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\begin{document}

\trnum{9201}
\date{March 1992\\(Minor corrections: 14 April 1992)}
\title{Essays on Darwinism \\1: Ontological Foundations}
\maketitle

\begin{abstract}

The problem tackled in this essay is to formulate ``Darwinian''
theory in a way which is divorced from any specifically
biological interpretation. For my purposes, the crucial
constraint on this abstract formulation of Darwinian theory is
that it must qualify as a realisation of {\em unjustified
variation and selective retention\/} in the sense of
D.T.~Campbell \cciteD{Campbell:BlindVariationInCreativeThought,%
Campbell:BlindVariationAsGeneralStrategy,%
Campbell:EvolutionaryEpistemology,%
Campbell:UnjustifiedVariation}---for it is (arguably) only
thus that Darwinian evolution can lead to the spontaneous growth
of ``knowledge''. I hope that such an abstract formulation might
help to clarify the application of the theory even in its native
biological setting; but, more importantly, it seems to me that
this exercise is a necessary pre-requisite for the valid
application of Darwinian principles in any domain {\em other\/}
than conventional, terrestrial, biology---such as the emerging
field of study commonly referred to as {\em Artificial Life}. In
attempting to solve the problem of formulating a satisfactory,
abstract, version of Darwinism, I introduce a novel ontological
category: the Similarity-lineage, or {\em S-lineage}. With the aid of
this hybrid category, which combines some aspects of both a class
and an individual, I argue that Darwinian evolution relies on
a process of S-lineage selection---which is to say that
S-lineages are, in a certain precise sense, the elusive
Darwinian {\em units of selection}.

\end{abstract}

\pagenumbering{roman}
\tableofcontents

\pagenumbering{arabic}
\section{Introduction}

\begin{quote}

With the work of Einstein, physicists were faced with a dilemma: If they
insisted on retaining Euclidian geometry, they would have to be content
with extremely complicated and variable laws; if they wanted laws
applicable anywhere in the universe, regardless of velocity, they would
have to abandon Euclidean geometry.  They opted for the second
alternative.  Evolutionary biologists are currently confronted by a
similar dilemma: If they insist on formulating evolutionary theory in
terms of commonsense entities, the resulting laws are likely to remain
extremely variable and complicated; if they want simple laws, equally
applicable to all entities of a particular sort, they must abandon their
traditional ontology.  

\citeAD{Hull:IndividualityAndSelection}

\end{quote}

The general notion of ``Darwinian evolution'' may be used very
loosely, with a wide variety of interpretations.  Considerable
confusion can result.  Thus I believe that it may be useful to
present a general, and abstract, discussion of the ``concept'' of
Darwinism, and that is the primary purpose of this essay.

This should not be interpreted as an attempt to probe the ``true
essence'' of Darwinism but rather as a stipulation of the kind of
thing to which I have decided to attach the label ``Darwinism''.  As
it turns out, my usage is somewhat distinct from, or at least
more precise than, the more generally accepted usages, though
naturally it overlaps considerably with them. I should emphasize
that this increased precision is not pursued for its own sake,
but because I have found it necessary for the treatment of
problems of interest. Nonetheless, some readers may wish to
challenge my choice of ``Darwinism'' to denote this thing; in
which case, I should refer them to Karl Popper:

\begin{quote}
If challenged by the question of whether a word one uses really
means this or perhaps that, then one should say: `I don't know,
and I am not interested in meanings; and if you wish, I will
gladly accept {\em your\/} terminology.'

\citeAD[Essay~8, p.~310]{Popper:ObjectiveKnowledge}
\end{quote}

We might expect that Darwinism could most easily be characterised
simply by reviewing it in its original, biological, context.
However, I am in agreement with the general position outlined
in the opening quotation above from Hull.  The orthodox categories
used in biology predate Darwinism; thus it would be hardly surprising
if they proved not to be the most suitable vehicles for the
expression of Darwinian theory, even within biology itself.  The
result is that, faced with the complexity of modern evolutionary
biology, it has become very difficult to distinguish issues which
are intrinsic to the notion of Darwinism from details or
artefacts arising from the particular realisation(s) of Darwinian
principles which (happen to) exist in the biological world as we
know it.\footnote{Implicitly, of course, I conjecture that some
  such distinction is indeed possible.}
I would even go so far as to say that the vastly more
comprehensive biological knowledge we now have, relative to that
available to Darwin, may actually serve to obscure rather than
illuminate his original insight.

The presentation I give here has been substantially influenced by
the ideas of Hull
\cciteD{Hull:IndividualityAndSelection,Hull:UnitsOfEvolution},
and Dawkins
\cciteD{Dawkins:SelfishGene,Dawkins:ExtendedPhenotype}.
Nonetheless, I present an essentially {\em original\/} proposal
(complete with its own jargon). I take this step with some
reluctance, but no apology. I shall argue elsewhere
\cciteAD{McMullin:GenicAndOrganismic} that the analyses
of Hull and Dawkins harbour some subtle and intricate flaws; but
it seems to me that {\em that\/} argument cannot even be coherently
stated without first establishing an independent ontological
foundation to support it.

Quite separately, there is also a significant overlap between my
formulation and the {\em Axiomatic Darwinism\/} introduced by
M.B.~Williams \cciteD{Williams:ConsequencesOfEvolution} (although I
became aware of this connection only after the essay was largely
complete).  I review this in more detail in section~\ref{mbw}
below.

The important issue in the framework I present is, of course, its
substance---the ontological categories I introduce, and the
relations between them;  the specific terminology is secondary.
However, I should warn in advance that at least some of my
terminology may seem somewhat arcane. My only excuse is that it
was necessary to carefully discriminate all the required concepts
while still avoiding any existing terms which might have had
confusing or prejudicial connotations.  This is not easy.
In any case, I can do no less that accept the same assessment
criteria as Hull suggested for his own related work:
\begin{quote}
This reconceptualization of the evolutionary process is certainly
counter-intuitive; its only justification is the increased scope,
consistency, and power of the theory that results.  If the
terminology suggested in this paper cannot characterize the
evolutionary process more accurately and succinctly than the
traditional terminology, it should not (and will not) be adopted.

\citeAD{Hull:IndividualityAndSelection}
\end{quote}

\section{Metaphysics}

I shall use the following general metaphysical categories.  These
are based on categories proposed by
\citeAD{Hull:UnitsOfEvolution}, but with some qualifications.

\subsection{Individuals}

In Hull's words, an individual is a ``spatiotemporally localised
entity which develops continuously through time''.  That is to
say, it is an entity which occupies a connected region of
spacetime---it has a continuous (but generally finite) extension
in space and time. Implicitly, I am accepting ``space'' and
``time'' as ontological primitives requiring no further analysis.

Individuals must be reasonably discrete in both time and space.
In particular, they can (if necessary) be distinguished from each
other {\em solely\/} by reference to their extension in time
and/or space---no two individuals may have precisely the same
spatiotemporal extension. That is, individuals are unique
historical entities. I shall refer to the extension of an
individual in time as its {\em lifetime\/}, and say that this
lifetime is delimited by its {\em birth\/} and its {\em death}.

Hull adds that individuals should also exhibit ``internal
cohesiveness at any one time''.  I presume he introduces this
condition in order that completely arbitrary regions of
spacetime need not be recognised as individuals.  However, I see
no benefit in making this {\em criterial\/} for
individuality---in practice we are not so capricious in our
designation of individuals.  If an entity's {\em only\/}
characteristic is its extension in spacetime, we shall not be
interested in it anyway; otherwise, we may well be interested in
it, and argument about whether its characteristics include
``cohesiveness'' (whatever that may be) are hardly productive.
Indeed, Hull himself makes a similar point when he says that
``the only individuals which are scientifically important are
those which are acted upon by natural processes''. No more
stringent criterion is needed.\footnote{If, notwithstanding my
  arguments here, one insists on including ``cohesiveness'' in the
  definition of ``individual'' I suggest it can be best done by
  interpreting it in the relatively technical sense of {\em
  autopoiesis\/} \cciteAD{Varela:BiologicalAutonomy}.  I am, in
  fact, quite sympathetic to that view; I just feel that it would
  be an unnecessary complication for my immediate purposes here.}

Distinct individuals, although unique, may be said to be more or
less {\em similar\/} according to various criteria.  In
particular, they will be said to be ``identical'' if they differ
{\em only\/} in spatiotemporal location. This is, of course, a
purely metaphysical notion of identity---I do not imagine that it
could be operationally realised---and for that reason I shall
retain the scare quotes around it.  I take Hull to have the
converse of this concept in mind when he refers to the idea of
individuals being ``numerically'' distinct.

Note that we have here a specialised, technical, definition of
``individual''.  In particular, it must not be confused with the
common biological convention which uses ``individual'' almost
synonymously with ``organism''.

\subsection{Classes}

A class is a {\em spatiotemporally unrestricted} entity, which
can have individuals as members.  Thus, although a class consists
of, or is composed of, entities which are spatiotemporally
localised, the class itself is not---it exists outside the
spacetime framework.  Indeed, a class may be perfectly
well-defined even if it has no members.

For many purposes, classes can be thought of as mathematical {\em
sets\/} (with the stipulation that class membership must be
spatiotemporally unrestricted). In what follows I shall freely,
if loosely, use the concepts and terminology of set theory in
discussing classes.

Class membership cannot involve spatiotemporal relations
between individuals, as this would imply some kind of
spatiotemporal restriction (relative or absolute) on the class.

The definition of a particular class {\em introduces\/} a
relation between its members, of course; but this
relation will not be recognisable from, or dependent upon,
the spatiotemporal configuration of the individuals. To put it
another way, the class membership of an individual must not be
sensitive to its spatiotemporal location.  In particular, given
that one individual is a member of a class, it follows that all
``identical'' individuals (if any), regardless of spatiotemporal
location, must be (distinct) members of the class. These need
not, of course, be the {\em only\/} members of the class---i.e.\
classes are not restricted to consisting of ``identical''
individuals.

Apart from the requirement that it not involve any spatiotemporal
restriction, no other constraint is placed on mechanisms of class
definition---although, obviously, for any practical application
of the ideas discussed here, we would require that class
membership be defined {\em operationally}.

Classes, being spatiotemporally unrestricted, are evidently not,
themselves, individuals.  So classes and individuals are mutually
exclusive kinds of entity.  In what follows, an entity may be
described as a class or an individual, but never both.

\section{Actors}

Individuals may be related in various ways.  Of particular
interest for my immediate purposes are relations of {\em
descent}. Hull relies on the common sense idea of descent: that
individuals are related by descent when one entity ``causally
produces'' another.  Without disagreeing with Hull on this, I
should nonetheless like to define the notion a little more
carefully, as I shall ultimately want to consider certain
distinct kinds of relation as being examples of (distinct kinds
of) descent.  It is important at this point that the reader try
to put aside (for the time being) any preconceptions she may have
as to what exactly constitutes a relation of ``descent''.

I shall refer to an individual for which a relation of descent is
defined as an {\em actor}. It is important to emphasize that
individuals are not intrinsically or self-evidently actors---they
are actors only relative to some definition or interpretation of
what we mean by descent. Indeed, the same individual may be an
actor relative to more than one definition of descent.  So
whenever an individual is referred to as an actor below, some
particular definition of descent, applicable to that individual,
is always being assumed, even if this is not expressly stated.

While I have said that descent is a matter of definition, I do
not mean that any arbitrary relation defined on some class will
serve as a relation of descent.  There are certain
restrictions which it must conform to.

Descent in general is defined in terms of a more primitive
relation, which I shall call {\em immediate\/} descent. This
relation has the following characteristics:
\begin{itemize}
\item It is {\em binary}, i.e.\ it is a relation which
exists (or not) between {\em two\/} individuals.

\item It is asymmetric---it has a ``direction''.  I shall henceforth
speak of one individual being (immediately) descended {\em from\/}
another; the former will be termed the {\em offspring\/} and the latter
the {\em parent}.

\item In general, any particular definition of immediate descent
is applicable only to some particular set of individuals. I shall
assume that such a set will be spatiotemporally
unrestricted---i.e.\ it will constitute a class.  Thus, a
definition of immediate descent will incorporate, implicitly or
explicitly, the definition of a class of individuals for whom
this relation can be evaluated.  This class is then, by
definition, the class of {\em actors\/} relative to the given
definition of (immediate) descent.

\item I shall stipulate that an actor can be the parent of
another only if the birth of the second occurs {\em after\/} the
birth of the first. Informally of course, we mean that, in some
sense, the parent has ``causally produced'' the offspring, or at
least has been ``causally implicated'' in its production. It
follows that no actor may be a parent of itself.
This condition is somewhat similar to M.B.~Williams'
\cciteD{Williams:ConsequencesOfEvolution} {\em Axiom~$B1$}; although
she avoids any explicit spatiotemporal restriction, the effect is
essentially equivalent. My formulation is, in one sense, slightly
stronger in that we can already also deduce that no
actor can be a parent of (any of) its own parent(s), whereas
Williams provides a separate axiom ($B2$) to capture this. On the
other hand, Willams' formulation has the advantage that it does
not require an axiomatic conception of absolute time.

\end{itemize}

Now a definition of immediate descent can, and generally does,
depend on a variety of factors relating to the particular actors;
but regardless of what else it depends on, it is clear that if
one actor is immediately descended from another this fact
necessarily implies something about their relative spatiotemporal
locations. Thus, no set of individuals which is defined wholly or
partly in terms of immediate descent can function as the
definition of a class.\footnote{At the risk of compounding the
  confusion already engendered: a {\em definition\/} of immediate
  descent may be spatiotemporally unrestricted, and may serve to
  define a class (the class of individuals on which this
  definition of descent may be successfully evaluated---its
  actors); however, any particular {\em example\/} of immediate
  descent, in accordance with some such definition, necessarily
  contains spatiotemporal restrictions and cannot function wholly
  or partly as the definition of a class.}

Immediate descent requires ``similarity'' between parent and
offspring only in the following weak sense: they must both (by
definition) be members of the class on which this particular form
of immediate descent has been defined.  Thus immediate
descent must, in general, imply {\em some\/} kind of similarity
between parent and offspring (namely, enough to establish
that they {\em are\/} both actors relative to the given
definition).  To put it another way, immediate descent may be
compatible with more or less arbitrary {\em differences\/}
between parent and offspring. It is for this reason---the fact
that immediate descent, as I use it, does not {\em necessarily\/}
require a high ``degree'' of ``similarity'' (however that might
be defined) between parent and offspring---that I do not use
terms like ``replication'' or ``reproduction'' here.

I have expressed (or at least, constrained) immediate descent as
an asymmetric, binary, relation between individuals (actors
relative to this form of immediate descent).  That is, any two
actors definitely either are or are not related by immediate
descent; and if they are so related, one is definitely
identifiable as the parent and the other as the offspring. This
should not be read as restricting the number of such
relations any single actor may participate in: in general, an
actor may have zero, one, or many parents, and may have zero,
one, or many offspring. Particular definitions of immediate
descent may be more restrictive of course.

If an actor has no parents, I shall say that it has been {\em
created\/};  otherwise I shall say it has been {\em procreated}.
A created actor will also be called a primary actor, or simply a
{\em primary.}\footnote{Primaries thus constitute a sub-class of
  the class of actors, relative to some particular definition of
  (immediate) descent.}
Please note carefully that I use the term {\em creation\/} to
describe the birth of an actor, {\em other\/}
than by (some particular definition of) immediate descent.  {\em
This is without prejudice to the exact process actually
involved}.  For example, creation relative to one definition of
descent may actually correspond to {\em pro-}creation by some other
definition. Thus when I refer to creation I do not imply any
inherently supernatural or mystical process. I am aware of the
dangers of my usage, which already existed, perhaps even more
acutely, in Darwin's day. As Hull puts it:
\begin{quote}
  \ldots even though `creation' might well have been a code word
  for unknown natural processes, the use of this word, especially
  when it was coupled with all sorts of additional theistic
  references, was guaranteed to give just the opposite
  impression.\nopagebreak

  \citeAD{Hull:DarwinAndTheNatureOfScience}
\end{quote}

However, it seems to me that ``creation'' is still, in its
literal sense, the best word for the job; so, I will continue to
use it, while hoping that this advance warning will mitigate any
confusion.

If an actor has exactly one parent, I shall say it is a product
of {\em unimodal\/} procreation; if it has exactly two parents, I
shall say {\em bimodal\/} procreation etc.; if the number of
parents is unspecified, but definitely more than one, I shall
refer to {\em multimodal\/} procreation.  This is obviously
similar to the biological idea of asexual versus sexual
reproduction, but is more general in that it permits more than
two parents, and is neutral on the question of whether there are
any constraints on which actors may ``mate'' with each other---in
particular, whether there is any kind of differentiation of
actors into distinct and complementary ``sexes''.

A set of actors which are all related by immediate descent to a
certain parent are said to be {\em siblings}.

Given a relation of {\em immediate\/} descent, the
generalised relation of descent is simply the transitive form of
this.  That is, if one actor is a parent of a second, and it, in
turn, is a parent of a third, then the first is said to be an
{\em ancestor\/} of the third, and the third is said to be a {\em
descendant\/} of the first (and so on).

\section{Lineages}

Given a relation of descent, I shall define a {\em lineage\/}
as any set consisting of some identified {\em primary\/} actor
(called the {\em founder\/} of the lineage), together with all of
its descendants.\footnote{I diverge somewhat from Hull's
  \cciteD{Hull:UnitsOfEvolution} definition of ``lineage'' here:
  Hull does {\em not\/} require the founder of a lineage to be a
  primary (indeed, in Hull's terms, {\em every\/} actor may be regarded
  as a founder).  However, I will effectively recapture the
  required generality of Hull's version when I introduce the
  notion of an {\em S-lineage\/} in section~\ref{similarity}
  below. I may note in passing that the Hullean lineage is, in
  turn, largely equivalent to M.B.~Williams' terms {\em clan\/}
  and {\em subclan\/}
  \cciteAD{Williams:ConsequencesOfEvolution}.}
Thus, each actor in a lineage is connected to the founder by at
least one continuous chain of ancestors each of which is also a
member of the lineage.

Every lineage has exactly one founder, and every primary actor
founds exactly one lineage (possibly consisting only of itself).
Any given actor may be a member of an arbitrary number of
distinct lineages---specifically it will be a member of the
lineages founded by each of its primary ancestors. However, by
the same token, a primary actor is a member of just one lineage,
being the one it founds.

Under any form of unimodal (``asexual'') procreation every actor
is either itself a primary or is descended from one, unique,
primary ancestor. Thus, in this case, lineages will all be
disjoint (every actor will be a member of exactly one lineage).
However, where multimodal (``sexual'') procreation is possible,
lineages may, over time, converge to an arbitrary extent.  Note
that if it arises at any time that the membership of two lineages
has become equal, then the membership of the two lineages will
remain identical thereafter---effectively they have become one.

Lineages are, in the terms discussed previously, well-defined
{\em individuals\/}; that is, they are distinct entities which
are reasonably well localised in space and time (by virtue of
consisting of a definite set of actors, each of which is
separately well localised in space and time).  Although it is an
individual, a lineage is, of course, a distinct individual from
any of its members; indeed a lineage is, in general, a different
{\em kind\/} of entity from its members.  In particular, a
lineage will not be an actor---not, at least, with respect to the
same definition of descent as relates its own members. A lineage
may, of course, function as an actor relative to some (other)
definition of descent---in which case, there will exist ``higher
level'' lineages etc.

Subsets, intersections and unions of lineages are not generally
lineages in their own right (though they are always well-defined
individuals).

Although a lineage has been defined as a kind of set, the
``size'' of a lineage will not refer to its total number of
members (or its cardinality in conventional set theoretic terms), but
rather to the number of members {\em at any given time}.
In this sense, lineages may grow or shrink over time and this
will be reflected in their sizes, which will be, in general,
well-defined integer functions of time.

A lineage continues in existence for as long as some descendants
of the founder have further offspring, so that a lineage is {\em
potentially\/} immortal (even if the actors themselves
necessarily each have a limited lifetime). However, a lineage can
and does die if it happens that, at any time, the last remaining
member dies---which is to say that the size of the lineage
becomes zero.

\section{On Similarity}

\label{similarity}

I have been at some pains to emphasize that a relation of
descent, in itself, need say very little (if anything) about the
``similarity'' between parents and offspring.  We shall see that
this freedom is essential: Darwinism demands that {\em both\/}
similarity {\em and\/} variation be associated with descent in
certain circumstances.

Let there be a class of individuals which are actors relative to
some relation of descent. Let an arbitrary, but well-defined,
sub-class of this class of actors be termed a Similarity-class or
{\em S-class}. That is, S-classification is a formalisation of
the notion of ``similarity'' {\em within\/} some class of actors.
Note that any given actor can, in general, be a member of many
different S-classes.

Consider some actor which is a member of a particular S-class. We
shall say that if an offspring of this actor is {\em also\/} a
member of this S-class then this is an example of {\em
S-descent\/} (relative to the specified S-class).  S-descent is a
special case of ordinary descent in which the offspring is
``similar'' to the parent in whatever characteristic(s) determine
membership of the particular S-class.

Depending on our choice of S-class, S-descent may range from
being non-existent to universal.  Roughly speaking, the former
would correspond to an S-classification which is not heritable at
all (is independent of the given relation of descent), and the
latter would correspond to an S-classification which is
completely heritable (in the limit, the S-class is set equal to
the class of actors for the given relation of descent, which
classification is completely ``heritable'' by definition).

A Similarity-lineage or {\em S-lineage\/} (relative to some
S-class) is then defined in much the same way as an ordinary
lineage, except that the members must be related by {\em
S\/}-descent.  More strictly, we define an S-primary as a member
of the specified S-class, none of whose parents (if any) were in
the S-class; we define an S-descendant of an S-primary as any
actor connected to the S-primary by a continuous chain of
S-descent; and an S-lineage is then defined as a set consisting
of an S-primary together with all its S-descendants.  Thus, every
S-primary is an S-founder of an S-lineage. The S-size of an
S-lineage will refer to the number of members it has at any
particular time. The notion of an S-lineage is largely equivalent
to Williams' \cciteD{Williams:ConsequencesOfEvolution} {\em
Darwinian subcland (sic)}.

S-lineages will evidently be subsets of lineages, but will {\em
not\/} be lineages in their own right (an S-primary need not be a
primary in the absolute sense; for that matter, an S-lineage will,
in general, exclude some descendants of the S-founder---namely
all those which lack a continuous chain of {\em S\/}-descent from
the S-founder).

By the definition of S-descent, we can see that all actors making
up a single S-lineage must be members of the same S-class. This
is the peculiar manner in which the S-lineage idea is a sort of
hybrid of an individual and a class: it is (among other things) a
spatiotemporally delimited subset of a class. In what follows I
shall sometimes loosely speak of an S-lineage being ``of'' an
S-class as a shorthand for saying that all its {\em actors\/} are
of that S-class.

Note, however, that the converse to this idea is not true:
actors which are of the same S-class need not be members of the
same S-lineage.  Specifically, if there exists more than one
S-primary belonging to a given S-class, these will found multiple
S-lineages containing actors of the same S-class.

In the case of unimodal (``asexual'') procreation, all S-lineages
relative to a given S-class, being subsets of distinct
lineages proper, are necessarily disjoint.  In the case of multimodal
(``sexual'') procreation, on the other hand, S-lineages relative
to the same S-class may intersect; indeed, at some point in time
they may become completely converged, and must then remain
completely converged at all later times.

S-lineages relative to {\em different\/} S-classes may intersect
under both unimodal and multimodal procreation, provided only
that the S-classes themselves intersect---which is to say that
a single actor may be a member of many
different S-lineages of different S-classes; or conversely, if
two S-classes are necessarily mutually exclusive, then their
respective S-lineages (if any) cannot intersect.  Even if two
S-lineages relative to different S-classes become completely
converged at some time, they may subsequently diverge again.

The actors present in a system at any given time may be viewed as
distributed over one or more S-lineages relative to various
S-classes.  These S-lineages may grow and shrink over time, and,
on occasion, some may ``die'' (become extinct). To the extent
that there is an on-going process of {\em S-creation\/} (the
birth of new S-primaries) new S-lineages may also be initiated or
``born'' over time.

\section{Selection}

\label{selection}

I shall now consider the general notion of {\em selection,} as it
arises in Darwinian theory.  For reasons which will become
apparent, I shall refer to this as {\em S-lineage selection.}
Loosely speaking, S-lineage selection is a process in which one
S-lineage consistently grows, at the expense of another,
ultimately displacing it.\footnote{In this respect, my usage of
  ``selection'' is more restrictive than is usual in population
  genetics---where the word seems to be commonly used to mean
  {\em any\/} ``approximately deterministic'' change in S-size.
  The latter condition is detailed in section~\ref{determinism},
  but as only one among a number of conditions which are required
  for ``selection'' in my sense.}

The purpose of this section is to try to characterise as
precisely as possible the circumstances under which such a
process will actually occur. I shall concentrate on the
relatively simple case of selection between just two S-lineages:
the generalisation to more than two should be reasonably clear,
and will not be discussed explicitly.

\subsection{Determinism (sort of)}

\label{determinism}

Our first requirement may be informally stated as being that the
changes in the S-sizes of each of the S-lineage's in question
should be ``determined'' or ``predicted'' by their S-classes.
This is a difficult idea to pin down precisely, but I shall make
the attempt nonetheless.

Clearly, the changes in S-size may generally depend on a variety
of factors ``outside'' of the S-lineage of interest. These
include the effects of actors not in the S-lineage, and any
salient aspects of the common system in which the actors are all
embedded.  These things will be collectively referred to as the
{\em S-environment\/} (each S-lineage clearly defines its own
unique S-environment).  We can thus restate our requirement,
somewhat more precisely, as being that S-class should be
predictive of the rate of change of S-size (per unit time) for
any {\em given\/} S-environment.

It is important to see that this requirement is non-trivial; it
certainly need not hold for arbitrary S-classes. In principle,
even if all ``relevant'' aspects of the S-environment have been
stipulated, the S-class of the actors making up the S-lineage may
still not allow any prediction of the rate of change in S-size:
the characteristics upon which the S-classification are based may
bear no relationship whatsoever to rate of change in S-size. More
generally, these characteristics may have {\em some\/} effect on
the rate of change in S-size, but this effect may be swamped by
other characteristics which are not relevant to the
S-classification.  That is, two distinct S-lineages (relative to
the same S-class), subjected to the same S-environment, may still
exhibit radically different rates of change in S-size, simply
because the particular S-class does not capture other {\em
relevant\/} characteristics of the actors.

Having recognised that {\em arbitrary\/} S-class need not be
predictive of the rate of change in S-size, we can now ask what
it might mean to say that a particular S-class {\em is\/}
predictive in this way.  I shall try to answer by first giving a
somewhat contrived and restrictive example, and then indicating,
in outline at least, how it might be generalised.

\label{homogenous}

Suppose then that the S-class of an actor (together with the
S-environment) uniquely determines two probability functions: one
specifies the probability that the actor will die in a unit time
interval, and the other specified the probabilities that the
actor will have each possible number of S-offspring in a unit
time interval.\footnote{In the case of multimodal (``sexual'')
  procreation, we would require some convention such that a
  given S-offspring is counted only relative to one S-parent, so
  that we can subsequently infer information about the total
  change in S-size.  But this is a minor detail not affecting the
  general principles here.}
We stipulate that these probability functions are
stationary---i.e.\ do not vary with time.  Now, in this case,
S-class is clearly implying very strong and particular similarities
between actors of that class, which was not the case for
arbitrary S-class. It also implies that actors of this S-class do
not interact with each other---or, at least, do not interact in
any way which affects these probability functions (the functions
are independent for every actor).

It then follows that there will exist two derived probability
functions, which will be associated with any S-lineage relative
to this S-class.  The first is the probability function for the
number of actors in the S-lineage which will die per unit time,
and the second is the probability function for the number of
S-offspring which will be born per unit time.  Both of these will
depend on the current S-size.  I shall term the death {\em
rate\/}---which is to say the number of deaths per unit time,
divided by S-size---the {\em S-mortality}. Similarly, the birth
rate will be termed {\em S-fecundity}. S-mortality and
S-fecundity will be random variables. Under the conditions
described, their expected values will be independent of S-size
(they will, in fact, be equal to the expected values of the
corresponding probability functions for the individual actors).
Furthermore, their variances will be inversely related to S-size.

That is, S-class (together with S-environment) will predict the
probability functions for S-mortality and S-fecundity; and,
further, if S-size is ``sufficiently'' large, S-mortality and
S-fecundity will, in effect, be uniquely determined by S-class.
The rate of change of S-size is simply the difference between
S-fecundity and S-mortality (multiplied by current S-size).

This example shows the strongest case of what might be meant by
S-class being predictive of the rate of change in S-size.  I now
want to suggest, even if only in outline, how this case might be
generalised.

The essential extra idea is to allow for diversity or structure
{\em within\/} an S-lineage.  That is, we don't require that
every actor of the given S-class should have exactly the same
probability functions for death and numbers of S-offspring.
Instead we allow that different actors may have different
probability functions; but we insist that the distribution of
these distinct probability functions {\em should\/} be determined
by S-class.  Crudely speaking, we might envisage that the actors
are organised into ``social'' units, such that these social
units would all exhibit the same gross probability functions for
births and deaths, even though there is heterogeneity {\em
within\/} each social unit.  Provided the S-size of the S-lineage
is large enough to contain many such units we will still have the
situation that S-class is more or less deterministically
predictive of the rate of change of S-size.

Note the implication here that, {\em in general}, it is quite
misleading to think of S-fecundity and S-mortality as directly
reflecting ``corresponding'' characteristics of the constituent
actors.  For example, while any given actor can contribute to
S-fecundity by the obvious mechanism of bearing S-offspring
itself, it can also significantly influence S-fecundity by
helping (or hindering) other members of the S-lineage in
bearing offspring.  That is:
\begin{quote}
\em S-fecundity and S-mortality are characteristics of S-lineages, not
actors.
\end{quote}

In summary then, our first requirement for S-lineage selection is
that S-fecundity and S-mortality of an S-lineage should be
uniquely determined (at least ``probabilistically''---in the
sense of definite, stationary, probability distributions) by its
S-class and some specified aspects of its S-environment.  This
allows S-fecundity and S-mortality to have more or less arbitrary
dependencies on the S-environment; what it rules out (indeed,
almost the only thing it rules out) is the possibility that
S-fecundity and S-mortality depend on any aspects of the
S-lineage itself {\em other\/} than those implicit in the S-class
of its actors.

The only remaining variability in S-fecundity and S-mortality (if
any) is limited to probabilistic variation according to a
stationary probability function.  Which is to say that, provided
the size of the S-lineage is large enough, then (by appeal to the
so-called ``law of large numbers'') this remaining variability
can be made arbitrarily small.  I do not, of course, attempt to
give any particular figure for how large would be large
``enough''---that will depend on the particular case; it is
sufficient if, in principle, some such figure can always be
specified.

Technically, the requirement which has been stated is that,
provided the size of the S-lineage (denote it $s$) exceeds some
minimum (i.e.\ is large ``enough'') then it will satisfy, to an
arbitrarily close approximation, a deterministic differential
equation of the form:
\[ \dd{s}{t} = (F - M) s\]
where $F$ and $M$ denote S-fecundity and S-mortality
respectively, and are uniquely determined by S-class, in the
given S-environment.

\subsection{Malthus}

My next requirement for S-lineage selection is that the
S-lineages in question be ``capable'' of increasing more or less
exponentially in size provided sufficient raw materials or
resources are available.  This is the so-called ``Malthusian''
growth that originally inspired both Darwin and Wallace.

I have already stipulated that S-class, in combination with
S-environment, should determine S-fecundity and S-mortality (at
least if S-size is large ``enough''). I now require that this
dependence on S-environment be expressed in terms of
``resources''; and that provided these resources are sufficiently
``large'', it must be possible for S-fecundity to exceed
S-mortality ($F > M$ in the previous equation).  S-size will then
grow (exponentially in the first instance).  As S-size increases
however, and resources\footnote{Strictly, this should be
  interpreted in terms of fixed resource ``flows'' (per unit
  time).}
therefore become more scarce, S-fecundity will fall and/or
S-mortality will increase.  Eventually this will result in a
balance between the resource availability and S-size, such that
S-fecundity and S-mortality are just equal, and S-size remains
more or less constant.

\subsection{Competition}

Our third condition for S-lineage selection is that the two
S-lineages must {\em compete\/} for certain limited
resources---this is what \citeAD[Chapter~III]{Darwin:TheOriginOfSpecies}
termed the ``struggle for existence''.  This is actually quite
subtle and the requirements must be spelled out
carefully.\footnote{Indeed, at first sight it seems that
  \citeAD[p.~1]{Lewontin:UnitsOfSelection}, in particular, holds
  that resource limitation is {\em not\/} a necessary condition
  for selection; but I think this is because he uses
  ``selection'' in that generalised sense, noted earlier, of {\em
  any\/} (approximately) deterministic change in S-size,
  regardless of whether this involves the eventual elimination of
  any S-lineage. By contrast, I reserve ``selection'' to denote
  cases where some S-lineage(s) really are selected in the sense of
  displacing or eliminating some alternative(s).}

We require two things.  Firstly, the two S-lineages must both
rely on the same resource(s). Thus, an increase in the S-size of
either will have an adverse effect on the resource availability
of both.  This is a relatively straightforward requirement.

Secondly, and less obviously, we require that the two lineages
are, to at least some extent, {\em exclusive\/} of each other.  More
technically, if $s_1$ and $s_2$ are the respective S-sizes, we
require that:
\[ \dd{s_1}{s_2} < 0 \]

This requirement is trivially satisfied if the S-lineages in
question are disjoint: in that case an increase in one can, in
the face of fixed resource availability, occur only at the
expense of a reduction in the other.  In the case where the
S-lineages are not disjoint (i.e.\ under multimodal procreation,
and assuming that the S-class definitions are not ``inherently''
disjoint), this condition evidently may or may not be satisfied.

\subsection{Consistency}

Our third requirement for S-lineage selection is that there
should be a ``consistent'' bias in the changes in S-size, in
favour of one or the other.

That is, our previous requirements have established that the
changes in the S-sizes of the two S-lineages will be more or less
deterministic and must, instantaneously, be in opposite
directions.  However, that would still allow a situation in which
the S-sizes could stabilize (both rates of change become zero) or
could oscillate (for a while, one increases and the other
reduces, and then vice versa etc.).\footnote{Examples of this
  kind of phenomenon can arise through so-called {\em frequency
  dependent selection\/}---which is again something of a
  misnomer in my terms, as it implies, {\it inter alia,} that no
  single S-lineage need be actually {\em selected}.}

This implies two distinct things in turn.  Firstly, we must
require that the S-environments of the two S-lineages do not
change ``too much''.  That is, we can permit some variation
in the S-environments provided that such changes affect both
S-lineages ``equally'' (or, at least, in the same direction).  We
have already explicitly introduced a change in S-environments of
this kind in the form of the dependence on shared ``resources''.
The point now is to restrict changes in the S-environments to be
{\em only\/} of this general kind (at least for the duration of
the selection ``episode'').

Secondly, we must assume that whatever ``bias'' exists must
be consistent for arbitrary S-sizes of the two S-lineages. By
``bias'' here I mean, roughly, the difference in the two rates of
change (per actor in the S-lineage). If the first S-lineage
increases when its size is small compared to the second, it
should still increase even if (when) it becomes large compared to
the second.  I do not require that the bias should remain
strictly constant (be independent of the S-sizes), but just that
it should stay in the same {\em direction}.

It now ``almost'' follows that one S-lineage will, over time,
necessarily completely displace or eliminate the other; which is
to say that one of the competing S-lineages will be {\em
selected}.  I say ``almost'' because there is still one further
condition to be satisfied---but I shall return to that in the next
section.

Selection bias is, in some sense, a measure of the rate at which
selection is instantaneously taking place.  I have allowed that
it may vary; but the ``important'' aspects of it (its direction,
and approximate magnitude) may be indicated by giving its value
under some kind of ``standardised'' conditions.  Quite what
standardised conditions are most convenient depends on the
detailed dynamics of the particular system, and I shall not go
into that. I merely wish to note that it may be possible to
attribute standardised numbers to particular S-lineages (in
standardised S-environments) in such a way that, for any pair of
competing S-lineages, the difference in these numbers will serve
to predict the outcome of selection between them (which will
displace the other, and how quickly).  I shall refer to such
standardised numbers as {\em S-values}.

The notion of S-value is closely related to the concept of {\em
fitness\/} as it is commonly used in population genetics.
However I deliberately avoid the use of this latter word
because it has been used in a variety of distinct, sometimes
mutually contradictory, ways---even within Darwinian theory. This
problem has been documented in detail by
\citeAD[Chapter~10]{Dawkins:ExtendedPhenotype}, where he, too,
concludes that the word has become so overloaded that it is now
best avoided altogether, if at all possible.\footnote{My
  ``S-value'' is also essentially the same as the ``selective
  value'' described by
  \citeAD[Chapter~9]{Provine:SewallWright}---which is something
  of a happy coincidence: the ``S'' in ``S-value'' strictly
  stands for ``similarity'' (being derived from
  ``S(imilarity)-lineage''), rather than for ``selection''.}

\subsection{S-creation}

The final condition for S-lineage selection is that it should not
be ``compromised'' by {\em S-creation\/} events.

S-creation means the birth of an actor of a given S-class,
none of whose parent(s) were of that S-class.  S-creation results
in the foundation of a new S-lineage (it would typically correspond
to the occurrence of a ``mutation'' in conventional biological
terms).

I consider three cases here.

Firstly, S-creation could result in the foundation of an
S-lineage of some S-class which is different from the S-classes
of either of the pre-existing S-lineages, {\em and which does not
compete with them}.  If effect, this corresponds to the
foundation of an S-lineage which exploits a distinct
environmental ``niche'' (which may, or may not, mean that it must
compete with some S-lineages {\em other\/} than those originally
under consideration).  This may, or may not, have some effect on
the selection dynamics between the original two S-lineages; but
if it does have such an effect, this falls under the heading of
a significant ``environmental perturbation'', and we would no
longer be dealing with S-lineage selection {\it per se}. This
general case is very complex, and I shall not pursue it further
here; I shall return to it, briefly, in section~\ref{mountains}
below.

Secondly, S-creation could result in the foundation of an
S-lineage of some S-class different from the S-classes of either
of the pre-existing S-lineages, but which {\em does\/} also
compete with them (i.e.\ meets all the conditions already
identified for this).  Clearly, this new S-lineage will then
potentially interfere with the completion of the selection
episode between the original two S-lineages.  On the other hand,
it simply means that we must generalise our concept of selection
to allow selection from more than two candidates---it does not
fundamentally affect our arguments so far, and I shall not
consider it further.

The final case is where S-creation results in the foundation of
an S-lineage of the {\em same\/} S-class as one or the other of
the pre-existing S-lineages. This possibility introduces some
subtle, and fundamental, complications.

Clearly this new S-lineage will affect the dynamics of the
selection process: by definition it will also compete with one of
the two other S-lineages (the one of different S-class from
itself).  Equally, however, its interaction with the other
S-lineage (that of the same S-class as itself) will be quite
different, for they must both have the same S-value.  The exact
nature of the interaction between two such S-lineages of the same
S-class (or, more generally, between any two S-lineages of the same
S-value) depends on whether procreation is unimodal or
multimodal. In the former case, the two S-lineages must still
``compete'' in the sense that an increase in one must generally
be matched by a decrease in the other, but this variation will be
purely stochastic---there will be no consistent bias (this is
what we {\em mean\/} by saying they are of equal S-value). The
outcome will be that some one will displace the other in a
stochastic manner: this is a pure example of so-called {\em
random drift}. This {\em may\/} also be the outcome in the case
of multimodal procreation; but in that case there is also an
alternative scenario which is that the two S-lineages of the same
S-class may completely converge, so that they may {\em both\/}
survive indefinitely.

At face value, these possibilities make the process of S-lineage
selection a little more complicated, but do not seem to have a
fundamental significance.  From the point of view of the
selection dynamics it seems that we could ignore the distinction
between S-lineages of the same S-class---because the question of
which, if either, of these eventually ``wins out'' is entirely
capricious, and of no particular interest anyway.  That is, if
S-creation events of this sort are a common event, we should
arguably abandon our analysis in terms of S-lineages, and simply
look at the dynamics of the \mbox{(sub-)populations} of the two distinct
S-classes {\em without regard to their parentage}.

This is an innocent looking step, which is implicit in much of
modern population (!) genetics---and yet I feel that it critically
alters the conceptual structure of the theory. If the step is
taken at all (and I admit that, for {\em practical\/} purposes in
the application of the theory this may well be convenient), it
should be only with very careful and explicit acknowledgement of
the conceptual implications.

Once one moves to thinking simply in terms of competing
sub-populations of differing S-class, one loses the conceptual
foundation for the auto-catalytic (Malthusian) growth of these
sub-populations, and thus the foundation on which selection is
based.  Auto-catalytic growth follows from the idea that actors
of a given S-class give rise to more actors of that same
S-class---but such growth is therefore strictly growth in
S-lineages.  It may, of course, happen that the sizes of
sub-populations also show this kind of auto-catalytic
growth---but there is no particular reason why they should; if
they do, then this must appear as an extra, essentially
ungrounded, property.

Let me put this another way.  When we move to thinking in terms
of sub-populations, growth in the size of a sub-population
may, in fact, be as strongly, or more strongly, related to the
size of some {\em other\/} sub-population as to itself.  The
growth, or decay, of a sub-population may, in this case, be
nothing at all to do with ``selection'' in the sense I have been
using.

More generally, I will say that, if the qualitative dynamic
behaviour of the system {\em relies\/} on (rates of) S-creation
(i.e.\ on the generation of actors of one S-class by actors of a
different S-class) then the process is of a different sort to the
kind I am labelling (Darwinian) selection.  It is a central element
in what I term selection that the outcome should {\em not\/} rely
on any details of the process for {\em generating\/}
``variation'' (actors of different S-classes to their parents).

This is actually a very central issue in Darwinian theory.  While
it is commonly stated that Darwinism is based on ``random''
variation I think this is a positively misleading formulation.
If ``random'' is interpreted in the sense of the probability
calculus, then it implies a predefined and constrained event
space.  In that kind of scenario it is quite possibly (though not
necessarily) the case that the dynamics {\em do\/} rely (albeit
stochastically) on the probability function of the ``variation''
process, as much or more than on the properly (in my terms)
``selective'' process.

It seems to me that these issues may be well illustrated in the
work of Eigen and Schuster on so-called {\em quasi-species\/}
\cciteAD{Eigen:Hypercycle}.  Briefly, they investigated a model
system in which a specific process of stochastic variation was
assumed, and they analysed the dynamics on the basis of this.  The
results rely, not surprisingly, on the details of the stochastic
variation.  The important point, for my purposes, is that Eigen
and Schuster then (in effect?) propose a re-classification of the
actors, which ``hides'' this stochastic variation (yielding what
they call quasi-species); and they suggest that, insofar as
{\em selection\/} is a phenomenon in the system, it is between
these quasi-species, and does {\em not\/} rely on the details of
the stochastic variation, which may be, as it were, factored out.

This impinges on a long-running discussion within evolutionary
biology: whether or not the ``characteristics'' underlying
Darwinian selection are required to be ``particulate'' (or
Mendelian).

Neo-Darwinism is, roughly speaking, the synthesis of Darwin's
idea of evolution by natural selection, with Mendel's theory of
``particulate'' inheritance.  The conventional wisdom is that
this augmentation of Darwin's original theory with Mendelism was
essential to the former's theoretical integrity---that Darwinism
simply cannot function in the absence of something more or less
like Mendelian inheritance. I do not wish, for the moment, to
consider the validity of this view within conventional biological
evolution as such; my purpose here is to enquire whether, or in
what sense, it can reasonably be claimed that Darwinism cannot
function at all, in {\em any\/} system, in the absence of (some
abstract analog of) ``Mendelian'' inheritance.

While I am not aware of any extensive analysis of this question,
Dawkins (for one) has expressed a clear opinion:
\begin{quote}
  Darwinism post-Fisher is called neo-Darwinism.  Its digital nature is
  not an accidental fact that happens to be true of genetic information
  technology.  Digitalness is probably a necessary precondition for
  Darwinism itself to work.

  \citeAD[p.~115]{Dawkins:BlindWatchmaker}
\end{quote}

Dawkins' claim for the necessity of Mendelian (``digital'')
inheritance is based on contrasting it with so called
``blending'' inheritance---the idea than an offspring will be,
genetically, intermediate between its parents.  Blending
inheritance leads to the rapid (genetic) homogenisation of a
population, and, arguably, evolution by selection would
be impossible---for if variation is absent, or if such variation
as may spontaneously appear is diluted before its correlated
benefits can be exploited, then it seems that there is no
effective basis for selection to operate.  Discrete, Mendelian,
inheritance, by contrast, does not suffer from such difficulties.
Dawkins refers to Fisher's discussion of this
argument;\footnote{The original formulation of the argument goes
  back, of course, to Fleeming Jenkin, though he construed it as
  an argument against Darwinian evolution in its totality, rather
  than just against a particular mechanism of inheritance.  Given my
  own professional background, I cannot resist
  \mbox{(re-)quoting} the
  following remark made by Darwin's son, Francis (copied from
  \ociteAD[p.~81]{Hardy:LivingStream}):
    \begin{quote}
    The late Mr.~Fleeming Jenkin's review, on the `Origin of
    Species', was published in the `North British Review' for
    June~1867.  It is not a little remarkable that the
    criticisms, which my father, as I believe, felt to be the
    most valuable ever made on his views, should have come, not
    from a professional naturalist but a Professor of
    Engineering.
    \end{quote}
  While I shall still conclude by rejecting Jenkin's argument, I
  hope to show that this is a little more difficult, and more
  significant, than has commonly been supposed.}
but consider carefully the following comment made by Fisher
himself:
\begin{quote}
  The important consequence of the blending is that, if not
  safeguarded by intense marital correlation, the heritable
  variance is approximately halved in every generation.  To
  maintain a stationary variance fresh mutations must be
  available in each generation to supply the half of the variance
  so lost.  If variability persists, as Darwin rightly inferred,
  causes of new variability must continually be at work.

  \citeAD[p.~5]{Fisher:GeneticalTheory}
\end{quote}

Fisher then argues that the observed mutation rate is not, in
fact, sufficient, to maintain the inheritable variability
observed in real biological organisms (at least, not under
blending inheritance); that if, in fact, blending inheritance
were coupled with these observed mutation rates, then it could
not support effective evolution; but that, as it happens,
blending inheritance is false; and the Mendelian, particulate,
theory, when coupled with the given mutation rates, {\em is\/}
perfectly compatible both with the observed variability, and the
effective operation of selective evolution.

I am not convinced that this argument is entirely sound, even as
it stands---for example, mutation rates are, to some extent, an
outcome of evolution, so a thought experiment (which is, in
effect, what is offered by Fisher) that couples blending
inheritance with mutation rates produced by evolution based on a
different inheritance mechanism is of dubious validity; but I do
not propose to pursue this aspect.  Even granting that Fisher is
entirely correct here, his argument applies only to the
biological world as we know it. That is, to turn the passage
quoted above back against Fisher (and thus, implicitly, against
Dawkins), it could be read as an admission that, if a particular
system {\em does\/} exhibit a ``high'' rate of mutation, per
generation, then {\em some\/} kind of ``selective evolution'' may
become perfectly practical, {\em even\/} with a blending
mechanism of inheritance!  This result is entirely independent of
what may, or may not, happen to be the case for terrestrial
biology.

More generally, the Fisher/Dawkins argument only goes through in
the case of sexual reproduction (multimodal procreation): the
contrast between digital and blending inheritance only arises in
that case. By contrast, it would seem that some kind of
``selective evolution'' would be possible based on non-Mendelian
(``continuous'') inherited characteristics, even with ``low''
mutation rates, in the case of {\em asexual\/} reproduction. Of
course, various arguments might be made regarding the
significance of sexual reproduction for the ``ultimate'' outcome
of evolution.  In particular, it might conceivably be claimed
that a purely asexual form of reproduction would always be
limited in some kind of ``evolutionary potential'' as compared to
sexual reproduction---that, other things being equal, a system
incorporating sexual reproduction could always ``progress''
further.  But this question is very complex. Dawkins himself has
recently commented that the ``problem of what sex is good for is
still as tantalizing as ever''
\cciteAD[p.~274]{Dawkins:NewSelfishGene}. Clearly, no strong
argument for the ``necessity'' of digital inheritance could be
based on so weak a foundation.

In contrast to all this, I want to suggest that Dawkins was
essentially right in his original claim---that digital
inheritance is necessary for ``Darwinian'' evolution---but that
this is so for quite a different (and more fundamental) reason to
that which Dawkins himself cites.  It seems to me that any kind
of theory of inheritance of ``continuous'' characteristics
(whether sexual or not, blending or not) {\em must\/} (implicitly
or explicitly) incorporate some specific theory of the {\em
generation\/} of new variation (whether stochastic or
otherwise)---and it is precisely {\em this\/} which is
problematic.  While one can formulate systems of this kind which
would exhibit a kind of ``selective evolution'', the selection
process in question would necessarily rely on, or reflect the
characteristics of, the variational process.  I think this is
unavoidable in the case of continuous characteristics; and {\em
can\/} still occur even in the case of discrete or digital
characteristics.  The most we can say is that, if we want a
selection process in which the selection dynamics are effectively
decoupled from the variational dynamics then digital inheritance
is a necessary (though not, in itself, sufficient) condition for
this.  And, of course, it is precisely the latter kind of
decoupled selection process for which I am choosing to retain the
label {\em Darwinian}.

I do not deny the possibility that processes in which the
variational and selective dynamics are coupled may occur. The
important point, for my purposes, is that we {\em can\/}
formulate an evolutionary theory which decouples variation and
selection; that we can {\em only\/} do this with S-classes based
on digital characteristics; that such a theory is quite
distinctive and interesting in its own right; and that it
therefore deserves to be terminologically distinguished.  I feel
that this is the kind of theory which Darwin himself hit upon,
and that is why I suggest reserving the term {\em Darwinian\/}
for just this case; but, of course, nothing hinges on what word
we use---the distinction is the important thing.

\subsection{Natural or Artificial?}

I have not distinguished between ``natural'' and ``artificial''
(S-lineage) selection.  The fact is that, within the framework as
so far defined there is no distinction.  Such a distinction
relates solely to the {\em mechanism\/} or {\em process\/} which
maps the S-class of an S-lineage onto its S-value (in given
environmental conditions). S-lineage selection requires that some
such mechanism exist; but it places no restriction on how it is
realised.  I shall now add simply that, if this mechanism relies
on the intervention of some ``intentional agent'', then S-lineage
selection is {\em artificial,} and that otherwise it is {\em
natural}.

{\em Artificial\/} selection thus implies that some person
(``intentional agent'') must (somehow) examine certain
characteristics of the actors, characteristics which are
preserved through S-descent, and must then (somehow) control the
S-value of their S-lineages, based on these characteristics (for
example, by regularly culling all those actors with
``undesirable'' characteristics, thus increasing the S-mortality
of certain S-lineages relative to others).  This will cause
certain S-lineages to have greater S-value than others, and thus
(potentially at least) to displace or eliminate them.

{\em Natural\/} selection, on the other hand, means that,
although more or less ``intentional'' agents may be participating
in the system in various ways, they are not affecting the
relative success in procreation of the actors, or at least are
not affecting them in a way which is systematically related to
their S-classes---which is to say that they do not
``deliberately'' control the S-values of the S-lineages.

The essential question here is whether S-values are being
adjusted in accordance with some anticipatory\footnote{In the
  sense of \citeAD{Rosen:AnticipatorySystems}.}
or predictive model of the S-lineage selection process itself, in
order to ensure a particular outcome.  If so then selection is
artificial, otherwise it is natural.

\subsection{In Summary}

In summary, I have equated Darwinian Natural Selection to
S-lineage selection---a process whereby one S-lineage
deterministically displaces another because it can more
efficiently exploit the same resources.  It can occur only if
several conditions are satisfied:

\begin{enumerate}
\item S-class must be predictive of S-fecundity and S-mortality
(in the relevant S-environments).
\item In favourable environmental conditions (i.e.\ in the absence
of resource limitations) both S-lineages must be capable of
demonstrating exponential growth.
\item The growth of both S-lineages must be limited by the same
resources.
\item The S-lineages must be such that they cannot converge; that
is, an increase in the S-size of one must necessarily imply a
reduction in the S-size of the other.
\item The expected difference in the rates of change of S-size of
the two S-lineages must be in a consistent direction, regardless
of the exact S-sizes, and large enough to outweigh stochastic
variations.
\item S-creation must be relatively infrequent (compared to the
time scale required by the selection episode).  That is, the
selection process should be decoupled from the variation process.
This implies, in turn, that the characteristics underlying
S-classification, and inherited through each S-lineage, must be
discrete or digital.
\item These various conditions must {\em not\/} be met through
the intervention of an ``intentional'' agent.
\end{enumerate}

While I have focussed on the phenomenon of one S-lineage
selectively eliminating another, this is not to imply that this
is the only kind of phenomenon or behaviour which can arise in
the kinds of system I have discussed.  Thus, it should be clear
that the general framework also supports a variety of other
(``ecological'') interactions, for example:
\begin{itemize}

\item S-lineages may rely on completely independent resources, in which
case each can thrive (or go extinct) independently of the
other(s).

\item One S-lineage may effectively {\em be\/} a resource for
another---which is to say they are in a predator/prey, or perhaps a
parasite/host, relationship.

\item Several S-lineages may positively co-operate with each
other---where each assists or facilitates the success of the others,
which is to say, they are symbiotes.

\end{itemize}

Nonetheless, it is selection which is of central importance in
Darwinian theory, and it is this which I shall continue to
emphasize.

In principle, S-lineage selection can continue
indefinitely---either due to changes in environmental conditions,
or to the birth of new S-primaries.  In the latter case, if a
new S-lineage is more efficient at ultilising some resource, it
may selectively displace some prior S-lineage (which, presumably,
had been the most efficient up to that time).

This discussion of selection has been informal and qualitative.
It can, of course, be given an elaborate mathematical
basis---see, for example,
\cciteAD{MaynardSmith:EvolutionaryGenetics} or
\cciteAD{Eigen:Hypercycle}. This allows analysis, in the simpler
cases at least, of the circumstances in which S-lineage selection
will occur, how long it will take, and whether it will be partial
or total etc. However, these detailed issues are not essential to
my purposes here. The point I want to make is that S-lineage
selection arises because S-descent preserves S-class (thus
ensuring that S-lineages consist of actors which are all
members of one S-class), and because S-class {\em also\/}
characterises certain relevant attributes of the members
(specifically, whatever attributes have a significant effect on
the fecundity and mortality of the containing lineage), so that
the S-lineages can consequently display deterministic behaviours
of growth, competition and, ultimately, selection.

\section{Darwinian Evolution}

The notion of Darwinian {\em evolution\/} is already largely
implicit in my discussion of Darwinian {\em selection,} but I
should like to now make it fully explicit.

Clearly, in any system of actors where S-lineage selection takes
place, there will be a change, in time, of the characteristics of
various lineages which contain the S-lineages.  That is, even
though S-creation identifies a discontinuity in an S-lineage,
the resulting S-primary will, in general, still be an offspring
of some pre-existing actors; it will be a member of
some pre-existing lineage, though it also simultaneously founds a new
{\em S\/}-lineage within that lineage.  Within such a lineage
then, we may have multiple S-lineages competing with each other
and, over time, we may have multiple episodes of S-lineage
selection, so that the actors at a later time are representatives
of S-classes which are very different from the S-classes
represented at an earlier time.  Thus, lineages can be said to
{\em evolve\/} via a process of S-lineage selection.

{\em Darwinian\/} evolution is then a process of evolution by
natural selection between S-lineages, but with one further
constraint: the S-creation process must be ``unjustified''.

Now I have earlier stipulated that the S-lineage selection process
should be decoupled from the S-creation process (i.e. from the
generation of variation).  I am now adding a further, separate
and independent, constraint on the S-creation process. By
requiring that S-creation be ``unjustified'' I mean that it must
not involve the application of anticipatory or predictive models
of the S-value of the resulting S-lineage.

Note carefully that this constraint is not a blanket exclusion of
teleology (in the sense of anticipatory behaviour, or {\em
knowledge\/}).  It may well be that individual actors and/or
S-lineages incorporate knowledge of their environments; indeed,
this possibility is what motivated my examination of Darwinism in
the first place. The Darwinian constraint is simply that the
course of evolution must not be {\em directed\/} by anticipatory
models; all other aspects of system behaviour can involve
anticipatory models, or knowledge. Thus, the requirement is
either that the actors simply do not have anticipatory models for
evolution (the normal case), or if they {\em do\/} have such
models, that these models are not employed in order to {\em
control\/} evolution.

I shall refer to actors satisfying all the conditions for
participating in a process of Darwinian evolution as Darwinian
actors, or simply {\em  D-actors}.\footnote{\citeAD{Gould:Darwinism}
  has previously introduced the term ``Darwinian actor'' more or
  less as I use it here.}
I shall call a system comprising D-actors embedded in an
environment which permits Darwinian evolution to occur a {\em
D-system}.  That which evolves by Darwinian means is then,
precisely, a lineage of D-actors, or a {\em D-lineage}.

\section{Climb Every Mountain}

\label{mountains}

\begin{quote}
The evolutionary literature is literally jammed with
representations of selective surfaces whose construction is
obscure \ldots\ It should give pause to consider that for over
fifty years the majority of evolutionary biologists have believed
Wright's \ociteD{Wright:RolesInEvolution} diagrams of the
adaptive landscape to be among the most heuristically valuable
diagrams in all of evolutionary biology, yet to discover that the
surface as he conceived it is unintelligible.

\citeAD[p.~316]{Provine:SewallWright}
\end{quote}

The idea of a ``selective surface'' is roughly this.  Suppose
that we represent every possible S-class by a point in a finite
dimensional space.  Since S-value is associated with S-class, we
can associate an S-value with every point in the space.  Let us
introduce one further dimension to represent this. The result is
then a ``surface'' of S-value.  The significance of this surface
is that Darwinian evolution can be viewed as a process in which
S-lineages are progressively displaced by other S-lineages which
are at a ``higher'' level on this surface; which is to say that
Darwinian evolution can be viewed as a form of {\em hill
climbing}.

This view is startling, and perhaps even beautiful, in its
simplicity.  It is hard to resist the idea that this is a deep
and powerful insight into the nature of Darwinian evolution.
Unfortunately I want to suggest that, in all but the very
simplest of situations, this view is at best incoherent or, at
worst, entirely mistaken.

I should emphasize that my representation of the hypothetical
fitness surface here is not quite the same as any of the several
variations on the idea considered by
\citeAD[pp.~307--317]{Provine:SewallWright}.  Furthermore, while
I share Provine's general skepticism as to the utility of the
fitness surface concept, my objections to it are somewhat
different, and perhaps more deeply seated, than his.

A first objection is as follows. If the argument already given
for the discrete or digital nature of S-classification is
accepted, then it follows that the graph of S-value will be a set
of discrete, disconnected, points and not a ``surface'' (in the
sense of a continuum) at all.  This is true as far as it goes,
and underlies (at least roughly) Provine's criticism of Wright's
earliest published version of the surface;  but it seems to me
that this is really not too serious an objection. The discrete
nature of the selective ``surface'' can be admitted while still
retaining the basic image of Darwinian evolution as a hill
climbing process (albeit the steps must be viewed as necessarily
discrete).

By contrast, my substantive objection to the fitness surface idea
turns on the very notion of S-value, and whether a ``graph'' of
S-value against S-class is a coherent or helpful notion at all.

Recall that S-value was defined as a quantity relating two
S-lineages rather than as a property of any single S-lineage in
itself. This immediately raises the question of whether it is
meaningful to talk of a graph of S-value against S-class in any
general sense. My view is that this idea {\em can\/} make sense:
but only in circumstances that are so circumscribed as to be of
very limited interest, at best.  Even then, the notion of ``hill
climbing'' as such hardly arises.

In outline, I suggest that we can meaningfully speak of a
selective ``surface'' only in relation to a set of S-classes such
that S-lineages of these S-classes would all be pairwise
competitive (that is, each such pair meets {\em all\/} the
requirements for selection, which implies that they are all
reliant on the same resources and are all more or less exclusive
of each other); furthermore, the resulting S-values must satisfy
some kind of approximate, if not exact, transitivity relation (if
A can displace B, and B can displace C, it must be the case that
A can displace C).  If we can identify {\em all\/} S-classes
which would compete with each other for a particular set of
resources (i.e.\ all S-classes which might potentially compete to
occupy a particular environmental {\em niche\/}) then it is fair
to view Darwinian evolution as a process in which each S-lineage
occupying this niche may be progressively displaced by an
S-lineage of higher S-value.\footnote{All this relies on an
  assumption of continued environmental constancy, of course:
  otherwise the S-values themselves would be dynamically changing
  also.}

Even this is not ``hill climbing''.  Hill climbing as a notion
only makes sense in a space where the ``steps'' satisfy some
condition of ``locality''.  If such a locality condition holds,
then hill climbing will involve proceeding to the ``nearest''
peak from the initial point, and staying there (this may or may
not be a global peak). But I have been at pains to emphasize
that, in my presentation of Darwinian evolution, {\em no\/}
assumption is made about the mechanism or direction of the
variational process. Thus, it will be generally the case that,
for any particular representation of the S-classes in question
(i.e.\ any geometry for the space of S-classes to which the
selective surface will be added) variations will {\em not\/} be,
in any sense, ``local'', and the evolutionary trajectory,
although it would consistently transit through S-lineages of
higher S-value, would not proceed to the ``nearest'' peak, and
might (or might not) cease at any point (regardless of whether
that is a ``local'' peak---the only exception being the global
peak, where Darwinian evolution would still necessarily cease).

So, even in very restricted circumstances, the hill climbing view
is hardly meaningful.  In the general case, it becomes wholly
untenable.  Firstly, of course, the assumption of environmental
``constancy'' cannot be upheld over long evolutionary periods.
But secondly, and much more seriously, it seems to me that it is
quite mistaken to suppose that distinct environmental niches
should be equated to distinct peaks in some kind of (suitably
defined) selective surface.  On my analysis, distinct niches
would correspond rather to distinct S-class spaces---with
entirely distinct selective ``surfaces''.  While S-values in
these different spaces might be expressed in the same units, they
would be basically incommensurable.  S-lineages from different
such spaces would not compete with each other, and taking the
difference in their S-values is a nonsense.

Of course, in any real application of Darwinian theory we cannot
expect to see neatly delimited, constant, niches anyway. ``Pure''
selection (where an S-lineage actually completely displaces one
or more competitors) may be the exception rather than the rule.
Furthermore, any given D-actor may simultaneously be a member of
many S-lineages occupying distinct ``niches'' and competing with
rivals over radically different time scales.\footnote{In this
  way, it may be that some forms of so-called ``species
  selection'' should be seen as identical to conventional
  ``organism selection'' (they are both properly called S-lineage
  selection, with organisms as D-actors) except that the timescale
  of the selection episodes is much longer for the former.}

In the general case, niches exist and are defined only relative
to the S-lineages exploiting them.  In this sense, Darwinian
evolution must encompass a dynamics in which the distinct S-class
spaces associated with distinct niches will be
coupled to each other, and will themselves undergo some kind of
ongoing structural transformation (so that the notion of a static
S-class ``space'', of finite dimensionality, is, at best, only a
temporally limited approximation).
\citeAD[Essay~7]{Popper:ObjectiveKnowledge} has emphasized the
process whereby a ``new'' niche becomes defined as a crucial
aspect of Darwinian evolution (though he does not describe it
quite in the terms used here).
\citeAD{Lewontin:GeneOrganismEnvironment} has aptly suggested
that, in place of seeing Darwinian evolution as steadily
ascending some ``selective surface'', we should think rather in
terms of a walk on a trampoline; this is an improvement, though I
think it may still be a gross oversimplification.

\section{On Adaptation}

\label{adaptation}

I have made no mention of ``adaptation'', or of the growth of
knowledge (in the most general sense), in this abstract
formulation of Darwinism. At first sight this seems strange---for
it is typically just such phenomena which we wish to study (or
even {\em explain\/}) by invoking Darwinism.

I suggest that almost the best we can say about this is that
D-systems evidently realise a form of unjustified variation and
selective retention, in the sense of D.T.~Campbell
\cciteD{Campbell:BlindVariationInCreativeThought,%
Campbell:BlindVariationAsGeneralStrategy,%
Campbell:EvolutionaryEpistemology};\footnote{At least, I hope
  this is a correct assessment---since my formulation of the
  D-system concept was motivated precisely with this result in
  mind.}
as such, they may {\em permit\/} the growth of knowledge
(including ``inate'' knowledge, or ``adaptive complexity'') but
cannot {\em guarantee\/} it.

Thus, knowledge {\em may\/} spontaneously grow in a D-system, but
it also may {\em not\/}; indeed, knowledge previously gained may
even be lost. If knowledge ever does grow, it will do so only
when the following additional conditions are satisfied:
\begin{itemize}

\item S-creation (``blind variation'') must be such that at least
some S-lineages can have greater knowledge than any which already
exist.

\item Knowledge must be well correlated with S-value for at least
some S-classes---i.e.\ some knowledge must be heritable {\em
and\/} selectively advantageous.

\end{itemize}

It may seem unnecessary to stipulate, as an explicit constraint
or hypothesis, that additional knowledge confer selective
advantage---for surely an S-lineage which ``knows'' more than
its competitor(s) will inevitably have higher
S-value?  This sounds plausible, but in fact it would only hold
to the extent that knowledge has no {\em costs\/} associated with
it; since, in general, the embodiment of knowledge may well
impose costs upon S-lineages, it may be that the more knowledgable
are {\em not\/} favoured by selection.\footnote{It may be argued
  that the substantive distinction between the growth of
  knowledge in general and the growth of conscious scientific
  knowledge resides precisely in the fact that, in the latter
  case, there is a deliberate or conscious attempt to {\em
  ensure\/} that increases in knowledge confer selective
  advantage.  However, I shall not pursue that argument here.}

To put it another way, the only knowledge or adaptation whose
acquisition can be explained in Darwinian terms, is knowledge
which demonstrably serves the self-interest of the S-lineage
possessing it: if there are things that it is better for an
S-lineage not to know (since they do it no {\em net\/} good in
its battle with rival S-lineages) then so be it, and ignorance
will in fact be bliss. Even if, by chance, S-creation should give
rise to a new S-lineage which is considerably more knowledgable
than any of its predecessors, it will be promptly eliminated
unless this extra knowledge actually gives it an edge in resource
ultilisation over relevant competitors.

Thus, it is only if the attribution of ``D-system'' is coupled
with an adequate theory of why, in the particular system, the
more-knowledgable S-lineages are also the more successful, and a
theory of how more-knowledgable S-lineages can be spontaneously
born (without teleology), then this {\em complete\/} theory may
explain some particular episodes of the growth of knowledge (by
Darwinian processes) in this \mbox{(D-)system}.

We might be tempted, at this point, to seek some further
augmentation of the concept of D-system, which would {\em
guarantee\/} the spontaneous growth of knowledge---presumably by
guaranteeing that the additional conditions stipulated above will
be satisfied.  However, it seems to me that it would be a mistake
to embark on such a project. For the conditions, as described,
are only meaningful in the context of some {\em particular\/}
knowledge; given some such specification we might well be able to
establish that {\em that\/} particular knowledge could (or could
not) spontaneously emerge, by strictly Darwinian processes, in
some particular system; but we could never examine a system and
say simply that ``knowledge'' (of an unspecified nature) will
grow in it.  The best that we might hope for is the negative
formulation that a given system is such that the knowledge
embodied within it (if any) definitely {\em cannot\/} grow (for
example by demonstrating that no more-knowledgable S-lineages are
accessible via S-creation, or that no accessible increase in
knowledge is selectively advantageous). I doubt however that even
this negative conclusion could be confidently asserted for any
but the simplest systems.

More abstractly, I suggest that the search for a {\em general\/}
theory which could ``predict'' the growth of knowledge (whether
by Darwinian means or otherwise) would be tantamount to the
search for a {\em logic of induction,} and would be equally
futile. The strictly limited scope of the purely Darwinian
theory, and its close relationship to the problem of induction,
is discussed by \citeAD[Chapter~37]{Popper:UnendedQuest} (albeit
in the purely biological interpretation of Darwinism). He
suggests that Darwinism {\em per se\/} may be best viewed as a
{\em metaphysical\/} rather than a {\em scientific\/} theory.
This should not be taken as pejorative: Popper himself describes
the Darwinian theory as having been ``invaluable''; and this is
because it provides a very general {\em schema\/} according to
which particular, properly scientific, theories can be developed
and, perhaps, tested, in specific cases. It is a schema for that
most difficult class of problems: those involving the growth of
knowledge.  It is a virtually unique tool in this context because
it offers the {\em possibility\/} of viable, yet {\em
non-theistic,} theories.  As Popper says, ``\ldots theism was
worse than an open admission of failure, for it created the
impression that an ultimate explanation had been reached''.

I should add that Popper has, since his original analysis,
significantly qualified his views on the status of Darwinism---even
going so far as to describe his more recent work in this area as
a ``recantation'' \cciteAD{Popper:EmergenceOfMind}.  I take the
view that this modification of Popper's views is concerned, not
with the abstract ontology of Darwinism, as discussed in this
essay, but rather with the specifically biological application
of Darwinism. In any case, it seems to me that Popper's original
analysis has commonly been misunderstood, and that his
``recantation'' is actually more in the nature of a
clarification; as such, I claim that the interpretation I have
given above is consistent both with Popper's original analysis,
and with his later, clarified, version.  For example, Popper has
reiterated that he still believes that Darwinism (``Natural
Selection'') works as a research program. Of course, in the final
analysis, the arguments given in this essay must stand (or fall)
on their own merit, and not on the implied authority of Karl
Popper (or anyone else).

Note that it follows from the discussion that there is nothing in
the strictly Darwinian framework which explains (much less
guarantees or predicts) any particular {\em progress\/} (in the
sense of the growth of knowledge at least) in Darwinian
evolution. At its simplest, this can be recast as being a
consequence of the fact that S-creation and S-lineage selection
are devoid of anticipatory or teleological influences or
controls: there is thus no {\it a priori\/} direction in
Darwinian evolution, though we may be able to recognise
``progress'' after the event.  In particular, while knowledge may
grow for an indefinite period, there is never any guarantee that
growth will continue, or even that the knowledge already achieved
will not be lost; as Popper notes ``\ldots there is more than one
way in which all life on Earth might be destroyed'',
\cciteAD[p.~168]{Popper:UnendedQuest}.

Conversely, we must not suppose that, because no system can be
{\em guaranteed\/} to exhibit, or continue to exhibit, the growth
of knowledge, then all D-systems are ``equal''; they are not.
While it is futile to seek general conditions which would be {\em
sufficient\/} for the growth of knowledge, we may well be able to
identify further conditions which are necessary, or even which
are only beneficial in some cases (provided they are not
positively detrimental in others).

So: Darwinism is no ``ultimate explanation'' but it offers the
possibility of particular, local, and limited explanations. This
is not a criticism of Darwinism:  for if the analysis given here
is valid then ``ultimate explanation'' is not possible.
Darwinism may be ``wrong'' (in any particular case of the growth
of knowledge), and there may yet be preferable theories (or
research programs); but this preference will not be by virtue of
another theory being logically ``stronger'', for Darwinism is
already as strong as its problems permit.

\section{M.B.~Williams: Darwinian Axiomatics}

\label{mbw}

Mary B.~Williams has attempted a detailed, formal, {\em
axiomatisation\/} of Darwinian theory
\cciteAD{Williams:ConsequencesOfEvolution}.  Although I
originally developed my own framework, as presented above,
largely in ignorance of Williams' work, it transpires that there
are substantial similarities in our approaches (convergent
evolution perhaps?)---and some significant differences also.

Firstly, there are superficial differences of terminology and
notation, some of which have already been noted. Thus, for
example, Williams uses {\em Darwinian biocosm\/} where I use {\em
D-system\/}; or, again, Williams has {\em biological entity\/}
where I have {\em D-actor\/} etc.  At this level our terms are
largely interchangable.

In addition to a special technical vocabulary, Williams
introduces a formal mathematical notation, which is quite austere
and concise, because she wishes to present formal mathematical
{\em theorems\/} derived from her axiomatisation of Darwinism.
My treatment, on the other hand, is essentially {\em in\/}formal,
and more concerned with identifying an appropriate ontology---a
set of more or less revealing concepts which underlie
Darwinism---rather than the formal establishment of theorems.
Thus I have not introduced a separate concise, mathematical,
notation.

I now turn to more substantive issues.

Williams and I have had similar objectives in that we have both
explicitly attempted to formulate a more or less {\em abstract\/}
version of Darwinism---for the reason that this does not prejudge
which (if any) aspects of reality might be well modelled by it.
However, Williams' motivation is still primarily the application
of Darwinism in a biological context. In particular, she wishes
to argue for the possibility that the abstract Darwinian theory
might have more than one separate, concurrent, and mutually
compatible, interpretation even {\em within\/} the biological
world.  I do not dispute this possibility, but wish also to
emphasize the possibility of applying Darwinism outside areas
conventionally thought of as biological---such as the growth of
knowledge, especially in artificial systems.  This incidentally
explains why Williams continues to use some explicitly biological
allusions ({\em bio\/}cosm, {\em biological\/} entity etc.)
whereas I have generally tried to avoid this.

Williams emphasizes, as I do, that it is {\em lineages\/} (she
calls them {\em clans\/}) and not individuals which are selected
between. In Williams' presentation, my notion of S-descent and
S-lineage corresponds to the existence of subclans within
subclans, or {\em subclands}, as Williams puts it, which are
distinguished from each other by the fact that they selectively
compete with, and ultimately eliminate each other. This requires
the postulation of some attribute(s) which characterise a
subcland (i.e.\ only those offspring are included in a subcland
which inherit the attribute(s) characteristic of the subcland)
and which are such that they jointly determine the outcome of
selection between distinct subclands.  Williams conflates all the
relevant attributes into a single parameter which she terms the
``fitness''.  As already discussed, I have chosen to avoid the
word ``fitness'' since it is potentially confusing, and have
chosen instead to introduce the distinct term {\em S-value}.

It is worth emphasizing that Williams' fitness is directly
axiomatised ({\em Axiom~$D3$}) as a property of individual biological
entities (D-actors), whereas my S-value is specifically
introduced as a characteristic of an S-lineage (albeit implied by
the S-class of its D-actors).  I shall argue below
(section~\ref{units}) that although these two approaches are, to
some extent, interchangable, it is conceptually much simpler to
attach the idea of fitness (or S-value, or whatever) to
S-lineages rather than to D-actors. This is implicit even in
Williams' treatment for, having introduced fitness as a property
of biological entities (D-actors) she immediately (definitions
$D3$ and $D4$) introduces a notion of ``average'' fitness which
can be attached to subclands (S-lineages) and it is this which is
used exclusively in the subsequent development.  Even when
attached to S-lineages, there are further detailed differences
between Williams' concept of fitness and my S-value; however, for
the rest of this discussion I shall ignore them, and consider my
S-value to be more or less synonymous with Williams' fitness.

On another issue, Williams implies that selection can only
take place under some assumption of {\em environmental
constancy\/}---for D-actors may be classified alike, and this may
be inheritable, but it can be predictive of the outcome of
selection only if the offspring are faced with relevantly similar
environmental conditions. However, Williams does not actually
stipulate environmental ``constancy'' as such; instead she allows
that environmental changes may occur, but stipulates that
relative ``fitness'' must be more or less consistently maintained
by some subclands (howsoever they may be identified) for long
enough that selection can work itself through: this is the
substance of her complex and subtle {\em Axiom~$D5$}.  I suggest
that although this appears to allow for arbitrary environmental
variation, it actually amounts to a requirement that the
environment be constant {\em in some relevant (exploitable)
respects\/}---at least while selection is actually going on. In
any case, whatever about the fine print, the essential point
which I accept is that my notion of S-classification can only
make sense {\em relative\/} to some constraints on the
environment of the D-actors.

Williams emphasizes, as do I, that the Darwinian theory ``does
not depend on any particular theory of inheritance'', such as
Mendelism. The {\em only\/} substantive constraint on the nature
of inheritance is it must be able to preserve distinctions in
S-value through indefinite numbers of generations.  This is the
essence of my requirement that S-descent {\em preserve\/}
S-classification (coupled with environmental constancy). I have
argued that a requirement for inheritance of discrete or digital
characteristics flows from this. However, it must be carefully
noted that this still does not automatically constrain the
detailed implementation of inheritance---for example in the form
that it must involve the use of some localisable ``hereditary
material''.

My final comment on Williams' treatment is that, in her informal
discussion, she occasionally seems to claim more for it than is
actually present.  In particular, in several places, she
informally implies that her axiomatisation {\em predicts an
indefinite and progressive improvement in adaptation}.  Yet,
according to my discussion previously, if ``adaptation'' is
thought of in the substantive sense of ``knowledge'' (inate to
the D-actors and/or S-lineages), then Darwinism in itself {\em
cannot\/} predict its progressive improvement---indeed I have
argued that that may be too great a demand for {\em any\/} theory
to bear.

The resolution of this apparent paradox consists of a minor and a
major point.

The minor point is covered by Williams herself.  Although her
theory {\em does\/} correctly predict an indefinite progression
in S-value, this is so {\em only\/} because the axiomatisation
guarantees that S-lineages of higher S-value will {\em always\/}
be present, in the process of being selected ({\em Axiom~$D5$}). This is
effectively a somewhat arbitrary, and unanalysed, constraint on
S-creation.  It in no way rules out the possibility that, in any
real system, such S-lineages of higher S-value may cease to
appear at any stage, and S-value would then stagnate, at best. In
such a case, Williams would simply say that the system has ceased
to be {\em Darwinian\/} (i.e.\ has ceased to satisfy her axioms).
There is no sense in which Williams claims to have provided a
general specification of conditions which would guarantee that a
system would {\em always\/} (or even ever) bring forth S-lineages
of higher S-value; her analysis is limited to saying what will
{\em subsequently\/} happen if this condition is, in fact,
satisfied (namely that the S-lineages of higher S-value will fix,
and S-value will therefore progressively and indefinitely
increase).

The second and more substantive point concerns what Williams
means by ``adaptation''.  Williams does not give any explicit
explanation of her usage.  Moreover, it does not appear {\em at
all\/} in the formal terms of her axiomatisation (either as a
primitive or in a definition)---and, of course, it cannot then
appear in any of the formal theorems.  I suggest that Williams
(consciously or otherwise) uses the notion of adaptation as being
synonymous with (her technical version of) ``fitness'', which, in
turn, is comparable to my S-value.  In this case, it is clear
that her conclusion does not bear upon, and therefore cannot
conflict with, my analysis of the growth of knowledge.

If, on the other hand, Williams actually intends adaptation to
imply something analogous to my meaning of ``knowledge'', then I
claim simply that the statements she makes in terms of adaptation
(for example, that ``a continuing sequence of adaptive
modifications is ubiquitous in a Darwinian subclan'') do not
(indeed could not) follow from her axioms; and that, if this was
her intended claim, it could only mean that she overlooked the
fact that there is no {\em inherent\/} reason why adaptation
(still in the sense of inate knowledge) should be correlated with
fitness or S-value.

\newpage % This is a hack, to remove an even worse break at the
         % next page...
\section{The Unit of Selection}

\label{units}

The question I propose to address in this section is: what
entities, or kinds of entities, are the ``units of selection''?

This issue has received extended consideration, as summarised by,
for example, \citeAD{Hull:UnitsOfEvolution}.  Not unnaturally,
this discussion has been carried out mainly in the particular
context of biological evolution, but Hull remarks
that the issue is so fundamental that it deserves to be called
`metaphysical'.  Similarly, Dawkins has suggested that the
question is not simply one of empirically deciding which real
entities function as units of selection, but rather is a
``dispute about about what we ought to {\em mean\/} when we talk
about a unit of selection''
\cciteAD{Dawkins:ReplicatorsAndVehicles}.

Needless to say, given the intense scholarly attention which has
been devoted to this issue, I do not claim to have a definitive
resolution to offer.  However, I shall at least try to offer a
clear target for further criticism.

In particular, I embrace Dawkins' view that the substantive
question here is to elucidate ``the nature of the entity for
whose benefit adaptations may be said to exist''
\cite[p.~81]{Dawkins:ExtendedPhenotype}.\footnote{There is, of
  course, an implication here that we are talking exclusively
  about ``Darwinian adaptations'': that is, ``adaptations''
  (whatever they may be) brought about by Darwinian evolution.}
This, whatever it is, is the entity which I shall identify as the
``unit of selection''.

In my formulation of Darwinian evolution there are really only
two candidates for the \role\ of the unit of selection: the D-actor
or the S-lineage. I shall suggest that, in certain, simple,
cases, there may be very little difference between the two; and
that this explains why the two possibilities are commonly
confused. However, I shall conclude that, in the {\em general\/}
case, the two possibilities are very different; and that the {\em
correct\/} candidate is, quite unambiguously, the S-lineage.

On my analysis, if ``adaptation'' evolves at all as a result of
Darwinian processes (and, to reiterate yet again, there is never
any guarantee that it will) then it does so in the form of a
change (or accumulation of changes) which allows a new S-lineage
to selectively displace an old one because it has higher S-value.
That is, adaptations will be such as to favour an S-lineage
relative to its competitors.  This immediately suggests the view
that adaptations can be viewed as Darwinian {\em only\/} to the
extent that they are beneficial to the S-lineage exhibiting them
(at least for the duration of the selection episode): i.e.\ that
the S-lineage is the ``unit of selection''.  However, this needs
to be considered in somewhat more depth.

In the simplest case, S-value can be improved by effects which
are uniform and consistent for all members of the S-lineage.
This kind of effect is particularly easy to envisage, given that
all D-actors in a given S-lineage are already known to be
``similar'', in the sense of having the same S-class.

Ultimately, this effect would be manifested in one of just two
possible ways: if all the D-actors
uniformly\footnote{``Uniformly'' implying that each D-actor is
  associated with an independent random sample from a single
  probability distribution, as previously discussed in
  section~\ref{homogenous} above.}
live longer than the D-actors of the competing S-lineage(s), then
the S-mortality of the S-lineage will be less than its
competitors; or if the D-actors all uniformly have more S-offspring
(per unit time) then the S-fecundity of the S-lineage will be
greater than its competitors. This way of thinking leads to
associating S-value with individual D-actors, at least as a form of
shorthand.  This is always legitimate in the formal sense that,
as already noted, all D-actors in an S-lineage share a single
S-class---so that any attribute of an S-lineage (such as S-value)
can be imputed to an S-class, and thus, implicitly, to single
D-actors, considered as ``exemplars'' of their S-class.

To put it a slightly different way, from the point of view of any
single D-actor, it ``knows'' for sure that it is a member of its
own S-lineage(s). Thus, it can surely favour its own S-lineage(s)
by having as many offspring as it can (involving some kind of
trade-off between living as long as it can while procreating as
fast as it can). Again, this viewpoint estimates or measures
S-value by reference to the activities or attributes of any single
``typical'' member of the S-class.

The problem with this point of view is that there may be no such
thing as a ``typical'' D-actor of a given S-class. It is
perfectly possible for D-actors of the same S-class to be very
different in a wide range of characteristics or attributes; or,
for that matter, for any single D-actor to engage in a wide
variety of different behaviours in different times and
circumstances; we require only that these differences not be such
as to establish two distinct, competing, S-lineages with distinct S-values.

Indeed, the S-value of an S-lineage may actually critically rely
on the distribution of variations in certain characteristics
among the D-actors making it up, so that no single D-actor, {\it
qua\/} D-actor, would allow a determination of S-value---except in
the contrived sense that we should be able to determine its
S-class, and thus determine the ``typical'' distribution of
variations within an S-lineage made up of D-actors from that
S-class, and thus, finally, determine the S-value of the
S-lineage (relative to specified competitors).

We can envisage a variety of ways in which variation in the
characteristics of D-actors (within an S-class) could be
beneficial to an S-lineage.  For example:
\begin{enumerate}
\item If there is a diversity of resources or habitats, whose
availability varies, then an S-lineage which keeps its options
open---distributes its D-actors over these resources or
habitats---may well do better than competitors which lack this
flexibility (whose D-actors are more uniform).  This can work
without any assumption of interaction or mutual recognition
between the D-actors making up a single S-lineage.
\item Alternatively, if there {\em is\/} any mechanism whereby a
D-actor can identify other D-actors with which it shares an
S-lineage (with some degree of probability) then there may be
other opportunities for {\em cooperation\/} between D-actors,
perhaps involving specialisation or division of labour.
\end{enumerate}

In general, both these possibilities could equally be based on dedicating
certain D-actors to certain tasks on a lifetime basis, or on the use of a
structured life cycle with different specialisations on the part of a
single D-actor at different stages in its cycle; and, of course, the two
possibilities could both be exploited to greater or lesser extents in a
single S-lineage.

The notion of a D-actor being so constituted (in structure or behaviour)
that it functions for the benefit of other D-actors, at a cost to itself
(especially as measured in terms of the number of offspring it
produces), is referred to as {\em altruism}, and has historically been
viewed as somewhat problematic for Darwinian theory.  However, we can
now see that this derives from a blinkered view of selection: the view
that supposes that a D-actor can benefit its S-lineage(s) (improve its
S-value) {\em only\/} through direct ``benefit'' to itself.  In particular,
the second scenario mentioned above, where an S-lineage can benefit from
mutual co-operation between its members, clearly allows the possibility
of altruism (on the part of D-actors to other D-actors in a single
S-lineage) being {\em favoured\/} by a Darwinian process.

I have suggested that altruism could be favoured where there is a
mechanism for D-actors to recognise other D-actors sharing a single
S-lineage. This is a necessary condition.  By definition,
altruism involves costs to the D-actors engaging in it.  This can
conceivably provide a net benefit to the S-lineage {\em only\/}
if most of the benefit of the altruism is retained within the
S-lineage.  To put it another way, altruism is in permanent
danger of being subverted or exploited.  If one S-lineage
contains D-actors which are unconditionally altruistic, and another
is otherwise similar (and thus competing) but its D-actors succeed
in restricting their altruism (even with limited effectiveness)
to members of their own S-lineage, then the latter S-lineage
clearly gets more benefits from altruism than its competitor, and
will be favoured by selection.

Note that this argument would not generally go though if we
considered a putative S-lineage which abandoned altruism
entirely. Such an S-lineage could certain {\em initially\/} grow
at the expense of the original S-lineage of altruistic D-actors.
But recall our basic hypothesis that the altruism of the D-actors
was beneficial to the S-lineage.  The new S-lineage, of entirely
selfish D-actors, would, as it becomes more numerous, start
losing this benefit. There is no general answer as to the final
outcome of such a process, but it certainly does not follow that
the original S-lineage would necessarily be
eliminated.\footnote{That is, it is not generally the case that
  such a process would meet all the specified requirements for
  selection to go through; in particular, it may not meet the
  requirement of a {\em consistent\/} selective bias (independent
  of S-size).}
It would be equally possible that another different S-lineage
would arise which {\em restricted\/} its altruism: this would
then generally be capable of selectively eliminating {\em both\/}
of the others.

So altruism, if it arises, is expected to rely, to a greater or
lesser extent, on mutual recognition between the D-actors of a
single S-lineage.  There are two basic mechanisms which suggest
themselves for such recognition:
\begin{enumerate}

\item One's own ``close'' relatives (parents, sibling, offspring) are
likely to be members (and, to a lesser extent, their close relatives in
turn).

\item One may attempt to ``recognise'' other members of some
S-class.

\end{enumerate}

The former corresponds to what is conventionally referred to as
``kin selection'' in biological Darwinism,
\cciteAD{Dawkins:TwelveMisunderstandings}.  The latter is not
generally explicitly emphasized in biology, though Dawkins has
considered a variety of mechanisms which might come under this
general heading \cciteAD[Chapter~8]{Dawkins:ExtendedPhenotype}.
Notwithstanding this apparent lack of emphasis among biologists,
I speculate that benefits of altruism (in the broadest possible
sense) mediated by recognition of S-class and/or recognition of
close S-kin may be quite significant for Darwinian evolution
generally.  I say this simply on the basis that recognition of
this sort is clearly, in the general case, a difficult problem
that calls for complex sensory processing, i.e.\ the utilisation
of relatively complex anticipatory models of the world.  In terms
of the argument of section~\ref{adaptation} above, benefits (for
S-lineages), deriving from more and more effective S-class
recognition and discrimination, seem to provide one plausible
general basis for a sustained correlation between S-value and
knowledge, and thus for an extended period of growth of knowledge
through evolutionary time. However, intriguing as this
possibility may be, it is of limited relevance to my purposes
here, and I shall not pursue it further.

To return to the central idea of this section, which is the
notion of the S-lineage as the unit of selection, the
significance of this point can be put at its most stark by asking
what is the ``adaptive'' value {\em to a D-actor\/} of procreation
itself? Basically, there is none, and yet everything else that
might be an adaptation of a D-actor is typically identified in
terms of its contribution to procreation.  Conversely, there is
no problem when we think of the structure and behaviour of
D-actors, insofar as it has been adapted by Darwinian evolution, as
being consistently for the benefit of the S-lineage.  Which is
simply to say that S-lineages which have come about through
Darwinian evolution, are expected to be characterised by naked
and exclusive {\em selfishness\/}; altruism on the part of an
S-lineage (as opposed to the part of its component D-actors) could
only be classified as pathological in a Darwinian context.

The essential point is that while the adaptations of an S-lineage
{\em may\/} be clearly manifested in ``adaptations'' of the
D-actors (or, more generally, of the S-classes, identified in some
particular manner), this is an incidental effect, which may or
may not occur, rather than being an intrinsic feature of
Darwinian evolution in general.  The general and reliable feature
is the selfishness of the S-lineages themselves.  The failure to
realise this is manifested in the attempts to introduce entirely
counter-intuitive modifications to the notion of ``fitness'' of
a D-actor when it is discovered (as, for example, in the case of
kin selected altruism) that, without such a correction, entities
of lower ``fitness'' are apparently favoured by selection.  This
is the general idea of so-called ``inclusive fitness''.  Dawkins
has rightly pointed out that that is an inspired, but ultimately
misguided, stratagem \cciteAD{Dawkins:ReplicatorSelection}: the
supposed difficulty simply does not arise when one realises that
the D-actors are not the appropriate entities to ground the
analysis in the first place---the S-lineages are.

So I consider that the unit of selection is the S-lineage; and
that the implication of this is that a given phenomenon can be
considered as an outcome of Darwinian evolution if and only if it
is consistent with an unconditional selfishness on the part of
the S-lineage possessing or exhibiting it.  This is the doctrine
of {\em The Selfish S-lineage}.

\section{Conclusion}

The central concept which I have introduced is that of the
S-lineage as the unit of (Darwinian) selection.  The S-lineage is
a peculiar hybrid of individual and class---perhaps it should
best be regarded as an ontological category of its very own. It
is a bastardized entity and not at all pretty; but it seems to me
that this somewhat uneasy alliance of the permanent and the
ephemeral is an intrinsic aspect of Darwinian theorizing, and
might as well be pulled out into the open so that we may examine
it properly.  At worst, if I am thoroughly mistaken in this view,
I hope that I have at least presented my ideas sufficiently
clearly that they can now be criticised and corrected.

As indicated in the introduction, this essay has involved the
introduction of a range of new, and occasionally cumbersome,
terminology.  My claim is that this is justified by the fact that
it will provide an effective framework for the application of
Darwinian concepts outside the specific field of biology.
Admittedly, this claim can be properly tested only in subsequent
work.

This is a internal Technical Report, and the freedom of that
format has perhaps been abused in the protracted nature
of some of the discussion.  Nonetheless, I would greatly
appreciate comments and criticism of any sort.

\section*{Acknowledgements}

This essay is part of an ongoing attack on the problem of
realising the spontaneous growth of Artificial Knowledge by
Darwinian (or any other!) means.  In this pursuit, I have
benefited greatly from discussions with colleagues, particularly
Noel Murphy in DCU, and John Kelly of University College Dublin.
I am also indebted to the School of Electronic Engineering in DCU
(particularly through the agency of its Head, Charles McCorkell)
for continuing encouragement, not to mention material support.
All errors remain, of course, my own responsibility.

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