E S S A Y S & C O M M
Varieties of Limits to Scientific
Knowledge
An exploration into how limits can advance our understanding
S
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 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 f o r w o r k i n p r o gress, we m a y w o n d e r w h e -t h e r -t h e r e are no-t m o r e per-m a n e n t f r o n t i e r s o r h o l e s 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 in-t r i n s i c l i m i in-t s in-to s c i e n in-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 g e n e r a l l y a b o u t s u c h intrinsic limits? C a n we c o n -s t r u c t a f r a m e w o r k w i t h i n 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 point 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.
Scientific progress was c o n s i d e r e d virtually u n l i m i t e d , even u n s t o p p a b l e , a h u n -dred years 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 on 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 fac-tors, 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 diffi-cult 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 limita-tions 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,
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.
BY PIET HUT, DAVID RUELLE, AND
JOSEPH TRAUB
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, Switzer-land, 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 computa-tional 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.
w h i c h is likely to provide a n i n c r e a s i n g l y 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 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 ability to s i m u l a t e c o m p l e x s y s t e m s .
However, a b a r r i e r b l o c k i n g progress in o n e direction c a n also b e s e e n as an 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 . 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-pass t h e limits p o s e d by classical phys-ics. And totally different a v e n u e s s u c h as DNA c o m p u t i n g [7] m a y also offer n e w p o s s i b i l i t i e s . H i s t o r i c a l l y , a t t e m p t s t o prove that s o m e t h i n g is i m p o s s i b l e , as in t h e n o - g o t h e o r e m s in physics [8], h a v e 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 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-s e a r c h e r re-s to even e n t e r t h o re-s e areare-s, until s o m e o n e discovered a fatal flaw in s o m e of t h e u n s t a t e d a s s u m p t i o n s .
In general, a n y a t t e m p t to prove that 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 t i m e a n invitation to l o o k for l o o p h o 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 order to find ways a r o u n d t h e difficulties. As we argue in m o r e detail below, the lim-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 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], a n d a d e e p e r u n d e r s t a n d i n g o f l i m i t s 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 .
TRANSCENDING LIMITS
Examples From Mathematics
A
s p e c t a c u l a r f o r m o f intrinsic limit to k n o w l e d g e was t h e s h o c k i n g in-c o m p l e t e n e s s t h e o r e m a n n o u n in-c e d by Godel in 1 9 3 1 : in a f o r m a l a x i o m a t i c 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 par-ticular 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 k n o w n in m a t h e m a t i c s . However, n o w 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-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 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-vestigator.I n s t e a d o f trying to a n s w e r this q u e s -t i o n i m m e d i a -t e l y , le-t us -travel b a c k in 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 w h e t h e r a m o r e r e m o v e d h i s t o r i c a l p e r -s p e c t i v e m a y b e o f -s o m e h e l p . Let u-s t a k e , i n s t e a d o f H i l b e r t ' s p r o g r a m , P y t h a g o r a s ' p r o g r a m o f u n d e r s t a n d i n g 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 r e l a in-t i o n s o f in-t h e n a in-t u r a l n u m b e r s (we 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 is difficult to a s s e s s a n d n o t relevant for o u r e x a m p l e ) .
F o r a w h i l e , P y t h a g o r a s ' p r o g r a m s e e m e d to b e sailing a l o n g q u i t e nicely. He a n d his followers also built up an i m p r e s s i v e b o d y o f m a t h e m a t i c a l k n o w l -edge, a n d t h e y were able to explain quite a lot in a p p l i e d areas s u c h as m u s i c . But t h e n t h e y discovered that t h e s q u a r e root o f two "did n o t exist as a n u m b e r " (i.e., is n o t a rational n u m b e r ) , a d e e p l y s h o c k -ing result. W i t h s u c h a s i m p l e n u m b e r as t h e length o f the diagonal o f a square with s i d e s o f u n i t l e n g t h b e i n g " o f f l i m i t s , " 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 -p e c t e d l y limited.
O
f c o u r s e , n o w a d a y s n o b o d y would c o n s i d e r this result to b e in any way a limit. Rather, it is part o f thestruc-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].
Examples From Physics
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 strikes us as a severe limit in a physical t h e o r y : t h e fact that t h e finite s p e e d o f light p o s e s an u p p e r limit o n t h e s p e e d
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.
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 ) o b j e c t . Starting f r o m Newton's classical 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 m a g n i t u d e are allowed, t h e e x i s t e n c e of
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
c o m e s as a surprise a n d strikes o n e as a l i m i t a t i o n .
H o w e v e r , if w e h a d first l e a r n e d to think in t e r m s o f general relativity theory, a n d only afterward tried to u n d e r s t a n d 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 e r m s o f differential g e o m e t r y , t h e n a t u -ral l a n g u a g e o f g e n e r a l relativity, a n alto-g e t h e r different picture o f w h a t is l i m i t e d would h a v e e m e r g e d . In this a p p r o a c h , N e w t o n i a n gravity is full o f limits, in the 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 -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 s i m p l e r way that general relativity c a n b e 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].
Similarly, q u a n t u m m e c h a n i c s s e e m s s t r i k i n g l y i n c o m p l e t e f r o m t h e v i e w p o i n t o f c l a s s i c a l m e c h a n i c s . T h e u n c e r tainly relation, for e x a m p l e , p u t s u n e x -p e c t e d limits o n w h a t c a n b e m e a s u r e d 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 a n d start with a f o r m a l i s m t h a t is m o r e 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 -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 t h a t is g l a r i n g l y " i n c o m p l e t e " a n d 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 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 i n t e r f e r e n c e [13].
I
n addition 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 l o c a l l i m i t s m a y b e c i r c u m v e n t e d g l o -bally. W o r m h o l e s in s p a c e - t i m e are a n 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 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 -n e o u s l y over (seemi-ngly) large d i s t a -n c e s without violating t h e s p e e d limit i m p o s e d locally by s p e c i a l relativity [14].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), t h i s g e n e r a l l y m e a n s t h a t t h e t h e o r y 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 o c c u r s
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 -tion leads to singular solu-tions.
-(Physical)
World
translation
recipes
(Mathematical)
Model
els o f t h e world versus limits that applyto t h e w o r l d as we k n o w it t h r o u g h o b -s e r v a t i o n a n d e x p e r i m e n t a t i o n . W i t h i n the physical s c i e n c e s , the m o d e l s are gen-erally 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 areas, s u ch as archaeology, c o h e r e n t ex-p 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 h a v i n g t h e i r o w n l o g i c a l s t r u c t u r e , 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 -g u a -g e . A m o r e c a r e f u l c o n s i d e r a t i o n , 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 that is at t h e heart of any form o f s c i e n -tific activity.
S c i e n c e p r o p o s e s theoretical s c h e m e s to d e s c r i b e t h e "real world." T h e c o n n e c tion with t h e real world involves a n o p -e r a t i o n a l d -e s c r i p t i o n o f h o w o b s -e r v -e d facts fit into a theoretical s c h e m e . In o t h e r words, t h e c h a l l e n g e is to build i n c r e a s ingly m o r e a c c u r a t e (usually m a t h e m a t i -cal) m o d e l s o f t h e world, t o g e t h e r with 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-p e r i m e n t s and o b s e r v a t i o n .
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 u p p o s e , for i n s t a n c e , that we are in-m a t h e in-m a t i c a l t h e o r y is f a i r l y 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 r e c i p e s f o r m e a s u r i n g v o l t a g e s , r e s i s -t a n c e s , c a p a c i -t a n c e s , e-tc. in -the lab. Simi-larly, l a n d surveying h a s a s i m p l e t h e o r y ( b a s i c a l l y E u c l i d e a n g e o m e t r y ) , w h i c h 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 (Eu-c 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 v o l t m e t e r s , e t c . ( g e o d e s y i n s t r u m e n t s ) 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 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 a s s o c i a t e d t h e o r y . O n e c a n , h o w e v e r , never c o m p l e t e l y remove 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].Let us c o m e b a c k to limits a n d try to 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 : t h o s e in nature, t h o s e in t h e m o d e l s we 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 o u r t r a n s l a t i o n recipes. For d e f i n i t e n e s s ,
The three realms of physics.
we c a n take t h e e x a m p l e of physics, for 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 Figure 1.
W h a t m a k e s f u n d a m e n t a l progress in physics hard is o f t e n n o t so m u c h t h e in-h e r e n t difficulties tin-hat are e n c o u n t e r e d within specific m a t h e m a t i c a l m o d e l s , but 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 c h o i c e o f t h e p a i r { t r a n s l a t i o n r e c i p e , m o d e l } .
T h e q u e s t i o n of the status of the trans-lation r e c i p e s is s o m e t h i n g that is rarely a d d r e s s e d specifically in a c a d e m i c sci-e n c sci-e c o u r s sci-e s . T h sci-e i r vsci-ery sci-e x i s t sci-e n c sci-e is of-ten glossed over, a n d it is n o t even clear w h e t h e r they are part of p h y s i c s of m a t h e m a t i c s . If we start f r o m t h e side o f p h y s -ics, a n d ask w h i c h part o f Figure 1 applies to m a t h e m a t i c s a n d w h i c h part to phys-ics, we m i g h t draw a p i c t u r e as given in Figure 2a.
I
n this figure, the natural world a r o u n d us is a s s i g n e d to t h e d o m a i n o f phys-ics. Clearly, t r a n s l a t i o n recipes are n o t part of this natural world. Rather, they are d e s i g n e d by us in o r d e r to fit with o u r 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 they s h o u l d b e g r o u p e d u n d e r t h e gen-eral c a t e g o r y o f m a t h e m a t i c a l tools.But as s o o n as we start f r o m t h e o t h e r side o f Figure 1, f r o m t h e side o f m a t h 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 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 -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 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 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 f o u n d w i t h i n E u c l i d e a n g e o m e t r y : t h e y 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 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.
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 given in Figure 2 b .
F a c e d w i t h t h e s e a l t e r n a t i v e s , d e -p i c t e d in Figures 2 a a n d 2 b , n e i t h e r o f t h e m feels q u i t e right. But p e r h a p s t h e 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 n o t just t h e u n e a s y status of t h e transla-t i o n r e c i p e s transla-t h a transla-t r e s i s transla-t s c l a s s i f i c a transla-t i o n . W h e r e else c a n we look? T h e m a t h e m a t i 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 physical world?
O n e possibility, in a variation o n Kant, would b e to declare t h e physical world, as the realm of t h e "things in t h e m s e l v e s , " offlimits to direct scrutiny. T h e p h e n o m -e n a w-e d-eal with ar-e a c c -e s s i b l -e to us only t h r o u g h t h e particular ways we c h o o s e to investigate t h e m . So we might even argue that we h a v e to shift t h e l e g i t i m a t e ter-rain o f physics towards a middle position, as illustrated in Figure 3. T h i s m a y s e e m a bit e x t r e m e , b u t as we will s e e below, it c a n gives us an interesting h a n d l e on t h e status o f limits.
We are n o w in a p o s i t i o n to return to t h e q u e s t i o n that led us to this classifica-tion: h o w to distinguish b e t w e e n limits that s h o w up in our m o d e l s of t h e world versus limits that apply to t h e world it-self. In o t h e r words, in c o n t r a s t to scruti-nizing m o d e l s a n d recipes, w h a t c a n we 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 real world?
Q
u e s t i o n s a b o u t t h e real world do 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 :FI GURE 2A
MATHEMATICS
PHYSICS
(Physical)
World
translation
recipes
(Mathematical)
Model
A view from physics.
FI GURE 2R
PHYSICS
(Physical)
World
translation
recipes
MATHEMATICS
(Mathematical)
Model
A view from mathematics.
W e g e t to c h o o s e t h e m a t h e m a t i c a l m o d e l . To rigorously establish a scientific limit w e s h o u l d s h o w t h a t every m a t h
-e m a t i c a l m o d -e l that captur-es th-e -e s s -e n c -e of a scientific q u e s t i o n is u n d e c i d a b l e or i n t r a c t a b l e [4].
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 prin-c 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 -t i o n a l c o m p l e x i -t y o f a m a -t h e m a -t i c a l m o d e l , we do p e r m i t all p o s s i b l e algo-r i t h m s to c o m p e t e . )
In scientific p r a c t i c e , all t h a t we e n 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 -tations, t o g e t h e r with a p p l i c a t i o n s that in turn are i n s p e c t e d t h r o u g h further m e a -s u r e m e n t -s . T h e r e f o r e , if w e critically a-sk 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 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-ure 3. D o e s this leave t h e real world side of physics? Actually what is left
out-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 a b o u t t h e real w o r l d [ 1 6 ] . B u t t h e real world is t h e r e s o m e h o w a n d limits t h e p h y s i c a l l y a c c e p t a b l e p a i r s { r e c i p e , m o d e l } . U n a c c e p t a b l e pairs are falsified by e x p e r i m e n t s .
A CLASSIFICATION OF LIMITS
I
f o n e starts listing limits to knowledge, o n e s o o n finds that t h e i t e m s are het-e r o g het-e n het-e o u s but c a n b het-e fit in cathet-egorihet-es that we n o w discuss (we leave out psy-c h o l o g i psy-c a l psy-c o n s i d e r a t i o n s , s u psy-c h as t h e q u e s t i o n of to w h a t extent h u m a n intel-ligence m a y b e a limiting factor). T h e cat-e g o r i cat-e s o v cat-e r l a p s o m cat-e w h a t (i.cat-e., s o m cat-e i t e m s fit in several c a t e g o r i e s ) .1. The Curse of the Exponential
T h e r e are p r o b l e m s t h a t c a n b e solved 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 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 et i m e e v o l u t i o n o f a p h y s i c a l sys-t e m is a s s u m e d sys-to h a v e a g o o d m a t h e m a t i c a l m o d e l , b u t t h e a c 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 -tial growth o f e r r o r s ( i m p o r t a n t 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 -t i o n a n d -t h e a s -t r o n o m y o f -t h e so-lar 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 a c c u r a c y t h e e q u a t i o n s p r o p o s e d by physical t h e o r i e s m a y o f t e n b e p r a c t i c a l l y i n s u p e r a b l e .
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 -ceptably fast with t h e length of the 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 suggests a class o f in-t r a c in-t a b l e p r o b l e m s . On in-t h e o in-t h e r h a n d , m a n y c o n t i n u o u s p r o b l e m s h a v e b e e n proven e x p o n e n t i a l l y 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 s e t t i n g . F o r s o m e p r o b l e m s i n t r a c t a b i l i t y c a n b e b r o k e n by switching to a s t o c h a s -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 m a y w a n t to ask c a n b e shown to have n o answer in the f r a m e w o r k o f a given theory (wrong q u e s t i o n ) . In o t h e r words, t h e r e are structural limitations to t h e q u e s t i o n s 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 s t a n c e , o n e should n o t ask to specify sim u l t a n e o u s l y t h e position a n d sim o sim e n -t u m of a par-ticle. T h e work of Godel and Turing h a s s h o w n that it is a b a d idea to seek 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 arbitrary s t a t e m e n t is true or false.
T
h e r e is a c u r i o u s relation b e t w e e n 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 Turing 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 o r d s it grows v e r y very fast w i t h t h e length of t h e input: faster t h a n e x p o n e n -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.W h a t we n o w c o n s i d e r to b e w r o n g
[image:4.612.42.434.64.425.2]FI GURE
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 o r h o w s o m e a l g o r i t h m c o u l d solve all m a t h e m a t i -cal p r o b l e m s (Hilbert's d r e a m ) . 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 l i m i t a t i o n s t h a t m a k e s u c h q u e s -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 p r o g r e s s .
3. Questions of Questionable
Status
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 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. A s e e m i n g l y hard question (such as "what 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 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 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 w h e t h e r a limitation is f u n d a m e n t a l m a y t h u s l e a d to i m p o r t a n t t h e o r e t i c a l progress. Currently in this c a t e g o r y are p r o b l e m s r e l a t e d t o c o s m o l o g y a n d q u a n t u m gravity.
A
n o t h e r e x a m p l e c o n c e r n s t h e c o n -s i s t e n c y o f a r i t h m e t i c . G o d e l proved (1) that a r i t h m e t i c c a n n o t b e proved internally to b e c o n s i s t e n t a n d (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 plete. Actually, arithmetic could be i n c o n -s i -s t e n t , b u t a p r o o f o f c o n t r a d i c t i o n would t h e n p r o b a b l y b e extremely long, 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 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 e118]. T h u s even a very f u n d a m e n t a l a n d
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 -c e r n i n g t h e -c o n s i s t e n -c y o f a r i t h m e t i -c 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 q u e s t i o n a b l e nature than o n e might have liked.
4. Emergent Properties
A n o t h e r class of very difficult q u e s t i o n s , w h i c h s e e m to lead to f u n d a m e n t a l limi-t a limi-t i o n s of knowledge, arise in limi-t h e slimi-tudy o f e m e r g e n t p r o p e r t i e s . O n e c a n c o n -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 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 e r m s of the level below, s u c h as:
??? < 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.
(Physical)
World
PHYSICS
MATHEMATICS
translation
(Mathematical)
recipes
Model
•
From ontology to epistemology.
In p r a c t i c e , however, we are very lim-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 t e r m s o f t h e lower levels. In a d d i t i o n , e m e r g e n t p r o p e r t i e s in general are fuzzy. This in turn c r e a t e s difficulties in classi-fication. An e x a m p l e of p r o b l e m a t i c as-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 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 , 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 b e c a u s e w e k n o w very little a b o u t life f o r m s u n r e l a t e d to us.
Surprisingly m a n y s y s t e m s with large n u m b e r s of degrees of f r e e d o m allow de-s c r i p t i o n de-s in far de-s i m p l e r t e r m de-s , w i t h properties that are not at all obvious from the underlying s y s t e m . T h e s e e m e r g e n t p r o p e r t i e s point toward a higher level of o r g a n i z a t i o n , exhibiting less c o m p l e x i t y than could b e a n t i c i p a t e d . (Coarse graining is o n e e x a m p l e of a simplifygraining p r o 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 -ties.) But higher levels are not g u a r a n t e e d to have s i m p l e r b e h a v i o r (and not m u c h has b e e n f o u n d , for e x a m p l e , in fully de-veloped t u r b u l e n c e ) .
5. Limited Access to Data
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 sufficient data leads to severe l i m i t a t i o n s to knowledge. Historical l i m i t a t i o n s are in this c a t e g o r y [5]. We m a y b e able to in-fer m a n y things a b o u t the origin of life, b u t it a p p e a r s that we shall never have a detailed story of w h a t really h a p p e n e d . 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 t h e origin o f stars, of m a n k i n d , of lan-guages, etc.
6. Sample Size of One
Cosmology, t h e study of t h e origin o f t h e
u n i v e r s e a n d its overall s t r u c t u r e , is an e x a m p l e o f a discipline that deals with a 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 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-ficult, s i n c e a typical t h e o r y for t h e large-s c a l e large-s t r u c t u r e of t h e u n i v e r large-s e p r e d i c t large-s n u m b e r s t h a t h a v e a s t a t i s t i c a l u n c e r -tainty a t t a c h e d to t h e m [19].
No m a t t e r h o w well we will ever b e able to m e a s u r e t h e particular realization 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 is correct. Certainly any Popperian n o t i o n of falsifiability does not apply, as long as 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 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 again, it is always risky to declare s o m e -thing to b e "off limits," and recently various 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 -ing " m u l t i - v e r s e s " [20].
7. Technological Limitations
T h e y are practically i m p o r t a n t , but o n e should b e wary o f treating as f u n d a m e n -tal s o m e limits that c a n p e r h a p s b e over-c o m e by n e w ideas. For i n s t a n over-c e , physiover-cs at t h e P l a n c k energy a p p e a r s very m u c h 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 this d o e s not m e a n that we shall n e v e r u n d e r s t a n d what is going on in this e n -ergy region.
DISCUSSION
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 h e y s e e m to h a v e c a r v e d up t h e world. E x a m p l e s :
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 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-s t i t u t e , w h e r e thiIn-s w o r k w a In-s In-s t a r t e d , for
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 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 -d a t i o n .
• 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;
• cell walls in biology allow t h e build-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 ,
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 their (ir) rationality property;
• the u n c e r t a i n t y principle: the very fact that position a n d m o m e n t u m , for exa m p l e , exare n o t s i m u l t exa n e o u s l y m e exa -s u r a b l e -s h o w -s t h e u n i f i e d n a t u r e o f q u a n t u m reality.
E
a c h limit at first p r e s e n t s itself as a 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 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-lenge: 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 areas o f knowledge.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 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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10. 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: Complex-ity and Information. Cambridge UniversComplex-ity Press, Cambridge, 1998.
18. P. Cartier drew our attention to this possi-bility.
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 uni-verse and others. Addison-Wesley, Read-ing, MA: 1997.
