"Science knows only one commandment: contribute to science."Bertolt Brecht (1898-1956)
"Science commits suicide when it adopts a creed."Thomas Henry Huxley (1825-1895)
One of the main axioms of traditional science is the question of reversibility. Often scientists insist that all scientific laws must be reversible and this contrasts strangely with the equally fervent attachment given to the Second Law of Thermodynamics, which says that the universe is irreversible. Here we look at this inherent contradiction within science and try to use complexity ideas to show that it is only apparent and due largely to a misunderstanding of probability.
We will challenge the idea that both these claims are universal by utilising the concepts of attractor theory. We show that, from the perspective of modern dynamics, each property has only limited validity, being just part of a more complex world of evolutionary change and transient stability.
When we create a mathematical formula, in an attempt to match theory to practice, we create a mapping between one state (the output) and others (the inputs). We isolate the system into two independent parts for the purpose of prediction. Thus we have y = F(x,t), the output is a function of the input and time. This use of time suggests that we can reverse it and in principle the system will run perfectly well in the opposite direction, there is no preferred time axis. If we add in the conservation laws (of momentum and energy) so that global properties are invariant, we seem to have a situation that treats time as identical in form to spatial co-ordinates. This implies that we can reverse time just as easily and relativistically as we can reverse north and south, all directions are equivalent and none have special meaning.
Yet we know that a broken cup doesn't reform, a fired projectile doesn't return to the barrel. There is a conflict here between what our mathematics tell us and what occurs in the world. In such cases, if we wish our theories to be called science, we would be wise to question the assumptions of our model and its relation to reality. Here, the assumption that time reversal allows us to still perform experiments is invalid. The observer is not independent of the world observed and thus the (backward) measurement process would fail. Nor will reversal of just an isolated 'closed experiment' work, since the remaining world influences preclude such experimental (as opposed to theoretical) closure. Maths isn't Life, but a simplification of it, and both these aspects can take many alternative (perhaps incompatible) forms.
By concentrating on simplifications, and viewing systems as collections of conserved parts, we are forced into a philosophical stance that pretends that each particle is discrete and must be somewhere at any one time. This implies trajectories or world lines that are always separate, and it is a basic assumption of reversibility ideas that this is so. To reverse a system it is essential that the components stay the same and that we can keep distinguished all the individual states within the system.
This viewpoint is false, and takes no account of the creation or destruction which is a noticeable part of the world. Physicists tend to flee in the face of such issues to sub-atomic particles, claiming that these are fundamental, and as everything is made up of them then their properties must be inherent at higher levels, thus if particles are conserved then trajectories must be. Yet this classical assumption is also false, particles are not conserved, far from it. They spontaneously come into being and disappear in the mathematically best physical theory we have - quantum electrodynamics. Additionally even classical systems have unstable states, where multiple options (bifurcations) exist, trajectories are not, in general, distinct and isolated.
Using the concept of attractors from complexity theory allows us to show that this is the case. Imagine a golf course. We find a hole and it has a ball in it. We can assume that it followed a deterministic trajectory to the hole, but which one ? Obviously many such trajectories arrive at the same position, that is the whole idea of the game. It is impossible to reverse this system, not because of complicated conversions from micro to macro worlds as is sometimes claimed, but simply because the possible trajectories of the system merge in this instance, so reversal is indeterminate.
In complexity thinking this relates to a fixed or point attractor and many equilibrium systems have this form. Many other attractor types do generally allow for endless repetition and reversibility, the Gas Laws for example are based on the ergodic attractor, the Laws of Planetary Motion on the cyclic attractor. Even with the newer 'strange attractor' we often state that trajectories never cross, that they do not merge at any stage. Yet the world of the point attractor is not disjoint with these other worlds, the same system can follow different attractors at different times, so we can say that the assumption of discrete trajectories is just that, a simplification valid in some systems but not a necessary axiom of science (actual or theoretical).
As well as the convergence of trajectories implied by the concept of attractors, we must also consider the divergence of trajectories. This again is disallowed in classical physics, but is an essential part of biological thinking. The simplest example is perhaps the bacteria, here a single cell divides to form two offspring, which go their own ways. This continues ad-infinitum until the single original trajectory has become billions. This splitting process is also common when mutation changes the evolutionary trajectory of a species, forming a new option, and when we ourselves choose between any two or more options.
In fact all innovation in the world, all novelty, actually requires this bifurcation process, this doubling of the world lines to accommodate choice. It is not, as some would have it, a split between alternative worlds (as in the many-worlds theory of quantum mechanics) but a natural occurrence within our own world and not in the least problematical - providing we discard the outdated 'universal reversibility' assumption of mathematical Newtonian mechanics.
Given such merging and splitting states, we can ask if these are always irreversible and the answer must be no. Within the field of automata theory we see that the rules followed are much more general that those constraining the simplistic thinking of physics. In physics 'laws' apply equally to all, there are no exceptions, but in fact this need not be the case. In more complex systems (with memory) each individual entity can follow different laws, they can use internal 'transition laws' based on current states or past history to determine behaviour. Thus we can have systems that follow deterministic trajectories, those that follow attractors and those that bifurcate - along with those that do all of these at different times and in different contexts.
Automata can converge to a single entity, they can (given appropriate environmental conditions) split again, or they can maintain a constant form. No problems arise in allowing for the possibility that science is much more general than previously allowed, far richer than that needed to deal with the often oversimplified aspects treated generally in physics (and those sciences that follow similar assumptions). Complexity science allows for reductionist science as a sub-set, but by dropping unnecessary and misleading axioms (replacing them instead by local constraints) it can take science into promising new areas of our world, and also into many of the other worlds that remain as interesting possibilities in state space.
It is quite common in complexity thinking to regard simple animals (e.g. ants) as automata. In other words we can treat the behaviour of living organisms as if they were machines with limited or finite possible states. In many ways this is also the view from genetics, the genome specifies the transition possibilities for the animal, it constrains the production of protein types which in turn limits the possible states or attractors that can be formed. It seems clear from this that if we are to treat scientifically our biological (and by implication social) world then we must transcend the limitations of global physics, we must look to local (often indeterminate or stochastic) laws. As we have seen this is not so difficult to do.
Higher animals and our own species have of course many possible states, we also have the ability to learn (memorise) and thus to increase those states. We could be said potentially to be infinite state machines. This blurs, rightly, the difference between inorganic and organic systems, which we can see form a continuum from single state sub-atomic particles, through multi-state atoms, towards near infinite state humans. Available options relate to complexity and constraints, the more the complexity (number of parts) and the more varied the constraints (inhomogeneity) the more complicated the behaviour may become.
Indeterminism relates to probabilities, the idea that not all options are equally likely, what actually happens will depend on fate or chance (stochastic operations). Given the possibility of multiple transitions, of behavioural choice, we must now look at the relative probabilities of these choices in order to estimate how the system will behave. It can be shown that given a biased probability distribution for a single particle, then a large independent collection of such particles will be found to be almost exclusively in the most probable state (the law of large numbers). Thus we can say that obtaining variety in nature and society requires systems that have fairly evenly distributed probabilities or small numbers, such that significant proportions of the population will be in uncommon states.
Many treatments in physics and biology make the assumption that each state is equi-probable, that there is no bias in the selection of which atom will decay or which gene will mutate. Similarly many human systems operate on the basis of an equi-probable coin toss, a random trial. So overall we would expect a fairly wide distribution of states within any population. Additionally, many states are correlated, samples are not independent of each other, a prior choice affects later decisions (either to split, allowing the growth of a new, fitter, possibility as in evolution - positive feedback; or to merge, restricting variety as in majority democracy where unity in decisions is desired - negative feedback).
Due to these conflicting considerations, most complex systems have a spread of properties. Thus we can expect that there will also be a spread of trajectories present, people will fight against the crowd (splitting), decide to conform (merging) or maybe just do their own thing (isolating). This dynamic human organisation is very different from the uniformity assumptions of reversible physics, but does contain aspects of that world also. Hence we are able at times to make simplifying assumptions and treat certain properties of our world as capable of being modelled by reversible equations.
The ability to do this however should not blind us to the need to keep in mind that such cases are merely a small sub-set of the possible behaviours of our complex system. Life is richer than non-life, society richer than biology. As we climb the complexity ladder we also encounter nested levels of detail, emergent properties that can have very different characteristics to the parts that make them up. Predictable (mathematical) parts can give rise to unpredictable behaviour (e.g. in chaotic systems) and conversely unpredictable (atomic) parts can give deterministic properties (e.g. in cellular stability).
The difficulty in reconciling the reversibility of small scale physics (e.g. showing a film of particle motions in reverse is undetectable) with the irreversibility of large scale events (e.g. a gas cloud collapse to form a star looks wrong in reverse) relates not to the reversibility of fundamental laws (here gravity) but to the effects of mathematical simplification. What is improbable for a thousand particles may be fairly probable for a few, and what is improbable for unconstrained particles may be very probable for constrained ones.
Most work in physics deals with either small numbers of particles or with very large numbers. In the first case individual reversibility can be achieved in constrained systems if we ignore minor disturbances and use short time scales - we smooth out the inherent irreversibility by using simplified formulae. In the second case we can use statistical techniques based on unconstrained systems (e.g. gasses) to achieve the reversibility - we force conformity by ignoring fluctuation in the sub-structures. In the middle complexity world neither of these techniques are valid, there are too many closely interacting entities to discard the higher order terms in our equations, and there are too few entities to ignore the small scale patterning of the parts due to constraints (random or otherwise). Hence in complexity studies whether the system is of reversible or irreversible form must be contextual and there is no resultant conflict between these ideas.
Given a set of options of differing probabilities, we can examine the effect of taking one option rather than another. In traditional ergodic systems all options will occur, there is no intrinsic difference between them at equilibrium and thus the system will be expected to occupy state space in proportion to their probability. We see this often in equilibrium chemical reactions, where the rate equations give the relative bothway movement of the reactants. Yet this simple view does not hold for more complex reactions, where catalysis processes occur. Adding a spark to an oxygen/hydrogen mixture will cause an explosion, an irreversible reaction, the system then stores a bias due to the catalytic feedback element.
Chain reactions of this type are also familiar in biology, where a decision (e.g. left-handed molecules) gets locked in to the system at an early stage and then shapes or canalizes its future options. This aspect of navigating state space in preferred directions is what we call irreversibility. It is not that the system cannot, in theory, reverse but rather that the probabilities are changed by the decisions made. It is like stepping on to the down escalator in preference to the up - you can reverse this but at a considerable cost in resources in time and/or energy, way in excess of those needed for the initial choice. These critical points in a system relate to instabilities, choices that take the system into one valley or another within the fitness landscape, rather like entering a one-way street at a Y junction, a path towards an unknown attractor.
In evolutionary terms, whether the choice of direction is conscious or stochastic is irrelevant. All choices have consequences, they take us a step beyond our starting point. After 20 such bifurcations, the chance of returning to the start exceeds a million to one against. In complex systems all parts perform these wanderings, so it is likely that almost every encounter will be novel, a one-off event.
To provide any order in our world we need constraints, predictabilities that reduce the available options. Scientific laws are forms of such limitations and each has applicability to one aspect of our universe. But this universe has many dimensions, many contrasting facets, and it is unsurprising that such laws form only partial descriptions of the possibilities. Given any two such laws, we can expect situations to arise in life where neither apply, where only one does and also where both are valid. Reversibility requires discrete closed loops, irreversibility infinitely joining paths - and both features exist widely in our world.
Being unable to return to our starting point is not usually a disadvantage. The size of state space is such that we need this limitation in order to better explore it. Small systems are boring, they may be amenable to exact mathematical treatment but (like much in modern art) they reduce science to the trivial, they lose what makes discovery interesting - and that is novelty and depth. For those not fascinated by mathematics itself, and that is by far the majority, it is the creative bootstrapping made possible by feedback and irreversibility that makes life's street the exciting one-way journey it is.
The findings of the Second Law and the Reversibility of particle physics treat different aspects of the same world, to confuse either with universal properties is to impoverish our thinking. Systems can be reversible, they can run down, or they can increase in variety, each in different circumstances. We need not claim any of these possibilities are 'true', nor claim that they are essential for our science. Complexity theory can take them or leave them, recognizing that in themselves they are only some of the many constraints we impose on our world, in order to theorize about selected, interesting, aspects of our reality. It is the obsession with homogeneity, with contextless universal laws that has blinded scientists to the limits of their viewpoints, which do not apply to heterogeneous, non-equilibrium systems.