Complexity and Context-Dependency
Abstract. It is argued that given the
“anti-anthropomorphic” principle, that the universe is not structured for our
benefit, that modelling trade-offs will necessarily mean that many of our
models will be context-specific. The
context heuristic, that divides the processing into rich, fuzzy
context-recognition and crisp, conscious reasoning and learning is outlined. The consequences of accepting the impact of
this human heuristic in the light of the necessity of accepting
context-specificity in our modelling of complex systems is examined. In particular the development of “islands” or
related model clusters rather than over-arching laws and theories. It is suggested that by accepting and dealing
with context (rather than ignoring it) we can push the boundaries of science a
little further.
Introduction – The Exception
of Simplicity
Many times in the past, people have assumed that they occupy a special
place in the universe or that the universe is somehow arranged to suit them,
only for this to turn out to be wrong.
Examples include believing that the Earth was the centre of the Solar
System, or that homo sapiens had a special origin, different from than that of
other animals. Today many believe the
following:
That the universe happens to
be structured on grounds of sufficient simplicity that our brains are able to
analyze and comprehend it.
Or, to put it another way:
That our brains happen to have
evolved so as to be able to analyze/understand models adequate to the phenomena
we observe.
We may look down on other animals, perceiving that
they have a biased and limited understanding of the world, but somehow assume
that we don’t have analogous biases or limitations that we cannot somehow
overcome. Surely this is merely another
example of anthropocentric arrogance.
That we have had some notable successes at understanding our world and
even a systematic set of approaches that has been shown to be useful is not
sufficient evidence to assume a lack of limitations and biases.
This astonishing assumption takes many forms in
philosophy and discussions about the scientific method. One such is that somehow simplicity is a
guide to truth. That is, that simplicity
in a model or theory has advantages other than the obvious pragmatic ones
(pragmatic virtues are such as: being able to analyze/solve it; being able to
have good analogies with which to think about it; needing less data in order to
parameterize it; and being able to compute it). Elsewhere I have argued that
simplicity does not indicate truth [7].
Another version is that everything somehow must be simple if only we can find the right way of looking
at it, or formalizing it. It is true
that frameworks such as Newtonian Physics are relatively simple (though I doubt
many in Newton’s time would have thought so), and using this,
many useful models and reliable predictions can be obtained. However, to make it actually cover observed
cases and encountered problems one has to complicate it immensely in order to
apply it. Even its general scope and
simplicity in theory are questionable, with the (in
hindsight inevitable) complications to cover new situations (Einsteinian,
Quantum).
I am not going to spend time arguing the above points
here. Rather I will consider the case
under the anti-anthropocentric assumption, that much
of the world around us is organized in a way that is beyond adequate modeling
in a sufficiently simple and general manner for us to cope with[1]. Where we
accept that we have biases and limitations in our abilities to develop our
understanding of the phenomena that confront us and, by acknowledging these and
thinking about them, consider ways we might extend our scientific understanding
to the greatest possible extent. Under
this, admittedly pessimistic, view the phenomena that are simple enough for us
to understand in a scientific manner are the exception – the exception to be
sought and struggled for. Under this
view, we should make the greatest use of the strengths we have, and seek to
acknowledge and mitigate our limitations.
Under this view a “Science of Complexity” makes no more sense than a
“Science of Non-Red Things”, since both red objects and simple systems are the
exception rather than the rule.
Some Modelling Trade-offs
In any modelling enterprise, when one is trying to represent some
observed phenomena in a model (analytic, statistical, or computational) then
there are inevitably some compromises and trade-offs one has to make. Figure 1, below, illustrates one possible set,
that I will discuss below.
Fig. 1. Some modelling trade-offs
Clearly the available modelling trade-offs are never as
simple as Figure 1 suggests. However, it is rarely the case that one can obtain
all these four desiderata simultaneously,
that is find a model that is: formal, simple enough to analyze and completely
understand, that is valid in terms of its specification and results compared to
the evidence and is fairly general in its scope. I will briefly discuss each of these four criteria
in turn.
Validity. There are many ways in which a model may be
judged as valid. This depends somewhat
upon the goal for the modelling, that is how good it is for its intended
use. In order for a model to be useful
for any particular purpose it may have to be adequate in terms of a number of
different criteria. For example, to be
useful for prediction a model has to be: effectively initialisable in terms of
parameters and settings that can be measured or determined from available
evidence, computable in that one can somehow derive the predictions from the
settings, and the results need to correspond to previously unknown observations
to a sufficient degree and to a sufficiently reliable extent (given the purpose
of the model). To be useful for
establishing a possible explanation for some observed phenomena a model has to
be: formulated in terms of the kinds of processes that one wants the
explanation to be expressed in; to match the known outcomes that are to be
explained; and to computationally trace out how the outcomes arose from the
processes. However, whatever the
purpose, it is clear that it is fundamental to science that a model is as valid
as possible.
Formality. A formal representation is one that can be
precisely specified. Thus both
mathematical and computational models are formal (the later not being
analytic). Not all scientific models (in
the broadest sense) are formal, there are plenty of informal theories,
descriptions and arguments which are part of science. However formal models play a central role in
science, they form the ‘backbone’ upon which less formal aspects hang, allowing
an unambiguous frame of reference for meaning of terms and facilitating a
inter-scientist development which would be infeasible if based on less copiable
entities. It also provides a basis for
the systematic variation of possible explanations of the phenomena we
observe. Thus, after validity preserving
the formality of our models is of greatest importance.
Simplicity. Simple models are those that are easy to
build, test, understand and analyze[2]. If we
accept the anti-anthropocentric principle then this implies that we may have to
accept that the simplest adequate model (for a given purpose) might be very complex[3]. There
are things one can do about such irreducible complexity but discussing these is
not the prime purpose of this paper.
Some of the suggestions later in the paper will touch upon ways around
complexity. Here we just consider the
situation where we have exhausted simplification techniques, and have to look
elsewhere to progress our scientific understanding.
Generality. Generality is the extent of the scope of a
model, that is, how many different situations or cases a model will be usefully
applicable to. If one has a general
model one can easily produce less general models by fixing some of its
parameters or settings to those appropriate for some specific circumstances,
thus making it less abstract. Thus a
general model of gravitation might be simplified to one where gravity only
works in one direction (downwards) and with one value (9.8 m/s2), so
it is specific to localized areas on the surface of the Earth. However doing the reverse, adding in
processes and variables for fixed constants in a model, or by making a model more abstract, does not necessarily lead to a more general
model, since one does not necessarily know what processes one should add or which
abstractions one can make. Indeed, going
this way can lead to a loss of generality, since the wrong additions or
abstractions can mean it is not applicable anywhere.
Given that validity is essential to science, and
formality highly desirable we are left with generality as the “free” aspect we
can work on in order to progress scientific modelling in the face of overly
complex phenomena. That is, one way we
might be able to make do with sufficiently simple models is by reducing the
scope of the models – making them more context-specific. In other words, sacrificing generality to
gain some simplicity, in the context of obtaining to formal models that are
sufficiently valid for our needs. This
is worth it, for although generality is a highly desirable property of models,
like simplicity, it is a pragmatic virtue.
Without wide generality we still have science (albeit more fragmented,
resembling biology more than physics),
but without formality we can not have progressive science as we know it, and
without validity we don’t have science at all.
The Context Heuristic
Many aspects of human cognition and behaviour are context-dependent to
some degree, including: visual perception, choice making, language
comprehension [15], memory, reasoning [16], emotion, judging trustworthiness [21] and assessing reputation [20]. Thus,
it appears that the brain uses context-dependency as an effective heuristic for
dealing with the complexity of the world around it. In very broad terms, it appears that we
recognise context in a rich, automatic, imprecise, unconscious but reliable
manner. With each recognised context
there is an associated resource of learnt facts, terms, behaviours, norms etc.
that are accessible from the context.
This resource makes reasoning, modelling, communicating, understanding
etc. much easier since the context allows access to the most relevant
information in a structured manner.
Within a context we tend to employ more crisp, formal, deliberate and
conscious reasoning and learning. Thus
context seems to allow the integration of “fuzzy” pattern recognition with
“crisp” reasoning mechanisms.
This heuristic is contingent – that is, for it to work,
there are a number of conditions that have to hold.
1. The domain has to be usefully divisible into a number of contexts, that is each context must have clusters of related knowledge associated with them, clearly each cognitive context must cover a (maybe small) number of situations which have some underlying commonality [3].
2. The contexts have to be recognisable in a reliable manner (though not necessarily definable in precise terms).
3. Learnt knowledge must be associated with the context so that it is effectively retrievable when the context is encountered again.
For this heuristic to be most effective then the
following are should hold.
4. That the contexts can be learnt, so that what is a context can be co-learnt with others so that everybody will recognise the same context.
5. That different contexts can be associated with the same situation, so that the relevant context can be flexibly swapped for another applicable one that might be more helpful (e.g. in case that the wrong context has been chosen or decision making is not feasible [6]).
There are many advantages to this heuristic. Dividing the world up into a number of contexts
means that both explicit learning and reasoning happen within constrained sets
of knowledge, making both of these processes computationally feasible [6]. In
other words, context “solves” the frame problem [18], for without a way of delimiting the possible
knowledge one could apply efficient reasoning is impossible [13]. If one is faced with either an under- or
over-determination of knowledge with regard to a particular decision then a more
or less specific context (respectively) can be selected [6]. The
same problem can be thought of from a number of different cognitive contexts,
allowing different perspectives to be applied (for example it may help to
understand the otherwise surprising reaction of someone if one take the context
to be a competition rather than a discussion).
Since the heuristic is shared between people then this can facilitate
the co-determination of appropriate contexts which aids communication and
coordination. Context allows the
integration of fuzzy pattern recognition and crisp reasoning processes in a
coherent manner [11].
There is a difficulty with talking about or analysing
context, due to the rich and unconscious way it can be recognised. Despite the fact that everybody can reliably
recognise a context does not mean that the definition of a context is precisely
definable or even identifiable. Everybody
recognises the difference between the living and the inanimate, but it does not
mean that it is easy to precisely define life. Context
seems to be recognised automatically, so to a large extent, we may be unaware
of its identification – it simply may not be easily accessible to as
scientists. Thus although I talk about
contexts in this paper, as if they are well-defined entities, it may well not
be possible to unambiguously talk about the context for
any particular case being discussed. “The context” is an abstraction of the class of situations
that would be recognised as such and to which the same set of existing
knowledge would be brought to bear, but this does not mean that context can be
reified as a well-defined labelled entity.
Thus the statement “A is true in context C”
is not formalised as C®A as suggested in [17] since C cannot be usefully reduced to a
precise statement.
Given this analysis, it should be clear that
context-dependency is not the same as subjectivity. It might well be that all relevant people can
correctly identify the correct context and, given that, reliably assess the
truth of a statement, where its truth depends on that context. Subjectivity would mean that each person
might come up with a different assessment.
Thus the in vitro vs. in vivo
distinction is well understood in biology, and which is being assumed is
usually abundantly clear, even if not stated explicitly. A statement’s truth may be specific to the in vitro context without this either being subjective or
necessarily being true generally (e.g. also in the in vivo
context). Also the meaning of a
statement might well be context-dependent, without the meaning being subjective
for the same reasons, for example the idea of a “behavioural norm” does not
make sense when considering inanimate objects.
Thus context-dependent statements are not inherently unscientific. In the case that the same context can not be
reliably identified by all participants, but varies with each observer, the
context-dependency of a statement may reduce to a form of subjectivity.
However, due to the fact that it can be difficult to
identify and talk about the appropriate
context of any statement this can make statements more difficult to formally assess, as the assumptions it depends on in any
formal analysis might be “rolled up” into the context. Digging out the assumptions in relevant
contexts, can be part of trying to formally assess or test context-dependent
statements, or as part of an attempt to generalise out of a specific context to
a more general one.
Context and Scientific Modelling
Given the anti-anthropomorphic principle we
are not going to be able to represent all the possible causal process in our
models[4]. In any
complex (e.g. biological or social) case there are an indefinite number of
possible causes of any event. Indeed
elsewhere I have argued that the very notion of a cause only makes sense if
there is effectively a context to delimit it [8]. In any
case, we have to choose what to explicitly include in our models and what are
considered, irrelevant or unchanging [23]. Much
of the usefulness of a model comes from being able to use the same model in a
related, but different set of circumstances that the one it was initially built
for. In order to be able to do this one
needs to know that none of those factors that were not represented in the model
will make the model invalid in the new circumstances. Sometimes the applicability of a model is
described explicitly, but since the set of background assumptions is
indefinitely large, this is impossible to do completely (what [22] call “causal spread”). Hence some of these background assumptions
have to be ensured simply by the context – that is, there is enough recognizable commonality between the situation for which the
model was constructed and the situation in which it is later applied, that we
do not have to worry about more extreme counterfactuals. A common way that humans use is the
recognition of a common kind of extended (or cognitive) context[5]. This
is illustrated in Figure 2.
Fig. 2. Using a model to transfer knowledge from one
situation to another which share context
As long as we can recognise when
a model can be applied with a fair degree of reliability this is not a
problem. However it is easy to take a
model out of context, in which case the model may lose some or all of its
validity. A change in context could
involve a change in modelling purpose as well as the situation and sometimes that
is more difficult to recognise from, say, its description in an academic
paper. In this case there is a danger of
the modelling being wrongly applied to this different purpose – a tendency that
is particularly observed in the social sciences [10]. Sometimes
when a model is being used as an analogy [5] (albeit in formal or computational form) this
vulnerability to context change is masked, due to the context-sensitive manner in
which people apply analogies, adapting meaning and reference almost
automatically.
Of course, a model may still be useful when the
influence of the unrepresented, background factors effect the results but not systematically
or severely enough to completely invalidate them. This sort of “leakage” into a modelling
context is often labelled as (exterior) noise [9]. Often
we use a random “proxy” to stand in for this interference, for example, to test
the sensitivity of the results to that factor.
When what was thought to be “noise” turns out to have significant or
systematic effects on the significant aspects of the results of a model, then
this indicates that either the model needs expanding to include this aspect, or
that the context is misidentified.
Regardless or whether one believes that “in principle”
that a context-dependent approach to modelling is necessary, with many complex
phenomena it is necessary in order to make our modelling feasible. It may be that formal, general and valid
models exist for many phenomena but the models are either too complex for us to
deal with, or just be very difficult to discover. The reason we are forced to a
context-dependent approach does not ultimately matter – the outcome is the same.
Thus to a large extent, modelling practice and human
cognitive bias coincide and for similar reasons – it facilitates representation
and reasoning when the phenomena happen to be usefully divisible into
contexts. The same underlying trade-offs
drive both in circumstances where this is possible.
Some Consequences of Complexity and Context-Dependency
If one accepts that one might have to settle for models that are either
too complex to fully understand and/or are limited in scope to a specific
context then this will have consequences for how one might have to approach
using formal models.
If models are going to be more context-specific then
more care will need to be taken to articulate and document as much of the
information about the modelling context as possible. This is impossible to do completely, as has
been discussed above, but what is known should
to be made explicit. The context the
model is developed in should be described, and some indication of the range of
contexts in which the model might be applied in, possibly explicating some of
the commonality that is hypothesized to underlie its applicability. This could include some positive and negative
examples of its scope and maybe a rule of thumb to guide other
researchers. Talking about context is
far more common in the social and ecological sciences and it may be that
physicists and other complexity scientists will have to get used to this too.
It may well be that a set of models is the best way to
try and represent some phenomena, rather than any single model [12]. This
set might range over different aspects of the phenomena, consider it from
different viewpoints, capture it at different granularities, or represent it at
different levels of abstraction. This
already occurs to some extent, for example in the development of separate
explanatory and phenomenological models [2].
An example of this that of an ideal gas. The gas laws successfully model the relation
between these triples of data, so that at the same pressure the volume is
proportional to the temperature (measured from absolute zero). However the gas laws simply relate the
phenomena, they do not explain why they are so
related. A model of the gas composed of
a sparse collection of molecules, can explain the gas laws, when the speed of
the particles is related to the temperature.
Using some judicious approximations one can derive the gas laws from
this model. This model explains why
temperature, pressure and volume are related, but does not itself express the
relation between them – to infer these relations one needs extra
assumptions. The mechanics of the
particles in this model are consistent with Newton’s laws of motion and the
theory of the nature of temperature.
Thus, even in this simple case one has, at least, two separate models:
the phenomenological model and the explanatory model.
An ideal gas is an exceptionally
simple case which can be dealt with using judicious assumptions of randomness,
due to the lack of any significant interaction between its parts. In many other cases there is no adequate
analytic approach to the modelling of molecule interaction – the only way to do
this is via extensive simulation. This
approach is discussed in [14] and the relation between the models
illustrated in Figure 3.
Fig. 3.A layering of related models in chemistry
adapted from that in [14].
In this case the focus model concerns how a set of
atoms interact (e.g. how the molecules pack or move in a certain state), this
can be simulated using a representation of a set of atoms in 3D space using
model of the underlying atomic processes.
The simulation is run a great many times so that a summary of the
results can be obtained by averaging etc.
These results are further summarised into an equation that approximates
the results, which is then compared to many different sets of measurements of
the relevant natural phenomena. Thus
here we see a cluster of at least five related models describing different
aspects of the interacting molecules, such clusters will become the norm rather
than single models as we tackle increasingly complex phenomena. Each member of the cluster might be validated
in a different way (e.g. those in [1]) and related to the others in their cluster in
a variety of manners. We know little
about how to build, test and maintain such model clusters – this is something
that needs attention.
Conclusion – Facing up to Complexity and
Context-dependency
We may have to get used to the fact that there may not always be a
general framework that can be assumed to relate all models together in a formal
manner, but rather “islands” of model clusters that relate to each other, where
each cluster is related to its own context.
In other words, the theoretical landscape might just look a lot more
“patchy” with only local areas of coherence, composed of a mess of
context-specific models. Where possible, generality is desirable, but it will
not be always achievable, but that does not invalidate the models. Context-dependency does not stop it being
science but, acknowledged and dealt with, it may open up new approaches
allowing us to push the envelope of scientific knowledge a little further.
Acknowledgements
I would like to thank all those at the Centre for Policy Modelling,
those at the International Conferences on Modelling and Using Context, and
those who attend the informal Manchester complexity seminar series for many
discussions on these issues.
References
[2]
Cartwright, N. (1983). How the Laws of Physics
Lie. Oxford, Clarendon Pres.
[19]
Pearl, J. (2000) Causality. Cambridge University Press.