©William O. Straub, fl. 2004-2016 email:wostraub@gmail.com AfterMath Copyright Disclaimer Index photos courtesy ETH-Bibliothek, Zurich Bildarchiv Who Was Hermann Weyl? Wheeler's Tribute to Weyl (PDF) Old Stuff 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Math Tools Weyl's Spinor and Dirac's Equation Weyl's Conformal Tensor Weyl Conformal Gravity Weyl's 1918 Theory Weyl's 1918 Theory Revisited Weyl v. Schrodinger Why Did Weyl's Theory Fail? Did Weyl Screw Up? Weyl and the Aharonov-Bohm Effect The Bianchi Identities in Weyl Space A Child's Guide to Spinors Levi-Civita Rhymes with Lolita Weyl's Scale Factor Weyl's Spin Connection Weyl and Higgs Theory Weyl & Schrodinger - Two Geometries Lorentz Transformation of Weyl Spinors Riemannian Vectors in Weyl Space Introduction to Quantum Field Theory A Children's Primer on Quantum Entanglement Electron Spin Clebsch-Gordan Calculator Bell's Inequality The Four-Frequency of Light There Must Be a Magnetic Field! Non-Metricity and the RC Tensor Curvature Tensor Components Kaluza-Klein Theory The Divergence Myth in Gauss-Bonnet Gravity Schrodinger Geometry A Brief Look at Gaussian Integrals Particle Chart Einstein's 1931 Pasadena Home Today Uncommon Valor She did not forget Jesus! "Long live freedom!" Visitors since November 4, 2004: |
“Hermann Weyl and the
Unity of Knowledge” by John Archibald Wheeler. American Scientist, Vol.
74, July-August 1986, pp. 366-375. Adapted from W. Deppert, ed.,
Proceedings of
the Internationaler Hermann-Weyl-Kongress: Exakte Wissenschaften und
ihre
Philosophische Grundlegung (Peter Land), 1986. Reproduced with the kind permission of American Scientist, February
2004
Unity of
KnowledgeJohn Archibald WheelerHermann Weyl was -- is -- for many of us, and for me, a friend, a teacher, and a hero. A North German who became an enthusiastic American, he was a mathematical master figure to mathematicians, and to physicists a pioneer in quantum theory and relativity and discoverer of gauge theory. He lives for us today, and will live in time to come, in his great findings, his papers and books, and his human influence.
Among Weyl’s papers is his Columbia
University bicentennial lecture, “The Unity of Knowledge”
(1). Unity? This
world in which we live and have our being: what was it to Hermann
Weyl? Not to him did the poet’s words apply, The
world to him was no dreary cavern; no, a miraculous panorama, exciting
in him a
passionate desire to capture its beauty, its order, and its unity. What
can we
learn from his search for unity? Remarkable issues connected with the puzzle of existence confront us today in Hermann Weyl’s domain of thought. Four among them I bring before you here as especially interesting: (1) What is the machinery of existence? (2) What is the deeper foundation of the quantum principle? (3) What is the proper position to take about the existence of the “continuum” of the natural numbers? And (4) what can we do to understand time as an entity, not precise and supplied free of charge from outside physics, but approximate and yet to be derived from within a new and deeper time-free physics? In brief, why time? What about the continuum? Why the quantum? What is existence?
Between four issues of such different
flavor, can we hope at the end to see the glimmerings of a linkage? Let
me
anticipate. Let me propose that we shall find the unity-bringing theme
in
Weyl's own 1954 address on the unity of knowledge: “the realm of
Being is not
closed, but open.”
Let us plot our course backward from our
theme. The unity of knowledge? At the end. Just before the theme? Our
consideration of those four great questions. Before the four questions?
Some
developments in knowledge relevant to those questions: contributions of
Weyl;
further findings since his day; and the driving force, straight out of
the age
of enlightenment, that impelled Weyl -- and thinkers before and since
-- to
apply reason to, and hope to make progress with, such great issues. And
before
this brief survey of some Weyl-related knowledge? A glimpse of Weyl
himself. I
last knew Weyl after I last knew him. Day after day in Zürich in
late 1955 he
had been answering letters of congratulation and good wishes received
on his
seventieth birthday, walking to the mailbox, posting them, and
returning home.
December eighth, thus making his way homeward, he collapsed on the
sidewalk
and, murmuring “Ellen,” died. News of his unexpected death
reached Princeton by
the morning
I first knew Weyl before I first knew
him. Picture a youth of nineteen seated in a Vermont hillside pasture,
at his
family’s summer place, with grazing cows around, studying
Weyl’s great book,
Some
years ago I was asked, like others, I am sure, to present to the
Library of the
American Philosophical Society the four books that had most influenced
me.
Erect, bright-eyed, smiling Hermann
Weyl I first saw in the flesh when 1937 brought me to Princeton. There
I
attended his lectures on the Elie Cartan calculus of differential forms
and
their application to electromagnetism --eloquent, simple, full of
insights.
Little did I dream that in thirty-five years I would be writing, in
collaboration with Charles Misner and Kip Thorne, a book on
gravitation, in
which two chapters would be devoted to exactly that topic. At another
time Weyl
arranged to give a course at Princeton University on the history of
mathematics. He explained to me one day that it was for him an absolute
necessity to review, by lecturing, his subject of concern in all its
length and
breadth. Only so, he remarked, could he see the great lacunae, the
places where
deeper understanding is needed, where work should focus.
If I had to come up with a single word to
characterize Hermann Weyl, the man, as I saw and knew him then and in
the years
to come, it would be that old fashioned word, so rarely heard in our
day,
“nobility.” I use it here not only in the dictionary sense
of “showing
qualities of high moral character, as courage, generosity,
honor,” but also in
the sense of showing exceptional vision. Weyl’s eloquence in
pointing out the
peaks of the past in the world of learning and his aptitude in
discerning new
peaks in newly developing fields of thought surely were part and parcel
of his
lifelong passion for everything that is high in nature and man.
Weyl
belonged to the community of learning. He felt at home in the great
wide world
of thought. What that community is and what it means I first fully felt
in
Sunday morning walks through the woods around the Institute for
Advanced Study
with Hermann Weyl, Oswald Veblen, and two or three other colleagues
from other
disciplines. Topics ranged from art to the history of the Renaissance,
from
Helen Porter Lowe’s current translation of Thomas Mann to Hella
Weyl’s putting
Ortega y Gasset’s Spanish into German and English, from
mathematical logic to
Europe’s nightmare, from movements to men, and from ideologies to
ideas. To
pass the place where John von Neumann was developing his computer was
to see
Weyl’s face light up with the delight of a small boy within
touching distance of
fireworks.
Weyl’s new house was being built.
Sometimes the walks took us there to inspect the progress and note the
architecture. Outside, its shape and flavor were new to the Princeton
community. Inside, when it was finished, order, calm, and beauty were
the
themes. From this place of happy reading, reflection, and writing
Weyl’s
thought ranged out over Minerva’s many meadows.
Not least for Hermann Weyl among the
green pastures of the mind was art. His Vanuxem Lectures at Princeton
University appear in that widely read and many times reprinted book,
Art was for Weyl not only an activity
practiced by others. Art characterized also his own way of thinking,
writing,
and speaking: art in the
If the painter-inventor moves
things around on the screen of the mind to make a new and greater
thing, if
Henry Moore mentally abstracts a beautiful new form out of the whitened
bones,
ebony carvings, and rounded stones he chooses and lays before him each
morning
on his little worktable, so -- we can believe -- Hermann Weyl by
like artistry knew how to select from his rich storehouse the ideas
needed at
the moment and push them about, this way and that, until they fell
together in
a new, greater, and more wonderful idea. Art for him was ever green.
Literature for him was another nourishing
pasture of the mind, from Goethe and Gottfried Keller to Rilke and Mann
and
from Shakespeare and Coleridge to Ortega y Gasset and T. S. Eliot.
Through the green fields of history Weyl
walked, too, with the greats, from Thucydides and Pliny to Vico and
Ranke,
Burckhardt, and Hesse. Surely it was from Weyl that I learnt that
wonderful
statement of Burckhardt, in
Weyl knew as well as anyone that we
cannot know who we are and where we are going unless we know how we got
here. “All history in the proper sense,” he reminded us,
“is concerned with the development of one singular phenomenon:
human civilization on earth” (1). What it takes to do history, he
adds, is not science, not mathematics, not a measuring device, but
interpretation. Interpretation “springs from the inner awareness
and knowledge of myself. Therefore the work of a great historian
depends on the richness and depth of his own inner experience.”
How he would have applauded the motto of the distinguished historian
Jack Hexter: “If you would read history, write
history.” Participation is as necessary for understanding, Weyl
recognized, as
understanding is for participation.
In the
field of economics it was enough, to satisfy Weyl's wish for
understanding, to
know Winfield Riefler and Oskar
Morgenstern. The scene comes vividly to mind of a cocktail party where
Weyl was
a guest, and guests, too, were Morgenstern and von Neumann, working at
that
time on their great investigation, embodied in their book
Colleagues? Weyl knew that nobody can be
anybody without somebodies around. Teatime was for him a central point
of the
day, an opportunity, as Oppenheimer later put it, to “explain to
each other
what we do not understand.” And cocktail parties? None do I
remember as richer
in interesting colleagues and lively conversation about fascinating
subjects
than those given by Hermann Weyl and his wife, Hella.
Weyl himself, so universally respected
and beloved, at home in so many fields of thought, made one think of
those
words of Frederick II about Leibniz, in effect: “Founder of the
Prussian
Academy? He already was an academy!” Weyl was an academy.
For Weyl to update in 1947 his great 1927
survey of philosophy, mathematics, and natural science was an enormous
undertaking, understandable only in the light of his passion for truth
and
unity. The marvelous discoveries in the thirty years since his death
would
surely have filled him with joy and reanimated his wish, harder now
than ever
for anyone to achieve, to capture for us all the larger unity of these
findings.
What
is the role of electricity in the geometry of spacetime? Weyl came back
to this
topic in paper after paper and book after book. The perspectives that
he and
his successors opened up are probably explored today, under the names
of “gauge
field theory” and “grand unified field theory,” by
more investigators than
there were in the entire physics community at the time of Weyl’s
first paper in
the field. Weyl invented the gauge concept in 1918. By 1928 he had
reformulated
and restated the idea in the way it is still understood today:
“gauge
invariance corresponds to the conservation of charge as a coordinate
invariance
corresponds to the conservation of energy-and-momentum” (7).
In 1950 Weyl referred to the 1921 idea of T. F. E. Kaluza that gauge
invariance, might be connected with the possible presence in nature of
a fifth
dimension (8). He noted that O. Klein added to Kaluza's idea the
further
conception, arising out of quantum theory, that the fifth-dimensional
part of the geometry is curled up into a very tight radius of
curvature. This
pregnant idea, students of modem particle theory know, has itself taken
fifty
years to bloom. Today the gauge field at each point in spacetime is
envisaged
as running, not around the one-dimensional rim of a tiny circle, but
around an ultra-small cavity of much higher -- perhaps six units
higher -- dimension. The variations of the field in the several
directions describe not only the electromagnetic field but also the
fields
associated with neutrinos and all the rest of elementary particle
physics. A
particle mass itself corresponds in effect to an organ-pipe resonance
of
the geometry in this tiny world -- or, in mathematical terms,
“fiber” --
attached to each point of spacetime. Weyl, were he still living, would
rejoice
in the rich modem development of elementary particle physics and be
contributing himself, surely, to putting the theory of gauge fields and
fiber
bundles into a wider and deeper mathematical and physical framework.
Another insight Weyl gave us on the
nature of electricity is topological in character and dates from 1924
(9). We
still do not know how to assess it properly or how to fit it into the
scheme of
physics, although with each passing decade it receives more attention.
The idea
is simple. Wormholes thread through space as air channels through Swiss
cheese.
Electricity is not electricity. Electricity is electric lines of force
trapped
in the topology of space.
Niels Bohr, presented at one point with
this thought, immediately asked, “but will not any wormhole pinch
itself off?”
Subsequent calculations showed that, in the context of classical
theory, indeed
it will (10). Further consideration, however, has made it clear that
one should
deal not with classical wormholes but with quantum fluctuations in the
geometry
of space, wormholes at the unbelievably small Planck scale of distances
(on the
order of 10
How are we to assess Weyl’s proposal,
thus updated? That is a question not for today but for tomorrow. To
deal with
it is to work at the frontier between quantum theory and general
relativity, in
that realm often called “quantum geometrodynamics,” one of
the most challenging
fields of research of our day.
It
is sometimes said today that no progress has been made in uniting
general
relativity and quantum theory. No statement could be further from the
truth. We
have possessed for years the appropriate wave equation. We have also
known,
through the work of Uhich Gerlach (15), that the predictions of this
wave
equation go over, in the so-called correspondence-principle limit,
into those of classical geometrodynarnics. In the past year we have
seen
applications; of this wave equation, by James Hartle and Stephen
Hawking (16),
to important issues of cosmology. But this whole field of investigation
is so
new and strange that we have still some way to go before we have a
proper
notion what to make of Weyl’s idea that “electricity is
field trapped by
topology.” But disregard it? Only at our peril.
In astrophysics generally and cosmology
in particular a fantastic growth has taken place in the years since
Weyl’s
death, thanks to the existence of redshifts in stellar spectra (which
show --
he was the first truly to explain -- that the universe is expanding)
and
a marvelous armament of telescopes, of not one kind alone, but four:
x-ray,
optical, infrared, and radio. Among all developments in astrophysics,
surely
none would have gripped his imagination more than the black hole, a
star that
has undergone complete gravitational collapse.
Already in the earliest days of “gravitation-as-geometry”
Weyl taught us a useful new view of a “mass.” (4) Draw a
sphere around it.
Stretch this sphere out in time into a fattened world line, a world
tube. What
happens to a star duster, a whole collection of masses interacting
gravitationally with one another? Viewed in spacetime in Weyl’s
way, it becomes
a twisting, writhing pattern of world tubes. To predict that pattern,
he
explained, we don't have to look inside the tubes.
Weyl’s way of looking at mass has a
special relevance for a black hole -- itself invisible -- that
happens to be paired with a visible companion. Weyl would be delighted
today to
find that the black hole is not simply an object on pencil and paper,
that it
really exists. We already have striking evidence for two black holes in
the
range of 10 to 20 times the mass of the sun (17, 18). Each is invisible
itself.
Each by its powerful pull swings about it in a tight, swift orbit a
normal star
which we can and do see. We see also x-rays. They do not come from the
black hole itself. They come from gas spewed out onto the black hole
from its
normal neighbor. The inwardstreaming gas is crunched up to high
density, and
therefore enormous temperature, by the powerful pull of the completely
collapsed object. Hence the x-rays. They fluctuate in intensity from
millisecond to millisecond because of random variations in the density
of the
gas falling in, as the smoke coming out of a factory chimney fluctuates
in its
darkness from second to second. In addition to those two black holes of
stellar
mass there also exists at the center of the Milky Way, according to
ever
stronger evidence (19), a black hole with mass about three and a half
million
times the mass of the sun.
Never has curved, empty space come more
spectacularly to man’s attention than it does in black hole
physics. Never has
any branch of physics been developed more fully and more richly on the
basis of
purely geometric reasoning. And never more insistently than in black
hole physics
have we been driven to the very frontiers between gravitation theory
and
quantum theory.
Jacob
Bekenstein found himself forced in 1973 to conclude that the surface
area of the
so-called horizon of a black hole not only is analogous to entropy, it
is
entropy; the surface gravity not only is analogous to temperature, it
is
temperature (20). This conclusion seemed so preposterous to Brandon
Carter and
Stephen Hawking that they set out to disprove it. Along the way Hawking
discovered that marvelous process (21), now known by his name, through
which a
black hole -- indeed endowed after all with a Bekenstein temperature --
can evaporate particles and radiation (less than one watt, however,
from a
black hole with the mass of the sun, and less in proportion as the mass
is
greater).
Black hole physics has led to one great
discovery that has in it not one word of black hole physics. I refer to
the by
now famous formula of William Unruh (22). It tells us that an
accelerated
detector, located in cold empty space, will nevertheless, by reason of
that
very acceleration, experience and register a temperature, a temperature
proportional to the product of Planck’s quantum constant and that
acceleration.
This result generalizes Bekenstein’s conclusion about the
temperature at the
“surface” of a black hole.
Unruh’s formula ties together three great
domains of physics. One is relativity. The second is quantum theory.
The third,
heat physics or thermodynamics or statistical mechanics, is also at
bottom a
part of information theory. I know no more beautiful discovery in
recent years,
nor any that connects more instructively three fields of endeavor dear
to
Weyl’s thinking and writing.
A brave new proposal of Bekenstein is now
the subject of exploration by more and more investigators (23): that
there is
no device whatsoever that will store a given number of bits of
information
which does not have a product of mass and linear dimensions expressed
in
appropriate units-which is at least as great as that number of bits.
Information is not dreamlike nothingness. What an incentive to put
“information” into the center of our thinking about
physics! And to ask, is
information theory the foundation for all we see and know?
Information
theory, Weyl knew, is central to the gene, the machinery of life, and
evolution. “The mighty drama of organic evolution,” as Weyl
called it, was for
him no domain of thought to be left to a few specialists, but a vital
topic of
concern to every thinking person. He gave it an important place in 1947
when he
updated his
How Weyl would have delighted in the DNA
Of Francis Crick and James Watson. How fascinated he would have been by
the
“life machine” of Manfred Eigen and his Göttingen
colleagues, showing as it
does the stupendous variety of life forms that can develop, and
emphasizing the
role of what we can only call “accident” in deciding which
shall come into
being. How eagerly he would have studied those twin discoveries of John
J.
Hopfield: nature’s method of “proofreading” molecules
and DNA, without which
life would quickly end in disaster; and insights into how the coupling
of
neuron to neuron powers memory and memory search (25, 26).
In what way did life begin? Weyl
describes the idea of A. I. Oparin that organic molecules, formed by
chance
processes and accumulated in favorable spots, in time built up -- in
the
absence of enzymatic breakdown -- to the concentration at which
self-duplicating
molecules could form (27). He would have been interested in the
proposal that
droplets were the centers of accumulation and the place of origin of
life (28),
and in the theory, more widely studied today, that it was day which
served as
the organizing material (29).
The concept that life fills every
ecological niche was not new to Weyl; but how interested he would have
been in
the 1978 finding of those students of evolutionary history, V.
Salvini-Plawen
and Ernst Mayr, that the eye -- the “window of the mind” --
originated over and over again, independently, at least forty times
(30). What
would he have said to the thesis of Homer Smith, so vividly expressed
in his
Pulitzer Prize-winning book, Kamongo, that man himself may, from the
standpoint of the future development of the evolutionary tree, mark a
blind
alley of life? In contrast, Weyl himself, discussing teleology, remarks
that
“the temptation of an interpretation in terms of an overall plan
of evolution
is almost irresistible” (31). Could he have known that the
distinguished
physiologist Christian Bohr, father of Niels Bohr, while wholeheartedly
accepting and supporting the Darwinian theory of evolution at a time
when that
position was not popular, nevertheless also felt that evolution,
understood
deeply, would prove to be compatible with ultimate purpose?
Teleology, come into physics? How? “Is it
conceivable,” Weyl asks, “that immaterial factors having
the nature of images,
ideas, ‘building plans,’ also intervene in the evolution of
the living world as
a whole?”
The
question about teleology that Weyl put to himself we have more
temptation than
ever to put to ourselves in this day, when the deeper lessons of
quantum theory
are so widely perceived.
The idea is old that the past has no existence except in the records of
today. In our time this thought takes new poignancy in the concept of
Bohr’s
elementary quantum phenomenon and the so-called delayed-choice
experiment (32). Ascribe a polarization, a direction of
This circumstance is one of the more
striking information-theoretic aspects of what we call existence.
Moreover, the delayed-choice elementary quantum phenomenon is well
established by theory and, within the last few years, by experiment
(33, 34,
35). Any view of existence which does not reckon with quantum theory
and the
elementary quantum phenomenon is medieval. Quantum theory marks the
summit of
The
man who ranged so far in his thought had mathematics as the firm
backbone of
his intellectual life. Distinguished as a physicist, as a philosopher,
as a
thinker, he was above all a great mathematician, serving as professor
of
mathematics from 1913 to 1930 at Zurich, from 1930 to 1933 at
Göttingen, and at
the Princeton Institute for Advanced Study from October 1933 to his
retirement.
What thinkers and currents of thought guided Weyl into his lifework:
mathematics, philosophy, physics?
“As a schoolboy,” he recalls, “I came to
know Kant’s doctrine of the ideal character of space and time,
which at once
moved me powerfully” (37). He was still torturing himself, he
tells us, with
Kant’s
Allow here a pause for a brief song of
thanksgiving. Shall we dedicate it to that Prince of Hanover whom the
English-speaking
world knows as George II? Or shall we praise instead the advisers at
whose
instance that ruler initiated, within a few years of each other, two
now famous
communities of learning? Those centers, both precious to Hermann Weyl,
are
known today as Göttingen (1734) and Princeton (1746). The
advisers, blessed be
their names-Christian Wolff, the disciple of Leibniz in the
German-speaking
world, and a group of concerned men of learning in the colony of New
Jersey had
for the two schools a common purpose: to testify to the glories of
creation by
looking at them, investigating them, and teaching about them.
That goal transmutes itself into ever new
language with the arrival of ever new generations, In our century this
powerful
Göttingen-Princeton tradition, springing straight out of the age
of
enlightenment, has nourished deep learning -- a happy, intense,
livelong-day
search by great thinkers for beauty, order, and understanding.
What way of work did Weyl adopt? Three
ways we know to make advances: the ways of the mole, the mutt, and the
map. The
mole starts at one point in the ground and systematically goes forward.
Great
science has been done by people so guided. The mutt sniffs around and
is led on
from one due to another. Great physics is done that way. But the third
method
of advance is marked by the mapmaker, the philosopher who conceives the
overall
picture, has a feeling for how things fit together, and finds his way,
by that
sense of fitness, to where new truth lies. That was Weyl.
To Kant, to the Göttingen of Gauss and
Riemann, and to Weyl the number one example of mapmaker, of
philosopher, of
guide in the enterprise of discovery was Leibniz. “Among the
heroes of
philosophy,” Weyl tells us, “it was Leibniz above all who
possessed a keen eye
for the essential” (38). He goes on to remind us that
“Leibniz planned to span
Europe by a network of academies, centers of research which he expected
to
become a strong combine for the promotion of enlightenment” (39).
Hermann Weyl
and his longtime Princeton colleague, Kurt Gödel, shared an
interest in
Leibniz’ notes on his project of a
If philosophy, the map, displayed the
goals, mathematics -- in the shape of Hilbert -- showed the
arriving Göttingen student the way. Weyl tells us the impression
made upon him
by Hilbert’s irresistible optimism, “his spiritual passion,
his unshakable
faith in the supreme value of science, and his firm confidence in the
power of
reason to find simple and clear answers to simple and clear
questions.” No one
who in his twenties had the privilege to listen to Weyl’s
lectures can fail to
turn around and apply to Weyl himself those very words. Neither
can anyone who reads Weyl, and admires his
style, fail to be reminded of Weyl’s own writing by what he says
of the
lucidity of Hilbert: “It is as if you are on a swift walk through
a sunny open
landscape; you look freely around, demarcation lines and connecting
roads are
pointed out to you before you must brace yourself to climb the hill;
then the
path goes straight up, no ambling around, no detours” (40).
Electrified by Leibniz and Kant, and
under the magnetic influence of Hilbert, Weyl leaped wholeheartedly, as
he
later put it, into “the deep river of mathematics.” That
leap marked the
starting point of his lifelong contributions to ever widening spheres
of
thought.
Out of Weyl’s thinking, out of his
speaking, out of his writing -- and out of work since his day --
what guidance can we now discover on our quartet of questions: the
mechanism of
existence, the origin of the quantum, the problem of the continuum, and
the
deeper foundations of the idea of time?
Existence,
the preposterous miracle of existence! To whom has the world of opening
day
never come as an unbelievable sight? And to whom have the stars
overhead and
the hand and voice nearby never appeared as unutterably wonderful,
totally
beyond understanding? I know no great thinker of any land or era who
does not
regard existence as the mystery of all mysteries.
But is the quantum a mystery, too? We
know that the way the quantum theory works is no mystery. It is
expounded in a
hundred texts. But from what deeper principle does its authority and
its way of
action derive? What central concept undergirds it all? Surely the magic
central
idea is so compelling that when we see it, we will all say to each
other, “Oh
how simple, how beautiful! How could it have been otherwise? How could
we have
been so stupid so long?” But what is the decisive clue to it all
that we of
today are missing?
The continuum of natural numbers: who
could dispense with them who works with matter and motion, particles
and
fields, space and time? Indeed, as Weyl reminds us, “Classical
analysis, the
mathematics of real variables as we know it and as it is applied in
mathematics
and physics, has simply no use for a continuum of numbers of different
levels
[integers, rational fractions algebraic numbers, etc.].” Yet, he
goes on to
say, “[L. E. J.] Brouwer made it clear, as I think beyond any
doubt, that there
is no
Time? The concept did not descend from heaven, but from the mouth of
man, an early thinker,
his name long lost. Time today is in trouble. Time ends in a big bang
and
gravitational collapse. Quantum theory denies all meaning to the
concepts of
before and after in the world of the very small. Most of all, time has
not yet been brought into submission to the rule of physics. It is fed
in from
outside physics. It has someday to be derived from inside physics --
when
physics is deep enough to measure up to the task.
Four puzzles? Four clues. Shall we look at
them one by one?
Let
us begin on puzzle number one, existence, with what sounds at first
hearing as
a far-away note, a solitary and pregnant passage from a 1919 paper of
Hermann Weyl on general relativity. There he points out the first of
the by now
famous large-number coincidences of physics in a single sentence, which
we can be forgiven for taking apart into three: “It is a fact
that pure numbers
appear with the electron, the magnitude of which is totally different
from one.
For example, the ratio of the electron radius to the electron's
gravitational
radius is of the order of 10 This coincidence between two enormous
numbers of very different origin was called in a paper twelve years
later by
Fritz Zwicky “Weyl’s hypothesis” (43). But not
everybody reads the literature.
Later it and further such relations were called
“Eddington’s large- number
coincidences” (44), then, later, “Dirac’s large
numbers” (45), and so they are
widely known today. But it all began with Weyl.
What feature of nature lies behind these
large number coincidences? What fixes these and other dimensionless
constants of
physics? Advocates of anthropic principle, nowadays investigated by
more and
more physicists and astrophysicists, propose a perspective-shattering
answer: not only is man adapted to the universe, the universe is
adapted to
man. Imagine a universe in which one or another of the fundamental
dimensionless constants of physics differs from this world’s
value by a few
percent one way or the other. The consequences for the physics of stars
so
multiply themselves up -- according to the analyses of numerous
investigators (46) -- that man could never have come into being in such
a
universe.
The anthropic principle superficially looks like a tautology:
we’re
adapted to the universe because we’re adapted to the universe.
However, closer
study by Brandon Carter (47) shows that the idea leads to an amazing,
concrete,
and someday testable prediction: it sets a limit of a few hundred
million
years, at most, on the time that the earth will continue to be an
inhabitable
planet. This prediction is derived from our knowledge of evolutionary
biology
and from modem statistical analysis. A simple mathematical expression,
called
Carter’s inequality, relates the likely duration of life on Earth
in the future
to the number of improbable evolutionary steps required in the past for
the
emergence of intelligent life.
Is the machinery of the universe so set
up, and from the very beginning, that it is guaranteed to produce
intelligent
life at some long-distant point in its history-to-be? And is this
proposition testable by the Carter prediction? Perhaps. But how should
such a
fantastic correlation come about between big and small, between
machinery and
life, between future and past?
Some who investigate the anthropic
principle put forward the notion of an ensemble of universes,
distinguished one
from another by different values of the dimensionality and the
dimensionless
constants of physics. In the overwhelming majority of cases, they
argue,
intelligent life is and always will be impossible. We belong, on this
view, to
one of the rare exceptions, a universe where awareness can and does
develop.
We can reject some of these ideas without
rejecting everything. We can forgo the notion of an ensemble of
universes as
outside the legitimate bounds of logical discourse. We can nevertheless
examine
the anthropic principle itself as an attractive working hypothesis
attractive
because it exposes itself, by its predictive power, to destruction in
the sense
of Karl Popper (48) and because it makes sense out of numbers that
would
otherwise have no rationale. But without multitudes of universes to
experiment
on, to bungle and to ruin à la David Hume, with solely this one
and only
universe to work with, how can history ever have made things come out
right, ever
given a world of life, ever thrown up a communicating community of the
kind
required for the establishment of meaning? In brief, how can the
machinery of
the universe ever be imagined to get set up at the very beginning so as
to
produce man now? Impossible! Or impossible unless somehow --
preposterous
idea -- meaning itself powers creation. But how? Is that what the
quantum is
all about?
To ask this question is to look at the
puzzle of existence from a new perspective; to see a thread of
connection with
puzzle number two, the how-come of the quantum.
Machinery
of existence for us means laws of physics under the overarching
governance of
the quantum principle; in brief, laws and the quantum. How can the
quantum ever
be understood as powered by meaning? Or laws of physics, by meaning?
Weyl reminds us that “postulation of the
external world does not guarantee that such a world will arise ... from
the
phenomena ... through the cognitive work of reason ... which attempts
to create
concordance. For this to take place,” Weyl emphasizes -- updating
Kant’s
concept of reality as “that which is
connected with perception according to laws” -- “it is
necessary that the world
be governed throughout by simple elementary laws” (49). Those laws, so beautiful, so
necessary for an understandable,
meaningful world, and on first inspection so full of structure, turn
out every
one of them on closer look to be built in large measure on tautology,
mathematical
identity, the most elementary statement of algebraic geometry: the
principle
that the boundary of a boundary is zero. Electromagnetism, in the shape
of
Maxwell’s equations, seen in four-dimensional perspective, falls
apart
into two divisions, one equivalent to the statement that the
one-dimensional
boundary of the two-dimensional boundary of a three-dimensional
region is identically zero; the other, that the two-dimensional
boundary
of the three-dimensional boundary of a four-dimensional region
likewise identically vanishes. The concept of the vanishing of the
boundary of
a boundary both in its 1-2-3 and in its 2-3-4 forms is used
again in gravitation physics and yet again in the Yang-Mills or
chromodynamic
or string theory -- of one or another degree of sophistication --
of elementary particle physics. How strange, we say at first. And then
we ask
ourselves, how could it have been otherwise?
Surely -- big bang and
gravitational collapse advise us -- the laws of physics cannot have
existed from everlasting to everlasting. They must have come into being
at the
one gate of time, must fade away at the other. But at the beginning
there were
no gears and pinions, no corps of Swiss watchmakers to put things
together, not
even a pre-existing plan. If this assessment is correct, every law of
physics
must be at bottom like the second law of thermodynamics,
higgledy-piggledy
in character, based on blind chance. Physics must be in the end law
without
law. Its undergirding must be a principle of organization which is no
organization at all. In all of mathematics, nothing of this kind more
obviously
offers itself than the principle that the boundary of a boundary is
zero. That
this principle should pervade physics, as it does -- is that the only
way
that nature has to signal to us a construction without a plan, a
blueprint for
physics that is the very epitome of austerity?
No. A second sign directs the seeker for
the plan of existence still more clearly to austerity: the quantum.
What is the thread that connects mystery
number two, the quantum, with puzzle number one, the machinery of
existence? Does the very concept of existence imply that there must be a world sitting “out there”? That was the view of many a great thinker before the advent of quantum theory -- and of Einstein himself to the end of his days. Nothing made him more unhappy than the thought that the observer-participator has anything to do with the establishment of what one is accustomed to call reality. In the last talk he ever gave, some months before his death, to my seminar on relativity, he explained how he had come to relativity and what relativity meant to him, but went on to express his discomfort with quantum theory: “If a person, such as a mouse, looks at the universe, does that change the state of the universe?” And to the visitor defending quantum theory to him in his study he objected against its probability features in the words, “God does not play dice” (50).
Weyl, in contrast, spoke up for the
physics community when he stated, “Quantum theory is incompatible
with the idea
that a strictly causal theory of unknown content stands behind it....
The
reasons for the passage from classical to quantum physics are no less
compelling than those for the relinquishment of absolute space and time
by relativity theory; the
success, if measured by the empirical facts made intelligible, is
incomparably
greater” (51).
No one saw deeper into the central point
of quantum theory than Niels Bohr, lifelong friend of Hermann Weyl; and
no one
stated its importance more strongly than he in his last interview, a
few hours
before his unexpected death: “They [certain philosophers] have
not that
instinct that it is important to learn something and that we must be
prepared
to learn something of very great importance ... They did not see that
it [the
complementary description of nature as it is seen in quantum theory]
was an
objective description-and that it was the only possible objective
description” (52).
Only possible? There is not a single
sight, not a single sound, not a single sense impression which does not
derive
in the last analysis from one or more elementary quantum phenomena.
Objective? Not until the observing sense,
or observing device -- by its geometry, its layout, and its adjustment
--
has chosen the question to be asked, and by its registration has made a
record
long enough lived to produce internal or external action, has an
elementary
quantum phenomenon taken place that contributes to the formation of
what we
call reality. No other way do we know to build this reality. Existence?
How
else is it brought into being except through elementary quantum
phenomena?
Unbelievable! The number of bits of
information that anyone can accumulate in a lifetime is incredibly
small
compared to the richness we know to be there in the great wide world.
Even if
we include more than a single observer-participator and count up the
contribution of every member of the meaning-establishing community, of
every
observer-participator past, present, and future, what hope is there for
deriving existence out of quantum acts? Of them there is at most a
countable
infinity. In contrast, existence seems to present us everywhere with
continuous
infinities: a continuous infinity of locations for particles, a
continuous infinity of field strengths, a continuous infinity of
degrees of
freedom of dynamic space geometry.
Issue number two, the how-come of
the quantum, here shows some thread of connection with issue number
three, the
continuum. How close is that connection?
The
continuum of natural numbers, Weyl taught us, is an illusion. It is an
idealization. It is a dream. With numbers of ever increasing
mathematical
sophistication we can approach that infinity ever more closely; but we
commit a
folly if we think we can ever get there. That, in poor man’s
language, is the
inescapable lesson of Gödel's theorem and modem mathematical logic.
Do we not have to say that the notion of
a physical world with a continuous infinity of degrees of freedom is an
equal
idealization, an equal folly, an equal trespass beyond strict logic?
Do we not do better to recognize that
what we call existence consists of countably many iron posts of
observation
between which we fill in by an elaborate papier-mâché
construction of
imagination and theory? The artist paints in the faces of five angels,
with
diminishing size, followed by a row of dots of still further decreasing
size,
stretching out into a line that runs to the horizon; but the beholder
believes
himself to see an infinitude of angels. When Bohr tells us that quantum
theory
gives us the only objective description of nature of which one can
possibly
conceive, is he not also telling us that no description can make sense
which is
not founded upon the finite?
For the advancing army of physics,
battling for many a decade with heat and sound, fields and particles,
gravitation
and spacetime geometry, the cavalry of mathematics, galloping out
ahead,
provided what it thought to be the rationale for the natural number
system.
Encounter with the quantum has taught us, however, that we acquire our
knowledge in bits; that the continuum is forever beyond our reach. Yet
for
daily work the concept of the continuum has been and will continue to
be as
indispensable for physics as it is for mathematics. In either field of
endeavor, in any given enterprise, we can adopt the continuum and give
up
absolute logical rigor, or adopt rigor and give up the continuum, but
we can't
pursue both approaches at the same time in the same application.
Adopt rigor or adopt the continuum? These
ways of speaking should not be counted as contradictory, but as
complementary.
This complementarity between the continuum and logical rigor we accept
and
operate with today in the realm of mathematics. The hard-won power thus
to
assess correctly the continuum of the natural numbers grew out of
titanic
struggles in the realm of mathematical logic in which Hermann Weyl took
a
leading part (53). The level of synthesis achieved by now in
mathematics is
still far beyond our reach today in physics. Happily the courageous
outpost-cavalry
of mathematical logic prepares the way, not only for the main cavalry
that is
mathematics, but also for the army that is physics, and nowhere more
critically
so than in its assault on the problem of existence.
Time, among all concepts in the world of
physics, puts up the greatest
resistance to being dethroned from ideal continuum to the world of the
discrete, of information, of bits. Of physics the heart is dynamics,
and of
dynamics the heart is time. That time parameter we treat today,
however,
as provided for us free of charge from outside, as our forebears
regarded
elasticity a hundred years ago. In our day we have learned that there
is no
such thing as elasticity in the space between the electron and the
nucleus.
Elasticity, thanks to solid state physics, has been reduced from
primordial and
precise to secondary, approximate, and derived. Time today requires a
like
reduction.
Reduce time? The idea of reduction is
old. Weyl reminds us that “the doctrine of the subjectivity of
sense qualities
has been intimately connected with the progress of science ever since
Democritus laid down the principle, ‘Sweet and bitter, cold and
warm, as well
as the colors, all these things exist but in opinion and not in
reality; what
really exist are unchangeable particles, atoms, which move in empty
space’ ”
(54). In accordance with this view of Democritus, we understand green
today as
a characteristic frequency of 5.7 x 10
But time: how is time to be reduced to
more primitive concepts? Reduced from the continuum to something built
on bits?
And along with the reduction of time how are we to understand that
puzzling
conservation of “I” from decade to decade, so vividly
expressed by Weyl in his
quotation from
Of all obstacles to a thoroughly
penetrating account of existence, none looms up more dismayingly than
“time.”
Explain time? Not without explaining existence. Explain existence? Not
without
explaining time. To uncover the deep and hidden connection between time
and
existence, to close on itself our quartet of questions, is a task for
the
future.
A
great problem, we know, means great hope, according to that
time-honored
doctrine of “no discovery without a paradox.” A still
better formula has
emerged out of the science of this century: no great advance without a
double
mystery, a double paradox, a double problem, two dues that can be
played off
against each other to yield the answer. Fortunate are we to have before
us two
such mighty mysteries as time and existence -- each linked with two
other
great questions, the quantum and the continuum.
Last year was the hundredth birthday not
only of Hermann Weyl but also of Niels Bohr. Their double drumbeat
summons us
to a great undertaking, tells us that we can and must achieve four
victories:
Explain
existence by the same idea that explains the quantum.
Through
this larger vision of existence and the quantum, recognize that the
continuum
of that physical world out there and the bit-by-bit means by which
alone
we can define that world are not contradictory, but complementary.
Reduce time into subjugation to physics. As we face these stirring challenges, a warm memory gives us courage. Hermann Weyl has not died. His great works speak prophesy to us in this century and will continue to speak wisdom in the coming century. If we seek a single word to stand for the life and work of Hermann Weyl, what better word can we find than passion? Passion to understand the secret of existence was his, passion for that dear, luminous beauty of conception which we associate with the Greeks, passionate attachment to the community of learning, and passionate belief in the unity of knowledge.
1. H. Weyl. 1968. Address on the unity of
knowledge delivered at the [1954] Bicentennial Conference
of Columbia University. Item no. 165 in K. Chandrasekharan, ed.,
2. H. Weyl. 1928.
3. H. Weyl. 1952.
4. H. Weyl. 1918.
5.
H. Weyl. 1949.
6. H. Weyl. 1950. Elementary proof of a minimax
theorem due to von Neumann. In ref. 1, vol.
4, item 151.
7. H. Weyl. 1955. Commentary in ref. 1
8. H. Weyl. 1950. 50
9. H. Weyl. 1924. 10.
R.
W. Fuller and J. A. Wheeler. 1962. 11. J. A. Wheeler. 1957. 12. J. A. Wheeler. 1968. Superspace and
the nature of the quantum geometrodynamics. In C.
M. De Witt and J. A. Wheeler, eds., 13
C. W. Misner, K S. Thorne, and J. A. Wheeler. 1973. 14.
G.
W. Gibbons and S. W. Hawking. 1977. 15.
U.
Gerlach. 1968. 16. J. Hartle and S. W. Hawking. 1983. 17.
R.
Giacconi, P. Gorenstein, H. Gursky, and J. R. Walter. 1967. 18. P. Cowley, D. Crampton, J. B.
Hutchings, R. Remillard, and J. E. Penfield. 1983. 19. J. H. Oort. 1977. 20. J. D. Bekenstein. 1973. 21.
S.
W. Hawking. 1975. 22.
W,
G. Unruh. 1976. 23. J. D. Bekenstein. 1980. 24.
Ref.
5, p. 296. 25. J. J. Hopfield. 1980. 26.
J. J. Hopfield. 1982. PNAS 79:2554-58. J. J. Hopfield. 1982. 27.
Ref. 5, pp. 299-300.
28.
Early
proposal of L. Onsager. See related discussion in C. Tanford. 1980. 29.
For a
review see G. Millot. 1979. Sci. 30.
V.
Salvini-Plawen and E. Mayr. 1978. 31. Ref. 5, p. 300.
32.
See the
chapter by J. A. Wheeler in A. R. Marlow, ed. 1978. 33.
A.
Aspect, J. Dalibard, and G. Roger. 1982. Phys. Rev. Lett. 49:1804-07. 34.
C.
Alley, 0. Jakubowicz, C. A. Steggerda, and W. C. Wickes. 1984. In
ref. 32, Kamefud-d et al., pp. 1584A. 35.
T. Hellmuth, H. Walther, and A. G. Zajon.-. 1985. Realization of a
Mach-Zehnder “delayed-choice"
interferometer. Report presented at the June 1985 Finnish symposium,
Foundations of Modem Physics: 50 Years
after EPR. 36.
Ref. 5, p. 216. 37.
H. Weyl. 1954. Lausanne lecture reprinted in ref. 1, vol 4, item 166.
The
quotation comes from p. 631. 38.
Ref.
5, p. 2. 39.
H.
Weyl. 1945. Two lectures given at Princeton University and reprinted in
ref. 1,
vol. 4, item 157; see p. 558. 40.
C.
Reid. 1970. 41. H. Weyl. 1946. Mathematics and logic. Preface to a review of “My Philosophy of Bertrand Russell.” Am. Math. Mon. 53:2-13. 42.
H.
Weyl. 1919. 43.
F.
Zwicky. 1939. 44.
A.
S. Eddington. 1923. 45.
P.
A. M. Dirac. 1937. 46.
See
J. D. Barrow and F. J. Tipler. 1986. 47.
B.
Carter. 1983. 48.
K. Popper. 1963. 49.
Ref. 5, p. 125. 50. J. A. Wheeler. 1979. Mercer Street and other
memories. In P. C. Aichelburg and R.
Sexl, eds. Albert Einstein: his Influence on
Physics, 51.
Ref.
5, p. 253. 52.
Quoted in E. M. MacKinnon. 1982. 53.
H.
Weyl et al. 1918. 54.
Ref. 5, p. 110. See also p. 112 for ideas of Leibniz like those of
Democritus. 55.
Ref. 5, pp. 237-38.
Note: Any transcription errors present in this document are the responsibility of weylmann.com and not American Scientist. |