brian
BRIAN GREENE, a graduate
of Harvard and Oxford, joined the physics faculty of
Cornell University in 1990. In 1996 he became
professor of physics and mathematics at Columbia
University.
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
Mathematics
& the non-scientist
1
Mathematics & the non-scientist
Someone
who seeks
access to the most advanced thought about the way the
physical world operates needs to have mathematical
sophistication. Mathematics is the language of the
scientists who work at the advanced edge of knowledge
about the macro- and micro-worlds. Lacking
mathematics, one is condemned to a kind of shadow
knowledge. He is at the mercy of writers who attempt
to approximate in words what their mathematics
precisely expresses. He has to content himself with
the metaphors they offer in place of formulae.
Alas, my only window on
the state of knowledge in physics is the English
language. This used to make me self-conscious; now,
it is just another of the legion of limitations that
has made itself apparent in the course of living a
certain number of decades.
When I was a freshman in
college in 1949-1950, science in the form of
introductory biology briefly attracted me. Dr. Paul
Wagner, our professor at Ursinus, was in love with
the body of descriptive knowledge that he commanded.
His love affair with his subject radiated out even to
callow kids who did not have a clue about the true
nature of science. Mercifully for science, that brief
flirtation ended when my introductory biology course
ended--it was the last science course I would ever
take in college. By going no farther, I avoided
college mathematics and the opening that it would
have given me into the operation of the physical
world.
Despite my limitations,
in the years after formal studies ended I developed a
wish to know something more about scientific insight
into natural phenomena. I finally could no longer
ignore what my orientation in the humanities long had
permitted me to downplay: scientific knowledge and
the technology that it supported shaped the template
of modern existence. They undergirded the
intellectual and cultural milieu of that which then
commanded my main interest, the expression of
contemporary sensibility in artistic forms.
Acknowledgment of this led me to a handful of texts
designed for people like me--those who could read a
fairly challenging paragraph but who could not think
in terms of the advanced forms of mathematics. Some
of those books are still on my shelf, evidence,
perhaps, that they constituted a thin lifeline to
some sort of understanding in a field whose precise
language I did not command:
Edward Neville da Costa
Andrade. An Approach to Modern Physics. Garden
City, NY: Doubleday & Company, Inc., 1957.
Originally published by G. Bell and Sons, Ltd.,
London, England, 1956.
Lincoln Barnett. The
Universe and Dr. Einstein. With a foreword by
Albert Einstein. New York: Mentor Books, 1958.
Originally published 1948.
Sir James Jeans. The
Growth of Physical Science. New York: Fawcett
World Library, 1958. Originally published by
Cambridge University Press 1947.
Stephen F. Mason. A
History of the Sciences. Toronto, Canada:
Collier-Macmillan, 1962.
Alfred North Whitehead. Science
and the Modern World. Lowell Lectures, 1925.
New York: Mentor Books. Originally published by The
Macmillan Company 1925.
end part 1 go to
part 2
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
An imagined mentor in
science
2
An imagined mentor in science
Whitehead was my favorite. I became for a
while something of a Whitehead junky. In addition to
the above book, I remember reading An
Introduction to Mathematics (1910), a little
book that sought to define mathematics for
non-mathematicians; The Aims of Education
(1928); some parts of Process and
Reality (1929); Adventures of Ideas
(1933); Modes of Thought (1938). Once I went
to the library at Penn and took down the landmark
work by Whitehead and Bertrand Russell, Principia
Mathematica (1910-1913). This was an act of
devotion, like that of a common monk venerating the
bones of a saint; it was certainly not an attempt to
enter into the thought that transformed mathematics
into modern symbolic logic.
Lucien Price published a
book titled Dialogues of Alfred North Whitehead
in 1954. In it Price recorded his informal
conversations with the aging Whitehead and Mrs.
Whitehead in their Cambridge home near Harvard. The
conversations started in 1934 and ran until
Whitehead's death in 1947. Price's Whitehead was
better, in my view, than Boswell's Johnson. Reading
Price's book elevated Whitehead to a level a little
beyond reality in my mind; yet he was warmly
accessible. The book embedded the image of a very
civilized man thinking and talking masterfully about
ideas in a way that calmed fear and elevated human
possibilities. Whitehead did this with the daring of
his intellect. He was willing to tackle the broadest
issue, the profoundest problem, and make some sense
of it. By the time Price had his evening
conversations with the Whiteheads, which he would
hasten home to record, the mathematician had long
since become the philosopher.
In the intimacy of his
record, Price gave me an imagined mentor. I fancied
that Whitehead would tolerate my ignorance and
encourage my willingness to hazard an original
thought. That this virtual mentor revolutionized
modern mathematics at an early age made my sense of
closeness to him all the more precious. If I had
missed the chance to master mathematics, the language
of science, at least I could feel an affinity with
the master of mathematics. Feeling this way, I was
able to address the significance of modern science a
little less gropingly. For some years, our college
librarians conducted surveys of faculty members to
determine their favorite books. I never named any
other book than Lucien Price's Dialogues of
Alfred North Whitehead.
It was possible, of
course, to gain a rough acquaintance with the sweep
of scientific thought from books such as those listed
above. Even a mathematically challenged reader could
get from them the import of the system of gravitation
devised by Newton and the idea of warped time
advanced by Einstein. He could get a rough sense of
the uncertainty principle and of quantum mechanics.
From Whitehead, in Science
and the Modern World, it was even possible to
glimpse the nature of mathematics as the language of
science, even if it was not possible to learn its
operations. He succeeded in persuading me of the
essential role of mathematics in the history of
thought. To omit mathematics from that history, he
said, would be like omitting Ophelia from the story
of Hamlet. "Ophelia is quite essential to the
play, she is very charming--and a little mad. Let us
grant that the pursuit of mathematics is a divine
madness of the human spirit, a refuge from the
goading urgency of contingent happenings."
(22)
The writing of this
essay has led me to re-read, after many years, the
chapter on mathematics in Science and the
Modern World. It brought back from a receding
past the original feeling of lightness that this text
aroused in me. I would never be a mathematician but I
would forever after acknowledge that the
interpretation of the world of sense in which I lived
required a mathematician. How simply and clearly
Whitehead explained the significance of mathematical
abstraction in a real world: "Mathematics
supplied the background of imaginative thought with
which men of science approached the observation of
nature." (32) It took the working out of
abstract ideas, devoid of particularity, in the form
of mathematics to light up the minds of the early
modern scientists as they sought to account for the
workings of nature. And that relationship of
abstraction and concreteness persisted, as Whitehead
says, from Pythagoras's day to his own--and it
persists to the present moment.
end part 2 go to
part 3
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
String theory promises
answers
3 String theory
promises answers
Brian Greene happened to be on Charlie Rose's
television show when I tuned in one night a couple of
months ago. It would have been just another
promotional gig for a newly published book--the bread
and butter of Rose's show--except for the luminous
clarity of Greene's talk. He was promoting The
Elegant Universe, but he was exhibiting his
passion for the pursuit of the truth about the way
the physical world really operates and doing so in
language that I could understand. If he could speak
so clearly, I thought, his book must be readable. So
I bought it, figuring that one more book on modern
science would not hurt me and might add something.
Indeed, it added a first
acquaintance with superstring theory, or M-theory,
the "theory of everything." Greene took the
time from his rising career to write this book so
that those without training in mathematics or physics
could gain access to the forefront of physics
research. Greene is as seriously engaging in print as
he was in his chat with Charlie. He gives an account
of his own passionate personal involvement in string
theory research--dealing with tears in the fabric of
space. Toward the end, he allows himself to express
the excitement of the chase for the "final"
theory that will explain matter.
Greene gives a lucid
review of modern physics. His discussion focuses on
the evolving understanding of space and time. This
allows him to set up the familiar unresolved
incompatibility of Einstein's theory of relativity
and the newer quantum mechanics. Relativity theory
shows how large matter works; quantum mechanics shows
how small matter works; but neither shows how all
matter works. When relativity theory is applied to
the micro-world, it yields weird results, just as
quantum theory yields weird results when applied to
the macro-world.
By Greene's account, the
new superstring theory (or string theory for short)
appears to be the mathematical means by which
scientists will resolve that incompatibility. String
theory has already helped us understand why the
theories of the large and the small do not work
together. It deals with a newly theorized form for
the ultimate matter--tiny strings in perpetual
vibration. Neither relativity nor quantum mechanics
has the power to do that.
String theory aims to be
the unified theory of the universe that
Einstein never developed. It rejects the older notion
that "the fundamental ingredients of
nature" are "zero-dimensional point
particles." Rather, they are "tiny
one-dimensional filaments." Physicists call them
strings--I found the image of tiny vibrating rubber
bands useful, even though I know how wildly
approximate such a metaphor is. String theory is the
mathematical system that seeks to be compatible with
this concept of the ultimate characteristic of
matter. Greene says of string theory that it
"harmoniously unites quantum mechanics and
general relativity, the previously known laws
of the small and the large, that are otherwise
incompatible." (422)
end part 3 go
to part 4
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
Strings in
many dimensions
4
Strings in many dimensions
Greene
has a talent
for creating understandable visual metaphors that
allow a lay reader at least to glimpse the knowledge
that string theorists are pursuing in mathematical
language.
Readers of
non-specialized books about relativity already are
familiar with drawings of the "warps and
ripples" of bent space. We have seen the
graceful curves of bent space caused when a
"bowling ball" drops onto a flat-surfaced
grid and causes a three-dimensional indentation.
Greene extends such
graphic explanation for quantum mechanics and string
theory. In the case of quantum mechanics, he shows
the surface of space-time not to be a smooth grid but
a rough helter-skelter of matter. He depicts the
agitation of matter at the smallest level in quantum
mechanics as a "violent quantum foam."
(128) It appears in his illustration as crazy cones,
holes, and loops on the otherwise orderly surface
grid--a vision worthy of Disneyland. When you see
this perpetually frenzied space, you have a better
understanding why Heisenberg's 1927 Uncertainty
Principle works: it tells you where an electron is at
a given instant--sort of--by establishing a ratio
between its ever-changing location and its velocity.
(112) You have a better understanding why Richard
Feynman's "sum over paths" approach to
quantum mechanics works: it determines the
destination of an electron by the "combined
effect of every possible way of getting there."
(111)
(For some crazy reason,
these subtleties of quantum theory brought to my mind
the visage of Bill Clinton as he was schooling the
Starr snoops with his subtle insight: their case
depended, he said, on what "the definition of is
is.")
But the most powerful
graphic aid comes from Greene when he depicts the
tiniest world, the world of strings. To get the gist
of string theory, you have to understand that it
posits a universe made of many more dimensions than
the three-dimensional one that we can commonly
imagine. And these dimensions curl up in newly
hypothesized spatial shapes so small that they remain
untouched by experiment.
Specifically, Greene
introduces us to Calabi-Yau spaces or shapes
that represent six dimensions, beyond the three
familiar extended dimensions of length, breadth, and
depth. (The name honors the mathematical work done by
Eugenio Calabi of the University of Pennsylvania and
Shing-Tung Yau of Harvard before string theory
developed. [207])
A Calabi-Yau space as
Greene approximates it on the two-dimensional page is
an elegantly complicated little ball, with multiple
loops of striated intertwining string-like strips,
each with differing shadings from light to dark. He
tells us that string theory allows for "many
tens of thousands" of variations of Calabi-Yau
shapes.
The extradimensional
geometry in Calabi-Yau spaces holds the key to
resolving the mathematical difficulties met when the
theory of relativity (which works for the
macro-world) applies weirdly to the micro-world and
quantum mechanics (which works for the micro-world)
applies weirdly to the macro-world. The weird
"nonsense" answers to their equations are
avoided when string theorists--who admit the extra
dimensions depicted in Calabi-Yau spaces--work their
mathematics. Particle masses and charges begin to
work similarly for both macro- and micro-worlds when
string theorists operate their equations.
The reason for this
compatibility is that, according to string theory, "the
universe is made up of tiny strings whose resonant
patterns of vibration are the microscopic origin of
particle masses and force charges....As a string
moves about, oscillating as it travels, the
geometrical form of the extra dimensions [as depicted
in the Calabi-Yau spaces] plays a critical role in
determining resonant patterns of vibration. Because
the patterns of string vibrations appear to us as the
masses and charges of the elementary particles [dealt
with in relativity and quantum mechanics], we
conclude that these fundamental properties of the
universe are determined, in large measure, by the
geometrical size and shape of the extra
dimensions."
Greene concludes that
trenchant paragraph as follows: "That's one
of the most far-reaching insights of string
theory." (206) Thus it appears that we can
close the gap between the theory of relativity and
quantum mechanics by tying them together with tiny
vibrating pieces of string.
So,
are we there? Does
Greene announce that he and his fellow string
theorists have found the theory of everything? Not
quite; not yet. It appears that there is not one
string theory. There are five string theories--Type
I, Type IIA, Type IIB, Heterotic-O, and Heterotic-E.
They yield many solutions--too many to be definitive.
So, string theorists have decided that the five
string theories are only approximate versions of the
ultimate equations of string theory. (285) They are
working on the assumption "that all five
string theories are actually part of a single,
unified framework, tentatively called M-theory."
(287)
Graphics again help
Greene convey this notion to the lay person. Picture
each of the five theories as the separate,
disembodied tips of a five-pointed starfish. They
appear to be isolated from one another. Then overlay
the body of a whole starfish so that it covers the
five tips and includes them in its single form.
That's M-theory. All five theories connect in one
grand M-theory. And that is what the string theorists
are busy trying to work out now. (M-theory turns out
to have eleven dimensions at this point, whereas the
Calabi-Yau shapes had only six. [287])
Duality is the situation
that is making this work possible. Duality involves
two or more theories that appear to be completely
different. It gives rise, despite their apparent
difference, to identical physical consequences. (415)
All five of the string theories, it turns out, are
dual to some other one. It is that finding that makes
Greene and his colleagues believe that "M-theory
provides a unifying substrate for pulling together
all five string theories." (286)
Greene concludes, "The
search is not over, but through superstring theory
and its evolution into M-theory, a cogent framework
for merging quantum mechanics, general relativity,
and the strong, weak and electromagnetic forces has
finally emerged." (386)
end part 4 go to
part 5
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
Ascent toward Truth?
5 Ascent toward Truth?
At
the very end,
Greene quotes Jacob Bronowski (from his 1973 book, The
Ascent of Man): "in every age there is a
turning point, a new way of seeing and asserting the
coherence of the world." (387) Adopting the
"ascent" trope of Bronowski's title, Greene
affirms that these turning points thrust upward. They
represent advance of human knowledge. He feels that
he and his generation of scientists are doing their
part, "contributing our rung to the human ladder
reaching to the stars." (387).
Knowledge of the
universe thus accumulates, he assumes. Einstein's
theory of relativity parochialized Newton's theory of
gravitation but did not invalidate it. Quantum
mechanics built beyond Einstein's theories but left
them intact for their purposes. String theory
elaborates on the unresolved contradictions of
relativity and quantum mechanics without making them
inoperative.
If one adopts a
postmodernist angle toward modern science, Greene's
ladder metaphor demands scrutiny. Does string theory
take us one rung closer to the top of the ladder,
where we finally will see Truth? Or, as the
postmodernists would say, is it just a new chapter in
a "metanarrative" that in principle cannot
ever move us upward toward a final sighting of real
Truth? These questions may be boring and unanswerable
but they are not irrelevant to the attitudes we adopt
toward our situation in the universe. Those attitudes
in turn influence the degree of humility or hubris we
bring to the task of social and personal development.
The postmodern
perspective makes us ask if any theory in principle
can represent the truth of the real world in a
direct, mirror-like manner. In 1996, physicist Alan
Sokal sought to defend the orthodox idea that science
does mirror reality. He set out to demolish
postmodernists' criticism that denied the universal
truth of scientific research. To get at his target,
he wrote a parody that purported to be a serious
critique of science by a postmodernist. He submitted
it straight-faced to Social Context, a
juried postmodernist journal. The editors accepted
and published it. Sokal then exposed his own hoax in
another journal, Lingua Franca.
By showing up what he considered the sham
intellectual standards of the editors who accepted
his text, he sought to illustrate that postmodernists
err when they deny the literal reality of modern
scientific truth. Sokal of course believed the
opposite of what his bogus article argued. The furor
over "the Sokal hoax" for a while burned up
the web and more traditional media. I
documented it in
the Postmodern Programme at Sixth Avenue.
At the time, my
sympathies were with the editors who were sucked in
by Sokal's hoax. But I also felt that I understood
Sokal's allegiance to an idea of objective truth--it
was, after all, the idea at the heart of my
education. Reading Greene's The Elegant
Universe has allowed me to think again about
the issue raised by Sokal. I think, on reflection,
that I was right when I originally felt ambivalent
about the issue!
With the latest reports
on superstring theory dancing in my head, I can
assert that scientists of course are making up
mathematical stories that cannot pretend to represent
the universal truth of things. But I can also say
that the findings of physical theory in modern times
have, in practice, proven how closely it can
approximate the real world. If it did not, the
familiar marvels of science and technology would
never have come into being. From an operational point
of view, modern science has a dependable bead on
reality.
I would say today that
modern science is right to strive to represent
the real world in theoretical--that is,
mathematical--terms. The idea of mathematics advanced
by my old imagined mentor Whitehead helps here. The
paradox of mathematics, in Whitehead's mind, was
"that the utmost abstractions are the true
weapons with which to control our thought of concrete
fact." (S&TMW, 34) The
abstractions do not control concrete fact; they
control our thought of it. By controlling
thought of the real world, the abstractions of
mathematical theory give humankind a tool for
survival. The question is a pragmatic one. What
works? The truth is that modern mathematics works
better than witchcraft or creationism in enabling
humankind to survive and prevail. But that is short
of a formal representation of universal truth.
I think we will have to
insist on sort of having it both ways--Sokal's way,
sort of (the truth of the world exists and modern
mathematics has the power to get at it and express it
somewhat); and postmodernists' way, sort of
(difference inevitably gives the signifier a
dimension of its own).
Modern science allows us
to have an accurate awareness of an incompletely
perceived reality. Postmodernists complain when
people ignore the incompleteness and infer a complete
picture. They get even more upset when people then
infuse that picture with the aura of the Absolute and
worship it as Truth. For their part, when
postmodernists deny the absoluteness, sometimes they
also deny the pragmatic working of our scientific
tools and sound ridiculous.
end part 5 go to
part 6
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
Truth a moving target
6
Truth a moving target
After
thinking over
my reading about physical science through the years
and getting an (elegant) update from Brian Greene, I
am ready to talk now about truth as a moving target.
That differs from talking about our ascent up a
ladder toward perfect Truth. We may hit truth with
our intellectual weaponry but it will move on and
away; it will not remain where we momentarily pinned
it. The implication of this metaphor is that we never
will touch the final truth; we will gain something
useful, though, whenever we hit it, however
temporarily it stays in front of us.
Much as I appreciate the
lucidity of Brian Greene's book, I was tempted to
register a minor objection to his too-easy acceptance
of the idea that he is helping humankind take a step
up the ladder of understanding. But I paused to see
that he also subscribes to Bronowski's phrase,
"a new way of seeing." Perhaps that notion
of novelty--the shot of an arrow at a moving
target--undercuts the old ladder metaphor and takes
precedence. Yes, let's settle on that as the final
Greene note. I'll cease quibbling with his metaphors.
Bronowski himself in his
book thirty years ago gave us what might still be the
right response to the unanswerable questions about
truth: "One aim of the physical sciences has
been to give an exact picture of the material world.
Our achievement of physics in the twentieth century
has been to prove that that aim is
unattainable." ( Bronowski, 353). He explained
this by likening the method of physics to the method
of art--a picture explores a face; it does not fix
it. And he concluded:
"There is no
absolute knowledge. And those who claim it, whether
they are scientists or dogmatists, open the door to
tragedy. All information is imperfect. We have to
treat it with humility. That is the human condition;
and that is what quantum physics says. I mean that
literally." (353)
I have a hunch that the
revelations of string theory of the past fifteen
years, as Greene reports on them, would not cause
Bronowski to change that opinion. Even if string
theory in the end gives every answer to every
question about our universe, it probably will not
tell us whether there is one universe or many.
That reminds us again of
the preeminent importance of the style
of thought. Elegance is the important thing. Treasure
Brian Greene because he understands that.
end part 6 end of
essay
Parts
of the essay 1 Mathematics & the
non-scientist | 2 An imagined mentor in
science
| 3 String theory promises answers | 4
Strings in many dimensions | 5 Ascent
toward truth? | 6 Truth a moving target |
