THE UNKNOWN
AND THE
UNKNOWABLE
Exploring
the boundaries
BY
J O S E P H
of scientific
F.
knowledge
T R A U B
n u m b e r o f s t a t e m e n t s c a l l e d a x i o m s , w h i c h a r e a c c e p t e d as
t r u e a n d " s e l f - e v i d e n t , " as w e l l as s o m e e q u a l l y
unobjec-
t i o n a b l e rules for d e r i v i n g o n e s t a t e m e n t f r o m
another,
called rules of inference. A r i t h m e t i c furnishes s o m e e x a m ples: t h e a x i o m o f a r i t h m e t i c t h a t m a k e s , say, 2 + 0 e q u a l
to 2 (and l i k e w i s e , for t h e a d d i t i o n o f z e r o to a n y natural
n u m b e r ) ; a n d t h e logical rule that enables o n e t o infer, f r o m
t h e p r o p o s i t i o n if p then
q and the proposition p
that
}
q.
G o d e l s h o w e d t h a t i f a s y s t e m as r i c h as a r i t h m e t i c is c o n Rene Magritte,
T h e Invisible W o r l d ,
1954
s i s t e n t , it c a n n o t b e c o m p l e t e . T h a t is, t h e r e a r e s t a t e m e n t s
w i t h i n a r i t h m e t i c t h a t c a n n o t b e p r o v e d t r u e o r false w i t h i n t h e s y s t e m ; a r i t h m e t i c is u n d e c i d a b l e .
W h e n G o d e l a n n o u n c e d h i s r e s u l t , its i m p a c t o n l o g i c
IMPOSSIBILITY: T H E LIMITS OF SCIENCE
a n d m a t h e m a t i c s w a s s e n s a t i o n a l a n d p r o f o u n d . It s o l v e d ,
AND THE SCIENCE OF LIMITS
by John D. Barrow
Oxford University Press, 1998
279 pages; $25.00
i n o n e b r i l l i a n t flash, t h e s o - c a l l e d d e c i s i o n p r o b l e m o f t h e
great nineteenth-century
7
German mathematician
David
Hilbert: Devise a mechanical procedure that could determine whether or not any given mathematical
statement
w a s t r u e . ( G o d e l ' s s h o c k i n g a n s w e r w a s , it c o u l d n o t
be
d o n e . ) G o d e l ' s r e s u l t a l s o d o o m e d , i n o n e fatal t h r u s t , t h e
fondest hopes of t u r n - o f - t h e - c e n t u r y logicians: that a x i o m
7
systems, m o d e l e d o n Euclid's geometry , could
W H A T REMAINS TO B E DISCOVERED
by John Maddox
Martin Kessler Books, The Free Press, 1998
434 pages; $26.00
theorem
s e e m e d , at least t o m a n y p e o p l e , t o i m p o s e a f u n d a m e n tal l i m i t o n h u m a n u n d e r s t a n d i n g . F o r if u n d e r s t a n d i n g
itself w a s a m a t t e r o f r e d u c i n g a p r o p o s i t i o n t o a t h e o r e m
provable
I
"capture"
all t h e t r u t h s o f a r i t h m e t i c . E v e n m o r e , G o d e l ' s
from
certain
"self-evident"
truths, then
what
G o d e l h a d d o n e was to p l u m b o n e of the intrinsic limits
N T H E EARLY
European
1 99()S
scientist
I MENTIONED TO A SENIOR
that
I was
interested
u n d e r s t a n d i n g the intrinsic limits to
to understanding.
in
N o w w h a t d o e s t h i s h a v e t o d o w i t h s c i e n c e ? It is e s s e n -
scientific
t i a l t o k e e p i n m i n d t h a t G o d e l ' s t h e o r e m is a b o u t t h e f o r -
k n o w l e d g e . H e replied that such limits h a d b e e n
m a l m a n i p u l a t i o n o f s y m b o l s ; t h a t is, it is a b o u t
mathe-
established b y K u r t Godel. T h a t interpretation of Godel's
m a t i c s . S c i e n c e c e r t a i n l y u s e s m a t h e m a t i c s , b u t s c i e n c e is
f u n d a m e n t a l c o n t r i b u t i o n to logic seems to be a c o m m o n l y
also v e r y
h e l d b e l i e f , b u t it is s i m p l y n o t s o .
u n d e r s t a n d i n g t h e u n i v e r s e a n d e v e r y t h i n g i n it. E x a m p l e s
It is e a s y t o u n d e r s t a n d w h y o n e m i g h t c o m e t o t h i n k
that G o d e l ' s result w o u l d h a v e implications for science. In
different
from
mathematics.
Science
o f scientific questions a b o u n d : H o w d o children
is
about
acquire
language? W i l l h u m a n activities cause m a j o r global changes,
1 9 3 1 t h e C z e c h l o g i c i a n , t h e n a y o u n g p r o f e s s o r at t h e U n i -
a n d w h a t w i l l b e t h e effects o f t h o s e c h a n g e s o n t h e o c e a n
versity of V i e n n a , published a scholarly paper
summariz-
levels, o n agriculture a n d o n biodiversity? H o w d o p h y s i -
i n g his doctoral investigations i n t o t h e p r o p e r t i e s o f formal
c a l p r o c e s s e s i n t h e b r a i n g i v e rise t o s u b j e c t i v e e x p e r i e n c e —
logical systems. S u c h systems are m a d e u p o f s o m e
t h a t is, h o w d o s u c h p r o c e s s e s e x p l a i n c o n s c i o u s n e s s ?
finite
January
'I chruayy
199 9 • T H E
SCIENCES
Can
3^
t h e h e a l t h y , active lives o f p e o p l e b e p r o l o n g e d b y , say, a
t h a t t h e g o l d e n a g e is o v e r a n d o n l y t h e m o p p i n g u p is left.
factor of t w o or three? H o w
A t t h a t t i m e , i n c i d e n t a l l y , t h e m a n u s c r i p t w a s t i t l e d The
d i d life o r i g i n a t e o n
earth?
Ends
W i l l t h e u n i v e r s e e x p a n d f o r e v e r , o r w i l l it c o l l a p s e ? Is t h e r e
of Science,
life e l s e w h e r e i n t h e u n i v e r s e ? Is it i n t e l l i g e n t ?
i n t e r e s t i n g . A p p a r e n t l y t h e p u b l i s h e r s e t t l e d o n The End
N o t e t h a t , at l e a s t o n t h e l a c e o f it, t h e r e a r e n o m a t h e matical m o d e l s that
formalize
t h e r e l e v a n t a s p e c t s of
the
Science,
a n a m b i g u o u s p h r a s e t h a t , t o m y m i n d , is f i r m o r e
of
in h o p e s o f r e a p i n g t h e s a m e success that has a t t e n d -
ed a slew of " T h e E n d of Y o u - N a m e - I t "
books.
With-
H o r g a n writes very well indeed, but I was astonished by
out such m o d e l s , the threat allegedly posed to science by
t h e a m o u n t o f a t t e n t i o n his b o o k r e c e i v e d in t h e general
world, w i t h i n w h i c h those questions can be asked.
G o d e l ' s t h e o r e m c a n n o t e v e n get off t h e g r o u n d . Y e t m a t h -
m e d i a . Its m e s s a g e a b o u t s c i e n c e is b a s i c a l l y
ematical m o d e l s , o f c o u r s e , are indispensable to the p h y s -
w h e r e a s t h e scientists w i t h w h o m I a m in t o u c h are vital-
i c a l s c i e n c e s : N e w t o n ' s l a w s of m e c h a n i c s p r o v i d e a g o o d
l y e x c i t e d b y t h e i r w o r k . T h e r e is m o r e t o b e d o n e
e x a m p l e . Is it p o s s i b l e , t h e n , t o u p t h e a n t e f r o m
e v e r , a n d w e c a n n o t w a i t t o g e t o n w i t h it.
Godel's
analysis o f m a t h e m a t i c s a n d p r o v e i m p o s s i b i l i t y results in
science?
That
is,
can
one
pessimistic,
than
T h e investigation o f t h e u n k n o w a b l e has l o n g b e e n t h e
province of philosophy and epistemology, and the
ques-
t i o n s it r a i s e s h a v e e n g a g e d s u c h p e n e t r a t i n g t h i n k e r s
as I m m a n u e l
Kant
and
Ludwig
W i t t g e n s t e i n . B u t w h a t is u n k n o w a b l e to o n e
generation
is a n o t h e r g e n e r a t i o n ' s m e r e
technical
challenge:
to
Aristotelians in the M i d d l e
Ages, the regions
beyond
the
celestial
moon
were
spheres reaching to heaven,
as u n a p p r o a c h a b l e
as
what-
ever might have c o m e
before
the big bang seems to
cos-
mologists today. A m o r e c o n structive v i e w p o i n t recognizes
t h a t w h a t is k n o w n o r c a n b e
k n o w n about the world
change
with
time.
It
can
makes
sense, then, to think a b o u t the
unknowable
as o n e
w h a t adjustable)
map
of what
mory,
the
(some-
region on a
R a l p h E.
president
Go-
of
the
Alfred P. Sloan F o u n d a t i o n in
N e w York City and the
mer
for-
senior vice president
of
s c i e n c e a n d t e c h n o l o g y at I B M ,
calls t h e t r i p a r t i t e d i v i s i o n of
knowledge. T o the
unknow-
able, G o m o r y adds the k n o w n
and
t h e u n k n o w n . T h e k n o w n is t a u g h t at s c h o o l s a n d u n i v e r sities a n d e x h i b i t e d i n s c i e n c e m u s e u m s . T h e u n k n o w n is
Yves 'L'anouy, The C e r t i t u d e
o f t h e N e v e r S e e n , 1933
the frontier, a territory that m a y s o m e d a y b e c o m e
known
a n d s o is n o t , i n p r i n c i p l e , u n k n o w a b l e .
establish t h e u n k n o w a b l e in science, t h e b o u n d s o f possible scientific k n o w l e d g e ? T h e t w o b o o k s u n d e r
J o h n M a d d o x ' s What
Remains
to Be Discovered
review,
and J o h n Bar-
CEENTISTS ARE FASCINATED BY T H E
UNKNOWN,
. b u t for t h e m o s t part t h e y h a v e b e e n
content
p r o b e m a n y o f t h e issues r e l e v a n t t o t h a t
' to leave the c o n t e m p l a t i o n of the u n k n o w a b l e
question. For the reader, they c o m p l e m e n t each other splen-
to philosophers. T h a t m a y be changing. For o n e thing, the
didly, a n d , t a k e n t o g e t h e r , t h e y offer an i n t e l l e c t u a l j o u r -
intellectual climate s e e m s right for tackling s u c h questions.
n e y to t h e v e r y heart o f t h e scientific enterprise.
G o d e l ' s success has led to o t h e r impossibility results in m a t h -
r o w ' s Impossibility,
Investigating the limits of science should n o t b e confused
g a n , The End
of Science.
W h e n H o r g a n sent m e the m a n u -
l y i m p o s s i b l e i n G o d e l ' s s e n s e , b u t w h o s e d i f f i c u l t y is t h o u g h t
his thesis: t h a t s c i e n c e has m a d e s u c h e x t r a o r d i n a r y
THE
SCIENCES
Fur-
t h e r m o r e , the study o f h a r d p r o b l e m s — p r o b l e m s n o t strict-
s c r i p t for c o m m e n t s , 1 t o l d h i m t h a t I t o t a l l y d i s a g r e e d w i t h
40
e m a t i c s , as w e l l as i n t h e o r e t i c a l c o m p u t e r s c i e n c e .
Hor-
with the intent of the 1996 b o o k by the writer J o h n
• January/February
L
l
l J<) J
7
progress
t o g r o w e x p o n e n t i a l l y fast—-has l e d t o t h e h i g h l y
i d e a o f classifying p r o b l e m s b y t h e i r c o m p u t a t i o n a l
fruitful
com-
plexity. T h e difficulty o f a p r o b l e m can b e m e a s u r e d b y t h e
W h a t is s p a c e m a d e o f ? W h a t d o e s e n e r g y h a v e t o d o w i t h
cost of the c o m p u t e r
m a t t e r ? W h a t is life? B u t i n 1 9 0 0 n e i t h e r h e n o r a n y o n e e l s e
resources—say,
the length
of
run
c o u l d h a v e f o r e s e e n t h e e m e r g e n c e of q u a n t u m m e c h a n i c s ,
t i m e — - n e e d e d t o s o l v e it b y t h e fastest m e t h o d p o s s i b l e .
computer
a n d it is h i g h l y u n l i k e l y t h a t a n y o n e w o u l d h a v e r e c o g n i z e d
science cannot be directly applied to science, the m o d e s of
h o w i m p o r t a n t t h e q u e s t i o n of g r a v i t y w a s t o b e f o r t h e s c i -
A l t h o u g h t h o s e ideas f r o m m a t h e m a t i c s a n d
t h o u g h t m i g h t be transferable. For e x a m p l e , suppose
you
ence of the twentieth century. N o r could
nineteenth-cen-
c o u l d list all t h e f o r m a l m o d e l s t h a t c a p t u r e t h e e s s e n c e o f
tury M a d d o x have foreseen that D N A molecules m a d e
a s c i e n t i f i c q u e s t i o n . I f y o u c o u l d p r o v e t h a t all t h e
o f j u s t f o u r c h e m i c a l u n i t s w o u l d b e r e v e a l e d as t h e f o n t o f
mod-
els a r e u n d e c i d a b l e — t h a t , b y a n a l o g y w i t h t h e a l g o r i t h m s
up
t h e fantastic variety w i t h w h i c h t h e living w o r l d a b o u n d s .
f o r s o l v i n g a h a r d p r o b l e m , n o n e o f t h e m o d e l s is l o g i c a l -
M a d d o x ' s t e x t is d i v i d e d i n t o t h r e e p a r t s : " M a t t e r , " " L i f e "
ly s i m p l e e n o u g h t o b e d e c i d a b l e — - t h e n t h e a n s w e r t o t h a t
and " O u r W o r l d . " " M a t t e r " explores the origins o f the u n i v e r s e a n d t h e e l e m e n t a r y p a r t i c l e s o f m a t t e r , as w e l l as t h e
r
scientific q u e s t i o n w o u l d b e u n k n o w a b l e .
S e c o n d , a n d p e r h a p s m o r e i m p o r t a n t , m o v i n g t h e effort
p r o s p e c t s f o r w h a t p h y s i c i s t s call a t h e o r y o f e v e r y t h i n g . " L i f e "
of distinguishing b e t w e e n the u n k n o w n and the u n k n o w -
also t u r n s t o o r i g i n s , as w e l l as t o s u c h t o p i c s as t h e r o l e o f
able from p h i l o s o p h y to science c o u l d lead to a great e n r i c h m e n t of science. W h y s h o u l d a scientist pay any a t t e n t i o n
t o w h a t c a n n o t b e k n o w n ? P a r t l y , b e c a u s e w h a t is a l l e g e d ly
unknowable
may
help
articulate
what
is
IF PEOPLE BECOME
(merely)
the kingdom of the
unknowable
may illegitimately
annex parts
of the knowable
unknown.
u n k n o w n . D o e s c o s m o l o g y really h a v e n o t h i n g t o say, e v e n
i n p r i n c i p l e , a b o u t c o n d i t i o n s b e f o r e t h e b i g b a n g ? O r is
that a legitimate inquiry i n t o an u n k n o w n b u t
DISCOURAGED,
knowable
aspect of the universe? A clear perspective o n the b o u n d a r i e s o f t h e u n k n o w a b l e m a k e s it less l i k e l y t h a t p e o p l e w i l l
b e c o m e discouraged, thereby permitting the k i n g d o m
of
t h e selfish g e n e a n d t h e h u m a n g e n o m e p r o j e c t . " O u r W o r l d "
the u n k n o w a b l e illegitimately to a n n e x parts of the k i n g -
concludes the b o o k with expositions on the nature of c o n -
d o m of the knowable
sciousness, m a t h e m a t i c s , a n d M a d d o x ' s t h o u g h t s o n
unknown.
B a r r o w e l o q u e n t l y m a k e s t h e case for s u c h a p e r s p e c t i v e :
avoid-
ing future calamities. In each c h a p t e r M a d d o x surveys w h a t
is k n o w n a n d c o n c l u d e s w i t h q u e s t i o n s a b o u t t h e u n k n o w n .
It w o u l d b e easy t o w r i t e s u c h a scientific success s t o r y . B u t w e
h a v e a n o t h e r tale t o tell: o n e t h a t tells n o t o f t h e k n o w n b u t o f
t h e u n k n o w n : o f t h i n g s i m p o s s i b l e ; o f limits a n d b a r r i e r s w h i c h
c a n n o t b e crossed. P e r h a p s this s o u n d s a little p e r v e r s e . S u r e l y
t h e r e is little e n o u g h t o say a b o u t t h e u n k n o w n w i t h o u t d r a g g i n g in t h e u n k n o w a b l e ? B u t t h e i m p o s s i b l e is a p o w e r f u l a n d
p e r s i s t e n t n o t i o n . . . [ t h o u g h ] its p o s i t i v e r o l e has e s c a p e d t h e
critics' a t t e n t i o n . O u r goal is t o u n c o v e r s o m e o f t h e limits o f s c i e n c e : to see h o w o u r m i n d s ' a w a r e n e s s o f t h e i m p o s s i b l e gives a
n e w p e r s p e c t i v e o n reality.
M a d d o x energizes that simple outline b y regaling the reader w i t h stories. F o r e x a m p l e , t o m a k e t h e p o i n t that a d a p tation t h r o u g h natural selection can lead to extinction, h e
relates t h e tale o f t h e d o d o , a l a r g e , flightless b i r d that, a l o n g
w i t h t w o r e l a t e d s p e c i e s , o n c e m a d e its h o m e o n i s l a n d s m
t h e I n d i a n O c e a n . T h e a n c e s t o r s o f t h e d o d o c o u l d fly: h o w
else c o u l d t h e b i r d s h a v e p o p u l a t e d t h r e e w i d e l y
spaced
islands? B u t n a t u r a l s e l e c t i o n later f a v o r e d i n d i v i d u a l s that
gave up
flying,
t h e r e b y saving h u g e m e t a b o l i c costs.
How
could natural selection have anticipated that a time w o u l d
F o r all t h o s e r e a s o n s , t h e p r o m i s e o f g a i n i n g r e a l i n s i g h t
b y s t u d y i n g t h e m a p o f w h a t is k n o w n ,
unknown
and
u n k n o w a b l e in s c i e n c e has g i v e n rise t o a g r o u n d s w e l l
of
interest. R e c e n t w o r k s h o p s h a v e b r o u g h t t o g e t h e r leading
c o m e w h e n flight w o u l d o n c e a g a i n b e c o m e a n a d v a n t a g e —
s u c h as w h e n , i n t h e s i x t e e n t h c e n t u r y , E u r o p e a n
sailors
b e g a n h u n t i n g t h e d o d o a n d its r e l a t i v e s f o r f o o d ?
F o r M a d d o x , e x p a n d i n g t h e s c o p e of scientific
edge
p u t e r scientists a n d m a t h e m a t i c i a n s w h o h a v e a s t r o n g i n t e r -
u n k n o w n i n t o t h e k n o w n offers t h e b e s t m e a n s p e o p l e h a v e
est i n i d e n t i f y i n g t h e u n k n o w a b l e i n t h e i r o w n f i e l d s . I n
for c o p i n g w i t h t h e u n c e r t a i n t i e s o f t h e f u t u r e — a n d
t h e p a s t f o u r y e a r s , f o u r w o r k s h o p s h a v e b e e n h e l d at t h e
p r e p a r i n g t o a v o i d its m o s t p o t e n t t h r e a t s . F o r t h o s e o f u s
Santa Fe Institute m N e w M e x i c o , s p o n s o r e d b y the Alfred
already c o n c e r n e d about nuclear proliferation, the destruc-
P. Sloan Foundation,
tion of biodiversity, overpopulation,
a n d a fifth w o r k s h o p w a s h e l d
Abisko, the Arctic research station ol the R o y a l
at
Swedish
A c a d e m y of Sciences. Barrow, w h o s e b o o k explores
the
varieties o f t h e u n k n o w a b l e , w a s a p a r t i c i p a n t at o n e o f t h e
is a n i m m e n s e l y
practical matter.
knowl-
physicists, e c o n o m i s t s , c o g n i t i v e scientists, biologists, c o m -
Converting
the possibility
global climate c h a n g e will trigger a n e w ice age, or
arrival o f tropical diseases in t e m p e r a t e climates,
gives us s o m e m o r e t h i n g s t o w o r r y
the
for
that
the
Maddox
about.
S a n t a F e I n s t i t u t e w o r k s h o p s a n d at A b i s k o . M a d d o x ' s b o o k
O n e calamity c o u l d transpire m the relatively near future,
is b e s t r e a d as a w a l l c h a r t o f t h e k n o w n a n d t h e u n k n o w n .
w i t h i n a c e n t u r y or t w o . T h e w e s t e r n Antarctic ice shelf
m a y b e u n s t a b l e a n d c o u l d t h e r e f o r e b e c o m e d e t a c h e d as
S EDITOR
I N C H I E F O H 'HI I E P R E S T I G I O U S
tific j o u r n a l Nature
SCIEN-
for n e a r l y t w e n t y - t h r e e y e a r s ,
M a d d o x is e m i n e n t l y q u a l i f i e d f o r t h e m i l l e n n i al task h e h a s s e t h i m s e l f . H e b e g i n s b y i m a g i n i n g h o w h i s
b o o k might have been written a century ago. T h e
hypo-
thetical
asked:
nineteenth-century
Maddox
might
have
a result of global w a r m i n g . M a d d o x writes:
T h i s mass o f ice, r e a c h i n g for several h u n d r e d s o f k i l o m e t e r s o u t
t o sea f r o m t h e solid c o n t i n e n t , . . . s u p p o r t s h u n d r e d s o f c u b i c
k i l o m e t e r s o f ice. If this b l o c k o f ice s h o u l d b e c o m e d e t a c h e d
f r o m A n t a r c t i c a , p e r h a p s b e c a u s e m e l t i n g ice l u b r i c a t e s its j u n c t i o n w i t h t h e r o c k b e n e a t h , t h e w h o l e mass c o u l d b e c o m e t h e
January/Vvbruary
1999
.
THE
SCIENCES
41
w o r l d ' s largest i c e b e r g . In t h a t case, t h e r e w o u l d b e an i n c r e a s e
o f s o m e five m e t e r s (sixteen feet) in global sea level. T h e m a p o f
the world w o u l d change quickly.
T u r i n g p r o v e d t h a t , for an a r b i t r a r y string o f p r o g r a m m i n g
c o d e , it is i m p o s s i b l e t o tell i n a d v a n c e w h e t h e r t h e p r o c e d u r e s it s p e c i f i e s o n t h e m a c h i n e w i l l h a l t o r n o t . I n p a r t i c -
A s e c o n d p o t e n t i a l h a z a r d c o u l d lie h u n d r e d s o f g e n e r a -
ular, o n e c o u l d n e v e r k n o w in a d v a n c e w h e t h e r a n y m e c h a n -
tions in the future: the h u m a n g e n o m e could b e c o m e unsta-
i c a l p r o c e d u r e f o r l i s t i n g all t r u e m a t h e m a t i c a l
theorems
b l e . T h a t is, e n o u g h d e f e c t s c o u l d a c c u m u l a t e m t h e g e n o m e
w o u l d ever c o m e to an end. In T u r i n g ' s hands,
Hilbert's
that further survival of the species w o u l d be threatened. As
d e c i s i o n p r o b l e m l e a d s at o n c e t o a n u n d e c i d a b l e q u e s t i o n .
M a d d o x and m a n y others infer that since c o m p u t e r s c a n -
M a d d o x p o i n t s o u t , t h e r e is a m p l e p r e c e d e n t t o r t h e p r o c e s s ,
recorded a m o n g the remains of extinct plants and animals.
n o t d e c i d e w h e t h e r a m a t h e m a t i c a l s t a t e m e n t is t r u e
U n l i k e the Antarctic catastrophe, h o w e v e r , genetic insta-
false, s u c h q u e s t i o n s c a n n o t b e r e l e g a t e d t o m a c h i n e s . B u t
b i l i t y , it s e e m s t o m e , d o e s n o t p o s e a n y
t h a t i n f e r e n c e is n o t v a l i d . O n e m i g h t as w e l l s a y t h a t s u c h
insurmountable
or
p r o b l e m s . T h e h u m a n species has e n o u g h t i m e to deal w i t h
questions should not be relegated to h u m a n
mathemati-
g e n e t i c p r o b l e m s as t h e y c o m e a l o n g .
c i a n s e i t h e r . T h e y a r e e q u a l l y i n c a p a b l e of
determining
M a d d o x lets r e a d e r s h a v e his v i e w s o n c e r t a i n issues in
w h e t h e r a r b i t r a r y s t a t e m e n t s a b o u t , say, a r i t h m e t i c , a r e t r u e
n o u n c e r t a i n t e r m s . O n a n u m b e r of t h o s e p o i n t s , h o w e v -
o r false. T h e q u e s t i o n is n o t a m a t t e r o f t h e s u p e r i o r i t y o f
er, I find h i m
h u m a n or machine.
u n c o n v i n c i n g . A b o u t t h e fears
regarding
recent developments m genetics, b o t h within the research
c o m m u n i t y a n d o u t s i d e it, h e w r i t e s d i s m i s s i v e l y :
IF THE A N T A R C T I C
Reflection
o n how^ G o d e l ' s a n d T u r i n g ' s
impossibility
results m i g h t b e relevant to h u m a n creativity has led t o a
ice shelf breaks away, major cities will be
M e r e l y t h e availability o f p r e n a t a l d i a g n o s i s suggests t o s o m e t h a t
c o m m u n i t i e s in w h i c h t h e t e c h n i q u e s are a l r e a d y available are
a b o u t to practice eugenics of the kind lgnorantly advocated in
G e r m a n y in t h e 1 9 3 0 s . O t h e r s fear t h e i m m i n e n c e o f t h e " d e s i g n er b a b y " — - a n i n d i v i d u a l e q u i p p e d w i t h s u p e r i o r faculties, b o t h
i n t e l l e c t u a l a n d physical. B o t h fears are g r o u n d l e s s . T h e r e are
i m p o r t a n t ethical q u e s t i o n s t o ask, b u t t h e s e are n o t t h e m .
inundated.
host of fascinating questions n o t addressed by
Maddox.
M a n y m a t h e m a t i c i a n s , for i n s t a n c e , b e l i e v e t h e y c r e a t e b y
a p r o c e s s o f i n t u i t i o n — b u t j u s t w h a t is i n t u i t i o n ? D o e s it
enable mathematicians to spot truths that c a n n o t be reached
t h r o u g h dogged, axiomatic argument? C a n o n e create a
mathematical
model of intuition,
o n e that captures
the
essence of the h u m a n m i n d ? A n d h o w d o such questions
P e r h a p s M a d d o x h a s g o o d r e a s o n s for t h i n k i n g t h o s e fears
tie i n t o t h e q u e s t for a scientific e x p l o r a t i o n o f c o n s c i o u s -
to be groundless. T h e reader, t h o u g h , w o u l d have
been
ness? T h e v e r y fact t h a t a n analysis o f t h e i m p o s s i b l e p r o -
dismissing
f o u n d l y e n r i c h e s s u c h q u e s t i o n s is c o m p e l l i n g e v i d e n c e t h a t
b e t t e r s e r v e d if h e h a d p r o v i d e d t h e m . M e r e l y
science has m u c h to gain f r o m this line o f t h o u g h t .
t h e fears w i l l n o t m a k e t h e m g o a w a y .
A
SECOND
MATTER
ON
WHICH
I
FIND
^"T^
M A D -
HAT
I
d o x u n c o n v i n c i n g is h i s a c c o u n t o f G o d e l ' s
t h e o r e m a n d o t h e r , related results, a n d their
_JL
POTENTIAL
FOR
TE1E E N R I C H M E N T
OF
s c i e n c e is w h a t B a r r o w h a s s e t o u t t o e x p l o r e
i n Impossibility.
A professor o f a s t r o n o m y at
i m p l i c a t i o n s for w h a t c o m p u t e r s a n d m a t h e m a t i c i a n s can a n d
t h e U n i v e r s i t y of Sussex in E n g l a n d , B a r r o w writes w i t h
c a n n o t d o . O n e o f t h e s t r o n g e s t impossibility results w a s
g r e a t s c h o l a r s h i p , e l e g a n c e a n d w i t . H e classifies t h e l i m i t s
p r o v e d j u s t f i v e y e a r s after G o d e l ' s , b y t h e E n g l i s h p o l y m a t h
to
Alan M . T u r i n g . T u r i n g devised a pencil-and-paper s c h e m e
imposed by the nature of the brain, some by
that i n c o r p o r a t e d j u s t a f e w s i m p l e rules. B u t w i t h his s c h e m e ,
s o m e are t e c h n o l o g i c a l , a n d s o m e arise f r o m G o d e l ' s t h e -
h e asserted, o n e c o u l d carry o u t any m e c h a n i c a l p r o c e d u r e ,
orem and from computational
o r a l g o r i t h m , o f t h e k i n d e n v i s i o n e d b y H i l b e r t for g r i n d i n g o u t all p o s s i b l e t r u e m a t h e m a t i c a l t h e o r e m s . T h e s c h e m e
h a s b e c o m e k n o w n as a T u r i n g m a c h i n e , a n d it is a b s t r a c t ly e q u i v a l e n t t o a n y d i g i t a l c o m p u t e r , n o m a t t e r h o w p o w e r f u l , i n t h e s e n s e t h a t it c a n p e r f o r m a n y c o m p u t a t i o n t h e
latter m a c h i n e can b e p r o g r a m m e d t o accomplish.
Some procedures on a Turing machine—or,
ly, o n t h e l a t e s t - m o d e l P C — l e a d
equivalent-
quickly through
a few
m a c h i n e cycles a n d t h e n halt. " F i n d t h e s q u a r e r o o t o f 3 (to
science
according
to
their
source:
some
limits
are
cosmology,
theory.
C o n c e r n i n g the brain, he writes:
O u r m i n d s w e r e n o t d e s i g n e d w i t h s c i e n c e in m i n d , n o r d i d e v o l u t i o n p r i m a r i l y fit t h e m for t h a t p u r p o s e . . . . O n t h e face of it,
t h e r e is n o r e a s o n w h y w e s h o u l d possess t h e c o n c e p t u a l ability
t o m a k e sense o f t h e w a y t h e U n i v e r s e w o r k s . It w o u l d r e q u i r e
a c o i n c i d e n c e o f c o s m i c p r o p o r t i o n s if t h e U n i v e r s e w e r e c o m p l i c a t e d e n o u g h t o g i v e n s e t o life, y e t s i m p l e e n o u g h for o n e
species t o u n d e r s t a n d its d e e p e s t s t r u c t u r e after j u s t a f e w h u n d r e d years o f s e r i o u s scientific i n v e s t i g a t i o n .
f i v e d e c i m a l p l a c e s ) a n d p r i n t t h e a n s w e r , " is a g o o d e x a m -
It c o u l d b e , f o r i n s t a n c e , t h a t w e h u m a n s b e l i e v e O c c a m ' s
ple. O t h e r p r o c e d u r e s , s o m e w r i t t e n in just a f e w characters
razor, w h i c h holds that t h e best theories are t h e simplest,
o f p r o g r a m m i n g c o d e , will n e v e r halt: t h e standard j o c u l a r
because those are the only theories o u r p u n y brains can
example conies from the instructions on a bottle of sham-
c o m p r e h e n d . Y e t I t h i n k it p o s s i b l e t h a t o n e o f h u m a n k i n d ' s
p o o : "Lather, rinse, repeat." B u t s o m e c o m p u t e r p r o g r a m s
greatest i n v e n t i o n s , t h e c o m p u t e r , c o u l d b e a t o o l for t r a n -
fall i n t o n e i t h e r c a t e g o r y . T h e y d o n o t s e e m t o h a l t , b u t t h e r e
s c e n d i n g t h e p u n y - b r a i n l i m i t a t i o n . N e e d l e s s t o say, s u c h
is n o c l e a r r e a s o n t o t h i n k t h e y w i l l g o o n f o r e v e r ,
a v i e w is h i g h l y s p e c u l a t i v e .
42
THE
SCIENCES
• January/February
19 99
either.
e n c e . A s I p o i n t e d o u t e a r l i e r , G o d e l s h o w e d t h a t if a f o r -
W h a t of B a r r o w ' s c o s m o l o i n c a l a n d t e c h n o l o g i c a l l i n i its? T h e y a r e r e l a t i v e l y p r a c t i c a l c o n s t r a i n t s , t h e k i n d s
of
m a l s y s t e m is as r i c h as a r i t h m e t i c , it m u s t b e i n c o m p l e t e .
limits that are i m p o s e d by the i m m e n s e universe in w h i c h
B u t , as B a r r o w n o t e s , s i m p l y r e m o v i n g t h e m u l t i p l i c a t i o n
w e h a p p e n to live, a n d b y t h e trifling r e s o u r c e s w e are e v e r
o p e r a t i o n f r o m a r i t h m e t i c g i v e s a s m a l l e r s y s t e m k n o w n as
great
Presburger arithmetic. C o d e l s h o w e d that Presburger arith-
questions about the beginning, the end and the structure
m e t i c is c o m p l e t e . It is p o s s i b l e , B a r r o w o b s e r v e s , t h a t t h e
likely to c o m m a n d . In B a r r o w ' s v i e w m a n y o f t h e
o f t h e u n i v e r s e a r e u n a n s w e r a b l e in t h a t p r a c t i c a l
" i n n e r l o g i c o f t h e p h y s i c a l u n i v e r s e " is r o o t e d m a s i m p l e r
sense.
B e c a u s e t h e s p e e d o f l i g h t is f i n i t e , t o r i n s t a n c e , t h e u n i -
l o g i c t h a n full a r i t h m e t i c . I n t h a t c a s e , G o d e l i a n
v e r s e is p a r t i t i o n e d i n t o r e a l m s t h a t a r e o u t o f c a u s a l c o n -
pleteness m i g h t n o t apply.
incom-
t a c t w i t h o n e a n o t h e r . F u r t h e r m o r e , if t h e t h e o r y o f t h e
T h e s a m e c o n c l u s i o n h o l d s for t h e c o n j e c t u r e s a n d t h e -
i n f l a t i o n a r y u n i v e r s e is v a l i d , i n f o r m a t i o n a b o u t t h e s t r u c -
ories o f c o m p u t a t i o n a l c o m p l e x i t y , a n d their p o t e n t i a l for
ture o f t h e visible u n i v e r s e before
defining the limits o f science. A l t h o u g h I greatly
inflation
took
place
Rene Magritte,
has b e e n
wiped
clean—forever
beyond
scientific
Clairvoyance,
admire
1936
ken.
B a r r o w ' s b o o k — a n d h e r e , I m u s t i m m e d i a t e l y confess that
T e c h n o l o g y w i l l also d e t e r m i n e t h e limits t o w h a t w e c a n
I a g r e e d t o w r i t e a n e n d o r s e m e n t for t h e b o o k after r e a d -
e v e n t u a l l y d i s c o v e r . B a r r o w p o i n t s o u t t h a t scientists
m a k e precise predictions a b o u t the universe u n d e r
can
condi-
tions that w e c a n n o t even r e m o t e l y a p p r o a c h via direct e x p e r i m e n t . W h a t r e a l l y t a k e s p l a c e , f o r i n s t a n c e , w h e n m a t t e r is
h e a t e d t o t e m p e r a t u r e s m o r e t h a n fifteen o r d e r s o f m a g n i t u d e greater t h a n any a c h i e v e d o n earth? B u t such limits are
fluid.
N e w t e c h n o l o g y is u n p r e d i c t a b l e , a n d t h e r e is a l w a y s
i n g it m m a n u s c r i p t f o r m — I a m n o t e n t i r e l y h a p p y
his discussion o f c o m p u t a t i o n a l
T
HE
with
complexity.
COMPUTATIONAL
COMPLEXITY
OF
l e m is a m e a s u r e o f t h e m i n i m u m
A
PROB-
amount
of
c o m p u t a t i o n a l r e s o u r c e s n e e d e d t o s o l v e it. E v e n
i f a m a t h e m a t i c a l p r o b l e m is d e c i d a b l e , it m i g h t b e
intractable.
Barrow
describes
the
com-
t h e p o s s i b i l i t y t h a t it w i l l e n a b l e s o m e e a r l i e r l i m i t t o b e s i d e -
putationally
traveling
s t e p p e d (in t h i s c a s e , t h r o u g h a s t r o n o m i c a l o b s e r v a t i o n s ) .
salesman p r o b l e m (TSP) to illustrate t h e idea. S u p p o s e a sales-
B a r r o w offers a n e x t e n d e d a n d s u b t l e d i s c u s s i o n o f t h e
m a n m u s t m a k e calls i n s o m e s e t o f c i t i e s . H o w d o e s h e p l a n
possible i m p l i c a t i o n s o f G o d e l ' s t h e o r e m for t h e limits t o sci-
h i s i t i n e r a r y s o t h a t h e visits e a c h o f t h e m j u s t o n c e , m i n i -
January/February
[999
• THE
SCIENCES
43
m i z e s t h e total d i s t a n c e h e travels, a n d r e t u r n s at t h e e n d o f
a n s w e r to t h e p r o t e m - f o l d i n g p r o b l e m has b e e n called t h e
h i s j o u r n e y t o t h e c i t y w h e r e h e s t a r t e d ? It t u r n s o u t t h a t as
H o l y Grail of m o l e c u l a r biology.
r
t h e n u m b e r o f c i t i e s o n t h e list g r o w s , t h e p r o b l e m b e c o m e s
i n c r e a s i n g l y difficult;
the n u m b e r
grows
with
exponentially
of route
combinations
In nature the folding takes place quickly-—in a b o u t a
s e c o n d . Y e t t h e k n o w n m o d e l s o f t h e p r o c e s s a r e so c o m plex that they c a n n o t be s i m u -
the
n u m b e r o f cities in t h e salesman's
lated even o n the most
tour. N o solution m o r e
ful
efficient
than simply e n u m e r a t i n g
possible
combinations
r o u t e s is k n o w n
power-
Aviezri
S.
the
F r a e n k e l , a m a t h e m a t i c i a n at t h e
travel
W e i z m a n n Institute of Science in
all
of
supercomputer.
(though no
R e h o v o t , Israel, has s h o w n
one
one formulation
has p r o v e d s u c h a solution does n o t
of the
that
problem
exist). H e n c e , t h o u g h t h e c o m p u -
is j u s t as c o m p l e x as T S P .
tational
protein folding "should" b e c o m e
complexity
unknown,
of
TSP
it is t h o u g h t t o
is
Hence
e x p o n e n t i a l l y h a r d e r as t h e l e n g t h
grow
exponentially with the n u m b e r of
of
c i t i e s . T S P is s a i d t o b e
grows. N a t u r e should not take a
computa-
the
chain
second;
tionally intractable.
of
it s h o u l d
amino
take
acids
perhaps
millions of years.
C o m p u t e r scientists h a v e m a d e
How
a remarkable discovery about TSP.
can such dissonances
be
prob-
e x p l a i n e d ? T h e possibilities recall
lems, s o m e of t h e m of great p r a c -
the relation b e t w e e n Godel's t h e -
t i c a l i m p o r t a n c e , all p o s s e s s e s s e n -
orem
tially
example, there m a y be other m a t h -
It t u r n s o u t t h a t n u m e r o u s
the
same
computational
and
scientific
limits.
For
a r e all
ematical models of protein folding
t r a c t a b l e o r all i n t r a c t a b l e , a n d t h e
t h a t fit t h e o b s e r v e d b e h a v i o r , y e t
c o m m o n b e l i e f a m o n g e x p e r t s is
are computationally tractable.
c o m p l e x i t y as T S P . T h e y
the
t h a t t h e y a r e all i n t r a c t a b l e . B a r -
that
TSP
is
Or
in-
t r a c t a b l e m a y t u r n o u t t o b e false.
r o w w r i t e s t h a t i n p r a c t i c e it is
Richard
extremely hard to p r o v e that any
g i v e n p r o b l e m is i n t r a c t a b l e .
conjecture
E. Prince, V e n u s a n d M a r s ,
1998
P e r h a p s e v o l u t i o n has
selected
for p r o t e i n s t h a t fold easily. A l t e r -
"At
p r e s e n t , " h e r e m a r k s , to s u b s t a n t i a t e his p o i n t , " t h e r e are
natively, nature m i g h t b e u n d e r t a k i n g a massive n u m b e r of
n o m o r e than a b o u t a thousand p r o b l e m s " in the same c o m -
parallel c o m p u t a t i o n s , t h o u g h t h e m e c h a n i s m s o n e
p l e x i t y class as T S P . I n m y v i e w , B a r r o w h a s i t b a c k w a r d .
p r o p o s e for s u c h activity r e m a i n h i g h l y s p e c u l a t i v e .
I f i n d it s u r p r i s i n g t h a t s o m a n y p r o b l e m s h a v e b e e n p r o v e n
t o h a v e t h e s a m e c o m p u t a t i o n a l c o m p l e x i t y as T S P .
might
W h a t e m e r g e s f r o m s u c h a n a n a l y s i s is a p o s s i b l e e x p e r i m e n t : H o w d o e s t h e t i m e that n a t u r e takes to fold a p r o -
Another shortcoming of Barrow's account of computa-
tein d e p e n d o n t h e l e n g t h o f t h e s e q u e n c e o f a m m o acids?
t i o n a l c o m p l e x i t y is t h a t it is c o n f i n e d t o c o m b i n a t o r i a l p r o b -
T h e t i m e d e p e n d e n c e is n o t e x p o n e n t i a l , b u t is it s u p e r -
l e m s , w h i c h i n v o l v e discrete variables (the n u m b e r o f cities
l i n e a r , s u b l i n e a r o r e v e n c o n s t a n t — t h a t is, i n d e p e n d e n t o t
in t h e salesman's tour) rather t h a n c o n t i n u o u s variables. As
t h e l e n g t h o f t h e a m i n o acid strand? E x p e r i m e n t a l
I n o t e d , t h e i r c o m p l e x i t y is u n k n o w n ; o n e m u s t s e t t l e i n s t e a d
surements might help to construct a predictive theory.
f o r s t a t e m e n t s s u c h as " p r o b l e m a is e q u i v a l e n t t o p r o b l e m
/ ? / ' o r " i t is c o n j e c t u r e d t h a t p r o b l e m x is m o r e
complex
than p r o b l e m y." M a t h e m a t i c a l m o d e l s in the sciences, h o w ever, often i n v o l v e c o n t i n u o u s functions o f real variables.
s
O REFLECT JUST A M O M E N T ABOUT THE NEW
IC. It s h o w s t h a t t h e r e a r e d i s s o n a n c e s
mea-
LOG-
between
complexity theory, supercomputer simulation and
Differential equations involving distance a n d t i m e p r o v i d e a
n a t u r e . T h o s e d i s s o n a n c e s l e d t o a p r o p o s a l for a n e x p e r i -
typical
m e n t t h a t c o u l d h e l p c o n s t r a i n t h e s e a r c h for
example.
The
computational
complexity
of
the
improved
n u m e r i c a l s o l u t i o n o f s u c h p r o b l e m s is s t u d i e d i n t h e f i e l d
m o d e l s : i n s h o r t , t o n o r m a l s c i e n c e . P e r h a p s t h a t is a s m a l l
of information-based
success, o r j u s t t h e p o t e n t i a l for a small success. B u t I t h i n k
complexity,
a n d it is o f t e n
known.
T h e r e is n o n e e d t o s e t t l e f o r c o n j e c t u r e s .
W
it b e g i n s t o m a k e t h e c a s e t h a t , f r o m t i m e t o t i m e , s c i e n tists c a n s t e p b a c k f r o m t h e i r h a n d - t o - h a n d s t r u g g l e s w i t h -
HAT,
THEN,
tween
the
IS
THE
conjectures
and
in t h e i r disciplines a n d profitably e x p l o r e a b i g g e r p i c t u r e .
of
T h e parsing of the u n k n o w n , the intractable, the u n k n o w -
and
able, so a p p e a l i n g t o t h e p o p u l a r i m a g i n a t i o n , a n d so e n t e r -
bio-
t a i n i n g l y a n d p e r s u a s i v e l y c a r r i e d o u t in t h e s e t w o b o o k s ,
results
computational complexity theory,
science? T h e protein-folding p r o b l e m
BE-
CONNECTION
of molecular
chemistry provides an enlightening example. T h e p r o b l e m
is a w o r t h y m i s s i o n t o r s c i e n c e itself.
•
JOSEPH
Armstrong
is e a s y t o s t a t e : G i v e n a l i n e a r s e q u e n c e o f a m i n o a c i d s , i n t o
what three-dimensional
configuration
will the
sequence
F.
TRAUB
fold? T h e i m p l i c a t i o n s o f t h e a n s w e r for b i o l o g y are m a n -
computer
ifold, b e c a u s e in p r o t e i n c h e m i s t r y t h e f u n c t i o n o f a m o l -
Press has just
e c u l e is a l m o s t e n t i r e l y d e t e r m i n e d b y its s h a p e . T h u s t h e
INFORMATION.
44
THE
SCIFNCFS
• January
/ February
1999
is the Edwin
science at Columbia
published
Howard
University.
Cambridge
his most recent booh,
professor
of
University
COMPLEXITY AND