The Language of Life

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NOT FOR QUOTATIOK WITHOUT PERMISSION CF THE ALTEOR

THE LANGUAGE OF

LIFE

David Berlinski

April 1985 CP-85-2C

C o l l a b o r a t i v e F k p e r s report work which has not been performed solely' a t the International Institute f o r Applied Systems Analysis and which has received only SIcited review. Views o r opinions expressed hereir. do not necessarily represent those of the Insti- tute, i+h Nzti0r.d Member O r g ~ ~ i z a t i o n s , o r other organizatio~s supporting the work.

INTERNATIONAL IYSTTTVE FOR A.?PLID SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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FOREWORD

This p a p e r r e p r e s e n t s t h e written version of a l e c t u r e given a t IIASA in Sep- tember 1984 under t h e auspices of t h e Science & Technology and t h e Regional Issues p r o j e c t s . In i t s c u r r e n t form i t will a p p e a r a s a c h a p t e r in t h e forthcoming IIASA book, C o m p l e x i t y , Language a n d Life: Mathematical Approaches, J . Casti and A. Karlqvist. eds.

Boris S e g e r s t a h l Leader Science & Technology Program

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ABSTRACT

This p a p e r e x p l o r e s t h e idea t h a t life comprises a language-like system. The arguments a r e c a r r i e d out against t h e background of t h e neo-Darwinian t h e o r y of evolution. The principal conclusion is t h e dilemma t h a t if life i s a language-like system, t h e n c e r t a i n concepts a r e missing from t h e Darwinian paradigm; if not, then Darwinian thought i s suspicious in t h e s e n s e t h a t i t s principles do not naturally apply t o cognate disciplines.

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Dnvid B e r l i n s k i i s a Researcher a t the I n s t i t u t d e s Hautes E'tudes S c i e n t i f i q u e , B u r - s u r - y v e t t e . P a r i s , France.

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T h e L a n g u a g e of Life

David Berlinski

In t h e spring of 1984, I delivered two lectures a t IIASA under t h e title The L a n g u a g e of L i f e . Dianne Goodwin was kind enough t o p r e p a r e a verbatim tran- s c r i p t of my talks; I have used t h e months since t h e n to purge t h e written record of what I said of i t s incoherence, vagrant inaccuracies, and general slovenliness.

This c h a p t e r is a t once long and t e r s e

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a n unhappy combination, and one t h a t makes severe demands of t h e r e a d e r . Many arguments a r e highly compressed and must b e elaborated before t h e y a p p e a r convincing. I have not hesitated to make use of mathematical concepts in expressing myself; but I draw no mathematical conclusions. I thus r u n t h e risk of alienating t h e general r e a d e r even as I anta- gonize t h e mathematician. For t h e s e reasons, i t may b e helpful if in this introduc- tion I endeavor to place this c h a p t e r in a somewhat wider personal and intellectual context.

As i t stands, The L a n g u a g e of Life r e p r e s e n t s a d r a f t of one-third of a larger work entitled L a n g u a g e , Life a n d Logic. Another p a r t of t h a t more ambi- tious project was delivered a t IIASA two years ago a s a s e t of lectures. The written record of those lectures, which I hope to publish separately a s a working p a p e r , is entitled C l a s s i f i c a t i o n a n d its D i s c o n t e n t s .

My aim in L a n g u a g e , Life a n d Logic is to explore a c e r t a i n complicated com- mon ground t h a t holds between language, on t h e one hand, and t h e graphic a r t s , on t h e o t h e r . These a r e t h e classic systems of representation of t h e human imagi- nation. In both, t h e r e is a curious division between t h e system's syntactic and semantic s t r u c t u r e s : a theory, for example, consists of a finite s e t of sentences, t h e sentences of words; paint and then pigment comprise a painting; and y e t , words and sentences, paints and pigments, manage, somehow, t o cohere and. then, in a miraculous a c t of self-transcendence, to make contact with a distinct and dif- f e r e n t e x t e r n a l world. The problems of theoretical biology, i t might seem, have nothing much t o do with issues t h a t arise in t h e philosophy of language o r t h e philosophy of a r t . Not so. A gene comprises a linear a r r a y of nucleotides t h a t under certain conditions e x p r e s s e s a protein o r s e t of proteins. The proteins, in t u r n ,

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2 D. B e r l f n s k t

are organized to form a s t r u c t u r e as complicated as a moose o r a mouse. The nucleotides are plainly alphabetic o r typographic in character; t h e organism itself is rich, complex, complete, continuous, unlike an alphabet. How is i t , then, t h a t such typographic s t r u c t u r e s as DNA manage t o express so much t h a t is not typographic a t all? This is a question quite similar t o questions t h a t might be raised about language itself, o r works of t h e graphic arts; and when i t is pursued, certain metaphors and quite peculiar images begin drifting from one subject to t h e other.

There is t h e notion of meaning, of course, which is common t o language, art, and life; but also the idea t h a t life is itself a language-like system; o r t h a t art is organic. The relations of satisfaction, representation, and expression. while for- mally distinct. of course, nonetheless display points of contact. In order t o explain how i t is t h a t a painting may represent a face, f o r example, one has recourse to t h e notion of a metaphor, a concept from t h e philosophy of language and linguistics; to make sense of gene expression, one deals in concepts such as code, codon, information, and regulation. In a general way, a theory, a painting, and a gene belong t o t h e class of i n t e r p r e t e d o r sigmficant typographic objects. I t is for this reason t h a t i t has seemed t o m e profitable to explore some of the con- cerns of theoretical biology and t h e philosophy of art and language in a single volume.

Within t h e context of this c h a p t e r , my aim is to explore t h e ramifications of a controlling metaphor: t h e idea t h a t life comprises a language-like system. I do this against t h e background of t h e neo-Darwinian theory of evolution

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global and comprehensive scheme of thought in theoretical biology. My argument a t its most general is constructed as a dilemma: if life is a language-like system.

then certain concepts are missing from Darwinian thought; if not. then Darwinian thought is suspicious in t h e sense t h a t its principles do not naturally apply to cognate disciplines. The intellectual p a t t e r n t o this chapter is thus one of movement between two unyielding points, a kind of whiplash.

Part One establishes t h e historical and contemporary background to D d n i a n thought; and makes t h e argument t h a t much of biology cannot b e reduced to physics. In Part Two. I consider t h e confluence of certain concepts:

distance in t h e m e t r i c spaces of organisms and of strings, metric spaces in phase.

complexity, simplicity, Kotmogorov complexity. the ideas of a weak theory, and a language-like system. Part Three plays off concepts of probability against t h e hypothesis t h a t molecular biological words are high in Kolmogorov complexity

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with results t h a t are inconclusive. In Part Four, I examine evolution o r biological change as a process involving paths of proteins. The discussion is set in t h e mathematical contexts of ergodic theory and information theory. In many respects, t h e classical concepts of information and entropy are most natural in discussing topics such as t h e generation of protein paths b y means of stochastic devices; but t h e r e is a connection between Kolmogorov complexity and entropy in t h e sense of information theory. which remains to be explored. Almost all of Part Four represents a tentative exploration of concepts that require, and will no doubt receive, a f a r fuller mathematical treatment.

Many of t h e points t h a t I make in this paper I first discussed with M. P.

Schutzenberger in Paris in 1979 and 1980. Indeed, it was our intention and hope t o publish jointly a monograph on theoretical biology.

This

has not come t o pass. Still,

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The L a n g u a g e 0 f L t . e 3

to t h e e x t e n t t h a t my ideas a r e interesting, t h e y a r e his; t o t h e e x t e n t t h a t t h e y a r e not. t h e y are mine.

John Casti read t h e penultimate d r a f t of this essay and discovered any number of embarrassing e r r o r s . I am grateful for his s t e r n advice, which I have endeavored t o heed.

PART ONE

A

System of Belief

The natural thought t h a t theoretical biology comprises a kind of intellectual Lapland owes much to t h e idea t h a t biology itself is somehow a derivative science, an analogue to automotive engineering o r dairy management, and, in any case, devoid of those special principles t h a t lend to t h e physical o r chemical sciences t h e i r striking mahogany lustre. This is t h e position f o r which J.J.C. Smart (1963) provided a classic argument in Philosophy a n d S c i e n t i f i c Realism. [l] Analytic philosophers, for t h e most p a r t , agree t h a t nothing in t h e nature of things com- pels them t o learn organic chemistry; Feyerabend. Putnam, and Kuhn have won- d e r e d whether a n y discipline can properly b e reduced t o anything a t all; and, then, whether anything is e v e r scientific, a t least in t h e old-fashioned and hon- orific sense of t h a t t e m . [ 2 ] Naive physicists

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t h e only kind

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are all too happy to hear t h a t among t h e sciences physics occupies a position of prominence denied, say, t o urban affairs o r agronomy. The result is r e d u c t i o n i s m from the top d o w n , a c r u d e but still violently vigorous flower in t h e philosophy of science. The physicist o r philosopher, with his e y e fixed on t h e primacy of physics, thus needs t o sense in t h e o t h e r sciences

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sociology, neurophysiology, macrame, whatever

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intimations of physics, however faint. This is easy enough in t h e case of biochemis- t r y : chemistry is physics once removed; biochemistry, physics a t a double distance. Doing biochemistry, t h e theoretician is applying merely t h e principles of chemistry to living systems: like t h e Pope, his is a reflected radiance.

In 1831, t h e German chemist Uriel Wohler synthesized urea, purely an organic compound

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t h e chief ingredient in urine, actually

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from a handful of chemicals t h a t h e took from his stock and a revolting mixture of dried horse blood. It was thus t h a t organic chemistry was created: an inauspicious beginning, but important, nonetheless, if only because so many European chemists were con- vinced t h a t t h e attempt to synthesize an organic compound would end inevitably in failure. The daring idea t h a t all of life

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I am quoting from James Watson's t e x t - book (1965), The Molecular Biology of the Gene

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will ultimately b e understood in terms of t h e "coordinative interaction of large and small molecules" is now a com- monplace among molecular biologists, a fixed point in t h e wandering system of t h e i r theories and beliefs. The c o n t r a r y thesis, t h a t living c r e a t u r e s go quite beyond t h e r e a c h of chemistry, biochemists regard with t h e alarmed contempt t h e y reserve f o r ideas t h e y a r e p r e p a r e d t o dismiss but not discuss. Francis Crick, for example, devotes fully a t h i r d of his little monograph, Of Molecules a n d Men, to a denunciation of vitalism almost ecclesiastical in i t s forthrightness and

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u t t e r lack of detail.[3] Like o t h e r men, molecular biologists evidently derive some satisfaction from imagining t h a t t h e orthodoxy they espouse is ceaselessly under attack.

Theoretical biologists still cast t h e i r limpid and untroubled gaze over a world organized in its largest aspects by Darwinian concepts; and s o do high-school instructors in biology

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hardly a group one would think much inclined to the idea of t h e survival of the fittest; but unlike t h e theory of relativity, which Einstein introduced to a baffled and uncomprehending world in 1905, the Darwinian theory of evolution has never quite achieved canonical status in contemporary thought , however much like a cold wind over water its influence may have been felt in economics, sociology, o r political science. Curiously enough. while molecular gene t- ics provides an interpretation for certain Darwinian concepts

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those differences between organisms that Darwin observed but could not explain

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t h e Darwinian theory resists reformulation in terms e i t h e r of chemistry o r physics. This is a point apt t o engender controversy. Woodger, Hempel, Nagel; and Quine cast reduction as a logical relationship: given two theories, t h e first may d i r e c t l y b e reduced to the second when a mapping of its descriptive apparatus and domain of interpretation allows t h e first to b e derived from t h e second. I am ignoring details. now. The standard and, indeed. t h e sole example of reduction successfully achieved involves t h e derivation of thermodynamics from statistical mechulics. In recent years, philosophers have come to regard direct reduction with some unhappiness. There are problems in t h e interpretation of historical terms: t h e Newtonian concept of mass. for example; and theories t h a t once seemed cut from the same cloth now appear alarmingly incommensurable. Kenneth Schaffner has provided a somewhat more elaborate account of reduction: his definition runs t o five points.[4] By a corrected t h e o r y , he means a theory logically revived to bring i t into conformity with current interpretations: Newton upgraded. for example.

His

general scheme for reduction. then, is this:

(1) ^All of t h e terms in the corrected theory must be matched to terms in t h e reducing theory

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a requirement of c o m p l e t e n e s s .

(2) The corrected theory must be deducible f rom t h e larger theory, given t h e existence of suitable reduction functions

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a requirement of d e r i v a b i l i t y . (3) The corrected theory must indicate why t h e original theory w a s incorrect

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a requirement of epistemological i n s i g h t .

(4) The original theory must b e explicable in terms of t h e reducing theory

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^a

requirement of c o g e n c y .

(5) The original and corrected theories must resemble each o t h e r

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a requirement of intellectural s y m m e t r y .

In t h e case of theoretical biology. to speak crisply of deriving, say, molecular biology from biochemistry is r a t h e r like endeavoring to cut s t e e l with butter:

t h e r e is a certain innocence to t h e idea that molecular biology has anything Like a discernable logical structure. What one actually sees is a m a s s of descriptive detail, a bewildering plethora of hypothetical mechanisms. much by way of anecdo- tal evidence, a f e w tiresome concepts, and an a r r a y of metaphors drawn from physics, chemistry. information theory, and cybernetics. The definition of reduction just cited is, in addition. incomplete. its flagrant inapplicability aside. In Men- delian genetics, t h e concept of a gene is theoretical. and genes figure in t h a t

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The L a n g u a g e of L fJe ⁵ theory as abstract entities. To what should they be pegged in molecular genetics in order to reduce the first theory to the second? DNA, quite plainly, but how much of the stuff counts as a gene? "Just (enough) to act as a unit of function, "

argues Michael Ruse, a philosopher whose commitment to prevailing orthodoxy is a model of steadfastness.[5] The functions that he has in mind a r e biochemical: the capacity to generate polypeptides; but to my way of thinking, the reduction achieved thus is illicit. In biochemistry, the notion of a unit of function is otiose, unneeded elsewhere. To the extent that molecular genetics is biochemistry, it does not reflect completely Mendelian genetics; to the extent that it does. it is not biochemistry, but biochemistry beefed-up by extrinsic concepts, a conceptual padded shoulder. What holds in a limited way for molecular genetics holds in a much larger way for molecular biology. Concepts such as code and codon, information. complexity, replication, self -organization, stability. negative entropy (grotesque on any reckoning), transformation, regulation, feedback, and control

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the stuff required to make molecular biology work

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a r e scarcely biochemical: the biochemist following some placid metabolic pathway need never appeal to them.

Population genetics, to pursue the argument outward toward increasing generality, is a refined and abstract version of Darwin's theory of natural selection. applied directly to an imaginary population of genes: selection pressures act directly on the molecules themselves, a high wind that cuts through the flesh of life to reach its buzzing core. Has one achieved anything like a reduction of Darwinian thought to theories that a r e e s s e n t i a l l y biochemical, or even vaguely physical? Hardly.

The usual Darwinian concepts of fitness and selection appear unvaryingly in place.

These a r e ideas, it goes without saying, that do not figure in standard accounts of biochemistry, which very sensibly treat of valences and bonding angles, enzymes and metabolic pathways, fats and polymers

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anything but fitness and natural selection. To Schaffner's list of five, then, I would add a sixth: n o r e d u c t i o n b y m e a n s of i n f l a t i o n

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a contingent and cautionary restriction that, for the time being a t least, enforces a s t e r n separation between biology and mathematical physics.

The Darwinian theory of evolution is the great, global organizing principle of biology, however much molecular biologists may occupy themselves locally in determining nucleotide sequences, synthesizing enzymes, or cloning frogs

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Those biologists who look forward to the withering away of biology in favor of biochemis- t r y and then physics a r e inevitably neo-Darwinians, and t h e fact that this theory

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t h e i r theory

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is impervious to reduction they count as an innocent incon- sistency. If mathematical physics offers a vision of reality a t its most comprehensive, the Darwinian theory of evolution, like psychoanalysis, Marxism, or the Catholic Faith, comprises, instead, a s y s t e m of b e l i e f . Like Hell itself, which is said to be protected by walls that a r e seven miles thick, each such system looks especially sturdy from the inside. Standing a t dead center, most people have con- siderable difficulty in imagining that an outside exists a t all.

The Historical Background

Charles Darwin completed his masterpiece, O n t h e O r i g i n of S p e c i e s , in 1859. He was then forty-nine, ten years younger than the century, and not a man inclined to hasty publication. In the early 1830s, he had journeyed to the islands

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of t h e South Atlantic as a naturalist aboard H.M.S. The Beagle. The stunning diversity of plant and animal life t h a t he saw t h e r e impressed him deeply. Prevailing biological thought had held t h a t each species is somehow fixed and unalterable.

Looking backward in time along a line of dogs, i t is dogs all t h e way. Five years in t h e South Atlantic suggested otherwise t o Darwin. The great shambling tortoises of t h e Galapagos, surely t h e saddest of all sea-going creatures, and countless subspecies of t h e common finch. seemed t o exhibit a p a t t e r n in which t h e spokes of geographic variation all radiated back to a common point of origin. The detailed sketches t h a t Darwin made of t h e Galapagos Finch. which h e later published in Oh the O r i g i n of Species, show what caught his eye. Separated by only a f e w hun- dred m i l e s of choppy ocean, each subspecies of t h e finch belongs to a single family; and yet. Darwin noted. one group of birds had developed a short. stubby beak;

another, living northward, a long, pointed. r a t h e r Austrian s o r t of nose. The variations among t h e finch were hardly arbitrary: birds that needed long noses got them. By 1837. Darwin realized t h a t what held for t h e finch might hold for t h e rest of life and this, in turn, suggested t h e dramatic hypothesis that f a r from being fixed and frozen. t h e species t h a t now swarm over t h e surface of t h e Earth evolved from species that had come before in a continuous, phylogenetic, saxophone-like slide.

What Darwin lacked in 1837 w a s a theory to account for speciation, but t h e great ideas of fitness and natural selection evidently came t o him before 1842, for by 1843 h e had prepared a version of his vision. and committed it to print in t h e event of his death. H e then s a t on his results in an immensely slow, self-satisfied.

thoroughly constipated way until news reached him t h a t A.R. Wallace was about to make known h i s theory of evolution. Wallace. so f a r as I know. had never traveled to t h e South Atlantic, sensibly choosing, instead, to collect data in t h e East Indies, and. yet, considering t h e same problem that had earlier vexed Darwin. h e had hit on precisely Darwin's explanation. The idea t h a t Wallace might hog t h e glory w a s too much for t h e melancholic Darwin: h e lumbered into print just months ahead of his rival; but in science. as elsewhere, even seconds count.

The theory t h a t Darwin proposed t o account for biological change is a conceptual mechanism of only t h r e e parts. It involves, in t h e first instance, t h e observation t h a t Living creatures vaxy naturally. Each dog ^isa member of a common species and thus dog-like to t h e bone; but every dog is doggish in his own way:

some are fast, others slow, some charming. and others bad from t h e first, suitable only for crime. Darwin wrote before t h e mechanism of genetic transmission w a s understood, but h e inclined t o t h e view t h a t variations in t h e plant and animal kingdoms arise by chance. and are then passed downward from fathers t o sons.

The biological world, Darwin observed, striking now for t h e second point to his three-part explanation. is arranged so t h a t what is needed for survival is gen- erally in short supply: food, water, space, tenure. Competition thus ensues, with every living thing scrambling to get his share and keep it. The struggle f o r life favors those organisms whose variations give them a competitive edge. Such is t h e notion of f i t n e s s . Fast feet make for fitness among t h e rabbits, even as a feathery layer of oiled down makes t h e Siberian swan a f i t t e r foul. A t any time. those creatures fitter than others are more likely t o survive and reproduce. The win- nowing in life effected by competition Darwin termed n a t u r a l selection.

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The L a n g u a g e of Ltfe 7

Working backward, Darwin argued t h a t p r e s e n t forms of life, various and wonderful as t h e y a r e , arose from common ancestors; working forward, t h a t biological change, t h e transformation of one species to a n o t h e r , is t h e result of small increments t h a t accumulate. s t e p b y inexorable s t e p , across t h e generations, until natural selection r e c r e a t e s a species entirely. The Darwinian mechanism is both random and determinate. Variations occur without plan o r purpose

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t h e luck of t h e draw; but Nature, like t h e House, is aggressive; organized t o cash in on t h e odds.

The Central Dogma

Everything t h a t lives, lives just once. To pass from f a t h e r s t o sons is t o pass from a copy to a copy. This is not quite immortality, even if c a r r i e d on forever, but i t counts for something. a s every p a r e n t knows. The higher organisms reproduce themselves sexually, of course, and every copy is copied from a double template.

Bacteria manage t h e matter alone, and so do t h e cells within a complex organism, which often continue t o grow and reproduce a f t e r t h e i r host has perished, unaware, for a brief time, of t h e gloomy c a t a s t r o p h e taking place around them. I t is possible, I suppose, t h a t each bacterial cell contains a tiny copy of itself, with t h e copy carrying y e t another copy; biologists of t h e early eighteenth c e n t u r y , i r r i t a t e d and baffled b y t h e mystery of i t all, actually thought of reproduction in t h e s e terms: peering into crude, brass-rimmed microscopes, t h e y persuaded themselves t h a t on t h e thin, stained glass, t h e y actually saw a homunculus; t h e more diligent proceeded t o s k e t c h what t h e y seemed t o see. The theory t h a t emerged had t h e g r e a t virtue of being intellectually repugnant. Much more likely, a t least on t h e grounds of reasonableness and common sense, is t h e idea t h a t t h e bacterial cell contains what Erwin Schroedinger called a code script

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a s o r t of cellular s e c r e t a r y organizing and recording t h e gross and microscopic features of t h e cell.

Such a code s c r i p t would b e logically bound t o double duty. As t h e cell divides in two, i t , too, would have to divide without remainder, doubling itself t o accommodate two bacterial cells where formerly t h e r e was only one. Divided. and thus doubled without loss, each code s c r i p t would require powers sufficient t o organize anew t h e whole of each bacterial cell. The code s c r i p t t h a t Schroedinger (1945) antici- pated in his moving and remarkable book, What is Life?

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h e wrote in t h e 1940s

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t u r n s out t o b e DNA, a long and sinewy molecule shaped r a t h e r like a spiral in two strands. The s t r a n d s themselves are made of stiff sugars, and stuck in t h e sugars, like beads in a sticky string, a r e c e r t a i n chemical bases: adenine, cytosine, gua- nine, and thymine: A, C, T, and G , in t h e now universal abbreviation of biochemists.

I t is t h e alternation of t h e s e bases along t h e backbone of D N A t h a t allows t h e molecule t o s t o r e information.

One bacterial cell splits in two: each is a copy of t h e first. All t h a t physically passes from cell t o cell is a s t r a n d of DNA: t h e message t h a t each generation sends faithfully into t h e f u t u r e is impalpable, a b s t r a c t almost, a kind of hidden hum against t h e coarse w e t plops of reproduction, gestation, and b i r t h itself. James Watson and Francis Crick provided t h e c o r r e c t description of t h e chemical struc- t u r e of DNA in 1952. They knew, as everyone did, t h a t somehow t h e bacterial cell

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in replication sends messages t o each of its immediate descendents. They did not know how. A s i t turned out, t h e chemical s t r u c t u r e of DNA. once elaborated. suggests irresistably a mechanism both for self-replication and t h e transmission of information. In t h e ceU itself, strands of DNA a r e woven around each other and by an ingenious t w i s t of biochemistry matched antagonistically: A with T. and C with G. A t reproduction. t h e ceU splits t h e double s t r a n d of DNA. Each half floats for a time, a gently waving genetic filament; chemical bonds a r e then repaired as each base fastens t o a new antagonist, one simply picked from the ambient broth of the cell and clung to, as in a single's bar. The process complete, t h e r e are now two strands of double-stranded DNA where before t h e r e w a s only one.

What this account does not provide is a description of t h e machinery by which the genetic code actually organizes a pair of new cells. To the biochemist, the bacterial cell appears as a kind of small sac enclosing an actively throbbing biochemical factory; i t s products a r e proteins chiefly

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long and complex molecules composed, in t h e i r turn, of twenty amino acids. The order and composition of the amino acids along a given chain determines which protein is which. The bacterial cell somehow contains a complete record of t h e right proteins, as w e l l as the instructions required t o assemble them directly. The sense of genetic identity that marks E. Coli a s E. Coli and not some o t h e r bug must thus be expressed in the amino acids by means of information stored in t h e nucleotides.

The four nucleotides, w e now know, are grouped in a triplet code of 64 codons o r operating units. A particular codon is composed of t h r e e nucleotides. The amino acids are matched t o t h e codons: C-G-A, for example, t o arginnine. In the transla- tion of genetic information from DNA to t h e proteins, t h e linear ordering of t h e codons themselves serves t o induce a corresponding linear ordering first onto an intermediary, messenger RNA, and then onto t h e amino acids themselves

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^{this via}

yet another messenger, transfer RNA. The sequential m n g e m e n t of t h e amino acids finally fixes t h e chemical configuration of t h e cell.

Molecular biologists often allude to t h e s t e p s so described as the C e n t r a l Clogmu, a queer choice of words for a science.

The dour Austrian monk. Gregor Mendel, founded t h e science of genetics on purely a theoretical notion of a gene, which h e likened to a bead on a string. In DNA. one looks on genetics bare: the ultimate unit of genetic information is t h e nucleotide. All that makes for difference. and hence for charm, in the natural world. and which is not t h e product of culture. art, artifice, accident, o r hard work, all this, which is brilliantly expressed in maleable flesh, is a matter of an ordering of four biochemical l e t t e r s along two ropey strands of an immemorial acid.

The Central Dogma describes genetic replication; but t h e concepts that it scouts plainly illuminate Darwinian theory from within. Whether as the result of radiation o r chemical accident, letters in t h e genetic code may be scrambled; one l e t t e r shifted for another; e n t i r e codons replaced, deleted, o r altered. These are genetic mutations: a r b i t r a r y , because unpredictable; and yet enduring, because they are variations in t h e g e n e t i c message. The theory by which Darwin proposed to account for t h e origin of species and the nature of biological diversity now admits of expression in a single English sentence. Evolution, o r biological change.

so the revised, t h e neo-Darwinian theory, runs, is t h e result of natural selection working on random mutations.

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The Language o,fLi,fe

PART Two

Evolutionary Theories

The popular view of evolution tends t o b e a tight s h o t on a tame subject: t h e dinosaur, who did not make i t ; t h e s h a r k , who did; b u t t h e maturation of an organism is itself much like t h e evolution of a species; only our intimate acquaintance with its precise and unhesitating c h a r a c t e r suggests, misleadingly, I think, t h a t t h e two processes differ in degree of freedom. Psychology, economics, urban affairs, anthropology, political science, and history also describe processes t h a t begin in a s t a t e of satisfying and undemanding simplicity, and e n d l a t e r with everything complex, unfathomable, chaotic. The c o n t r a s t to physics is sobering.

The dynamics of evolutionary theories are often divided into two conceptual stages. In economics, t h e r e a r e macro- and micro-economic theories, aggregate demand versus t h e t h e o r y of t h e firm: within linguistics, language a t t h e continuous level of speech. and language some levels below, d i s c r e t e , a matter of t h e concatenation of words o r morphemes. Biology, too, is double-tiered: above, t h e organism prances; unseen, below, a t a s e p a r a t e level, i t s life is organized around t h e alphabetic nucleotides.

Metric Spaces

By a metric s p a c e S I mean a space upon which a function

has been defined. assigning t o each pair of points s , s ' in S a nonnegative real number

-

t h e d i s t a n c e d ( s , s ')

-

and satisfying t h e usual axioms:

d ( s , s ' ) = d ( s l , s ) ; (9.2)

d ( s , s t )

+

d ( s P , s") l d ( s , s") ^, (9.3)

Double metrics

The d i s t a n c e between o r g a n i s m s

The disciplines of comparative anatomy and systematic zoology classify c r e a t u r e s into ever-larger s e t s and sets of sets: individuals (dogs, say), species, genera, families, o r d e r s , classes. phyla. taxa. and kingdoms. The classification itself forms an algebraic lattice, with individuals acting as t h e system's atoms.

Comparative anatomists and zoologists bring an exquisitely refined and elaborate intuition t o t h e task of sorting t h e various biological c r e a t u r e s into appropriate categories: t h e obvious cases leap t o t h e eye; a t t h e margins of t h e system, where t h e whale resides, difficult matters a r e . decided by r e f e r e n c e t o historical and comparative anatomy, parallel s t r u c t u r e , common organization, biological t r a i t s , and, often, levels of biological achievement. If t h e image of a lattice is f o r t h e

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moment taken literally, then each level of t h e lattice, from t h e atoms upward, comprises a set o r ensemble: of individuals. in t h e first instance. of sets of individuals, in t h e second. An ensemble a t any distinct level of t h e lattice, I assume, satisfies equations (9.1)-(9.3). and counts thus as a metric space.

Ths d i s t a n c e between s t r i n g s

DNA is a string drawn from a four-letter alphabet; proteins are strings of fixed length composed of 20 amino acids; as such, both strings belong t o a wider family of string-like objects: computer programs written in a given language, t h e sentences of a natural language. formal systems; and acquire b y osmosis a distinct conceptual and mathematical structure. I t makes little difference whether strings of DNA o r strings of amino acids a r e taken as fundamental; and. in any case. I often alternate between t h e two. By an alphcrbet A I mean a fixed and finite col- lection of elementary entities called mrds; by t h e u n i v e r s e of s t r i n g s over a finite alphabet, t h e set of all finite sequences A* whose elements lie in A .

The natural distance between words W

=

wl...w,, V

=

vl...v, ( W , V E A ) is In]

+

[nl

-

²^xlkl, where k is t h e maximum of t h e length of a word

U =

ul...ul, which is a subword both of W and V. For example. let W

=

cadbabbd. V

=

xcaaba.

An appropriate

U

is

U =

caab; hence ~ ( w . v

=

8

+

6

-

²^x^{k .}

Grantham (1974) has proposed a definition of distance in a Euclidean m e t r i c space of proteins based on properties of composition. polarity, and volume; but t h e theory of evolution suggests that changes in biological strings come about through mutations

-

random flash points a t which l e t t e r s a r e scrambled. Some strings may change in a large-hearted way, with whole blocks of letters wheeling and shifting like cavalry horses; but t h e least mechanism to which these operations may b e resolved is t h e simple one of erasure and substitution

-

deletion and insertion. The elementary processes of evolution a t t h e molecular level lend t o t h e natural metric a certain simple plausibility in t h e face of fancy competition. T

=

A * , then. is a t y p o g m p h i c metric spcrce; d T , its natural distance.

Metric spaces in phase

M and

P ,

suppose, are two metric spaces; g : M ^-+

M*

assigns t o each point p in M a distinct point

F

in

W .

M and

M*

_arein p h a s e under g if g acts roughly to preserve distances: for any (

>

0, t h e r e exists a 9

>

0, such t h a t for all p and q i n M

g is thus u n i j o m l y c o n t i n u o u s on M ; rp is, of course, a function of .$. I t often happens that a particular mapping between metric spaces is especially natural

-

f o r reasons that are not mathematical. The English alphabet. for example, makes for two metric spaces: strings of letters, sets of words. Strings of letters are close if they agree in spelling; words if they agree in meaning. Small typographic changes give rise to large differences in meaning: these metric spaces a r e not in phase. This observation is often regarded as a paradox in t h e context of theoretical biology. In an important and influential article, King and Wilson recount evidence showing t h a t chimpanzee and human polypeptide sequences a r e more than

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Th.e L a n g u a g e ofLtfe ¹¹ 99 p e r c e n t identical; t h e s p e c i e s appear f u r t h e r a p a r t than a comparative analysis of t h e i r polypeptide chains might otherwise suggest.[6]

Complexity

Complexity and simplicity, like Yin and Yang, a r e metaphysical duals; e x c e p t for a vagrant connection t o intuition, i t hardly makes a difference what is called which. Mathematicians and philosophers are i n t e r e s t e d in complexity for t h e i r own ends; so are theoretical biologists, who in t h e i r b e t t e r moments a r e quite capable of evincing a sense of Heraclitian awe when confronted with t h e intrica- cies of t h e protozoan swim bladder. Simple counting principles often seem as if t h e y might provide a general scheme f o r the,measurement of complexity. Suppose t h a t X is a nonempty set of objects and t h a t A , B, C,

. . .

are constructed from t h e elements of X by c e r t a i n specified operations

-

concatenation, for example. Can w e not t h e n say t h a t t h e complexity C ( z ) of any object is a measure of t h e number of i t s distinct elements and t h e s e p a r a t e and specifiable relations between them?

C ( z ) would b e a monotonically increasing function of t h e square of t h e number of distinct elements in any given construction. Simple, no? And intuitively satisfying?

Apparently not. Label t h e p a r t s of an ordinary watch in an obvious alphabetic fashion; and t h e binary relations between i t s p a r t s a s well. The watch when working, l e t me suppose, h a s a complexity measured a t C ; but so, t h e n , does t h e watch when not working

-

when not assembled. in f a c t , binary relations being free f o r t h e asking. Examples of this s o r t , when extended and made precise, suggest ultimately t h a t any complex object belongs t o an embarrassingly large equivalence class of objects precisely equal in point of complexity.

Statistical mechanical complexity

A system of identical particles moving within a fixed, bounded, and finite volume of space constitutes a c o n f i g u r a t i o n ; never having seen t h e blue smoke from a cigar spontaneously collect in but one c o r n e r of a warm room, t h e thought- ful physicist

-

pipe, slippers. Beagle-eyes, an a i r of e a r n e s t confusion

-

concludes t h a t not all configurations are equally probable; y e t if t h e r e are N configurations Pr(Nc)

=

Nc / N

-

this f o r e a c h i

.

This incompatibility between what one sees and what one gets is known a s Boltzmann's p a r a d o z , an unhappy name if only because no real paradox is forthcoming; but an unhappiness nonetheless. Distinct conf i- gurations, Boltzmann argued, may b e grouped into s t a t e s ; what t h e altogether more elegant Gibbs called ensembles. Within thermodynamics

-

statistical mechanics from above

-

t h e e n t r o p y S of a system a p p e a r s perpetually in t h e ascendancy and tends inexorably t o a maximum; statistically, Boltzmann reasoned, S is thus proportional t o

S = k log W ; (9.4)

where k is Boltzmann's constant, and W a measure of those configurations compati- ble with a given s t a t e

-

complezions a s t h e y a r e called in old-fashioned t e x t s . Configurations a r e alike in point of probability: not s o complexions; t h e probabil- i t y of finding a mechanical system in a given s t a t e is proportional t o t h e number of

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distinct complexions realizing t h a t state. A t equilibrium, the complexions a r e a t a maximum; and so, too, t h e entropy. which functions as a kind of ectoplasmic measure of r a n d o m n e s s o r d i s o r d e r .

C o m p l d t y under a classification

Statistical mechanics has a good point t o its credit, and implies a second.

Certain s t a t e s of a physical system may be multiply realized; their number, if counted, makes for a measure of sorts. What is measured within statistical mechanics is plainly not complexity; t h e description of entropy as disorder serves only t o explain t h e whole business to t h e baffled undergraduate. with t h e explanation rapidly withdrawn by t h e time he e n t e r s graduate school. Still. I am struck by t h e extent to which t h e mathematical definition of entropy is made possible by an enterprising reorganization of t h e way in which mechanical systems are classified;

in assessing complexity, a concept with a brutish family resemblance to disorder, the classification may well come first.

An example? Of course. I shall pass glowing colored slides about shortly. Con- sider t h e set of aLl functions

f

:Rn

-.

R. Those smooth functions whose critical points are nondegenerate a r e known as Morse f u n c t i o n s and a r e a t once open, dense, and locally stable in C"(Rn, R). Any Morse function may be expressed in canonical form: if z is a critical point of f , t h e r e exists a number k such t h a t in a neighborhood of z , and a f t e r a suitable change in coordinates,

Such is Morse's lemma. Their mathematical docility suggests that t h e Morse functions are simple, if anything is; but t h e Morse functions are simple because t h e y are Morse functions, and not Morse functions because they a r e simple; simplicity is a derivative quality, like color, contingent upon a classification. and unremarked otherwise.

The concept of a degenerate singularity makes for a simple classification on t h e space of smooth functions c m ( R n , R); but a set of objects may be simple under a classification even if t h e classification is itself unpleasantly complex. Writing some years ago, Smale asked whether t h e r e exists a least Baire s e t

U

in t h e space of all dynamical systems Dyn(M) on a compact manifold M , whose elements might be qualitatively described "by discrete numerical and algebraic invariants".[?] The question as posed admitted of a simple answer: no. What is needed, Smale later con- cluded, is a sequence of nested subsets

UC

CDyn(M)l, where k is relatively small.

UC

open, and

Uk

dense. As i increases. more of Dyn(M) is swallowed; as i decreases, stability and regularity properties come t o t h e fore. I t is for

U1

that Axiom A is satisfied, nonwandering sets are finite, and the transversality condition is m e t .

U l

thus consists of "the simplest. best-behaved, nontrivial class of dynamical systems"; but nothing in Smale's organization of Dyn(M) is simple a t all.

A set is absolutely simple under a classification if it is a t once open, dense, and locally stable; under this definition simplicity does not come in degrees.

Often. suitable sets t u n out to be merely of t h e first Baire category, t h e best one can do; sets t h a t a r e dense need not be stable, and vice versa. First category sets and sets of measure zero coincide in t h e case of countable sets; but not beyond.

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The Language of LVe ¹³ From t h e point of view of statistical mechanics, simplicity and complexity a r e concepts t h a t involve configurations; complexity under a classification is a matter of routine: what is complex is singular, unusual. These notions may b e brought into alignment

-

but only for a certain class of objects. An object A is d i s s e c t i v e only when i t may b e decomposed to a finite stock of p a r t s in a finite number of s t e p s . The mammalian e y e is a dissective s t r u c t u r e ; so is t h e whole of a mouse. a moose, o r a mole; but curves and concepts, t h e r e a l numbers, t h e coast of Britain, sea-green sea-waves, and, p e r h a p s , t h e e n t i r e bizarre universe of element'ary particles, a r e indissective. A dissective object is thus composed of i t s p a r t s taken together under a c e r t a i n distinctive relationship. Say t h a t A is composed of al, a z ,

...

,an under R. By a relational alternative t o R I mean a single permutation of t h e p a r t s of A . If A , for example, contains but two p a r t s , a and b , say, under t h e relationship R (a, b ), R (b , a ) is a relational alternative t o R

-

t h e only one in fact.

Given R , I denote by R* t h e full s e t of all relational alternatives t o R. If A is dissective i t is R* t h a t forms i t s complexion class: t h e s e t of all s e t s of i t s p a r t s under all and only t h e i r relational alternatives.

An elementary partition of a complexion class splits t h e class a s a whole into equivalence classes; relative t o a partition, complexity and simplicity a r e a t t r i - butes of equivalence classes, and are judged simply by size. To t h e e x t e n t t h a t [Ec] is larger than [E,], i t is simpler a s well; and vice versa. Almost all s t r u c t u r e s in theoretical biology may b e dissected t o a finite. although very large, base; in this sense, biological complexity and simplicity have pliant finite measures.

The mammalian eye, for example, is a dissective s t r u c t u r e . I t s p a r t s (on one level of dissection, a t least) a r e proteins, which a r e arranged in various delicate and precise ways. I am ignoring, now, any dynamic considerations and thinking instead of t h e mammalian e y e a s a s t a t i c object. The complexion class t o t h e mammalian e y e consists of all and only those rearrangements of proteins t h a t comprise relational alternatives t o t h e mammalian e y e itself.

What makes an e y e distinctively an eye, r a t h e r than some assembly of jelly- like proteins, is obviously t h e f a c t t h a t i t is capable of sight. This invocation of function sounds an unavoidably Aristotelian note; but without some concept such as function o r purpose, theoretical biology loses much of i t s point. Let m e partition t h e relational alternatives t o t h e mammalian e y e into equivalence classes on t h e simple basis of function. In t h e full complexion class, those s t r u c t u r e s t h a t a r e capable of sight f a l l t o one side; and those t h a t a r e blind and s t a r e sight- lessly, fall to t h e o t h e r . Complexity and simplicity a p p e a r a s matters of relative size: t h e larger t h e equivalence class, t h e simpler t h e s t r u c t u r e s . Given t h e deli- cacy of t h e mammalian e y e , most of i t s relational alternatives will b e incapable of sight; like t h e Morse functions, t h e s e complexions a r e simple s t r u c t u r e s ; but again, simple because t h e y a r e sightless, and not sightless because t h e y a r e simple.

Complexity

in

strings

Of t h e 2" binary sequences of length n , some, such a s

o , o , o , o , o

,...

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seem simpler than others,

f o r example; yet t h e most natural probability distribution over t h e space of n - place binary strings assigns to both t h e same probability: 2-. It goes against t h e grain. mine. a t any rate, t o reckon (9.6) as likely as (9.7). especially when n is large; but nothing in t h e sequences themselves indicates obviously the point of distinction.

The goal of science, Ren4 Thom has suggested. is t o reduce t h e arbitrariness of description; substitute data f o r description. and t h e apothegm gains my assent.

A law of nature is data made compact:

F =

m a , said once and for all. the whole of an observed o r observable world compressed into just four symbols. A series of observations compactly described is rational; if rational. not random. This curious but compelling chain of deductions prompted Kolmogorov t o argue t h a t randomness in binary sequences o r strings might be measured by t h e degree t o which such strings admit of a simpler description.[8] In following this line, Kolmogorov took t h e first s t e p toward severing information theory from i t s unwholesome connection to the theory of probability. If S is a binary string its length is measured in bits:

an n-place binary string is n bits long. By a s i m p k r d e s c r i p t i o n of S, Kolmo- gorov meant a string D s h o r t e r than S such that D describes S by acting as t h e input to a fixed computer t h a t generates S. Strings t h a t cannot be compactly described are complaz. r a n d o m , o r i n . o m a t i o n - r i c h ; strings t h a t can, are not;

of these adjectives, only t h e second preserves even a vagrant connection between t h e concept t h a t it connotes and what is being measured. This r a t h e r inelegant idea makes plain t h e felt difference between a string of n 0s. and a mixed string.

Sequence (9.6). for example, may be expressed by a program. speaking loosely, whose length is log2n

+

C. If n

=

32, log2n = 5 : t h e relevant instruction is simply to write o r compute 0 '2 times. C measures what little is needed to c a r r y out t h e instructions; 32

-

5

=

27, t h e compactness of t h e program. The shortest pro- gram that computes a mixed sequence such as sequence (9.7). by way of contrast.

may w e l l be close to 32 bits in length: to compute t h e sequence, t h e computer must f i r s t store i t precisely.

The details? They have been changing since Kolmogorov first spoke, oracle- like, on t h e subject in a note published in 1967; like a snake engulfing an egg, t h e theory of recursive functions is engaged in swallowing algorithmic information theory, a development t h a t I deplore, but accept as inevitable. Consider the s e t of all n-place binary strings A* over a binary alphabet A and l e t T M be a fixed computer

-

a Turing machine. say; g is a general input-output function on ?"M mapping strings onto strings. The complezity of a string S of length n is the Length of the shortest binary string D t h a t generates S under T M by means of ^{g .}What- ever the complexity of S, D will plainly be maximally complex. and. hence.

entirely random. Otherwise. i t would not be t h e shortest description of S.

All

fin- i t e length strings quite obviously have a finite measure of complexity; and only fin- itely many distinct strings of t h e same length have the same finite measure of complexity. Quite surprisingly. t h e decision problem for complexity is recursively unsolvable; this result follows almost directly from t h e unsolvability of the halting problem for Turing machines. Like t r u t h , randomness is a property that remains ineluctably resistant to recursive specification.

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The Language of Life 15

If all else fails, a binary sequence of length n may b e generated b y a binary sequence of length n : t h e r e a r e 2" such algorithms, and 2"

-

²algorithms s h o r t e r than this. On any reasonable interpretation of complexity, algorithms within a fixed integer k of n itself must b e reckoned random o r complex o r nearly so. Thus 2n -k -1

-

2 / 2n algorithms have a complexity less than n

-

^k^;and are hence nonrandom o r simple. If k

=

10, this ratio is roughly 1 in 1000; of 1000 binary sequences of length n , only one can b e compressed into a program more than t e n bits s h o r t e r than itself. Hence:

Theorem 9.1 The s e t of random sequences of length n in t h e space A* of all binary sequences of length n is generic in A * .

These random sequences a r e simple under a classification because t h e y a r e typi- cal, but complex in a s t r o n g e r and more absolute sense because t h e y a r e random o r information-rich. In this context, genericity is a finite measure of size. The number of purely random strings grows exponentially with n , of course. If most binary sequences a r e random, t h e appearance of sequence (9.6) prompts a natural stochastic surprise: sequences such a s (9.7) a r e what one expects. The definition of Kolmogorov complexity may b e directly extended t o recursively enumerable s e t s ; s e t s of strings especially, and hence languages.

Language-like

Systems

When i t comes t o language, t h e r e is syntax and semantics. Phonetics is t h e province of t h e specialist; pragmatics remains a pale albino dwarf. To semantics belongs t h e concept of meaning; t o syntax, t h e concept of a well-formed formula or a grammatical sentence. The r e f e r e n c e t o logic is happy if only because i t highlights t h e f a c t t h a t language-like systems go beyond t h e natural languages.

Any language no doubt e x i s t s primarily to convey meaning; b u t meaning in mathematics is a matter of a model

-

an extrinsic object.

The construction of strings within a language-like system involves con- catenating o r associating simpler strings: any finite string may b e dissected t o a finite s e t of least elements. Going up, concatenation; going down, finite dissection;

retrograde motion of this s o r t suggests t h a t language-like systems on this level b e r e p r e s e n t e d algebraically as semigroups. Let A b e any nonempty s e t of objects

-

words, for example, o r l e t t e r s , o r numbers. A has t h e s t r u c t u r e of a semigroup if t h e r e e x i s t s a mapping A x A ⁺A such t h a t f o r alI a , 6 , and c in A

In English words go over t o sentences from left t o right; in Hebrew, from right t o left; b u t in any case, one s t e p a t a time. Let A b e a finite s e t of words now, with words understood implicitly as t h e least elements of a natural language; and let A*

b e t h e s e t of all finite sequences ( a l,..., a n ) whose elements lai,

...,

^{a n}j lie in A . To endow A* with t h e s t r u c t u r e of a semigroup, i t suffices t o define an associative mapping A* ^xA* ^-rA* ^:easy enough. If

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and

then

where

A* is a t once a free-semigroup over a finite alphabet and a u n i v e r s a l Language:

no sequences a r e left out.

Almost a l l language-like systems a r e large in t h e sense that they have many distinct strings. Meditating on the matter in the late 1950s. and regularly thereafter. Noam Chomsky argued t h a t every natural language is infinite by virtue of its recursive mechanisms

-

conjunction and alternation. for example

-

^and.

simultaneously, that such mechanisms are recursive by virtue of t h e fact that every natural language is infinite. Both halves to this argument. taken together, describe a closed circle in space. Whatever the truth, language-like systems. if they are infinite, a r e countably infinite and no bigger.[9]

Going further toward a definition of a language-like system involves t h e bad- lands beyond triviality. Linguistics, t h e French linguist Maurice Gross once provo- catively remarked, admits of but a single class of crucial experiments. Native speakers of a given language a r e able to determine whether a given sentence is grammatical. Experiments of this sort exist because no language-like system encompasses the whole of a s e t of strings drawn on a finite alphabet

-

a curious and interesting jbct. which t h e sheer concept of communication might otherwise not suggest. The distinction between grammatical and ungrammatical strings induces a primitive classification on a language-Like system; and reflects an even stronger principle of fastidiousness: the vast majority of language-like strings a r e not grammatical a t all and represent syntactic gibberish. The fastidiousness of Language-Like systems is yet again a fact: it would b e easy. if unrewarding, t o design an artificial language in which most strings were grammatical. From the point of view of grammar, t h e strings of a natural language a r e complex under the classification of strings into grammatical and ungmmmatical sets. With the strings arrayed in front of the mind's bleak and rheumy eye, in ascending order, by length, with sets of strings stacked like an inverted pyramid, the grammatical strings in a language-like system appear as nothing more than a thin smudge; they a r e thus complex under this classification because they a r e singular, unusual.

The origins of this bit of natural history a r e to be discovered, no doubt, in the algorithmic properties of t h e human brain: in order t o store a natural language.

the brain must first represent i t

-

in the form of recursive rules, for example.

This suggests that language-like systems a r e low in point of Kolmogorov complexity; and from this point of view, s i m p l e .

A natural language, I have already observed, realizes two metric spaces (cf. p 240): but the informal example that I gave involved the concept of meaning, and

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The Language of Lire 17

not grammar. No matter: t h e point c a r r i e s over t o t h e case a t hand

-

and comprises t h e t h i r d of t h r e e queer natural f a c t s t h a t nothing in t h e concepts of grammar o r communication obviously implies. Thus, l e t T b e a typographic metric space of strings under t h e natural metric; t h e same s e t of strings comprises a second metric space u n d e r t h e degenerate distance function d * : if s and s ' are both grammatical, d * ( s , s')

=

0 ; if not, d* ( s , s f )

=

^m.These are t h e natural and (degenerate) g r a m m a t i c a l metric spaces of a language-like system. In a language- like system, natural and grammatical metric spaces a r e plainly n o t in phase.

Two models of generation

Linguistics is a rebarbative, h a i r s h i r t of a subject; and grammar a vexing p r o p e r t y . Linguists, f o r reasons of t h e i r own, a r e often i n t e r e s t e d in t h e weakest of generative devices t h a t specify all and only t h e sentences of a natural language.

R e p r e s e n t a t i o n by g r a m m a r

A p h r a s e s t r u c t u r e g r a m m a r is a quadruple G

=

(A, T , S, P), where A is some finite alphabet of symbols; T, a distinguished s u b s e t of A

-

t h e s e t of so- called terminal symbols; S, a distinguished initial symbol; and P, a f i n i t e s e t of production rules of t h e form u ⁺v ; u is a nonempty s e t of nonterminal symbols, and v some specified s t r i n g of c h a r a c t e r s . The s e t of all strings of terminal symbols constitutes a p h r a s e s t r u c t u r e l a n g u a g e

-

a p r o p e r s u b s e t of t h e s e t of all strings A* defined over A

.

By a c o n t e x t - p e e production rule. I mean one in which u may occur in any c o n t e x t

-

in effect, a rule in which u figures in isolation. Correspondingly, t h e r e a r e context-free grammars.

Example 9 . 1 L e t A

=

( a , b ) , T

=

( a , b ) , and P b e t h e two rules S + a b ; S ⁺US.

This grammar generates all and only t h e strings of t h e form a n b ".

R e p r e s e n t a t i o n by s y s t e m s of e q u a t i o n s

Consider t h e context-free grammar G whose production rules are S ⁺&a, and S ⁺c , where T

=

( a , c ), and S is an initial symbol. Let t h e variable f range over terminal symbols. The action of t h e production rules may b e mimicked by an equation:

where addition is construed a s s e t t h e o r e t i c union. For G ,

Replacing S by ^SO

=

c ,

s(') =

a c a

+

c

.

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D. Berltnskt

This process repeated ultimately yields a system of equations

t =n

s ( ~ ) ⁼

a n c a n

+ - . +

^{a c a}

+

c

= C

a t c a i

.

t =m

A t t h e Limit, t h e solution s ( - ) )=

C

a f c a ' is given by a formal power series in

t

*

noncommutative variables. [lo]

A language-like system has f i r m a l s u p p o r t when each and every string in t h e system may b e described by a single algorithm; only for context-free languages may grammars and systems of equations be balanced against each other.

~ l s e w h e r e , t h e situation is darker. There is a sense, however, in which these two representations exhaust t h e possibilities for t h e description of structured and infinitary objects; and correspond. in t h e Metaphysical Large, to t h e alternatives confronting an imaginary Deity in creating t h e observable world.

Weak Theories

The vitalist believes t h a t life cannot b e explained in terms of physics o r chemistry. In t h e nineteenth century, in Germany and France, a t least, his w a s t h e dominant voice before Darwin; and natural philosophers. such as Cuivier o r von Baer, o r Geoffrey St. Hilaire, dismissed mechanism with a kind of troubled confidence that suggests. in retrospect, a combination of assurance and wistful- ness. Orthodoxies have subsequently reversed themselves with no real gain in credibility. David Hull, in surveying this issue. concludes that neither mechanism nor vitalism is plausible, given t h e uninspiring precision with which each position is usually cast.[ll] D'dccord. To t h e e x t e n t t h a t t h e refutation of vitalism involves t h e reduction of biological to physical reasoning, t h e effort involved appears to m e misguided. and reflects a discreditable, almost oriental, desire for t h e Unity of Opposites. On t h e standard view of reduction, t h e sciences collapse downward until they hit physics: Rez-da-Chausee; but our intellectual experience is divided: mathematics. physics, biology, t h e social sciences. Each science extends sideways for some time and then simply stops. The a r d e n t empiricist, surveying t h e contemporary scene, might w e l l incline to scientific polytheism, with mathematics under t h e influence of an austere Artin-like figure, and biology directed by a God much like Wotan: furious, bluff. subtle, devious, and illiterate.

Still, t h e philosopher of science is bound to wonder why so many philose phers have remained partial to t h e reductionist vision, and hence to mechanistic thought in biology. David Armstrong. J.J.C. Smart, Michael Ruse, and even t h e usually cagey W.v.0. Quine, call on elegance t o explain their attachment. Were t h e sciences irreducibly striated, one set of laws would cover physics, another biology, and still a third, economics and urban affairs, with t h e whole business resembling nothing so much as a parfait in several lurid and violently clashing colors. This is an aesthetic argument, and none t h e worse for that, but surely none t h e b e t t e r