittersweet. Man had gained control over nature,
but only by radically separating himself from it. In the mere span of ten
generations, he had for the first time in history awarded himself the dubious
honor of being able to blast the entire planet, himself included, to
smithereens. The earth's heavens were so choked with fumes that birds were
abandoning existence; the lakes so clogged with greasy sludge that some of them
would spontaneously catch fire; the oceans so dense with insoluble chemical
Jello that fish were buoyed to the surface like Styrofoam on mercury; and the
rains that fell to the earth in some places would corrode sheet metal.
yet, during the span of ten generations, a second revolution in science was
forming. Nobody guessed, or could have guessed, that this revolution, when it
finally culminated around 1925, would signal the surpassing of classical
physics—its boundaries, meta-boundaries, and meta-meta-boundaries. The whole
world of classical boundaries shattered and fell before the likes of Einstein,
Schroedinger, Eddington, deBroglie, Bohr, and Heisenberg.
you read the accounts of this twentieth-century revolution in science given by
these physicists themselves, you can't help being struck by the awesome nature
of the intellectual upheaval that occurred in the brief span of a single
generation, 1905–1925, dating from Einstein's relativity theory to
Heisenberg's uncertainty principle. The classical boundaries and maps of the
old physics literally fell apart. In 1925, Whitehead stated, "The progress
of science has now reached a turning point. The stable foundations of physics
have broken up. . . . The old foundations of scientific thought are becoming
unintelligible. Time, space, matter, material, ether, electricity, mechanism,
organism, configuration, structure, pattern, function, all require
reinterpretation. What is the sense of talking about a mechanical explanation
when you do not know what you mean by mechanics?" And Louis deBroglie
said, "On the day when quanta, surreptitiously, were introduced the vast
and grandiose edifice of classical physics found itself shaken to its very
foundations. In the history of the intellectual world there have been few
upheavals comparable to this."