I t is a peculiarity of molecular biological strings t h a t , like t h e elements of a natural language, t h e y realize two spaces. These a r e spaces with distinct and dif- f e r e n t metrics: t h e r e is no reason t o suppose t h a t t h e y a r e in phase. Evolution a s
a process works most directly on biological organisms, which must perish o r per- severe in t h e face of circumstance. To t h e e x t e n t t h a t evolution is a process b y which organisms converge over time t o some local (or global) optimal, t h e
processes of convergence t h a t are sketched broadly in life must have some sub- stantial echo a t t h e molecular biological level. where words and strings hold sway.
The relationship between m e t r i c spaces t h a t this p a t t e r n exemplifies is quite gen- eral
-
t h e province, in fact, of Weizenbaum theory. Thus letM
and N be two metric spaces, each with its own natural metric; points inM
are labeled t l ,t z . ...
,-
points in N, e l , e z ,... . e n ; j : M + N is a mapping between points in M
and
t n
points in N
-
a bijection, t o make matters trivially simple.M
and N are arbitrary, and admit of obvious specification:(I) M
is a typographic m e t r i c space; N, t h e space of biological organisms (see p 240).( 2 )
M
is a typographic m e t r i c space under t h e natural m e t r i c on words; N , the same space under distance defined in terms of meaning o r grammar (see p p 245-248).( 3 ) M is a typographic m e t r i c space; N, a space of algorithms.
Thus j might map linear sequences of DNA o r proteins, o r sets of such sequences, onto organisms, o r sets of organisms; equally, j might map a linear string of l e t t e r s onto a sentence, with a fixed meaning in a natural language; o r onto an algorithm in a given computer language such as Algol; then, too, j might map fixed strings in an assembly language onto a computer program. In each of these cases, j does not preserve metrics;
M
and N are not necessarily in phase.In addition to t h e natural metric on
M ,
t h e r e exists an induced metric d N ( ( ) onM
defined by t h e following relationship:The Weizenbaum experiment
To specify a Weizenbaum experiment, it is necessary to provide
M
with a p m - bability t r a n s i t i o n s y s t e m P r determining for each point t inM
t h e probability t h a tt
will change tot
'; and a n i n i t i a l probability d i s t r i b u t i o n Pro. A d i s - t i n g u i s h e d element e* E N is fixed from t h e first. Within t h e context of molecu- lar biology, transition probabilities are focused on relatively nearby strings-
thisbecause point mutations result in string-Like changes of a short typographic dis- tance. In a biological Weizenbaum experiment, this f a c t is respected to t h e e x t e n t t h a t t h e typographic metric space and t h e probability transition system are mutu- ally in accord: probabilities follow typographic neighborhoods. Elsewhere, proba- bilities and distances are adjusted accordingly.
A point t o is selected in accordance with t h e initial probability distribution Pro over M . The distance d N ( 0 ) from j
( t
O ) to e* is measured; t h e system engaged for i=
1. 2 , 3....
; as tt moves t o t i , the distance d N ( t ) between j ( t t ) and e* is recorded. The outcome of t h e Weizenbaum experiment is t h e sequenced ~ ( ~ ) * d ~ ( l ) * - - - s d ~ ( n )
.
The Weizenbaum experiment is successful if:
The Language o/L.tre 33
Condition W For d N ( 0 ) a t an a v e r a g e d i s t a n c e from e* t h e sequence [ d N ( t ) { converges t o a neighborhood of 0.
Condition W, when met, implies that t d N ( t ) ] is both stable and oriented. The graph of a sequence of points constitutes a trajectory; t h e s e t of trajectories in N that a r e a t once stable and oriented is of measure zero. A successful Weizenbaum experiment thus establishes t h a t Pr(M) c a n n o t be arbitrary with respect t o its i n d u c e d metric structure. In particular, points t h a t a r e f a r in t h e i n d u c e d metric have small transition probabilities: those probabilities that count must be concentrated on nearby objects
-
nearby in t h e sense of t h e induced metric. On t h e other hand, transition probabilities over molecular biological strings a r e , on t h e neeDarwinian theory, focused on neighborhoods t h a t a r e nearby in a natural metric.It is perhaps for this reason that. with t h e exception of life itself, no one has ever seen a successful Weizenbaum experiment.
Eigenvalues of natural selection
In Darwinian thought, t h e effects of randomness a r e played off against what biologists call t h e c o n s t r u c t i v e effects of natural selection, a mechanism t h a t philosophers have long regarded with sullen suspicion. Wishing t o know why a species t h a t represents nothing more than a persistent snore throughout t h e long night of evolution should suddenly (or slowly) develop a novel characteristic, t h e philosopher will learn from t h e definition of natural selection only t h a t those characteristics t h a t a r e relatively fit a r e relatively fit in virtue of t h e fact t h a t they have survived, and t h a t those characteristics t h a t have survived have sur- vived in virtue of t h e fact t h a t they are relatively fit. This is not an intellectual exercise calculated t o inspire confidence.
Natural selection is a force-like concept; and, as such, acts locally if i t acts a t all. Mathematicians often assume t h a t evolution proceeds over a multidimen- sional fitness surface, something that resembles a series of hills and valleys; a great deal t h a t is theoretically unacceptable is often hidden in a description of its topology. But I am anticipating my own argument. In speaking of locality, I mean t o evoke t h e physicists's unhappiness a t action a t a distance. Strings t h a t a r e f a r apart should be weak in mutual influence; this is a spatial constraint. Then again, no string should be influenced by a string t h a t does not yet exist. This is a tem- poral constraint, a rule against d e f e r r e d success. The historical development of a complex organ such as t h e mammalian e a r involved obviously a very long sequence of precise historical changes. Comparative anatomy suggests t h a t t h e reptilian jaw actually migrated earward in t h e course of evolution. It is very difficult to under- stand why each of a series of partial changes in t h e anatomy of t h e reptilian jaw should have resulted in a net increase in fitness befire t h e advent of t h e mam- malian ear. Certain genes within t h e bacterial cell, t o take another example, "are organized into larger units under t h e control of an operator, with the genes linearly arranged in t h e order in which t h e enzymes to which they give rise a r e utilized in a particular metabolic pathway". [23] The genetic s t e p s required to organize an operon cluster do not "confer any selective advantage t o t h e pheno- type so t h a t individual s t e p s a r e independentU.[23] The rule against deferred
success functions as a prophylactic against t h e emergence of teleological o r Aris- totelian thought in theoretical biology. @3]
I have pictured evolution on the molecular level as a process involving paths;
natural selection a c t s to induce a statistical drift on some paths, and not others;
those paths involving a positive gain in fitness are favored. A t any particular time, a t any particular place, one has an ensemble E of protein strings, embedded, so t o simply t h e finite-state stationary process with identically distributed terms men- tioned in t h e example already discussed; and may be represented as a linear a r r a y
What are t h e chances, one might ask (with a marked lack of breathlessness in my own case). t h a t a system of this s o r t
-
a pure Bernoulli process-
could con- verge on a p a r t i c d m sentence of English? Following Mannfred Eigen. let us sup- pose t h a t t h e sentence in question is TAKE ADVANTAGE OF MISTAKES. so that k is 23; this is t h e t a r g e t s e n t e n c e-
S .Even here, poised between irrelevance and imprecision, delicate and impor- tant biological questions arise.@4] Thus. while it makes sense of s o r t s t o say t h a t
In any event. nothing in Eigen's own example quite indicates why a stochastic system with a target sentence, however defined. should stop when it has reached
Stochastic device. target sentence. fitness function, and evaluation measure.
taken as a quartet, comprise an Eigen s y s t e m . The enterprising Professor William R. Bennett J r has calculated that an Eigen system would require a virtually infin- i t e amount of time t o reach even a simple target sentence
-
a number roughly a trillion times g r e a t e r than t h e life of t h e universe In t h e same spirit. MurrayThe L a n g u a g e oj'LVe 35 lessly slow Eigen system already described. Under t h e advanced Eigen system, fit- ness is no longer an all o r nothing affair; f thus takes values, l e t us say, between local property; an evaluation measure so constructed would plainly b e responding t o signals s e n t from t h e Beyond, a clear c a s e of action a t a distance. The problem of discovering a t a r g e t sentence remains unchanged, hopeless. In f a c t , this is pre- cisely what t h e advanced Eigen system actually measures, since an a r b i t r a r y sen- t e n c e in which A a p p e a r s in t h e second position is judged fit only because i t is closer t o t h e t a r g e t s e n t e n c e than i t might otherwise be. When t h e matter is care- fully explained, theoretical biologists understand a t once t h a t t h e very concept of a t a r g e t sentence constitutes a b e e r y and uninvited guest in evolutionary thought.
a b s t r a c t characterization of all t h e English sentences. Of these, t h e r e are infin- Crick (1966) OJMolecules a n d Men (Seattle: University of Washington Press).
[4] I discuss reductionism from t h e perspective of atomistic theories in Berlinski (forthcoming) The Aise of Di f f e r e n t i d Topology (Boston: Birkhaeuser Boston). S e e also Kenneth Schaffner (1967) Approaches to reduction. P h i l o s o p h y of Science 34 (1): 137-47.
[5] Michael Ruse has argued f o r his thoroughly incoherent position in Ruse (1973) The P h i l o s o h y of Biology (London: Hutchinson). The concept of evolution w a s . of Grasse remarks 'however complicated i t s molecular structure, is in my view aber- rant."
[6] M.C. King and A.C. Wilson (1975) Evolution a t t w o levels in humans and chimpanzees.
Science 88 (4184).
171
S. Smale (1980) Z?ze Mathematics of Erne (New York: Springer).[8] A. Kohogorov (1967) Logical basis f o r information theory and probability theory.
IRZE T r a n s a c t i o n s o n I M o r m a t i o n Z?zeory IT
-
14 (5). I have patterned my dis- cussion on: G.J. Chaitin (1974) Information-theoretic computational complexity.IEEE
T r a n s a c t i o n s o n I W o r m a t i o n h e o r y . IT-
20 (1). The interested r e a d e r should consult Chaitin's o t h e r papers. and relevant papers by Solovay. Chaitin's bibliography may be consulted f o r details.[9] See, f o r example, Noam Chomsky (1972) Language a n d Mind (New York: Harcourt, Brace Javonovich).
[lo]
The idea of representing context-free languages by means of a system of equations in noncommutative variables is due to M.P. Schutzenberger. See M. Gross (1972) although staid, contains a competent discussion of many of these issues.The L a n g u a g e of L i f i 37
[I21 S e e , f o r example, L. Lofgren (1975) On t h e formalizability of learning and evolu- tion, in Suppes. Henkin, Joja, and Mosil (Eds) Logic, Methodology a n d P h i l o s o p h y o f s c i e n c e (Amsterdam: North-Holland).
[13] J. Monod (1971) Chance a n d N e c e s s i t y (New York: Alfred Knopf).
[14] S e e P e t e r Medawar (1977) The Life Sciences (London: Wildwood House).
[15] Murray Eden (1967) Inadequacies of neo-Darwinian evolution a s a scientific t h e o r y , in P . Moorhead and M. Kaplan (Eds) Mathematical C h a l l e n g e s to N e o - & m i n i s m (Philadelphia: The Wistar Institute P r e s s ) .
[I61 R.M. Thompson (1981) M e c h a n i s t i c a n d Non-Mechanistic Science (Lynbrook, N e w York: Bala Books).
[17] H.P. Yockey (1977) A calculation of t h e probability of spontaneous biogenesis by information t h e o r y . J o u r n a l of Theoretical B i o l o g y 67.
[I81
S e e K. P e t e r s e n (1983) E r g o d i c T h e o r y (Cambridge: Cambridge University P r e s s ) f o r details.[I91 My discussion follows t h a t of A.I. Khinchine (1957) Mathematical F o u n d a t i o n s of I n f o r m a t i o n T h e o r y (New York: Dover Publications).
[20] N. Chomsky and G. Miller (1963) Finitary models of language use, in Luce, Bush, and Galanter (Eds) Handbook of Mathematical P s y c h o l o g y (New York: John Wiley &
Sons).
[21] Eden. o p . c i t .
[22] The idea of t h e Weizenbaum experiment i s due t o M.P. Schutzenberger.
[23] Eden, op. c i t .
[24] See, f o r example, Eigen (1971) Self-organization of m a t t e r and t h e evolution of biological macromolecules. Die N a t u r w i s s e n s c h q t Y e n 10. Together with Ruth Winker, Eigen h a s r e c e n t l y (1981) published a popular account of his thought under t h e t i t l e The L a w s of t h e Game (New York: H a r p e r & Row).
References
Grantham, R. (1974) Science, 185, 62.
Schroedinger, E. (1945) What is Life? (New York: Macmillan).
Smart, J. J.C. (1963) P h i l o s o p h y a n d S c i e n t i f i c R e a l i s m (London: Routledge and Kegan Paul).
Watson, J.D. (1965) Molecular B i o l o g y of t h e Gene (New York: Benjamin) p 67.