Varieties of limits to scientific knowle

ESSAYS

&

COM M

Varieties of Limits to Scientific
Knowledge
An exploration into how limits can advance our

S

understanding

tudying the limits o f k n o w l e d g e is an effective way to i n c r e a s e k n o w l e d g e . C h a r t ing t h e b o u n d a r y b e t w e e n w h a t we do a n d do n o t k n o w c a n e n r i c h s c i e n c e . A

BY PIET HUT, DAVID RUELLE, AND
JOSEPH TRAUB

r e c e n t burst o f activity, c e n t e r e d a r o u n d t h e n o t i o n of limits to scientific knowl-

edge, h a s led to a series o f w o r k s h o p s , p a p e r s , a n d b o o k s [1-5]. In this paper, we take a
critical look at the n o t i o n of limits, a n d the role t h e y play in scientific r e s e a r c h .
At any given t i m e , our b o d y of scientific k n o w l e d g e h a s l i m i t a t i o n s that reflect t h e
t e m p o r a r y e n d p o i n t of investigations so far. In general, we say that scientific r e s e a r c h
is c o n d u c t e d at t h e "frontiers" o f w h a t is k n o w n . E x p a n d i n g t h e frontiers is i n d e e d an
apt m e t a p h o r for t h e growth of scientific knowledge. In c o n t r a s t to t h o s e e v e r - e x p a n d i n g o u t e r l i m i t s to s c i e n tific k n o w l e d g e , t e m p o r a r y
m a r k e r s for w o r k - i n - p r o -

Any area of knowledge is structured by an
intricate interplay of limits intrinsic to that area.
Limits have two opposite functions: setting apart
and joining. An apparently restrictive limit may
actually reveal itself as liberating. This article
discusses the role that limits play in the real
world, In our mathematical idealizations and in
the mappings between them. The article also
proposes a classification of limits and suggests
how and when limits to knowledge appear as
challenges that can advance knowledge.

gress, we m a y w o n d e r w h e t h e r t h e r e are not m o r e perm a n e n t frontiers or holes
that in p r i n c i p l e c a n n o t b e
filled in. In o t h e r words, we
are i n t e r e s t e d in t h e q u e s tion o f w h e t h e r t h e r e are int r i n s i c l i m i t s to s c i e n t i f i c
k n o w l e d g e . W h a t c a n we say
m o r e generally about such
intrinsic limits? C a n we c o n struct a framework within
w h i c h to discuss t h e s e types
of limits? T h e p r e s e n t p a p e r

p r e s e n t s an initial a t t e m p t to c o m e to grips with t h e s e q u e s t i o n s , a n d to p o i n t out s o m e
p o s s i b l e steps towards a c l a s s i f i c a t i o n o f limits.
S c i e n t i f i c progress was c o n s i d e r e d virtually u n l i m i t e d , e v e n u n s t o p p a b l e , a h u n dred y e a r s ago. S u c h a positive o u t l o o k h a s given way, in this century, to a m o r e s o b e r
reflection o n various factors that m a y limit further growth of scientific knowledge. E c o n o m i c c o n s i d e r a t i o n s as well as c o n c e r n for the e n v i r o n m e n t are two i m p o r t a n t factors, a n d w e m a y b e r e a c h i n g t e c h n o l o g i c a l limits as well. W h i l e it is n o t o r i o u s l y difficult to predict t e c h n o l o g i c a l b r e a k t h r o u g h s (scientists, by n a t u r e c o n s e r v a t i v e , do n o t
h a v e a g o o d track record in this r e s p e c t ) , there s e e m to b e s o m e f u n d a m e n t a l limitations that s t a n d in t h e way of further progress. O n e e x a m p l e is t h e finite s p e e d o f light,

© 1 9 9 8 J o h n Wiley & S o n s . Inc., Vol. 3, No. 6
CCC 1 0 7 6 - 2 7 8 7 / 9 8 / 0 6 0 3 3 - 0 6

Piet Hut is Professor

in the School
of
Natural Sciences at the Institute
for
Advanced
Study in Princeton.
While
his
main research
area is
theoretical
astrophysics,
he is frequently
invovled
in multidisciplinary
collaborations,
from geology, paleontology
and
cognitive
science to particle

physics
and
computer
science. He is
currently
involved
in a Tokyo-based
project
aimed at developing
a special
purpose
computer
for simulations
in stellar
dynamics,
with a speed of 1
Petaflops.
David Ruelle is a
mathematical
physicist

whose interests have
ranged
widely from quantum
field theory
and
statistical
mechanics
to chaos
theory
etc. He has worked in Belgium,
Switzerland, the United States, and is
currently
at the IHES in France. He is the
author
o /C h a n c e a n d C h a o s
(Princeton
University Press, 1991). Joseph
Traub is
the Edwin Howard Armstrong
Professor

of Computer
Science at
Columbia
University and External
Professor
at the
Santa Fe Institute.
He started
his
pioneering
research
on what is now
called information-based
complexity
in
1959. His current research
includes
mathematical
finance,
the

computational complexity
of economic
models,
and the implications
of
impossibility
theorems
from theoretical
computer
science and mathematics
for
science.
His latest book is C o m p l e x i t y a n d
I n f o r m a t i o n , Cambridge
University
Press,
1998.

C O M P L E X I T Y

33

an u p p e r b o u n d o n p o s s i b l e v e l o c i t i e s

w h i c h is likely to provide a n i n c r e a s i n g l y

I n s t e a d o f trying to a n s w e r this q u e s -

severe b a r r i e r for f u r t h e r d e v e l o p m e n t s

t i o n i m m e d i a t e l y , let us travel b a c k in

c o m e s as a surprise a n d strikes o n e as a

in c o m p u t e r b u i l d i n g , a n d t h u s in o u r

t i m e , by a c o u p l e t h o u s a n d y e a r s , to s e e

limitation.

ability to s i m u l a t e c o m p l e x s y s t e m s .

w h e t h e r a m o r e removed historical per-

H o w e v e r , if w e h a d first l e a r n e d to

s p e c t i v e m a y b e o f s o m e h e l p . Let us

think in t e r m s o f general relativity theory,

in o n e d i r e c t i o n c a n also b e s e e n as an

take, instead of Hilbert's

program,

a n d only afterward tried to u n d e r s t a n d

i n v i t a t i o n to e x p l o r e o t h e r d i r e c t i o n s .

Pythagoras' program of understanding

N e w t o n i a n g r a v i t y a n d m e c h a n i c s in

T h u s , q u a n t u m c o m p u t a t i o n [6] m a y by-

t h e w o r l d in t e r m s o f p r o p e r t i e s a n d in-

t e r m s o f differential g e o m e t r y , t h e n a t u -

pass t h e limits p o s e d by classical p h y s -

t e r r e l a t i o n s o f t h e n a t u r a l n u m b e r s (we

ral l a n g u a g e o f g e n e r a l relativity, a n alto-

ics. And totally different a v e n u e s s u c h as

refer to t h e usual tale; h i s t o r i c a l a c c u r a c y

g e t h e r different picture o f w h a t is l i m i t e d

DNA c o m p u t i n g [7] m a y also offer n e w

is difficult to a s s e s s a n d n o t r e l e v a n t for

would h a v e e m e r g e d . In this a p p r o a c h ,

possibilities. Historically, a t t e m p t s to

our example).

However, a b a r r i e r b l o c k i n g progress

N e w t o n i a n gravity is full o f limits, in the
program

f o r m o f s e e m i n g l y arbitrary a n d c o m p l i -

t h e n o - g o t h e o r e m s in physics [8], h a v e

s e e m e d to b e sailing a l o n g q u i t e nicely.

c a t e d r e s t r i c t i o n s , c o m p a r e d to t h e far

b l o c k e d p r o g r e s s in c e r t a i n a r e a s for a

He a n d his followers also built up an i m -

s i m p l e r way that general relativity c a n b e

c o n s i d e r a b l e t i m e by d i s c o u r a g i n g re-

pressive body of m a t h e m a t i c a l knowl-

f o r m u l a t e d in g e o m e t r i c t e r m s [12].

s e a r c h e r s to even e n t e r t h o s e areas, until

edge, a n d t h e y were able to explain quite

Similarly, q u a n t u m m e c h a n i c s s e e m s

s o m e o n e discovered a fatal flaw in s o m e

a lot in a p p l i e d areas s u c h as m u s i c . But

strikingly i n c o m p l e t e from the view

of t h e u n s t a t e d a s s u m p t i o n s .

t h e n t h e y discovered that t h e s q u a r e root

point of classical m e c h a n i c s . T h e uncer-

In general, a n y a t t e m p t to prove that

o f two "did n o t exist as a n u m b e r " (i.e., is

tainly r e l a t i o n , for e x a m p l e , p u t s u n e x -

s o m e t h i n g is i m p o s s i b l e is at t h e s a m e

n o t a rational n u m b e r ) , a d e e p l y s h o c k -

p e c t e d limits on what can b e m e a s u r e d

t i m e a n invitation to l o o k for l o o p h o l e s ,

ing result. W i t h s u c h a s i m p l e n u m b e r as

s i m u l t a n e o u s l y . B u t if w e t u r n t h e t a b l e s

h i d d e n a s s u m p t i o n s in s u c h a proof, in

t h e length o f the diagonal o f a square with

a n d start with a f o r m a l i s m t h a t is m o r e

order to find ways a r o u n d t h e difficulties.

sides of unit length being "off limits,"

n a t u r a l l y s u i t e d to a q u a n t u m m e c h a n i -

As we argue in m o r e detail below, the lim-

m a t h e m a t i c s itself s e e m e d to b e u n e x -

cal d e s c r i p t i o n , it is c l a s s i c a l m e c h a n i c s

its i n h e r e n t in a b o d y o f k n o w l e d g e are

p e c t e d l y limited.

t h a t is g l a r i n g l y " i n c o m p l e t e "

prove that s o m e t h i n g is i m p o s s i b l e , as in

For a while, Pythagoras'

i n t i m a t e l y i n t e r w o v e n with t h e i n t e r n a l
structure of that b o d y of k n o w l e d g e [9],
and a deeper understanding of limits
therefore leads to a d e e p e r u n d e r s t a n d ing of t h e m a t t e r u n d e r c o n s i d e r a t i o n .

f course, nowadays n o b o d y would

p r o c e s s e s s u c h as q u a n t u m m e c h a n i c a l

c o n s i d e r this result to b e in any way

i n t e r f e r e n c e [13].

O

a limit. Rather, it is part o f the struc-

ture of m a t h e m a t i c s that y 2 is irrational,

just as y - 1 is an i m a g i n a r y n u m b e r [11].

TRANSCENDING LIMITS
Examples From Physics

I

n a d d i t i o n to t h e fact t h a t s o m e limits
are t h u s s e e n to b e in t h e eye o f the b e holder, it is quite p o s s i b l e that s o m e

local limits m a y b e c i r c u m v e n t e d glo-

Let us t a k e a t y p i c a l e x a m p l e o f w h a t

bally. W o r m h o l e s in s p a c e - t i m e a r e a n

s p e c t a c u l a r f o r m o f intrinsic limit

strikes us as a severe limit in a physical

e x a m p l e , t h r o u g h w h i c h it m i g h t b e p o s -

to k n o w l e d g e was t h e s h o c k i n g in-

t h e o r y : t h e fact that t h e finite s p e e d o f

sible to c o m m u n i c a t e n e a r l y i n s t a n t a -

completeness theorem announced

light p o s e s an u p p e r limit o n t h e s p e e d

n e o u s l y over (seemingly) large d i s t a n c e s

Examples From Mathematics

A

and

t h e r e b y l i m i t e d , in t h a t it s i m p l y o m i t s

without violating t h e s p e e d limit i m p o s e d

by G o d e l in 1 9 3 1 : in a f o r m a l a x i o m a t i c

locally by s p e c i a l relativity [14].

d e s c r i p t i o n of a part of m a t h e m a t i c s (if
not so s i m p l e as to b e u n i n t e r e s t i n g , a n d
if free f r o m c o n t r a d i c t i o n s ) , t h e r e are
s t a t e m e n t s t h a t c a n n e i t h e r b e proved
n o r disproved [10]. T h i s a p p l i e s in particular to f o r m a l i z a t i o n s o f a r i t h m e t i c s .
Godel's d i s c o v e r y t h w a r t e d Hilbert's
p r o g r a m o f r e d u c i n g all o f m a t h e m a t i c s
to m e c h a n i c a l p r o c e d u r e s a n d s e e m e d
to p l a c e s e v e r e l i m i t s o n w h a t c a n b e

Wormholes in space-time are
an example, through which
it might be possible to
communicate nearly
instantaneously over
(seemingly) large distances
without violating the speed
limit imposed locally by
special relativity.

Boundaries in Mathematics Leading
to Bridges to New Physics
W h e n physical t h e o r i e s lead to s o l u t i o n s
with singularities (for i n s t a n c e , infinities),
this generally m e a n s that the theory
b r e a k s down a n d that n e w p h e n o m e n a
appear, requiring a n e w theory. T h u s t h e
p h e n o m e n o n of breaking

of waves

occurs

w h e n a s i m p l e t h e o r y of wave p r o p a g a -

k n o w n in m a t h e m a t i c s . However, n o w

tion leads to singular solutions.

t h a t we are g e t t i n g u s e d to i n c o m p l e t e n e s s results, it is n o t c l e a r w h e t h e r we
really s h o u l d i n t e r p r e t t h e m as an indi-

of any m a t e r i a l (as well as i n f o r m a t i o n a l )

MATHEMATICS, PHYSICS, AND REALITY

c a t i o n o f limits, as s o m e t h i n g t h a t walls

o b j e c t . Starting f r o m Newton's classical

A natural c l a s s i f i c a t i o n o f limits to s c i e n -

o f f a r e a s o f k n o w l e d g e to t h e h u m a n in-

m e c h a n i c s , in w h i c h v e l o c i t i e s o f a n y

tific k n o w l e d g e offers itself: a distinction

vestigator.

m a g n i t u d e are allowed, t h e e x i s t e n c e of

b e t w e e n limits that show u p in our m o d -

34

C O M P L E X I T Y

© 1 9 9 8 J o h n Wiley & S o n s , Inc.

els o f t h e world versus limits that apply
to t h e w o r l d as we k n o w it t h r o u g h o b servation and experimentation. Within
the physical s c i e n c e s , the m o d e l s are generally of a m a t h e m a t i c a l n a t u r e . In o t h e r

(Physical)
World

areas, s u c h as archaeology, c o h e r e n t exp l a n a t i o n s for o b s e r v a t i o n a l data, while

translation

(Mathematical)
Model

recipes

having their own logical structure, are
typically n o t c a s t in m a t h e m a t i c a l l a n -

The three realms of physics.

guage. A more careful consideration,
however, throws d o u b t on t h e validity o f
t h e m o d e l /w o r l d d i s t i n c t i o n . Let us look
in s o m e detail at t h e m o d e l i n g p r o c e s s

we c a n take t h e e x a m p l e of physics, for

that is at t h e heart of any form o f s c i e n -

T h i s w o u l d s u g g e s t a c l a s s i f i c a t i o n as

w h i c h t h e s e t h r e e areas are i n d i c a t e d in

tific activity.

given in Figure 2 b .

Figure 1.

S c i e n c e p r o p o s e s theoretical s c h e m e s

Faced with these alternatives, de-

W h a t m a k e s f u n d a m e n t a l progress in

p i c t e d in Figures 2 a a n d 2 b , n e i t h e r o f

to d e s c r i b e t h e "real world." T h e c o n n e c -

physics hard is o f t e n n o t so m u c h t h e in-

t h e m feels q u i t e right. But p e r h a p s t h e

tion with t h e real world involves a n o p -

h e r e n t difficulties that are e n c o u n t e r e d

real p r o b l e m lies e l s e w h e r e . M a y b e it is

erational description of how observed

within specific m a t h e m a t i c a l m o d e l s , but

n o t just t h e u n e a s y status of t h e transla-

facts fit into a theoretical s c h e m e . In o t h e r

rather t h e c h a l l e n g e o f finding t h e p r o p e r

tion recipes that resists classification.

words, t h e c h a l l e n g e is to build i n c r e a s -

c h o i c e of the pair {translation recipe,

W h e r e else c a n we look? T h e m a t h e m a t i -

ingly m o r e a c c u r a t e (usually m a t h e m a t i -

model}.

cal) m o d e l s o f t h e world, t o g e t h e r with

T h e q u e s t i o n of the status of the trans-

cal m o d e l s as s u c h s e e m to b e least p r o b l e m a t i c . But w h a t a b o u t t h e status o f t h e

r e c i p e s to c o n n e c t t h o s e m o d e l s with ex-

lation r e c i p e s is s o m e t h i n g that is rarely

p e r i m e n t s and o b s e r v a t i o n .

a d d r e s s e d specifically in a c a d e m i c sci-

O n e possibility, in a variation o n Kant,

e n c e c o u r s e s . T h e i r very e x i s t e n c e is of-

w o u l d b e to declare t h e physical world,

physical world?

u p p o s e , for i n s t a n c e , that we are in-

ten glossed over, a n d it is n o t even clear

as the realm of t h e "things in t h e m s e l v e s , "

t e r e s t e d in e l e c t r i c c i r c u i t s : T h e i r

w h e t h e r they are part of p h y s i c s of m a t h -

off-limits to direct scrutiny. T h e p h e n o m -

mathematical

S

fairly

e m a t i c s . If we start f r o m t h e side o f p h y s -

e n a we deal with are a c c e s s i b l e to us only

s i m p l e b u t h a s to b e s u p p l e m e n t e d by

theory

is

ics, a n d ask w h i c h part o f Figure 1 applies

t h r o u g h t h e particular ways we c h o o s e to

recipes for m e a s u r i n g voltages, resis-

to m a t h e m a t i c s a n d w h i c h part to p h y s -

investigate t h e m . So we m i g h t even argue

t a n c e s , c a p a c i t a n c e s , etc. in the lab. Simi-

ics, we m i g h t draw a p i c t u r e as given in

that we h a v e to shift t h e l e g i t i m a t e ter-

larly, l a n d surveying h a s a s i m p l e t h e o r y

Figure 2a.

(basically Euclidean geometry), which
h a s to b e s u p p l e m e n t e d by t h e u s e o f
g e o d e s y i n s t r u m e n t s . Incidentally, t h e s e
e x a m p l e s call for t h e r e m a r k t h a t o n e
n e e d s t h e t h e o r y o f electric circuits (Euc l i d e a n g e o m e t r y ) to u n d e r s t a n d h o w

rain o f physics towards a middle position,
as illustrated in Figure 3. T h i s m a y s e e m

I

n this figure, the natural world a r o u n d

a bit e x t r e m e , b u t as we will s e e below, it

us is a s s i g n e d to t h e d o m a i n o f p h y s -

c a n gives us an interesting h a n d l e on t h e

ics. Clearly, t r a n s l a t i o n r e c i p e s are n o t

status o f limits.

part of this natural world. Rather, they are

We are n o w in a p o s i t i o n to return to

d e s i g n e d by us in o r d e r to fit with o u r

t h e q u e s t i o n that led us to this classifica-

voltmeters, etc. (geodesy instruments)

m o d e l s . O n e m i g h t t h e r e f o r e argue that

f u n c t i o n . W h a t w e call r e c i p e s m a y t h u s

tion: h o w to distinguish b e t w e e n limits

they s h o u l d b e g r o u p e d u n d e r t h e g e n -

b e m o d i f i e d or r e f o r m u l a t e d by use of t h e

that s h o w up in our m o d e l s of t h e world

eral c a t e g o r y o f m a t h e m a t i c a l tools.

versus limits that apply to t h e world it-

associated theory. One can, however,
never c o m p l e t e l y r e m o v e the o p e r a t i o n a l
c o n n e c t i o n b e t w e e n physical world a n d
m a t h e m a t i c a l m o d e l . B e c a u s e this o p e r a t i o n a l c o n n e c t i o n , w h i c h we call " r e c i pes," h a s n o purely m a t h e m a t i c a l f o r m u lation, it t e n d s to b e a bit m e s s y a n d swept
u n d e r t h e rug in m a n y f o r m u l a t i o n s o f
scientific t h e o r i e s [15].

But as s o o n as we start f r o m t h e o t h e r

self. In o t h e r words, in c o n t r a s t to scruti-

side o f Figure 1, f r o m t h e side o f m a t h -

nizing m o d e l s a n d r e c i p e s , w h a t c a n we

e m a t i c s , it is c l e a r t h a t t h e m a t h e m a t i -

say a b o u t t h e e x i s t e n c e o f limits in t h e

cal m o d e l s a n d f r a m e w o r k s a r e c o m -

real world?

pletely s e l f - c o n t a i n e d a n d do n o t e v e n
leave r o o m for t h e e x i s t e n c e o f t r a n s l a t i o n r e c i p e s . For e x a m p l e , t h e n o t i o n s o f
ruler a n d c o m p a s s m a y not b e part o f t h e
fields b e i n g m e a s u r e d by a surveyor, b u t

Let us c o m e b a c k to limits a n d try to

at l e a s t t h e y a r e m a d e f r o m p h y s i c a l

a s c r i b e t h e m to t h r e e d i f f e r e n t t y p e s :

m a t e r i a l . T h e y are c e r t a i n l y n e v e r to b e

t h o s e in nature, t h o s e in t h e m o d e l s we

found within Euclidean geometry: they

u e s t i o n s a b o u t t h e real world do

Q

n o t c o m e e q u i p p e d with a m a t h -

e m a t i c a l m o d e l . E x a m p l e s of q u e s -

tions a b o u t the real world i n c l u d e :

•

Will t h e r e b e m a j o r g l o b a l c l i m a t e
c h a n g e s d u e to h u m a n activities?

h a v e built, and t h o s e t h a t are i n h e r e n t in

are n e i t h e r a x i o m s , n o r t h e o r e m s , n o r

•

Will t h e u n i v e r s e s t o p e x p a n d i n g ?

o u r t r a n s l a t i o n r e c i p e s . For d e f i n i t e n e s s ,

a n y o t h e r f o r m o f m a t h e m a t i c a l entity.

•

How did life o r i g i n a t e o n Earth?

© 1 9 9 8 J o h n Wiley & S o n s , Inc.

C O M P L E X I T Y

35

t i m e e v o l u t i o n o f a p h y s i c a l sys-

FIGURE 2A

t e m is a s s u m e d to h a v e a g o o d
mathematical model, but the ac-

MATHEMATICS

tual c a l c u l a t i o n o f t h e t i m e e v o l u t i o n is l i m i t e d by t h e e x p o n e n -

PHYSICS

tial growth o f e r r o r s ( i m p o r t a n t

translation

(Physical)
World

(Mathematical)
Model

recipes

s p e c i a l c a s e s are w e a t h e r p r e d i c tion and the a s t r o n o m y of the solar s y s t e m ) . M o r e g e n e r a l l y t h e
p r o b l e m o f solving with s u i t a b l e
accuracy the equations proposed
by physical t h e o r i e s m a y o f t e n b e
practically insuperable.

A view from physics.

In c o m p u t a t i o n a l c o m p l e x i t y
theory, o n e finds that t h e r u n n i n g
t i m e o f any a l g o r i t h m for solving
certain p r o b l e m s i n c r e a s e s u n a c c e p t a b l y fast with t h e length of the

FIGURE 2R

i n p u t ( e x p o n e n t i a l l y or w o r s e ) .
For discrete p r o b l e m s , the c o n j e c ture P ^NP

PHYSICS

suggests a class o f in-

t r a c t a b l e p r o b l e m s . On t h e o t h e r
hand, many continuous problems

MATHEMATICS

h a v e b e e n proven e x p o n e n t i a l l y

translation

(Physical)
World

(Mathematical)
Model

recipes

hard in t h e n u m b e r of v a r i a b l e s
[17]. (Note, however, that t h e c o n j e c t u r e and t h e o r e m s c o n c e r n the
w o r s t - c a s e setting. For

some

problems intractability can be
b r o k e n by s w i t c h i n g to a s t o c h a s -

A view from mathematics.

tic a s s u r a n c e . )

2.

Asking the Wrong Questions

T h e r e are c a s e s in w h i c h a q u e s t i o n o n e
mathematical

side o f physics are o n t o l o g i c a l q u e s t i o n s

m a y w a n t to ask c a n b e shown to have n o

m o d e l . To rigorously establish a scientific

a b o u t t h e real w o r l d [ 1 6 ] . B u t t h e real

answer in the f r a m e w o r k o f a given theory

limit w e s h o u l d s h o w t h a t every

world is t h e r e s o m e h o w a n d limits t h e

(wrong q u e s t i o n ) . In o t h e r words, t h e r e

physically a c c e p t a b l e pairs

{recipe,

are structural limitations to t h e q u e s t i o n s

of a scientific q u e s t i o n is u n d e c i d a b l e or

m o d e l } . U n a c c e p t a b l e pairs are falsified

o n e m a y ask. In q u a n t u m m e c h a n i c s , for

i n t r a c t a b l e [4].

by e x p e r i m e n t s .

i n s t a n c e , o n e should n o t ask to specify si-

A CLASSIFICATION OF LIMITS

t u m of a particle. T h e work of Godel and

W e g e t to c h o o s e t h e

math-

e m a t i c a l m o d e l that captures the e s s e n c e

T h i s m a y b e a p o s s i b l e a t t a c k in princ i p l e , b u t it is far f r o m e v i d e n t t h a t it
could actually b e carried out. (Note, however, that in e s t a b l i s h i n g t h e c o m p u t a tional complexity of a m a t h e m a t i c a l

multaneously the position and m o m e n -

I

f o n e starts listing limits to knowledge,

Turing h a s s h o w n that it is a b a d idea to

o n e s o o n finds that t h e i t e m s are het-

s e e k an a l g o r i t h m d e c i d i n g w h e t h e r an

e r o g e n e o u s but c a n b e fit in categories

arbitrary s t a t e m e n t is true or false.

m o d e l , we do p e r m i t all p o s s i b l e algo-

that we n o w discuss (we leave out psy-

r i t h m s to c o m p e t e . )

c h o l o g i c a l c o n s i d e r a t i o n s , s u c h as t h e

In scientific p r a c t i c e , all t h a t we e n -

q u e s t i o n of to w h a t e x t e n t h u m a n intel-

c o u n t e r are m e a s u r e m e n t s a n d i n t e r p r e -

ligence m a y b e a limiting factor). T h e cat-

tations, t o g e t h e r with a p p l i c a t i o n s that in

e g o r i e s o v e r l a p s o m e w h a t (i.e., s o m e

turn are i n s p e c t e d t h r o u g h f u r t h e r m e a -

i t e m s fit in several c a t e g o r i e s ) .

s u r e m e n t s . T h e r e f o r e , if w e critically ask

h e r e is a c u r i o u s relation b e t w e e n

T

c a t e g o r y 1 a n d 2: T h e fact that t h e
h a l t i n g p r o b l e m for a T u r i n g m a -

c h i n e is u n d e c i d a b l e c a n a l s o b e e x pressed by saying that t h e halting t i m e is
n o t a c o n s t r u c t i v e f u n c t i o n . In o t h e r

w h a t p h y s i c s d e a l s w i t h , w e a r e really

1. The Curse of the Exponential

w o r d s it grows v e r y very fast w i t h t h e

forced to t h e c o n c l u s i o n d e p i c t e d in Fig-

T h e r e are p r o b l e m s t h a t c a n b e solved

length of t h e input: faster t h a n e x p o n e n -

ure 3. D o e s this leave t h e real world out-

in p r i n c i p l e , b u t are in fact i n t o l e r a b l y

tial, e x p o n e n t i a l of e x p o n e n t i a l , etc.

side of physics? Actually w h a t is left out-

h a r d . C h a o s is a n e x a m p l e , w h e r e t h e

36

C O M P L E X I T Y

W h a t we n o w c o n s i d e r to b e w r o n g
© 1 9 9 8 J o h n Wiley & S o n s , Inc.

FIGURE 3

q u e s t i o n s were not always s u c h .
It was o n c e r e a s o n a b l e to ask h o w
\ 2 c o u l d b e w r i t t e n as a r a t i o n a l
f r a c t i o n p/q

PHYSICS

or how s o m e algo-

r i t h m c o u l d solve all m a t h e m a t i -

MATHEMATICS

cal p r o b l e m s (Hilbert's d r e a m ) .

(Physical)
World

T h e u n d e r s t a n d i n g of s t r u c t u r a l
limitations that m a k e such ques-

translation

t i o n s u n a n s w e r a b l e is s c i e n t i f i c

(Mathematical)
Model

recipes
•

progress.

3. Questions of Questionable
Status

From ontology to epistemology.

In physics o n e often does not have
a c o m p l e t e l y reliable theory, and
it is thus u n c l e a r w h e t h e r a n a t u ral q u e s t i o n that a p p e a r s hard to a n s w e r

In p r a c t i c e , however, we are very lim-

u n i v e r s e a n d its overall s t r u c t u r e , is an

c o r r e s p o n d s to a f u n d a m e n t a l limitation.

ited in u n d e r s t a n d i n g t h e h i g h e r levels in

e x a m p l e o f a discipline that deals with a

A s e e m i n g l y hard q u e s t i o n (such as "what

t e r m s o f t h e l o w e r levels. In a d d i t i o n ,

s a m p l e size o f o n e . C o m p a r i n g t h e o r y

is t h e n a t u r e o f p h l o g i s t o n ? " ) m a y turn

e m e r g e n t p r o p e r t i e s in general are fuzzy.

a n d o b s e r v a t i o n thus b e c o m e s m o r e dif-

out to h a v e q u e s t i o n a b l e status, a n d c a n

This in turn c r e a t e s difficulties in classi-

ficult, s i n c e a typical t h e o r y for t h e large-

e v e n t u a l l y e v a p o r a t e . To i n v e s t i g a t e

fication. An e x a m p l e of p r o b l e m a t i c as-

s c a l e s t r u c t u r e of t h e u n i v e r s e p r e d i c t s

w h e t h e r a limitation is f u n d a m e n t a l m a y

p e c t s o f this f u z z i n e s s c a n b e f o u n d in

n u m b e r s that have a statistical uncer-

theoretical

d i s c u s s i o n s of t h e a n t h r o p i c p r i n c i p l e ,

tainty a t t a c h e d to t h e m [19].

progress. Currently in this c a t e g o r y are

w h i c h t e n d to b e i n c o n c l u s i v e , s i m p l y

No m a t t e r h o w well we will ever b e

p r o b l e m s r e l a t e d to c o s m o l o g y

b e c a u s e w e k n o w very little a b o u t life

able to m e a s u r e t h e particular realization

thus lead

to i m p o r t a n t

and

f o r m s u n r e l a t e d to us.

q u a n t u m gravity.

Surprisingly m a n y s y s t e m s with large

A

nother example concerns the con-

we live in, this m a y not e n a b l e us to c h e c k
w h e t h e r our theory of universe f o r m a t i o n

n u m b e r s of degrees of f r e e d o m allow de-

is correct. Certainly any Popperian n o t i o n

Godel

s c r i p t i o n s in far s i m p l e r t e r m s , w i t h

of falsifiability d o e s not apply, as long as

proved (1) that a r i t h m e t i c c a n n o t

p r o p e r t i e s that are not at all obvious from

w e c a n n o t c r e a t e o t h e r u n i v e r s e s with

b e proved internally to b e c o n s i s t e n t a n d

the underlying s y s t e m . T h e s e e m e r g e n t

w h i c h to c h e c k o u r t h e o r i e s . But t h e n

(2) that it is e i t h e r i n c o n s i s t e n t or i n c o m -

p r o p e r t i e s p o i n t toward a higher level of

again, it is always risky to d e c l a r e s o m e -

sistency

of a r i t h m e t i c .

plete. Actually, arithmetic could be i n c o n -

o r g a n i z a t i o n , exhibiting less c o m p l e x i t y

thing to b e "off limits," and recently vari-

sistent, but a proof of c o n t r a d i c t i o n

than could b e a n t i c i p a t e d . (Coarse grain-

ous ideas h a v e b e e n d e v e l o p e d c o n c e r n -

would t h e n p r o b a b l y b e e x t r e m e l y long,

ing is o n e e x a m p l e of a simplifying p r o -

ing " m u l t i - v e r s e s " [20].

p e r h a p s so l o n g t h a t it c o u l d n o t b e

cess that c a n u n c o v e r e m e r g e n t p r o p e r -

i m p l e m e n t e d in o u r p h y s i c a l u n i v e r s e

ties.) But higher levels are not g u a r a n t e e d

7. Technological Limitations

118]. T h u s even a very f u n d a m e n t a l a n d

to have s i m p l e r b e h a v i o r (and not m u c h

T h e y are practically i m p o r t a n t , but o n e

n a t u r a l q u e s t i o n s u c h as t h e o n e c o n -

has b e e n f o u n d , for e x a m p l e , in fully de-

should b e wary o f treating as f u n d a m e n -

c e r n i n g the c o n s i s t e n c y of a r i t h m e t i c

veloped turbulence).

tal s o m e limits that c a n p e r h a p s b e over-

q u e s t i o n a b l e nature than o n e might have

5. Limited Access to Data

at t h e P l a n c k energy a p p e a r s very m u c h

liked.

In s o m e areas of s c i e n c e , t h e a b s e n c e of

out o f p r e s e n t t e c h n o l o g i c a l r e a c h , but

c o m e by n e w ideas. For i n s t a n c e , physics

a p p e a r s u p o n r e f l e c t i o n to b e o f m o r e

sufficient data leads to severe l i m i t a t i o n s

this d o e s not m e a n that we shall n e v e r

4. Emergent Properties

to knowledge. Historical l i m i t a t i o n s are

u n d e r s t a n d what is going on in this e n -

A n o t h e r class of very difficult q u e s t i o n s ,

in this c a t e g o r y [5]. We m a y b e able to in-

ergy region.

w h i c h s e e m to lead to f u n d a m e n t a l limi-

fer m a n y things a b o u t the origin of life,

t a t i o n s of knowledge, arise in t h e study

b u t it a p p e a r s that we shall never have a

of emergent properties. One can con-

detailed story of w h a t really h a p p e n e d .

struct h i e r a r c h i e s of s y s t e m s w h e r e e a c h

Similar s t a t e m e n t s m a y b e m a d e a b o u t

level c a n in p r i n c i p l e b e u n d e r s t o o d in

t h e origin o f stars, of m a n k i n d , of lan-

t e r m s of the level below, s u c h as:

guages, etc.

DISCUSSION
i m i t s c o m b i n e two o p p o s i t e f u n c -

L

tions: setting apart a n d joining. T h e y

partition the world (in fact, all that

a p p e a r s , in any form) into s e p a r a t e areas,

in intricate and overlapping h i e r a r c h i e s .
??? < q u a n t u m field t h e o r y < a t o m s
< m o l e c u l e s < life.

© 1 9 9 8 J o h n Wiley & S o n s , Inc.

6. Sample Size of One

But, at t h e s a m e t i m e , they structure t h e

Cosmology, t h e study of t h e origin o f t h e

interrelationships and c o m m u n i c a t i o n

C O M P L E X I T Y

c h a n n e l s b e t w e e n t h e p i e c e s into w h i c h

t i o n , for r e s e a r c h o n l i m i t s to s c i e n t i f i c

t h e y s e e m to h a v e c a r v e d up t h e world.

k n o w l e d g e . W e t h a n k t h e S a n t a Fe In-

in p a r t by t h e N a t i o n a l S c i e n c e F o u n -

Examples:

s t i t u t e , w h e r e this w o r k w a s s t a r t e d , for

dation.

•
•

a river or m o u n t a i n s e p a r a t e s a n d
c o n n e c t s two areas;

REFERENCES

cell walls in b i o l o g y allow t h e build-

1.

up o f c o m p l e x hierarchical structures;
•

\ 2 w a s first s e e n as n o t - a - n u m b e r ,

2.

l a t e r as a d i f f e r e n t t y p e o f n u m b e r ,
w h e r e rationals a n d irrationals were
j o i n e d as n u m b e r s b u t s e p a r a t e d by

3.

their (ir) rationality property;
•

t h e i r h o s p i t a l i t y . J . F. T. w a s s u p p o r t e d

the u n c e r t a i n t y principle: the very fact
that p o s i t i o n a n d m o m e n t u m , for ex-

4.

a m p l e , are n o t s i m u l t a n e o u s l y m e a surable shows the unified nature of
q u a n t u m reality.
5.
a c h limit at first p r e s e n t s itself as a

E

barrier, as s o m e t h i n g that sets apart.
F u r t h e r i n s p e c t i o n t h e n shows the

6.

c o n n e c t i n g , bridging a s p e c t of that limit.

T h u s , e a c h limits provides us with a chal-

7.

l e n g e : to look past its o b s t a c l e c h a r a c t e r
to discover its key role in disclosing n e w

8.

areas o f knowledge.
9.

ACKNOWLEDGMENTS
T h i s w o r k w a s s u p p o r t e d in p a r t by a

10.

J. L. Casti and J. F. Traub: On limits.
Report, Santa Fe Institute, Santa Fe, NM,
1994.
J. Casti and A. Karlqvist: Boundaries and
barriers. Addison-Wesley, Reading, MA:
1996.
J. B. Hartle, P. Hut and J. F. Traub:
Proceedings, workshop on fundamental
sources of unpredictability, Complexity
3(1), 1997.
J. F. Traub: What is scientifically knowable? Twenty-fifth anniversary symposium, School of Computer Science,
Carnegie-Meilon University. AddisonWesley, Reading, MA, 1991.
R.E. Gomory: The Known, the Unknown,
and the Unknowable. Scientific American. 272: pp. 120,1995.
D. P. DiVincenzo: Quantum Computation. Science. 270: pp. 255-261,1995.
L M. Adleman: Molecular Computation
of Solutions to Combinational Problems.
Science. 266: pp. 1021-1024,1994.
M. Kaku: Quantum field theory. Oxford
University Press, Oxford, 1993.
P. Hut: Boundaries and barriers. J. Casti
and A. Karlqvist (Eds.) Addison-Wesley,
Reading, MA, 1996.
K. Gbdel: Ober formal unentscheidbare

Satze der Principia Mathematica und
verwandterSysteme, 1. Monatshefte. Math.
Phys. 48: pp. 173-198,1931.
11. R. Rosen: Boundaries and barriers. J. Casti
and A. Karlqvist (Eds.) Addison-Wesley,
Reading, MA, 1996.
12. C. W. Misner, K. S. Thorne and J. A. Wheeler:
Gravitation. W. H. Freeman, San Francisco,
1973.
13. D. Finkelstein: Quantum relativity. Springer,
New York: 1996.
14. C. W. Misner, K. S. Thorne and J. A. Wheeler:
Gravitation. W. H. Freeman, San Francisco:
1973.
15. D. Ruelle: Chance and chaos. Princeton
University Press, Princeton, N.J., 1991.
16. D. Finkelstein: Quantum relativity. Springer,
New York, 1996.
17. J. F. Traub and A.G. Werschulz: Complexity and Information. Cambridge University
Press, Cambridge, 1998.
18. P. Cartier drew our attention to this possibility.
19. M. White, D. Scott, and J. Silk: Anistrophies
in the Cosmic Microwave Background. Ann.
Rev. Astron. Astroph. 82: pp. 319-370,
1994.
20. M. Rees: Before the beginning: our universe and others. Addison-Wesley, Reading, MA: 1997.

g r a n t f r o m t h e Alfred P. S l o a n f o u n d a -

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