Berlinski paper presented 1985 at Applied systems analysis

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Page 1
NOT FOR QUOTATIOK
WITHOUT PERMISSION
CF THE ALTEOR
THE LANGUAGE OF
LIFE
David Berlinski
April 1985
CP-85-2C
Collaborative Fkpers report work which has not been performed
solely' at the International Institute for Applied Systems Analysis
and which has received only SIcited review. Views or opinions
expressed hereir. do not necessarily represent those of the Insti-
tute, i+h Nzti0r.d Member Orgizations,
or other organizatios
supporting the work.
INTERNATIONAL IYSTTTVE FOR A.?PLID SYSTEMS ANALYSIS
2361 Laxenburg, Austria
Page 2
Page 3
FOREWORD
This paper represents the written version of a lecture given at IIASA
in Sep-
tember 1984 under the auspices of the Science
&
Technology and the Regional
Issues projects. In its current form it will appear as a chapter in
the forthcoming
IIASA book,
Complexity, Language and Life: Mathematical Approaches,
J . Casti
and A. Karlqvist. eds.
Boris Segerstahl
Leader
Science
&
Technology Program
Page 4
Page 5
ABSTRACT
This paper explores the idea that life comprises a language-like
system. The
arguments a r e carried out against the background of the neo-
Darwinian theory of
evolution. The principal conclusion is the dilemma that if life is a
language-like
system, then certain concepts a r e missing from the Darwinian
paradigm; if not,
then Darwinian thought is suspicious in the sense that its principles
do not natur-
ally apply to cognate disciplines.
Page 6
Page 7
Dnvid Berlinski i s a Researcher at the Institut des Hautes E'tudes
Scientifique,
Bur-sur-yvette. Paris, France.
-
vii
-
Page 8
Page 9
The Language
of
Life
David
Berlinski
In the spring of 1984, I delivered two lectures at IIASA under the
title
The
Language of Life. Dianne Goodwin was kind enough to prepare a verbatim
tran-
script of my talks; I have used the months since then to purge the
written record
of what I said of its incoherence, vagrant inaccuracies, and general
slovenliness.
This chapter is at once long and terse
-
an unhappy combination, and one
that makes severe demands of the reader. Many arguments are highly
compressed
and must be elaborated before they appear convincing. I have not
hesitated to
make use of mathematical concepts in expressing myself; but I draw no
mathemati-
cal conclusions. I thus run the risk of alienating the general reader
even as I anta-
gonize the mathematician. For these reasons, it may be helpful if in
this introduc-
tion
I
endeavor to place this chapter in a somewhat wider personal and
intellectual
context.
As it stands, The Language of Life represents a draft of one-third of
a
larger work entitled Language, Life a n d Logic. Another part of that
more ambi-
tious project was delivered at IIASA two years ago as a set of
lectures. The written
record of those lectures, which I hope to publish separately as a
working paper, is
entitled Classification a n d
its
Discontents.
My aim in Language, Life a n d Logic is to explore a certain
complicated com-
mon ground that holds between language, on the one hand, and the
graphic arts,
on the other. These are the classic systems of representation of the
human imagi-
nation. In both, there is a curious division between the system's
syntactic and
semantic structures: a theory, for example, consists of a finite set
of sentences,
the sentences of words; paint and then pigment comprise a painting;
and yet,
words and sentences, paints and pigments, manage, somehow, to cohere
and. then,
in a miraculous act of self-transcendence, to make contact with a
distinct and dif-
ferent external world. The problems of theoretical biology, it might
seem, have
nothing much to do with issues that arise in the philosophy of
language or the phi-
losophy of art. Not so. A gene comprises a linear array of nucleotides
that under
certain conditions expresses a protein or set of proteins. The
proteins, in turn,
Page 10
2
D.
Berlfnskt
are organized to form a structure
as
complicated as a moose or a mouse. The
nucleotides are plainly alphabetic or typographic in character; the
organism itself
is rich, complex, complete, continuous, unlike an alphabet. How is it,
then, that
such typographic structures as DNA manage to express so much that is
not typo-
graphic at
all?
This
is
a question quite similar to questions that might be raised
about language itself, or works of the graphic
arts;
and when it
is
pursued, certain
metaphors and quite peculiar images begin drifting from one subject to
the other.
There is the notion of meaning, of course, which
is
common to language,
art,
and
life; but also the idea that life is itself a language-like system; or
that
art
is
organic. The relations of satisfaction, representation, and
expression. while for-
mally distinct. of course, nonetheless display points of contact. In
order to
explain how it is that a painting may represent a face, for example,
one has
recourse to the notion of a metaphor, a concept from the 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 to the class of interpreted or sigmficant
typographic objects. It
is for this reason that it has seemed to me profitable to explore some
of the con-
cerns of theoretical biology and the philosophy of art and language in
a single
volume.
Within the context of
this
chapter, my aim
is
to explore the ramifications of
a controlling metaphor: the idea that life comprises a language-like
system. I do
this against the background of the neo-Darwinian theory of evolution
-
the most
global and comprehensive scheme of thought in theoretical biology. My
argument
at 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 the sense that its principles do not
naturally apply to cog-
nate disciplines. The intellectual pattern to this chapter
is
thus one of movement
between two unyielding points, a kind of whiplash.
Part One establishes the historical and contemporary background to
D d n i a n thought; and makes the argument that much of biology
cannot be
reduced to physics. In Part Two. I consider the confluence of certain
concepts:
distance in the metric 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 the
hypothesis that molecular biological words are high in Kolmogorov
complexity
-
with results that are inconclusive. In Part Four, I examine evolution
or biological
change
as
a process involving paths of proteins. The discussion
is
set
in
the
mathematical contexts of ergodic theory and information theory. In
many
respects, the classical concepts of information and entropy are most
natural in
discussing topics such as the generation of protein paths by means of
stochastic
devices; but there
is
a connection between Kolmogorov complexity and entropy in
the 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 the points that 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 to
publish jointly a monograph on theoretical biology.
This
has
not come to pass. Still,
Page 11
The
Language
0fLt.e
3
to the extent that my ideas are interesting, they are his; to the
extent that they
are not. they
are
mine.
John Casti read the penultimate draft of this essay and discovered any
number of embarrassing errors.
I
am grateful for his stern advice, which I have
endeavored to heed.
PART
ONE
A
System of Belief
The natural thought that theoretical biology comprises a kind of
intellectual
Lapland owes much to the idea that biology itself is somehow a
derivative science,
an analogue to automotive engineering or dairy management, and, in any
case,
devoid of those special principles that lend to the physical or
chemical sciences
their striking mahogany lustre. This is the position for which J.J.C.
Smart (1963)
provided a classic argument in Philosophy and Scientific Realism.
[l]
Analytic
philosophers, for the most part, agree that nothing in the nature of
things com-
pels them to learn organic chemistry; Feyerabend. Putnam, and Kuhn
have won-
dered whether any discipline can properly be reduced to anything at
all; and,
then, whether anything is ever scientific, at least in the old-
fashioned and hon-
orific sense of that tem.[2] Naive physicists
-
the only kind
-
are
all too happy
to hear that among the sciences physics occupies a position of
prominence denied,
say, to urban affairs or agronomy. The result is reductionism from the
top down,
a crude but still violently vigorous flower in the philosophy of
science. The physi-
cist or philosopher, with his eye fixed on the primacy of physics,
thus needs to
sense in the other sciences
-
sociology, neurophysiology, macrame, whatever
-
intimations of physics, however faint. This is easy enough in the case
of biochemis-
try: chemistry is physics once removed; biochemistry, physics at a
double dis-
tance. Doing biochemistry, the theoretician is applying merely the
principles of
chemistry to living systems: like the Pope, his is a reflected
radiance.
In 1831, the German chemist Uriel Wohler synthesized urea, purely an
organic compound
-
the chief ingredient in urine, actually
-
from a handful of
chemicals that he took from his stock and a revolting mixture of dried
horse
blood. It was thus that organic chemistry was created: an inauspicious
beginning,
but important, nonetheless, if only because so many European chemists
were con-
vinced that the attempt to synthesize an organic compound would end
inevitably in
failure. The daring idea that all of life
-
I am quoting from James Watson's text-
book (1965), The Molecular Biology
of
the Gene
-
will ultimately be understood in
terms of the "coordinative interaction of large and small molecules"
is now a com-
monplace among molecular biologists, a fixed point in the wandering
system of
their theories and beliefs. The contrary thesis, that living creatures
go quite
beyond the reach of chemistry, biochemists regard with the alarmed
contempt
they reserve for ideas they are prepared to dismiss but not discuss.
Francis
Crick, for example, devotes fully a third of his little monograph, Of
Molecules and
Men, to a denunciation of vitalism almost ecclesiastical in its
forthrightness and
Page 12
utter lack of detail.[3] Like other men, molecular biologists
evidently derive some
satisfaction from imagining that the orthodoxy they espouse is
ceaselessly under
attack.
Theoretical biologists still cast their limpid and untroubled gaze
over a world
organized in its largest aspects by Darwinian concepts; and so do high-
school
instructors in biology
-
hardly a group one would think much inclined to the idea
of the survival of the
fittest;
but unlike the 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, or political science. Curiously enough. while
molecular gene
t-
ics provides an interpretation for certain Darwinian concepts
-
those differences
between organisms that Darwin observed but could not explain
-
the Darwinian
theory resists reformulation in
terms
either of chemistry or physics.
This
is
a
point apt to engender controversy. Woodger, Hempel, Nagel; and Quine
cast reduc-
tion
as
a logical relationship: given two theories, the first may
directly
be
reduced to the second when a mapping of its descriptive apparatus and
domain of
interpretation allows the first to be derived from the second. I am
ignoring
details. now. The standard and, indeed. the sole example of reduction
successfully
achieved involves the derivation of thermodynamics from statistical
mechulics. In
recent years, philosophers have come to regard direct reduction with
some unhap-
piness. There are problems in the interpretation of historical terms:
the
Newtonian concept of
mass.
for example; and theories that 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 to
five points.[4] By a
corrected theory,
he means a theory logically revived to bring
it into conformity with current interpretations: Newton upgraded. for
example.
His
general scheme for reduction. then,
is
this:
(1)
All
of the terms in the corrected theory must be matched to terms in the
reducing theory
-
a requirement of
completeness.
(2)
The corrected theory must be deducible
f
rom the larger theory, given the
existence of suitable reduction functions
-
a requirement of
derivabilit y.
(3) The corrected theory must indicate why the original theory was
incorrect
-
a requirement of epistemological
insight.
(4)
The original theory must be explicable in terms of the reducing theory
-
a
requirement of
cogency.
(5)
The original and corrected theories must resemble each other
-
a require-
ment of intellectural
symmetry.
In the case of theoretical biology. to speak crisply of deriving, say,
molecular
biology from biochemistry
is
rather like endeavoring to cut steel with butter:
there
is
a certain innocence to the idea that molecular biology has anything
Like a
discernable logical structure. What one actually sees is a mass of
descriptive
detail, a bewildering plethora of hypothetical mechanisms. much by way
of anecdo-
tal
evidence, a few tiresome concepts, and an array of metaphors drawn
from phy-
sics, chemistry. information theory, and cybernetics. The definition
of reduction
just cited
is,
in addition. incomplete. its flagrant inapplicability aside. In Men-
delian genetics, the concept of a gene is theoretical. and genes
figure in that
Page 13
The Language 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 are
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, informa-
tion. complexity, replication, self -organization, stability. negative
entropy
(grotesque on any reckoning), transformation, regulation, feedback,
and control
-
the stuff required to make molecular biology work
-
are 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 are essentially biochemical, or even vaguely
physical? Hardly.
The usual Darwinian concepts of fitness and selection appear
unvaryingly in place.
These are 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
-
anything but fitness and natural
selection. To Schaffner's list of five, then, I would add a sixth: no
reduction by
means of inflation
-
a contingent and cautionary restriction that, for the time
being at least, enforces a stern separation between biology and
mathematical phy-
sics.
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
..
Those
biologists who look forward to the withering away of biology in favor
of biochemis-
try and then physics are inevitably neo-Darwinians, and the fact that
this
theory
-
their theory
-
is impervious to reduction they count as an innocent incon-
sistency. If mathematical physics offers a vision of reality at its
most comprehen-
sive, the Darwinian theory of evolution, like psychoanalysis, Marxism,
or the
Catholic Faith, comprises, instead, a system of belief. Like Hell
itself, which is
said to be protected by walls that are seven miles thick, each such
system looks
especially sturdy from the inside. Standing at dead center, most
people have con-
siderable difficulty in imagining that an outside exists at all.
The Historical Background
Charles Darwin completed his masterpiece, On the Origin of Species, 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
Page 14
of the South Atlantic
as
a naturalist aboard
H.M.S. The Beagle.
The stunning diver-
sity of plant and animal life that he saw there impressed him deeply.
Prevailing
biological thought had held that each species
is
somehow fixed and unalterable.
Looking backward in time along a line of dogs, it
is
dogs all the way. Five years in
the South Atlantic suggested otherwise to Darwin. The great shambling
tortoises
of the Galapagos, surely the saddest of all sea-going creatures, and
countless sub-
species of the common finch. seemed to exhibit a pattern in which the
spokes of
geographic variation all radiated back to a common point of origin.
The detailed
sketches that Darwin made of the Galapagos Finch. which he later
published in
Oh
the Origin of Species,
show what caught his eye. Separated by only a
few
hun-
dred miles of choppy ocean, each subspecies of the finch belongs to a
single fam-
ily; and yet. Darwin noted. one group of birds had developed a short.
stubby beak;
another, living northward, a long, pointed. rather Austrian sort of
nose. The varia-
tions among the finch were hardly arbitrary: birds that needed long
noses got
them. By 1837. Darwin realized that what held for the finch might hold
for the
rest of life and this, in turn, suggested the dramatic hypothesis that
f a r
from
being fixed and frozen. the species that now swarm over the surface of
the Earth
evolved
from species that had come before in a continuous, phylogenetic,
saxophone-like slide.
What Darwin lacked in 1837 was a theory to account for speciation, but
the
great ideas of fitness and natural selection evidently came to him
before 1842, for
by 1843 he had prepared a version of his vision. and committed
it
to print in the
event of his death. He then sat on his results in an immensely slow,
self-satisfied.
thoroughly constipated way until news reached him that A.R. Wallace
was about to
make known
his
theory of evolution. Wallace. so far as
I
know. had never traveled
to the South Atlantic, sensibly choosing, instead, to collect data in
the East
Indies, and. yet, considering the same problem that had earlier vexed
Darwin. he
had hit on precisely Darwin's explanation. The idea that Wallace might
hog the
glory was too much for the melancholic Darwin: he lumbered into print
just months
ahead of his rival; but in science. as elsewhere, even seconds count.
The theory that Darwin proposed to account for biological change
is
a con-
ceptual mechanism of only three parts. It involves, in the first
instance, the
observation that Living creatures vaxy naturally. Each dog
is
a member of a com-
mon species and thus dog-like to the bone; but every dog is doggish in
his own way:
some are fast, others slow, some charming. and others bad from the
first, suitable
only for crime. Darwin wrote before the mechanism of genetic
transmission
was
understood, but he inclined to the view that variations in the plant
and animal
kingdoms arise by
chance.
and are then passed downward from fathers to sons.
The biological world, Darwin observed, striking now for the second
point to
his three-part explanation.
is
arranged so that 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 for life
favors those organisms whose variations give them a competitive edge.
Such
is
the
notion of
fitness.
Fast feet make for fitness among the rabbits, even as a feathery
layer of oiled down makes the Siberian swan a fitter foul. At any
time. those
creatures
fitter
than others are more likely to survive and reproduce. The win-
nowing in life effected by competition Darwin termed
natural selection.
Page 15
The Language of Ltfe
7
Working backward, Darwin argued that present forms of life, various
and
wonderful as they are, arose from common ancestors; working forward,
that biolog-
ical change, the transformation of one species to another, is the
result of small
increments that accumulate. step by inexorable step, across the
generations, until
natural selection recreates a species entirely. The Darwinian
mechanism is both
random and determinate. Variations occur without plan or purpose
-
the luck of
the draw; but Nature, like the House, is aggressive; organized to cash
in on the
odds.
The
Central Dogma
Everything that lives, lives just once. To pass from fathers to sons
is to pass
from a copy to a copy. This is not quite immortality, even if carried
on forever, but
it counts for something. as every parent knows. The higher organisms
reproduce
themselves sexually, of course, and every copy is copied from a double
template.
Bacteria manage the matter alone, and so do the cells within a complex
organism,
which often continue to grow and reproduce after their host has
perished,
unaware, for a brief time, of the gloomy catastrophe taking place
around them. It
is possible, I suppose, that each bacterial cell contains a tiny copy
of itself, with
the copy carrying yet another copy; biologists of the early eighteenth
century,
irritated and baffled by the mystery of it all, actually thought of
reproduction in
these terms: peering into crude, brass-rimmed microscopes, they
persuaded them-
selves that on the thin, stained glass, they actually saw a
homunculus; the more
diligent proceeded to sketch what they seemed to see. The theory that
emerged
had the great virtue of being intellectually repugnant. Much more
likely, at least
on the grounds of reasonableness and common sense, is the idea that
the bacterial
cell contains what Erwin Schroedinger called a
code
script
-
a sort of cellular
secretary organizing and recording the gross and microscopic features
of the cell.
Such a code script would be logically bound to double duty. As the
cell divides in
two, it, too, would have to divide without remainder, doubling itself
to accommodate
two bacterial cells where formerly there was only one. Divided. and
thus doubled
without loss, each code script would require powers sufficient to
organize anew
the whole of each bacterial cell. The code script that Schroedinger
(1945) antici-
pated in his moving and remarkable book, What is
Life?
-
he wrote in the 1940s
-
turns out to be DNA, a long and sinewy molecule shaped rather like a
spiral in two
strands. The strands themselves
are
made of stiff sugars, and stuck in the sugars,
like beads in a sticky string, are certain chemical bases: adenine,
cytosine, gua-
nine, and thymine: A, C, T, and G, in the now universal abbreviation
of biochemists.
It is the alternation of these bases along the backbone of DNA that
allows the
molecule to store information.
One bacterial cell splits in two: each is a copy of the first. All
that physically
passes from cell to cell is a strand of DNA: the message that each
generation sends
faithfully into the future is impalpable, abstract almost, a kind of
hidden hum
against the coarse
wet
plops of reproduction, gestation, and birth itself. James
Watson and Francis Crick provided the correct description of the
chemical struc-
ture of DNA in 1952. They knew, as everyone did, that somehow the
bacterial cell
Page 16
in replication sends messages to each of its immediate descendents.
They did not
know how.
As
it turned out, the chemical structure of DNA. once elaborated. sug-
gests irresistably a mechanism both for self-replication and the
transmission of
information. In the ceU itself, strands of DNA are woven around each
other and
by an ingenious twist of biochemistry matched antagonistically:
A
with T. and C
with G.
At
reproduction. the ceU splits the double strand of DNA. Each half
floats
for a time, a gently waving genetic filament; chemical bonds are then
repaired as
each base fastens to 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,
there are now
two strands of double-stranded DNA where before there was only one.
What this account does not provide
is
a description of the 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; its products are proteins chiefly
-
long and complex
molecules composed, in their turn, of twenty amino acids. The order
and composi-
tion of the amino acids along a given chain determines which protein
is
which. The
bacterial cell somehow contains a complete record of the right
proteins, as well as
the instructions required to assemble them directly. The sense of
genetic identity
that marks E. Coli as E. Coli and not some other bug must thus be
expressed in
the amino acids by means of information stored in the nucleotides.
The four nucleotides, we now know, are grouped in a triplet code of 64
codons
or operating units.
A
particular codon
is
composed of three nucleotides. The amino
acids are matched to the codons: C-G-A, for example, to arginnine. In
the transla-
tion of genetic information from DNA to the proteins, the linear
ordering of the
codons themselves serves to induce a corresponding linear ordering
first onto an
intermediary, messenger RNA, and then onto the amino acids themselves
-
this via
yet another messenger, transfer RNA. The sequential mngement of the
amino
acids finally fixes the chemical configuration of the cell.
Molecular biologists often allude to the steps so described as
the
Central
Clogmu, a queer choice of words for a science.
The dour Austrian monk. Gregor Mendel, founded the science of genetics
on
purely a theoretical notion of a gene, which he likened to a bead on a
string. In
DNA. one looks on genetics bare: the ultimate unit of genetic
information
is
the
nucleotide.
All
that makes for difference. and hence for charm, in the natural
world. and which
is
not the product of culture.
art,
artifice, accident, or hard
work, all this, which is brilliantly expressed in maleable flesh, is a
matter of an
ordering of four biochemical letters along two ropey strands of
an
immemorial
acid.
The Central Dogma describes genetic replication; but the concepts that
it
scouts plainly illuminate Darwinian theory from within. Whether as the
result of
radiation or chemical accident, letters in the genetic code may be
scrambled; one
letter shifted for another; entire codons replaced, deleted, or
altered. These are
genetic mutations: arbitrary, because unpredictable; and yet enduring,
because
they are variations in the genetic message. The theory by which Darwin
proposed
to account for the origin of species and the nature of biological
diversity now
admits of expression in a single English sentence. Evolution, or
biological change.
so the revised, the neo-Darwinian theory, runs, is the result of
natural selection
working on random mutations.
Page 17
The Language o,fLi,fe
PART
Two
Evolutionary Theories
The popular view of evolution tends to be a tight shot on a tame
subject: the
dinosaur, who did not make it; the shark, who did; but the maturation
of an organ-
ism is itself much like the evolution of a species; only our intimate
acquaintance
with its precise and unhesitating character suggests, misleadingly, I
think, that
the two processes differ in degree of freedom. Psychology, economics,
urban
affairs, anthropology, political science, and history also describe
processes that
begin in a state of satisfying and undemanding simplicity, and end
later with
everything complex, unfathomable, chaotic. The contrast to physics is
sobering.
The dynamics of evolutionary theories
are
often divided into two conceptual
stages. In economics, there are macro- and micro-economic theories,
aggregate
demand versus the theory of the firm: within linguistics, language at
the continu-
ous level of speech. and language some levels below, discrete, a
matter of the con-
catenation of words or morphemes. Biology, too, is double-tiered:
above, the organ-
ism prances; unseen, below, at a separate level, its life is organized
around the
alphabetic nucleotides.
Metric Spaces
By a metric space S I mean a space upon which a function
has been defined. assigning to each pair of points s
,
s
'
in S a nonnegative real
number
-
the distance d
(s
,
s
')
-
and satisfying the usual axioms:
d ( s , s') = d ( s l ,
s )
;
(9.2)
d ( s ,
st)
+
d ( s P ,
s")
l d ( s ,
s")
,
(9.3)
Double
metrics
The
distance between organisms
The disciplines of comparative anatomy and systematic zoology classify
creatures into ever-larger sets and
sets
of
sets:
individuals (dogs, say), species,
genera, families, orders, classes. phyla. taxa. and kingdoms. The
classification
itself forms an algebraic lattice, with individuals acting as the
system's atoms.
Comparative anatomists and zoologists bring an exquisitely refined and
elaborate
intuition to the task of sorting the various biological creatures into
appropriate
categories: the obvious cases leap to the eye; at the margins of the
system, where
the whale resides, difficult matters are. decided by reference to
historical and
comparative anatomy, parallel structure, common organization,
biological traits,
and, often, levels of biological achievement. If the image of a
lattice is for the
Page 18
moment taken literally, then each level of the lattice, from the atoms
upward,
comprises a set or ensemble: of individuals. in the first instance. of
sets of indivi-
duals, in the second. An ensemble at any distinct level of the
lattice, I assume,
satisfies equations (9.1)-(9.3). and counts thus
as
a metric space.
Ths distance between strings
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
to
a
wider
family of string-like objects: computer programs written in a given
language, the
sentences of a natural language. formal systems; and acquire by
osmosis a distinct
conceptual and mathematical structure. It makes little difference
whether strings
of DNA or strings of amino acids are taken as fundamental; and. in any
case. I
often alternate between the two. By an alphcrbet
A
I mean a fixed and finite col-
lection of elementary entities called
mrds;
by the universe of strings over a
finite alphabet, the 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
x
lkl, where k is the maximum of the 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 metric
space of proteins based on properties of composition. polarity, and
volume; but
the theory of evolution suggests that changes
in
biological strings come about
through mutations
-
random flash points at which letters are scrambled. Some
strings may change in a large-hearted way, with whole blocks of
letters wheeling
and shifting like cavalry horses; but the least mechanism to which
these opera-
tions may be resolved
is
the simple one of erasure and substitution
-
deletion and
insertion. The elementary processes of evolution at the molecular
level lend to the
natural metric a certain simple plausibility in the face of fancy
competition.
T
=
A*,
then.
is
a typogmphic metric spcrce; dT, its natural distance.
Metric spaces
in phase
M
and
P,
suppose, are two metric spaces;
g :
M
-+
M*
assigns to each point
p
in
M
a distinct point
F
in
W .
M and
M*
are
in
phase under
g if g
acts roughly
to preserve distances: for any
(

0,
there exists a
9

0,
such that for all
p
and
qinM
g is
thus unijomly continuous on M;
rp is,
of course, a function of
..$.
It often
happens that a particular mapping between metric spaces is especially
natural
-
for 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
are not in
phase. This observation is often regarded
as
a paradox in the context of theoreti-
cal biology. In an important and influential article, King and Wilson
recount evi-
dence showing that chimpanzee
and
human polypeptide sequences are more than
Page 19
Th.e Language
ofLtfe
11
99 percent identical; the species appear further apart than a
comparative
analysis of their polypeptide chains might otherwise suggest.[6]
Complexity
Complexity and simplicity, like Yin and Yang, are metaphysical duals;
except
for a vagrant connection to intuition, it hardly makes a difference
what is called
which. Mathematicians and philosophers
are
interested in complexity for their
own ends; so
are
theoretical biologists, who in their better moments are quite
capable of evincing a sense of Heraclitian awe when confronted with
the intrica-
cies of the protozoan swim bladder. Simple counting principles often
seem as if
they might provide a general scheme for the,measurement of complexity.
Suppose
that
X
is a nonempty set of objects and that A ,
B,
C,
.. . .
are
constructed from the
elements of
X
by certain specified operations
-
concatenation, for example. Can
we
not then say that the complexity C (z) of any object is a measure of
the number
of its distinct elements and the separate and specifiable relations
between them?
C (z) would be a monotonically increasing function of the square of
the number of
distinct elements in any given construction. Simple, no? And
intuitively satisfy-
ing?
Apparently not. Label the parts of an ordinary watch in an obvious
alpha-
betic fashion; and the binary relations between its parts as well. The
watch when
working, let me suppose, has a complexity measured at C; but so, then,
does the
watch when not working
-
when not assembled. in fact, binary relations being free
for the asking. Examples of this sort, when extended and made precise,
suggest
ultimately that any complex object belongs to 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 configuration; never having seen the
blue smoke
from a cigar spontaneously collect in but one corner of a warm room,
the thought-
ful physicist
-
pipe, slippers. Beagle-eyes, an air of earnest confusion
-
concludes
that not all configurations
are
equally probable; yet if there
are
N configurations
Pr(Nc)
=
Nc / N
-
this for each
i
..
This incompatibility between what one
sees
and
what one gets is known as Boltzmann's paradoz, an unhappy name
if
only because
no real paradox is forthcoming; but an unhappiness nonetheless.
Distinct conf i-
gurations, Boltzmann argued, may be grouped into states; what the
altogether
more elegant Gibbs called ensembles. Within thermodynamics
-
statistical
mechanics from above
-
the entropy
S
of a system appears perpetually in the
ascendancy and tends inexorably to a maximum; statistically, Boltzmann
reasoned,
S
is thus proportional to
S
= k
log W
;
(9.4)
where
k
is Boltzmann's constant, and W a measure of those configurations
compati-
ble with a given state
-
complezions as they are called in old-fashioned texts.
Configurations are alike in point of probability: not so complexions;
the probabil-
ity of finding a mechanical system in a given state is proportional to
the number of
Page 20
distinct complexions realizing that state. At equilibrium, the
complexions are at a
maximum;
and so, too, the entropy. which functions as a kind of ectoplasmic
mea-
sure of randomness or disorder.
Compldty
under
a
classification
Statistical mechanics has a good point to its credit, and implies a
second.
Certain states of a physical system may be multiply realized; their
number,
if
counted, makes for a measure of sorts. What
is
measured within statistical mechan-
ics is plainly not complexity; the description of entropy as disorder
serves only to
explain the whole business to the baffled undergraduate. with the
explanation
rapidly withdrawn by the time he enters graduate school. Still.
I
am struck by the
extent to which the mathematical definition of entropy
is
made possible by an
enterprising reorganization of the 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 the set of
aLl
functions
f
:Rn
-.
R. Those smooth functions whose critical
points are nondegenerate are known as Morse functions and are at 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
,
there exists a number
k
such that in a
neighborhood of
z ,
and after a suitable change in coordinates,
Such is Morse's lemma. Their mathematical docility suggests that the
Morse func-
tions
are
simple, if anything is; but the Morse functions are simple because
they
are Morse functions, and not Morse functions because they are 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
the space of smooth functions cm(Rn, R); but a
set
of objects may be simple under
a classification even
if
the classification
is
itself unpleasantly complex. Writing
some years ago, Smale asked whether there exists a least Baire set
U
in the 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 to the fore. It
is
for
U1
that Axiom
A
is
satisfied, nonwandering sets are finite, and the transversality
condition
is
met.
Ul
thus consists of "the simplest. best-behaved, nontrivial class of
dynamical sys-
tems"; but nothing in Smale's organization of Dyn(M)
is
simple at
all.
A
set is absolutely simple under a classification
if
it is at 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 the first Baire
category, the best one
can do;
sets
that are dense need not be stable, and vice versa. First category sets
and sets of measure zero coincide in the case of countable sets; but
not beyond.
Page 21
The
Language
of
LVe
13

From the point of view of statistical mechanics, simplicity and

complexity are
concepts that involve configurations; complexity under a
classification is a matter
of routine: what is complex is singular, unusual. These notions may be
brought
into alignment
-
but only for a certain class of objects. An object A is dissective
only when it may be decomposed to a finite stock of parts in a finite
number of
steps. The mammalian eye is a dissective structure; so is the whole of
a mouse. a
moose, or a mole; but curves and concepts, the real numbers, the coast
of Britain,
sea-green sea-waves, and, perhaps, the entire bizarre universe of
element'ary par-
ticles, are indissective. A dissective object is thus composed of its
parts taken
together under a certain distinctive relationship. Say that A is
composed of
al,
az,
....
,an
under
R.
By a relational alternative to
R
I mean a single permutation of
the parts of A . If A , for example, contains but two parts,
a
and b , say, under the
relationship
R (a,
b
),
R
(b
,
a ) is a relational alternative to
R
-
the only one in fact.
Given
R,
I denote by
R*
the full set of
all
relational alternatives to
R.
If A is
dissective it is
R*
that forms its complexion class: the set of all sets of its parts
under all and only their relational alternatives.
An elementary partition of a complexion class splits the class as a
whole into
equivalence classes; relative to a partition, complexity and
simplicity are attri-
butes of equivalence classes, and are judged simply by size. To the
extent that
[Ec]
is larger than
[E,],
it is simpler as well; and vice versa. Almost all structures
in theoretical biology may be dissected to 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 structure. Its parts
(on one
level of dissection, at least) are proteins, which are arranged in
various delicate
and precise ways. I am ignoring, now, any dynamic considerations and
thinking
instead of the mammalian eye as a static object. The complexion class
to the mam-
malian eye consists of all and only those rearrangements of proteins
that comprise
relational alternatives to the mammalian eye itself.
What makes an eye distinctively an eye, rather than some assembly of
jelly-
like proteins, is obviously the fact that it is capable of sight. This
invocation of
function sounds an unavoidably Aristotelian note; but without some
concept such
as function or purpose, theoretical biology loses much of its point.
Let me parti-
tion the relational alternatives to the mammalian eye into equivalence
classes on
the simple basis of function. In the full complexion class, those
structures that
are capable of sight fall to one side; and those that are blind and
stare sight-
lessly, fall to the other. Complexity and simplicity appear as matters
of relative
size: the larger the equivalence class, the simpler the structures.
Given the deli-
cacy of the mammalian eye, most of its relational alternatives will be
incapable of
sight; like the Morse functions, these complexions are simple
structures; but
again, simple because they are sightless, and not sightless because
they are sim-
ple.
Complexity
in
strings
Of the 2" binary sequences of length n, some, such as
o,o,o,o,o
,...
Page 22
seem simpler than others,
for example; yet the most natural probability distribution over the
space of
n-
place binary strings assigns to both the same probability: 2-. It goes
against the
grain. mine. at any
rate,
to reckon (9.6) as likely
as
(9.7).especially when
n
is
large; but nothing in the sequences themselves indicates obviously the
point of
distinction.
The goal of science, Ren4 Thom has suggested.
is
to reduce the arbitrariness
of description; substitute data for description. and the apothegm
gains my assent.
A
law of nature
is
data made compact:
F =
ma, said once and for all. the whole of
an observed or 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 to argue that
randomness
in binary sequences or strings might be measured by the
degree
to which such
strings admit of a simpler description.[8] In following this line,
Kolmogorov took
the
first
step toward severing information theory from its 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 simpkr description of
S,
Kolmo-
gorov meant a string D shorter than
S
such that D describes
S
by acting as the
input to a fixed computer that generates
S.
Strings that cannot be compactly
described are complaz. random, or in.omation-rich; strings that can,
are not;
of these adjectives, only the second preserves even a vagrant
connection between
the concept that
it
connotes and what
is
being measured. This rather inelegant
idea makes plain the 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:
the relevant instruction is sim-
ply to write or compute 0 2' times.
C
measures what
little is
needed to carry out
the instructions;
32
-
5
=
27, the compactness of the program. The shortest pro-
gram that computes a mixed sequence such
as
sequence (9.7).by way of contrast.
may
well
be close to
32
bits in length: to compute the sequence, the computer must
first
store it precisely.
The details? They have been changing since Kolmogorov first spoke,
oracle-
like, on the subject in a note published in 1967; like a snake
engulfing an egg, the
theory of recursive functions
is
engaged in swallowing algorithmic information
theory, a development that I deplore, but accept
as
inevitable. Consider the set of
all n-place binary strings
A*
over a binary alphabet
A
and let TM be a fixed com-
puter
-
a Turing machine. say;
g is
a general input-output function on
?"M
map-
ping strings onto strings. The complezity of a string
S
of length n
is
the Length
of the shortest binary string D that generates
S
under TM by means of
g .
What-
ever the complexity of
S,
D will plainly be maximally complex. and. hence.
entirely random. Otherwise. it would not be the shortest description
of
S.
All
fin-
ite length strings quite obviously have a finite measure of
complexity; and only fin-
itely many distinct strings of the same length have the same finite
measure of com-
plexity. Quite surprisingly. the decision problem for complexity is
recursively
unsolvable; this result follows almost directly from the unsolvability
of the halting
problem for Turing machines. Like truth, randomness is a property that
remains
ineluctably resistant to recursive specification.
Page 23
The Language
of
Life
15
If all else fails, a binary sequence of length
n
may be generated by a binary
sequence of length
n:
there are
2"
such algorithms, and
2"
-
2
algorithms
shorter than this. On any reasonable interpretation of complexity,
algorithms
within a fixed integer k of
n
itself must be reckoned random or complex or nearly
so. Thus
2n
-k
-1
-
2/ 2n
algorithms have a complexity less than
n
-
k
;
and
are
hence nonrandom or simple. If k
=
10, this ratio is roughly 1 in 1000; of 1000
binary sequences of length
n ,
only one can be compressed into a program more
than ten bits shorter than itself. Hence:
Theorem 9.1
The set of random sequences of length
n
in the space A* of all
binary sequences of length
n
is generic in A*.
These random sequences are simple under a classification because they
are typi-
cal, but complex in a stronger and more absolute sense because they
are random
or 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 are random, the appearance of sequence
(9.6)
prompts a natural
stochastic surprise: sequences such as
(9.7)
are what one expects. The definition
of Kolmogorov complexity may be directly extended to recursively
enumerable
sets; sets of strings especially, and hence languages.
Language-like
Systems
When it comes to language, there is syntax and semantics. Phonetics is
the
province of the specialist; pragmatics remains a pale albino dwarf. To
semantics
belongs the concept of meaning; to syntax, the concept of a well-
formed formula or
a grammatical sentence. The reference to logic is happy if only
because it
highlights the fact that language-like systems go beyond the natural
languages.
Any language no doubt exists primarily to convey meaning; but meaning
in
mathematics is a matter of a model
-
an extrinsic object.
The construction of strings within a language-like system involves
con-
catenating or associating simpler strings: any finite string may be
dissected to a
finite set of least elements. Going up, concatenation; going down,
finite dissection;
retrograde motion of this sort suggests that language-like systems on
this level be
represented algebraically as semigroups. Let A be any nonempty set of
objects
-
words, for example, or letters, or numbers. A has the structure of a
semigroup if
there exists a mapping A
x
A
+
A such that for alI
a ,
6, and
c
in A
In English words go over to sentences from left to right; in Hebrew,
from right to
left; but in any case, one step at a time. Let A be a finite set of
words now, with
words understood implicitly as the least elements of a natural
language; and let A*
be the set of all finite sequences
(a
l,...,
an)
whose elements
lai,
....,
an
j
lie in A .
To endow A* with the structure of a semigroup, it suffices to define
an associative
mapping A*
x
A*
-r
A*
:
easy enough. If
Page 24
and
then
where
A*
is
at once a
free-semigroup
over a finite alphabet and a
universal Language:
no sequences are left out.
Almost
all
language-like systems are large in the sense that they have many
distinct strings. Meditating on the matter in the late 1950s. and
regularly
thereafter. Noam Chomsky argued that 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 the
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, are countably infinite and no bigger.[9]
Going further toward a definition of a language-like system involves
the bad-
lands beyond triviality. Linguistics, the French linguist Maurice
Gross once provo-
catively remarked, admits of but a single class of crucial
experiments. Native
speakers of a given language are able to determine whether a given
sentence
is
grammatical. Experiments of this sort exist because no language-like
system
encompasses the whole of a set of strings drawn on a finite alphabet
-
a curious
and interesting jbct. which the 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 are
not grammatical at all and represent syntactic gibberish. The
fastidiousness of
Language-Like systems is yet again a fact:
it
would be easy.
if
unrewarding, to
design an artificial language in which most strings were grammatical.

From the

point of view of
grammar,
the strings of a natural language are 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
are thus
complex
under this classification because they are singular, unusual.
The origins of this bit of natural history are to be discovered, no
doubt, in the
algorithmic properties of the human brain:
in
order to store a natural language.
the brain must first represent it
-
in the form of recursive rules, for example.
This suggests that language-like systems are low in point of
Kolmogorov complex-
ity; and from this point of view,
simple.
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
Page 25
The Language
of
Lire
17
not grammar. No matter: the point carries over to the case at hand
-
and
comprises the third of three queer natural facts that nothing in the
concepts of
grammar or communication obviously implies. Thus, let T be a
typographic metric
space of strings under the natural metric; the same set of strings
comprises a
second metric space under the degenerate distance function d*: if
s
and
s'
are
both grammatical, d* (s
,
s')
=
0;
if not, d* (s
,
sf)
=
m.
These are the natural and
(degenerate) grammatical metric spaces of a language-like system. In a
language-
like system, natural and grammatical metric spaces are plainly not in
phase.
Two models of generation
Linguistics is a rebarbative, hair shirt of a subject; and grammar a
vexing
property. Linguists, for reasons of their own, are often interested in
the weakest
of generative devices that specify all and only the sentences of a
natural
language.
Representation by grammar
A phrase structure grammar is a quadruple G
=
(A, T, S,
P),
where A is
some finite alphabet of symbols; T, a distinguished subset of A
-
the set of so-
called terminal symbols; S, a distinguished initial symbol; and
P,
a finite set of
production rules of the form u
+
v ;
u is a nonempty set of nonterminal symbols,
and
v
some specified string of characters. The set of all strings of
terminal sym-
bols constitutes a phrase structure language
-
a proper subset of the set of all
strings A* defined over A
..
By a context-pee production rule. I mean one in which u may occur in
any
context
-
in effect, a rule in which u figures in isolation. Correspondingly,
there
are context-free grammars.
Example
9.1Let A
=
(a,b),T
=
(a,b), and
P
be the two rules S + a b ; S
+
US.
This grammar generates all and only the strings of the form an b ".
Representation by systems of equations
Consider the context-free grammar G whose production rules
are
S
+
&a,
and S
+
c
,
where T
=
(a, c
),
and S is an initial symbol. Let the variable f range
over terminal symbols. The action of the production rules may be
mimicked by an
equation:
where addition is construed as set theoretic union. For G
,
Replacing S by
SO
=
c
,
s(')
=
aca
+
c
..
Page 26
D.
Berltnskt
This process repeated ultimately yields a system of equations
t
=n
s ( )
=
ancan
+
- .
+
aca
+
c
=
C
atcai
..
t
=m
At the Limit, the solution
s(-)
)=
C
afca'
is
given by a formal power series in
t
*
noncommutative variables. [lo]
A
language-like system has firmal support when each and every string in
the system may be described by a single algorithm; only for context-
free
languages may
grammars
and systems of equations be balanced against each other.
lsewhere, the situation is darker. There
is
a sense, however, in which these two
representations exhaust the possibilities for the description of
structured and
infinitary objects; and correspond. in the Metaphysical Large, to the
alternatives
confronting an imaginary Deity in creating the observable world.
Weak
Theories
The vitalist believes that life cannot be explained in terms of
physics or
chemistry. In the nineteenth century, in Germany and France, at least,
his was
the dominant voice before Darwin; and natural philosophers. such
as
Cuivier or
von Baer, or 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 the uninspiring precision with which each position
is usually cast.[ll] D'dccord. To the extent that the refutation of
vitalism involves
the reduction of biological to physical reasoning, the effort involved
appears to
me misguided. and reflects a discreditable, almost oriental, desire
for the Unity of
Opposites. On the standard view of reduction, the sciences collapse
downward
until they hit physics: Rez-da-Chausee; but our intellectual
experience is
divided: mathematics. physics, biology, the social sciences. Each
science extends
sideways for some time and then simply stops. The ardent empiricist,
surveying
the contemporary scene, might well incline to scientific polytheism,
with
mathematics under the influence of an austere Artin-like figure, and
biology
directed by a God much like Wotan: furious, bluff. subtle, devious,
and illiterate.
Still, the philosopher of science is bound to wonder why so many
philose
phers have remained partial to the reductionist vision, and hence to
mechanistic
thought in biology. David Armstrong. J.J.C. Smart, Michael Ruse, and
even the usu-
ally cagey W.v.0. Quine, call on elegance to explain their attachment.
Were the sci-
ences irreducibly striated, one
set
of laws would cover physics, another biology,
and still a third, economics and urban affairs, with the whole
business resembling
nothing so much
as
a parfait in several lurid and violently clashing colors.
This is
an aesthetic argument, and none the worse for that, but surely none
the better
Page 27
The Language
of
Life
19
either. If elegance is inadequate as a motive, intellectual anxiety,
realized uncons-
ciously, is not.
Vitalism commences from the conviction that nothing in our experience
is
much like the life that ripples and bubbles so abundantly over an
entire planet,
and nowhere else, apparently. Now mathematical physics is not only the
pre-
eminent discipline of our time
-
it is where the laws are. Evolutionary theories in
biology are weak in the sense that they are not directly sustained by
the author-
ity of physics; and, worse, weaker still in being counterph ysical.
Thermodynamic
arguments count against the very existence of the structures that they
are meant
to explain. Fact heavy, law poor, such theories remain surprisingly
resistant to
confirmation. Were biology an aspect merely of physics, the sceptic
would get
short shrift: there, the answer to whether what works, works, is
simply that it
does.
Science is unavoidably general. To say that copper conducts
electricity is
weakly to imply the counterfactual conditional that were anything much
like
copper it would conduct electricity as well. It has often appeared to
philosophers
of science that specifying what it means for something
-
an
z,
say
-
to be much
like copper inevitably comes to claiming that, among other things,
z
conducts
electricity. Still, the similarity in structure between two domains of
discourse
-
computer prokpms and natural languages, for example
-
may be obvious on
grounds other than the fact that they share the same laws.
When I speak of a theory, I follow the logician's lead: a theory
consists of a
consistent set of sentences in a given language; the set-theoretic or
algebraic
structures in which a theory is satisfied comprise its models. Two
models that
share the same structure are isomorphic and hence elementarily
equivalent in the
sense that they satisfy the same sets of sentences. What I am after is
a weaker
notion entirely
-
partial similarity in structure. I know of no way, unfortunately,
to define this concept so that the definition applies equably to
biology, and, say,
geology; I suspect, in fact, that partial similarity in structure will
require a defini-
tion with indefinitely many separate clauses. Whatever the details,
similarity in
structure is bound to be a matter of degree, so that it makes sense of
sorts to say
of two models that they
are
at a certain distance, one from the other. In this way
a family
Mt
I,
i
=
1.2,.
..
..
of (possibly) f i r s t r d e rmodels may be given an appropri-
ate and empirical metric structure.
Suppose that
T
is a theory holding in
M;
and let AP be a model at some fixed
distance from
M.
By the symmetric diffeence
T/ F
of
T
I mean the number of
formulas
P
of
T
that fail to hold in M* when
T
is interpreted in
M*.
A theory
T
is general, I shall say, if for any
E

0,
there exists a
6

0
(a
function of
e,
of course) such that
Generality in my sense is a kind of stability; and as
D r
Johnson remarks, the
soul must ultimately repose in the stability of the truth.
To see an analogy between the operations of life, on the one hand, and
the
operations of language, on the other, is to raise the question whether
the laws of
biology have a natural and legitimate interpretation in linguistic
terms.
I
am myself
indifferent to the fate of the Darwinian theory, and perfectly
prepared to
believe, along with Wickramasinghe and the luckless Hoyle, that life
originated in
outer space. or that the Universe-as-a-Whole is alive and breathing
stertorously;
Page 28
20
D. Berltnskt
but
if
Darwinian theories work in life, they should work elsewhere
-
in
language-
like systems, I should think. Should they fail there, this may be
taken as evidence
for the inadequacy of Darwinian theories, or as evidence for the
inadequacy of the
analogy that prompted the comparison in the first place.
I stress this point
if
only because it has so often been misunderstood.
Life
as a language-Like
system
It was von Neumann who gave to the idea that life is Like language a
part of its
curious current cachet. The last years of his life he devoted to a
vast and clumsy
orchestration of cellular automata. showing in a partial fashion that
when properly
programmed they could, like abstract elephants, reproduce themselves.
Some
years before. McCulloch and Pitts had constructed a series of neural
nets
in
order
to simulate simple reflex action; Kleene demonstrated that their nets
had the
power of finite automata and were capable of realizing the class of
regular events;
von Neumann's automata had the full power of Turing machines. Michael
Arbib,
E.F.
Codd. G.T. Herman, A. Lindenmayer, and many others, have carried this
work for-
ward. with results that asymptotically approach utter irrelevance.[lZ]
Yet the
analogy between living systems and Living languages has not lost any
of its brassy
charm. There is information, of course, which
is
apparently what the genes store;
replication. coding; messages abound
in
the bacterial cell. with
E.
Coli.
in
particu-
lar, busy
as
a telephone switchboard. So striking has the appropriation of termi-
nology become, that some biologists now see the processes of life.
in
all
their
gran-
deur, as the effort of a badly protected and vulnerable bit of genetic
material to
keep taking for all eternity.
UnIike an argument. an analogy stands or falls in point of
plausibility; good
arguments
in
favor of bad analogies are infinitely less persuasive than bad argu-
ments in favor of good analogies. Certainly the proteins. to stick
with one class of
chemicals. may be decomposed to a finite base
-
the 20 amino acids. The precise,
delicate, dance-Like steps that are involved in their formation
suggest, moreover,
that they satisfy some operation
as
abstract
as
concatenation. On the other hand,
the number of possible proteins, although large,
is
finite; but one of the joys of
analogical reasoning
is
the vagueness with which the line between success or
failure may be drawn.
The grammatical strings of a language-like system are low in Kohogorov
com-
plexity, and so are not random. Such
is
the fastidiousness of a language-like sys-
tem. What of the proteins?
If
they are random, it makes Little sense to think of
them as biological words or sentences. Jacques Monod, whose
metaphysical attitude
toward biology suggested nothing so much
as
a kind of chirpy bleakness, drew
attention to the random character of the proteins in
La
husard et la necessite;
his argument has been gravely accepted by many molecular biologists.
[l3] In fact,
the evidence leading to his conclusion is fragmentary; the standards
of random-
ness to which he appealed, imprecise. Thus it struck Monod that
knowing, say, 249
amino acid residues in a chain 250 residues
in
length, one could yet not predict
the last member of the chain; much the same
is
true for English sentences, of
course; it is, in any case, simply untrue that protein strands exhibit
such wanton
degrees of freedom. Within protein chemistry, there are many instances
of what
appear to be strong internal regularities: palindromic patterns, for
example.
Page 29
The Language opL.tpe
21
Nonetheless,
I
am in sympathy with Monod to this extent: it is unlikely that the
analogy between life and language will be profitably pursued on the
atomistic level
of the nucleic acids or the proteins themselves.
PART THREE
Arguments
Good
and
Bad
The theory of evolution is haunted by an image and an observation: the
first,
that of the hapless chimpanzee, typewriter-bound, endeavoring, quite
by chance,
to strike off the first twenty lines of Hamlet's soliloquy; the
second, the comment
of an anonymous Jansenist logician, who remarked, quite sensibly,
"that it would
be sheer folly to bet even ten coppers against 10000 gold pieces that
a child
arranging at random a printer's supply of letters would compose the
first twenty
lines of Virgd's Ansicill. Image and observation do not quite cohere
into a single
argument: it is clear in neither case
how
the imagined stochastic experiment is to
stop. Still, the notion of randomness yet lies at the center of
evolutionary
thought, and there it sits, toad-like and croaking. On the simplest
and most intui-
tive conception of probability, what can occur is weighted against the
background
of what might occur: five diamonds: all other combinations of the
cards. In poker,
there are 2 598 960 five-card hands, but only 5 148 flushes. It is
their ratio that
one might expect to observe as cards
are
actually dealt; but in the longest of long
runs, the passage to the limit gives content to the intuitive idea
that a number of
successive trials will converge to a particular real number: 0.002,
for example, if
flushes are being counted.
One of the curiosities of the very notion of probability is the
inescapability
of the improbable. The laws of thermodynamics, to take a notorious
example, are
anisotropic: they go in one direction; downhill, as it happens, a
circumstance with
what appears to be overwhelming personal support. Statistical
mechanics provides
a brilliant and persuasive explanation for thermodynamic laws; yet
PoincarC
demonstrated, in an absurdly easy proof, that any statistical
mechanical confi-
guration, of whatever degree of implausibility
-
k molecules of gas, for example,
occupying 1/V of the total volume V of a finite and bounded container
-
is bound
to recur, in all its vividness, poignant symmetry, and complexity,
given enough
time. Physicists often explain the discrepancy between thermodynamics
and sta-
tistical mechanics by arguing that the time involved is very long. No
doubt.
The evolution of life on this planet is, as Darwin realized. not a
hurried
affair. Early on, Darwinian biologists got rid of the theological
limits
set
to the age
of the Earth by Bishop Ussher and others in the seventeenth century;
the scale
within which Darwinian evolution might have worked is bounded by
perhaps five
billion years. Nineteenth century biologists assumed that whatever
else one might
say about Darwinian biology, it would not fail for lack of time; this
thesis twentieth
century biologists have carried over intact.
Five billion years is apt to seem long if one is counting the minutes;
but it is
not long enough to sample on a point by point basis a space whose
cardinality is
roughly 1015
-
touching base with a new point at every second, say; and yet there
are 20'' possible proteins
-
a number larger by f a r than the expected life of the
Page 30
22
D.
Berlinski
universe measured in seconds. In a space of this size. the odds
against discover-
ing a specific protein
-
fishing it from
an
urn, say
-
are
prohibitive: 1 in 2 0 .
I spoke hastily just now of a speci* protein: if a n y protein will
do, the
odds
improve: in a uniform probability space (ai
1,
Pr(al
v
a2
.. . .
v
ai)
=I.
The dis-
tinguished British biologist Peter Medawar has seized upon this point,
and com-
menced happily to trot, but in what I think
is
the wrong direction.[l4] "Biolo-
gists," he writes. "in certain moods are apt to say that organisms
are
madly
improbable objects or that evolution
is
a device for generating high degrees of
improbability. I am uneasy about this entire line of thought," he
continues, "for
the following reason:
Everyone will concede that
in
the games of whist or bridge any one particular
hand is
just
as
unlikely
to
turn up as any other. If I pick up and inspect a par-
ticular hand and then declare myself utterly
amazed
that such a hand should
have been dealt
to me,
considering the fantastic odds against it, I should be
told by those who have steeped themselves
in
mathematical reasoning that its
probability cannot be measured retrospectively, but only against a
prior
expectation
....
For much the same reason, it seems
to me
profitless
to speak
of natural selection's 'generating improbability'
....
it is silly
to
be thunder-
struck by the evolution of organ A if w e should have been just
as
thunder-
struck by a turn of events that had led to the evolution of B or C
instead."
Medawar
is
roughly right about probability: the fallacy to which he refers
is
the error of retrospective specijkation; and consists precisely in
reading back
into an original sample space information revealed only on the
realization of a par-
ticular event. In poker, a deal distributes n hands of equal
probability: 1
in
2 598 960, as it happens. This sample space
is
retrospectively specified if one hand
in particular
is
contrasted with the full
set
of 2598 959 hands that remain. and
probabilities assigned to the partition so created; what appears
initially as one
among equiprobable events becomes under retrospective specification an
improb-
able event in a sample space of only two points. It
is
embarrassing for
an
author to
point such things out. Still. Medawar
is
wrong in the general conclusions that he
draws from this paragraph. Card sharps and statisticians are little
interested in
the set of
all
five-card sequences. In poker, sequences
are
initially partitioned
into equivalence classes of uneven size: a royal straight flush, of
which there
are
four, a straight flush, four
of
a kind. a full house, a straight, three of a kind. two
pairs, and, then, finally, whatever is left
-
the vast majority. There are four ways
to achieve a royal straight flush; many more ways in which to realize
a full house.
Since they
are
specified in advance, partitions
in
poker carry no taint of
retro-
spection; and plainly, in poker there
is
only a rough correlation between the
internal character of sequences within a partition and their payoffs:
what
is
important here, as elsewhere,
is
the classification, which
is
very largely arbi-
trary.
Medawar's argument, on its face, thus involves rather an uninspiring
mistake,
but it
is
not yet a mistake in evolutionary thought. The human eye, a chastened
Medawar might argue, turning his back on his own analogy between life
and the
cards, represents one arrangement of its constituents: any other might
have done
as
well.
In admiring the structure that results, we suffer from misplaced awe,
like
a toad contemplating a dog. Does this argument carry conviction eye-
wise? Is it
reasonable to suppose that any other arrangement of the eye's
constituents would
result in an eye? In anything at all? The question sounds
an
unavoidably Aristo-
Page 31
The Language
of
LVe
23
telian note: an eye is an organ with a specific function
-
sight, most obviously; an
eye-like configuration does not count as an eye unless it can see. To
frame the
discussion thus is to answer the question immediately, at least on the
level of
intuition; but what I have said must not be confused with an argument
in refuta-
tion.
Viable proteins
Linguistics is possible if only because human beings have strong and
reliable
intuitions about natural languages. The polypeptides are alien
strings, accessible
only through an arduous act of the biochemical imagination. Grammar
effects a
segregation of strings in a language-like system; beyond grammar,
aloof, untouch-
able, there is meaning; the two concepts do not coincide. Some
grammatical
strings, in a natural language, at least, are grammatical and
meaningless; others,
meaningful but ungrammatical; but meaning and grammar belong together,
yoked
pairs in the same corner of some dimly understood conceptual space. An
algebraic
system of strings in which no distinctions of meaning and grammar are
recognized
is profligate; and pointless because of its profligacy.
In a preanalytic sense, the concept of meaning indicates a kind of
coherence;
and has a usefulness of application in domains other than language. A
life well-
spent is meaningful: its parts and patterns are ordered; full with
life, biological
creatures are filled with meaning, a kind of blunt, irrefrangible
purpose; in death,
this meaning disappears, and what is left, the corpse and its grim
constituents,
appears all at once to lose the integrity of the creature itself, and
becomes,
instead, a thing among other things, an object merely. To the
vitalist, living
creatures instantiate some unique property that remains stubbornly
unseen else-
where
-
in the domain of objects studied by mathematical physics, for example;
in
death, this property vanishes, like a fluid evaporating. In
mechanistic thought,
the passage from life to death is rather like a phase transition, a
singularity of
sorts in the trajectory of the organism. a disabling and permanent
catastrophe,
that reflects, as it must. only a change in the constituents of the
organism, a vari-
ation in its underlying pattern. The concept of a complexion. which
figures in statistical mechanics, provides a useful measure of
meaning. The com-
plexion set to a biological organism represents those relational
alternatives of its
biological parts that correspond to living systems. The unalterable
fact that living
systems die and hence do not persist indicates that some of their
complexions fail
to preserve life and hence meaning; in fact, the number of meaningless
complex-
ions must be significant: most of the arbitrary rearrangements of a
complex organ
-
a mammal, say
-
result in nothing more than a botch
-
a circumstance with
which every surgeon is familiar. The Central Dogma of molecular
biology estab-
lishes a relationship between strings of nucleotides and strings of
proteins; to the
extent that the whole of a biological organism may be resolved into
its protein-like
parts, the Central Dogma establishes a larger, more indirect,
relationship be tween
molecular biological order and order in the larger sense of Life. This
relationship
has an inverse: if only certain forms of life have meaning, this, too,
is reflected, as
it must be, in the universe of molecular biological strings
-
on the level of string
ensembles, for example. If certain protein ensembles are meaningful,
and not oth-
ers, this suggests, but does not imply, that the same distinction is
palpable on the
level of the individual proteins themselves. The
term
viable I mean as a biological
Page 32
coordinate to the Siamese concepts of meaning and grammar; a protein
is
viable
only when it achieves a certain
minimum
level of biological organization and useful-
ness. What level? What kind of organization? Usefulness in what
respect and to
what degree? Who knows?
FULL
loads,
fair
loads,
jhir
samples
In a natural language. sentences decompose to words; words to letters.
Gram-
matical constraints hold weakly at the level of English words. The set
of
all
word-
like combinations of English letters of fixed length n. I shall say,
make up a
fill
load;
the
set
of
all
grammatical words, a jhir
load.
Within molecular biology. a
full
load corresponds to
all
possible proteins of normal length: a set whose cardinality
is
2 0 .
TO the
fair
loads in English correspond the viable proteins in molecular
biology. How large
is
the biological fair load? Again. who knows? Whatever its ulti-
mate size, those proteins that have already been synthesized in the
course of bio-
logical history are viable
if
anything
is:
nothing succeeds Like success. This set
is
a fair
sample
of a
fair
load. Its size Murray Eden calculates at 2 0 .
The
task
that he sets himself
is
the infinitely delicate one of drawing inferences about the
fair
load from its
fair
sample.
[I51
Between the fair sample of a
fair
load. and the
f a i r
load itself.
is
the differ-
ence between what is and what might be; between the
fair
load and the full load.
the difference between biology and mathematics. In English, the
difference
between the
fair
load and the full load
is
as
absolute as death. Any two words of
English thus resemble each other more than they
are
Likely to resemble a word
generated at random from the letters of the English alphabet. In the
case of the
polypeptides. Murray Eden writes:
Two hypotheses
suggest
themselves. Either functionally useful proteins are
very common
to
this space.
so
that almost any polypeptide one
is
likely
to
find
has
a
useful function
to
perform. or else the topology appropriate to this
protein space
is
an
important feature of the exploration: that is. there exists
certain strong regularities for finding paths through this space.
In asking whether'the viable proteins are common in the space of
all
polypep-
tides, Eden
is
asking, in effect, whether the fair sample
is
marked by discernable
statistical regularities. "We cannot now discard the first hypothesis.
" he adds.
"but there
is
certain evidence which seems to be against it: if all polypeptide
chains were useful proteins, we would expect that existing proteins
would exhibit
very different distributions of amino acids." Statistical tests appear
to show that
pairs of proteins are
drawn
from a common stock. His example involves the alpha
and beta human hemoglobin chain. One form of hemoglobin has 146 amino
acid resi-
dues, the other 140. The two chains may be set down, side by side, and
matched.
residue by residue. They agree at 61 points: there
are
76
points at which they
differ, and 9 points at which no match
is
possible because the chains are not of
the same length. It is plausible that one chain was derived from the
other. or that
both were derived from a common ancestor. What is curious about these
pairs of
proteins, however, is the fact that even though the chains do not
agree completely
in the order of their amino acids, they do agree
in
their
distribution;
reason
enough, Eden argues, to suppose that the proteins themselves are drawn
from a
statistically significant fair sample.
The criticism of this historically important argument. I leave as an
exercise.
Page 33
The Language ofLiJ'e
25
Delicate inferences
In What is
Life?,
Schroedinger argued that living systems must have
recourse to what he dubbed an "aperiodic crystal" in order to store
information.
Crystals are repetitive, regular, and information poor; the order of a
living system
is specific, irregular, information rich. There is a certain splendid
effulgence to
the vocabulary of theoretical biology that it would be uncharitable
not to cherish.
H.P.
Yockey identifies order with Kolmogorov complexity; and so does R.M.
Thomp-
son, a mathematician who in writing on theoretical biology alternates
between
information theory and a pious endeavor to communicate to the reader
his appre-
ciation for the many faces of Krishna.[l6] On the other hand, G.J.
Chaitin and
R.M. Bennett identify biological order with algorithmic simplicity. A
division of
intuition on so fundamental a point may suggest a degree of conceptual
confusion
approaching the schizophrenic.
If biological words
are
characterized by a high degree of Kolmogorov com-
plexity, could time and chance have combined to discover a structure
comparable,
say, to cytochrome
c
or any of the modern hemoglobin chains? This is the ques-
tion raised by the redoubtable H.P. Yockey: the problem as posed has
but two
parameters.[l?] In the beginning the primeval soup, which I always
imagine as
rather a viscous, Borscht-like fluid, contained perhaps
amino acid molecules.
There is, inevitably, an element of fantasy to all quantitative
calculations of this
sort. At each second, over the course of 1
X
10' years, an indefatigable stochastic
Deity arranges and then rearranges the
amino acid residues in sequences
whose length N
=
101. There are
such sequences. The odds against discovering any one in particular
thus stand at 1
in 2.535
x
10131. Not
all
residues. however,
are
equally probable. Save for a very
large set of strings of small probability, the number of sequences of
length N is
where
Here
pj
measures the probability of the j t h residue, and a
=
2, so that H is
measured in bits.
In the end
-
the details are not important to my argument
-
Yockey con-
cludes that
H
=
4.153 bits/ residue
;
(9.12)
the number of 101 place sequences is
"Information theory," he remarks, "shows that, in this case, the
actual number of
sequences is smaller than the total possible number by a factor of
lo5".
Now there
are, in all. 3.8 x
lo6'
families of cytochrome
c
sequences; in order to obtain any
one of them by chance, Yockey argues, it would be necessary to repeat
an elemen-
tary stochastic experiment 3.15
x
times on
lo8
separate planets "in order to
Page 34
have a reasonable expectation of selecting at least once a member of
the ensemble
of
3.8 x
cytochrome c sequences in only ten of them".

From nothing, nothing, the Darwinian doubters have always claimed; and

I
have been there with the best of them; but
this
argument, couched as it
is
largely
within the algorithmic theory of complexity stands on what seems to
me
dubious
ground:
(1)
A binary string is random to the extent that its shortest program
is
roughly
of the same length
as
the string itself; this definition trades only in counting
bits. Now, the impulse to assert that contemporary proteins are random
owes
much. I think, to the rather primitive idea that life, if complex,
requires com-
plex constituents or atoms; I have suggested something similar in
arguing that
the proteins inherit a grammatical distinction from the structures
that they
constitute. Kolmogorov complexity, however, is ill-defined on any
level of bio-
logical organization past the molecular; but even if a mammal or a
mollusk
could be represented as a binary string, nothing suggests that those
strings
would be high in Kolmogorov complexity. Quite the contrary. Life in
the large.
on the level of the organism itself,
is
organized with what appears to be brisk
algorithmic efficiency. Living creatures are simple in the sense of
Kolmogorov
complexity; but complex under the classification of their complexions.
In this
sense, they behave much as a language-like system. This observation
is
com-
patible with the thesis that protein strings are, nonetheless. high in
Koho-
gorov complexity; but it is compatible, too, with the contrary thesis
that pro-
tein strings reflect the complexity of life by means of their
organization
and not their complexity. Nothing in the concept of Kolmogorov
complexity
measures the algorithmic organization of a string or set of strings;
two
equally complex strings may well differ in their time
complezity
to the
extent that only one
is
polynomially bounded.
(2)
The difference between the space of available proteins. and the small
subset
actually chosen by evolution, makes for a trite contrast; yet what
lends to
cytochrome c its position of statistical distinction? "Because of the
very fun-
damental function of the cytochromes." Yockey writes. "
....
the histones and
other proteins, which
are
believed to be of very ancient and even pre-
cellular origin, one cannot relax the
specificity
requirement derived from
cytochrome c" [emphasis added]. In generous conversation. Yockey has
ampli-
fied this point by suggesting that the specific protein chains
necessary for
life correspond to the set of words in a language
-
fair
and not full loads; a
curious remark inasmuch
as
words in a natural language
are
low, and not high,
in Kolmogorov complexity. Still. I am sympathetic to the drift of this
Line; but
the difficulty goes beyond the problems of an imperfect analogy.
Certain
classes of proteins, Yockey argues.
are
necessary for life. Such are the
information-rich, complex strands; other strands are specific in the
limited
sense that they are statistically unlikely: "only a tiny fraction of
the (avail-
able) sequences
will
carry specificity." It follows by Theorem
9.1
(p 245)
that specificity and complexity are not the same thing: the
set
of complex
strands (of a given length)
is
in the majority; their emergence
is
probabilisti-
cally favorable. indeed, unavoidable. C ytochrome c, considered simply
as a
complex protein.
is
no more likely to appear than any other complex protein:
but no less likely either. Having discovered cytochrome c, quite by
chance,
Page 35
The
Language
of
LVe
27
Life might have made do with any other protein of comparable
complexity. If
by specificity, Yockey means statistical unlikelihood in a uniform
sample
space
-
the space of all complex proteins, for example
-
his surprise at the
emergence of cytochrome c is attributable to retrospective
specification; if
not, what then is specificity, the mysterious middle term to his
argument? If
the specific proteins have some independent description, Yockey does
not
provide it; and their size, apart from suggesting that it is low, he
does not
calculate.
Der Prozess
The evidence in favor of the thesis that proteins are random sequences
of
amino acids is exiguous; and random words may well be grouped into
nonrandom
sequences. This suggests that the close study of the statistical
properties of cer-
tain proteins may involve a kind of dense conceptual myopia, something
that
reflects a passionate absorption in minutiae. The process by which
evolution in
strings takes place, on the other hand, is macroscopic and global, an
energetic
probabilistic swarming over sample spaces that are never specified by
means of
mechanisms that are never clarified.
Biological paths
Life loiters over two metric spaces. The first is alphabetic; the
second, zoo-
logical. Evolution comprises a drama in the large, at the zoological
level; but the
Central Dogma requires that any change in the large be mirrored by an
alphabetic
change, and so the process is doubled as it is divided. To talk
blithely of evolution
in strings is to assume the completion of the two first steps in
biological evolution:
the emergence of life-like systems from inorganic matter; and the
adventitious
creation of the modern biological system of replication and genetic
information. An
explanation of these steps I cede to the forces of the Night: my more
limited con-
cern is with evolution as a process that takes place once the genetic
machinery is
throbbing moistly. In evolution at the molecular level, one amino acid
is dropped
from a protein string, another is inserted: make way!, move over!, get
out!, get
Lost!, to cast the operations in easily understood
terms;
even if the process is
more complicated, it may mathematically be resolved into discrete and
finite
steps. Whatever the details, proteins change over time; and the
changes leading
to their creation may be regarded as a path
P
=
p l , p z ,
....,
pn or protein
sequence. Suppose that A comprises the full stock of 20 amino acids;
A / , the
set
of all words of amino acids precisely 250 points in length; and A*,
the set of all
finite sequences drawn over A / . I assume
-
an assumption note!
-
that A* has
the structure of a language-like system under the binary and
associative operation
of protein concatenation. where concatenation has precisely its usual
linguistic
meaning.
Page 36
28
D. Berltnskt
Stochastic processes
Let S be a system and X the set of its states or configurations. State
transi-
tions are represented by a transformation T : X
+
X , an artifice expressing the
action of the system's laws of evolution. If Ts+c
=
TsTt, [Tt ER] is a
flow,
or
group
action
of R on X.
On the Darwinian theory, evolution
is
at
its
secret heart stochastic;
it
is
natural, therefore, to specialize the concept of a process to the case
in which X
is
a measure space, T a measure preserving transformation. This
is
the domain
chiefly of ergodic theory. Its underlying, indeed, fundamental, object
is a proba-
bility space (X,
B,
u), where X is a set of
states.
B
a a-algebra of measurable sub-
sets of X, and u a countably additive nonnegative set function on
B.
u(X) is. of
course, 1. Let T be an invertible injection from X onto X;
if
u ( T I E )
=
u
(E)
for
all
E in
B,
T
is
a
measure-preserving transformation;
the system (X,
B,
u
,
T), a
basic probability space.
[la]
By the
orbit
of a measure-preserving transformation T , I mean the extended
history of a single point
z
under T from the infinite past to the infinite future: a
trajectory from void to void. Artificially truncated at
z ,
the system is in an initial
state or condition. A real valued function
f
:X
+
R, whose values correspond to
f
( z ) ,
f
(%),
f
(!I%),
....
acts to measure a system along its orbit; the class of such
measurements is defined only to the extent that
f is
itself measurable:
is
thus the
time mean
of the system;
its space
mean:
systems
in
which the two coincide for every measurable function
are
ergodic.
Example
9.2
Let A be an alphabet of n symbols
al,
a2,
....,
a,,
with probabilities
pl, pz.
....
,
p,.
such that
pi

0, and Cpi

=
1.
The product space nz consists of
the set of
all
two-sided sequences in n
;
the various probabilities assigned to each
sequence induce a measure
u
on
nz.
The shin
transfirmation
Vz),
=
z,
+l
is
measure preserving; the system that results
is
a finite-valued stationary stochas-
tic
process with identically distributed terms.
Example
9.3
Let
A4
=
(at,) be
an
n
x
n stochastic matrix. Let
p
=
(pl....,pn) be
a row probability vector fixed by
M:
Keep the product space and shift transformation from
Example
9.2:
Uu
may be
extended to a countably additive measure on the algebra generated by
cylinder
sets; by the Caratheodory-Hopf theorem,
Uu
thus forms a measure on the Bore1
Fields of n
' .
Example
9.2
models, say, a doubly infinite series of coin flips. each with
probabil-
ity of one-half;
Example
9.3,
a regular Markov chain, where
p
measures the
a
Page 37
The
Language
of
Ltfe
29
priori probability of each symbol,
M ,
the transition probabilities from one symbol
to another.
Consider a source consisting of a finite alphabet A and an associated
string
of symbols,
....
zoz iz z
....
,
where each
z,
is an element of A. Symbols appear in
sequence with a fixed probability pt; if the probabilities
are
independent, the
average entropy per symbol is
n
H z -
C
P, log2 P,
a
t
=1
H
is at its maximum if each pt
=
I/
n . In general, the probability that a particu-
lar symbol appears in sequence may depend on symbols that have gone
before.
This is true if the source is a finite-state Markov device. Let A' be
the ensemble
of all doubly infinite sequences drawn on A ; the cross section on A'
of sequences
that coincide at a finite number of points at
=
zti,
where
tt
represents any set of
integers, is a cylinder set. Now if A contains
k
letters, the number of n-term
sequences over A is k
;
and each sequence is a cylinder in the larger space A'. It
is the cylinder that has a fixed probability Pr(C): the
set
of all n -term sequences
represents a finite probability space, k n points in size. The average
amount of
information per symbol sent out by a source of this sort is
the entropy of the source itself
H
=
lim
-I/
k
C
Pr (C) logzPr(C)
..
k
+-
C €Cb
The concept of a source may be specialized to the case of a measure-
preserving system under ergodic constraints.
[l9]
The
Shannon-Macmillan theorem
A source puts out sequences; at any given time, there will be only
finitely
many
-
An in fact, if A is a finite alphabet, and n is the length of each
sequence.
Finite length sequences are cylinders in the infinite probability
space determined
by the source; they inherit a probability structure. If n is
sufficiently large,
there exists an arbitrarily small
t
and
6

0
such that the n-term sequences may
be separated into two groups. For the first
for the second,
This is Shannon's theorem, a result in mathematics that appears to add
an author
in regular periods. In any case, sequences of the first group are
characterized by
Page 38
the fact that (l/n) logZPr(C) is arbitrarily close to
-
H. The probability of any
such sequence Ct is thus
z":
the number of such sequences is ZnH, and
-
2n
logs
a
comprises a very small share of the total number
an
-
of available
sequences: a happy result. In coding a channel of communication.
attention need
be directed only to a tiny sample of the output.
The Stochastic Structure
of
an
Evolutionary
Source
In considering evolution
as
a stochastic process, the object of study becomes
biological paths; and not biological words
-
stray proteins, say, or bits of nucleic
acids. The full set of paths in evolutionary space comprises an
infinitely Large set
of strings,
if
only because evolution appears unbounded as a natural language. The
sheer stress on the notion of mndomness in popular accounts of
evolutionary
thought suggests at first that something like a pure Bernoulli process
may under-
lie
the whole business, an extended coin flip by means of a coin with 20''
separate faces. This is obviously absurd. Evolution
is
a process by which an
ensemble of strings changes over time. Each string
is
composed of points
-
amino
acids. in fact; the probability that any particular point
will
change is arbitrary
but low: there
is
little likelihood that
aU
points in a string
will
change simultane-
ously. Transition probabilities in a neighborhood
N
of a set of proteins
E
are thus
concentrated
in that neighborbood.
If an ensemble of proteins occupies a certain finite set of states
A*,
its evo-
lution comprises a finite state
Markov process
-
a stochastic source satisfying
the hypothesis of the Shannon-Macmillan theorem.
Trapping problems
The entropy of a source
is
a measure of its stochastic character: H at its
maximum represents a high degree of uncertainty: all messages are
equally prob-
able. The hypothesis of the neo-Darwinian theory
is
that evolutionary sources are
largely random. What this means
is.
in fact. not entirely clear; but
it
surely implies
that H
is
relatively large. Let HM, thus be the
imagined entropy
of an evolu-
tionary source. "If the process of manufacturing messages". Chomsky
and Miller
remark, "were completely random. the product would bear little
resemblance to
actual utterances in a natural language". [ZO] Going backward,
if
the utterances of
a natural language are regular, their source
is
not random. To the extent that a
fair sample of evolutionary paths is regular, the fair load is regular
as weU.
A
source
specifically designed
to generate the fair load of protein paths has thus
an entropy H
In itself, this
is
neither controversial nor surprising;
if
the degree of protein
regularity
is
small, the difference between H and HMax is negligible;
if
Large. an
evolutionary source
over-generates.
The real issue is a matter of degree, a ques-
tion of finesse. Linguistics. of course, suggests that if H is very
much lower than
HMar, over-generation becomes inordinate; a stochastic source cannot,
in general.
converge on any
natural
language whose complexity
is
beyond the recursive
Page 39
The Language of Life
31
capacity of finite-state automata; but while life may be a language-
like system, it is
not necessarily like a natural language, Chinese, say, or even
Esperanto.
Hemoglobin chains
Statistical entropy is a measure of uncertainty; and a measure, too,
of the
number of alternative messages
-
my use of the word is metaphoric
-
that a sto-
chastic source may generate.
H
at a relative maximum indicates that a source may
send out multiple messages, a kind of energetic babble; at a relative
minimum,
H
is
constrained
-
by the rules of grammar, for example, or the laws of logic. In the
case of life, Murray Eden observes, path lengths between proteins are
most obvi-
ously limited by time, and evolution must be achieved within bounds
set by the
number of generations in the history of an organism. Meandering paths
between
proteins
are
temporally inaccessible. The alpha and beta hemoglobins. Eden
argues, were derived by a process of evolution, one from the other; a
path
between the two sequences must thus exist. Eden calculates that this
path at its
shortest requires something like 120 separate steps, where each step
involves a
specific point mutation.[21] The population size of hemoglobin
proteins
-
the fair
sample
-
is, he estimates,
lo6;
the
rate
of mutation lo*. Each step in this path
corresponds to a positive gain in fitness: movement upward along a
local gradient
of relative perfection. In a conservative sense, it would take roughly
2 700 000 gen-
erations to convert a population of
lo6
alpha hemoglobin chains to a population of
beta hemoglobin chains. So far, so good.
If certain paths, of whatever length, are inaccessible to life, a
stochastic
source is occluded. This is certainly what the hypothesis that life is
a language-
like system implies; it is implied, too, by the fact that contemporary
hemoglobin
chains exhibit relatively little variance: certain possible paths are
deficiently
viable, ungrammatical in a sense. Nature, in passing from one chain to
another,
has evidently rather a small target in mind.
2700000 generations for the evolution of a protein is short: twice
that
number is long. Let k be a point midway between these numbers. If
Hi3
comprises
an initial set of
lo6
hemoglobin chains.
Pm
is the full load of protein paths [Pi,],
whose initial terms
PI,
all lie within m3. The number of targeted protein paths in
Pm
is small: so much to do, so much to see. The number of targeted
protein paths
that reach their target within k generations is vastly smaller: so
much to do, so
little time. If the entropy of a stochastic source is great, the
untargeted meander-
ing paths are apt to be favored; by contraposition, this implies that
the source
entropy for an evolutionary system is rather low; constrained, in
fact, by the
choice of time and targets; but what expresses these constraints, the
Darwinian
theory does not say.
Weizenbaum
Theory
1221
It is a peculiarity of molecular biological strings that, like the
elements of a
natural language, they realize two spaces. These are spaces with
distinct and dif-
ferent metrics: there is no reason to suppose that they are in phase.
Evolution as
a process works most directly on biological organisms, which must
perish or per-
severe in the face of circumstance. To the extent that evolution is a
process by
which organisms converge over time to some local (or global) optimal,
the
Page 40
processes of convergence that are sketched broadly in life must have
some sub-
stantial echo at the molecular biological level. where words and
strings hold sway.
The relationship between metric spaces that this pattern exemplifies
is
quite gen-
eral
-
the province, in fact, of Weizenbaum theory. Thus let
M
and N
be
two
metric spaces, each with its own natural metric; points in
M
are labeled
t l ,
tz.
....
,
-
points
in
N,
el, ez,
....
..
en;
j : M
+
N is a mapping between points in
M
and
tn
points in N
-
a bijection, to make matters trivially simple.
M
and N are arbitrary,
and admit of obvious specification:
(I)
M
is
a typographic metric space; N, the space of biological organisms (see
p
240).
(2)
M
is a typographic metric space under the natural metric on words; N,
the
same space under distance defined in terms of meaning or grammar (see
pp
245-248).
(3)
M is
a typographic metric space; N, a space of algorithms.
Thus
j
might map linear sequences of DNA or proteins, or sets of such
sequences, onto organisms, or sets of organisms; equally,
j
might map a linear
string of letters onto a sentence, with a fixed meaning in a natural
language; or
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 the natural metric on
M,
there exists an
induced metric dN(()
on
M
defined by the following relationship:
The Weizenbaum experiment
To specify a Weizenbaum experiment, it
is
necessary to provide
M
with a
pm-
bability transition system
Pr determining for each point
t
in
M
the probability
that
t
will
change to
t
';
and an
initial probability distribution Pro. A dis-
tinguished element e*
E
N
is
fixed from the first. Within the context of molecu-
lar biology, transition probabilities are focused on relatively nearby
strings
-
this
because point mutations result in string-Like changes of a short
typographic dis-
tance. In a biological Weizenbaum experiment, this fact is respected
to the extent
that the typographic metric space and the 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 the initial probability distribution
Pro
over
M.
The distance
dN(0)
from
j
(t
O)
to
e* is
measured; the system engaged
for
i
=
1.
2, 3.
....
;
as
tt
moves to
ti,
the distance
dN(t)
between
j ( t t )
and
e*
is
recorded. The outcome of the Weizenbaum experiment is the sequence
d ( ) *
d ( l ) * - - - s
d ( n
)
..
The Weizenbaum experiment is
successful
if:
Page 41
The Language o/L.tre
33
Condition
W For dN(0) at an average distance from e* the sequence [dN(t){
converges to a neighborhood of
0.
Condition
W, when met, implies that tdN(t)] is both stable and oriented. The
graph of a sequence of points constitutes a trajectory; the set of
trajectories in
N
that are at once stable and oriented is of measure zero. A successful
Weizenbaum
experiment thus establishes that Pr(M) cannot be arbitrary with
respect to its
induced metric structure. In particular, points that are f a r in the
induced
metric have small transition probabilities: those probabilities that
count must be
concentrated on nearby objects
-
nearby in the sense of the induced metric. On
the other hand, transition probabilities over molecular biological
strings are, on
the neeDarwinian theory, focused on neighborhoods that are nearby in a
natural
metric.
It is perhaps for this reason that. with the exception of life itself,
no one
has ever seen a successful Weizenbaum experiment.
Eigenvalues of natural selection
In Darwinian thought, the
effects
of randomness are played off against what
biologists call the constructive effects of natural selection, a
mechanism that
philosophers have long regarded with sullen suspicion. Wishing to know
why a
species that represents nothing more than a persistent snore
throughout the long
night of evolution should suddenly (or slowly) develop a novel
characteristic, the
philosopher will learn from the definition of natural selection only
that those
characteristics that are relatively fit are relatively fit in virtue
of the fact that
they have survived, and that those characteristics that have survived
have sur-
vived in virtue of the fact that they
are
relatively fit. This is not an intellectual
exercise calculated to inspire confidence.
Natural selection is a force-like concept; and, as such, acts locally
if it acts
at
all.
Mathematicians often assume that evolution proceeds over a multidimen-
sional fitness surface, something that resembles a series of hills and
valleys; a
great deal that 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 to
evoke the physicists's unhappiness at action at a distance. Strings
that are far
apart should be weak in mutual influence; this is a spatial
constraint. Then again,
no string should be influenced by a string that does not yet exist.
This is a
tem-
poral constraint, a rule against deferred success. The historical
development of a
complex organ such as the mammalian ear involved obviously a very long
sequence
of precise historical changes. Comparative anatomy suggests that the
reptilian jaw
actually migrated earward in the course of evolution. It is very
difficult to under-
stand why each of a series of partial changes in the anatomy of the
reptilian jaw
should have resulted in a net increase in fitness befire the advent of
the mam-
malian ear. Certain genes within the bacterial cell, to take another
example,
"are
organized into larger units under the control of an operator, with the
genes
linearly arranged in the order in which the enzymes to which they give
rise are
utilized in a particular metabolic pathway". [23] The genetic steps
required to
organize an operon cluster do not "confer any selective advantage to
the pheno-
type so that individual steps are independentU.[23] The rule against
deferred
Page 42
success functions as a prophylactic against the emergence of
teleological or
Aris-
totelian thought in theoretical biology. @3]
I have pictured evolution on the molecular level as a process
involving paths;
natural selection acts to induce a statistical drift on some paths,
and not others;
those paths involving a positive gain in fitness are favored. At any
particular time,
at any particular place, one has an ensemble
E
of protein strings, embedded, so to
speak, in an underlying probabilistic structure, a measure-preserving
system. to
keep to the concepts already introduced. To this structure, natural
selection
is
grafted, and acts, presumably in virtue of a property that may be
represented by
the action of a real-valued, measure-theoretic function: thus
j
(z
),
j
(Tz
).
j
('z).
....
are
successive local calculations of fitness under the action of the
system's
transformation. the
eigenvalues
to the system. Suppose now we consider a
finite-state system consisting of an alphabet of 26 letters; and the
set of
sequences
k
places in length. There are. of course, 6 such sequences. Each
letter a$
E
A
occurs with a fixed and independent probability
pi.
The shift
transformation moves a given string one place to the left. In effect.
this system
is
simply the finite-state stationary process with identically
distributed terms men-
tioned in the example already discussed; and may be represented as a
linear array
of
k
squares. An initial probability distribution fixes the configuration
of the sys-
tem for the first (integral) moment; at each subsequent step, every
square
changes: the odds in favor of any particular letter appearing are
1/26. If doubly
infinite in extent, this system models the play of
k
26-sided dice continued from
the indefinite past to the indefinite future.
What are the chances, one might ask (with a marked lack of
breathlessness in
my own case). that a system of this sort
-
a pure Bernoulli process
-
could con-
verge on a particdm sentence of English? Following Mannfred Eigen. let
us sup-
pose that the sentence in question
is
TAKE ADVANTAGE OF MISTAKES. so that
k is
23; this
is
the
target sentence
-
S .
Even here, poised between irrelevance and imprecision, delicate and
impor-
tant biological questions arise.@4] Thus. while it makes sense of
sorts to say that
for every string, there
exists
a target
-
there would be many target sentences
-
it makes f a r less sense to say. as Eigen does. that there exists a
target for every
string
-
just one. in fact. Fixed in advance. a target so singular would seem
suspi-
ciously like a goal and hence
streng verboten
in evolutionary thought. How might
such a target be represented and by what means might its influence be
transmit-
ted to strings? These are not trivial questions.
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
its goal. This. however.
is
a trivial defect, easily made good by the construction of
an
evaluation measure.
Suppose. for the sake of simplicity, that fitness involves
only a mapping from strings to 0 and
1:
at
S,
j(S)
=
1,
elsewhere.
j
is 0. An
evaluation measure serves to size up strings in point of fitness
as
they appear: at
S,
where
J
(S)
=
1.
it orders the system simply to stop; at all other strings, the
command is to mush on.
Stochastic device. target sentence. fitness function, and evaluation
measure.
taken as a quartet, comprise an
Eigen system.
The enterprising Professor William
R.
Bennett Jr has calculated that an Eigen system would require a
virtually infin-
ite amount of time to reach even a simple target sentence
-
a number roughly a
trillion times greater than the
life
of the universe In the same spirit. Murray
Page 43
The Language oj'LVe
35
Eden has figured that life would require something like 1013 blubbery
tons of E.
Coli "if one expected to find a single ordered gene pair in 5 billion
years". The
trouble is not simply one of finding the right letters: it is also the
problem of not
losing them once they are found.
What more, then, is needed? The opportunity, Stephen Jay Gould
remarks,
for the system to capitalize on its partial successes. Curiously
enough, this is
Eigen's answer as well, a bizarre example of independent origin and
convergent
confusion. As Eigen works though his example, his system is designed
to retain
those random changes that fit the target sentence. Looking at the
record of
Eigen's own simulation. we
see
that quite by chance the letter A appears in the
first generation in the right place on the sequence. It stays intact,
Apish so to
speak, for the
rest
of the simulation. When an E pops up,
it,
too. gets glued to the
system.
The
result is an advanced Eigen system, and an improvement over the hope-
lessly slow Eigen system already described. Under the advanced Eigen
system, fit-
ness is no longer an all or nothing affair; f thus takes values, let
us say, between
0 and 1. Scanning every new string, the evaluation measure selects
those strings
si
such that
j'
(st
)

f (st
These
the system retains until it finds a string
superior in point of fitness.
The
result is a sequence of strings the ascends in
fitness.
At
S , as before, the system stops.
An advanced Eigen system may well reach a target system in rather a
short
time: unfortunately, in theoretical biology, as elsewhere, the
question is not
whether but
how.
To the extent that fitness is purely a local property, it is diffi-
cult to understand why every ascending sequence should necessarily
converge to
a neighborhood of 1, and hence indirectly toward S. A string that only
partially
conforms to S is locally no
fitter
than a string that remains resolutely unlike S.
On the other hand, if each of the ascending sequences converge to S it
is very
hard to see that fitness is a local property, and hard thus to
understand what it is
that an evaluation measure manages to measure. What is unacceptable is
the obvi-
ous and tantalizing idea that an evaluation measure judges fitness by
calculating
the distance between random strings and a target sentence: distance is
not a
local property; an evaluation measure so constructed would plainly be
responding
to signals sent from the Beyond, a clear case of action at a distance.
The problem
of discovering a target sentence remains unchanged, hopeless. In fact,
this is pre-
cisely what the advanced Eigen system actually measures, since an
arbitrary sen-
tence in which A appears in the second position is judged fit only
because it is
closer to the target sentence than it might otherwise be. When the
matter is care-
fully explained, theoretical biologists understand at once that the
very concept of
a target sentence constitutes a beery and uninvited guest in
evolutionary thought.
I
have taken the argument a step further by insisting that evaluation
measures
themselves be purely local.
Need
I
insist that the situation is made no better
if
instead of a specific tar-
get sentence
I
talk of systems
set
for success when they reach any sentence
whatsover? I suppose, since it may at first appear easier to design a
system that
by randomly changing letters, in what Eigen hopefully calls the
evolution game,
approximates an arbitrary English sentence instead of just one. The
illusion of
ease is ill-gotten, of course: a target sentence is a minor stand-in
for a major con-
cept. If no particular target sentence is fixed in advance, then any
sentence of
English. once reached, makes for success. Simply to stop, the system
must have an
Page 44
abstract characterization of all the English sentences.
Of
these, there are infin-
itely many.
A
system bouncing briskly from one set of random permutations to
another, no less than the linguist or logician, thus requires nothing
less than a
grammar
of the English language
if
it is not to keep babbling forever.
I have described grammars in terms of the notion of formal support;
these
concepts receive no definition in Darwinian theory.
Notes
[I]
See my review of Michael Ruse's The Philosophy of Biology, in
Philosophy of Sci-
ence 41, Number 4, December 1974, for
a
discussion of Smart's position, and
related issues.
[2]
See, for example. Thomas Kuhn (1970) h e Structure of S c i e n t
cRevolutions
(Chicago: University of Chicago Press),
a
book which has prompted
a vast
secon-
dary literature and much merited soul-searching among analytic
philosophers of
science.
[3]
'Those who
are
oppressed by their own reputations."
D r
Johnson remarks,
"will
perhaps not be comforted by hearing that their cares are unnecessary."
Francis
Crick (1966) OJMolecules and Men (Seattle: University of Washington
Press).
[4] I discuss reductionism from the perspective of atomistic theories
in Berlinski
(forthcoming) The Aise of Di fferentid Topology (Boston: Birkhaeuser
Boston). See
also Kenneth Schaffner (1967) Approaches
to
reduction. Philosophy of Science 34
(1): 137-47.
[5]
Michael Ruse has argued for his thoroughly incoherent position
in
Ruse (1973) The
Philosohy of Biology (London: Hutchinson). The concept of evolution
was.
of
course, in the European air for
at
least
a
century before Danvin wrote. European
biologists
are
yet unreconciled
to Darwin.
In this regard.
see
Pierre Grasse
(1977) Evolution of Living Organisms (New York: Academic Press). The
facts
of
molecular biology, it is worth stressing, are not in dispute: it is
their interpreta-
tion that
remains
clouded. The central role of DNA. in particular, has troubled
many thoughtful observers. 'To attribute such powers
to a
single substance",
Grasse remarks 'however complicated its molecular structure, is in my
view aber-
rant."
[6]
M.C. King and A.C. Wilson (1975) Evolution at
two
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 for information theory and
probability theory.
IRZE Transactions on IMormation Z?zeory
IT
-
14 (5). I have patterned my
dis-
cussion on: G.J. Chaitin (1974) Information-theoretic computational
complexity.
IEEE
Transactions on IWormation h e o r y . IT
-
20 (1). The interested reader
should consult Chaitin's other papers. and relevant papers by Solovay.
Chaitin's
bibliography may be consulted for details.
[9]
See, for example, Noam Chomsky (1972) Language and 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)
Mathematicd Models in Linguistics (New Jersey: Prentice-Hail) for
details.
I am
inclined
to
think that the Deity, in creating the observable world, hesitated
between programming or painting the whole business. As
a
programmer, he would
have chosen
a set
of recursive rules; as
a
painter,
a
system of simultaneous equa-
tions.
[I11
David Hull (1974) Philosophy of Biologicd Sciences (New Jersey:
Prentice-Hall),
although staid, contains
a
competent discussion of many of these issues.
Page 45
The
Language of Lifi
37
[I21 See, for example, L. Lofgren (1975) On the formalizability of
learning and evolu-
tion, in Suppes. Henkin, Joja, and Mosil (Eds)
Logic, Methodology and Philosophy
ofscience
(Amsterdam: North-Holland).
[13]
J. Monod (1971)
Chance and Necessity
(New York: Alfred Knopf).
[14] See Peter Medawar (1977)
The Life Sciences
(London: Wildwood House).
[15]
Murray Eden (1967) Inadequacies of neo-Darwinian evolution as a
scientific theory,
in P. Moorhead and
M.
Kaplan (Eds)
Mathematical Challenges to Neo-&minism
(Philadelphia: The Wistar Institute Press).
[I61 R.M. Thompson (1981)
Mechanistic and Non-Mechanistic Science
(Lynbrook,
New
York: Bala Books).
[17] H.P. Yockey (1977) A calculation of the probability of
spontaneous biogenesis by
information theory.
Journal of Theoretical Biology
67.
[I81
See K. Petersen (1983)
Ergodic Theory
(Cambridge: Cambridge University Press)
for details.
[I91 My discussion follows that of A.I. Khinchine (1957)
Mathematical Foundations of
Information Theory
(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 Psychology
(New York: John Wiley
&
Sons).
[21] Eden.
op. cit.
[22] The idea of the Weizenbaum experiment is due to M.P.
Schutzenberger.
[23] Eden,
op. cit.
[24] See, for example, Eigen (1971) Self-organization of matter and
the evolution of
biological macromolecules.
Die NaturwissenschqtYen
10. Together with Ruth
Winker, Eigen has recently (1981) published a popular account of his
thought
under the title
The Laws of the Game
(New York: Harper
&
Row).
References
Grantham, R. (1974)
Science,
185,
62.
Schroedinger, E. (1945)
What
is
Life?
(New York: Macmillan).
Smart, J. J.C. (1963)
Philosophy and Scientific Realism
(London: Routledge and Kegan
Paul).
Watson,
J.D.
(1965)
Molecular Biology of the Gene
(New York: Benjamin) p 67.

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