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logic> hierarchies - an introduction

What is a hierarchy? A shape, a geometry, an organisation, a form? A connection is a link, a relationship, between two things. A network is a lot of things connected. A hierarchy is then about levels of connectedness.

And normally the levels are ones of increasing scale. The small builds towards the larger scale. Cells make organs and bodies. Or alternatively the hierarchy is about a rising chain of command. Groups of soldiers are ruled by a sergeant, groups of sergeants by a captain, and so on.

At least this is the mechanical idea of a hierarchy – one constructed from the bottom upwards. Here we are going to take a very different logical view, one where hierarchies develop as a process of “1, 2, 3” .

That is they begin as a vagueness – a state of potential being. This initial fuzzy one-ness develops by a process of dichotomous separation towards a twoness. Then out of the mixing of the twoness comes the emergent threeness of a complex hierarchical system. We end up with a sandwich that has three logical components.

We shall see that this organic hierarchy has some special features. For instance, it is inherently dynamic – it is always expanding.

The middle ground of this hierarchy is also scale-free. The dichotomous separation creates divided scale and then the free mixing of what has been divided creates an internal – scale free or fractal - axis of symmetry. Features erupt over all scales in powerlaw fashion and so levels of scales become homogeneous and isotropic – lacking in distinctive scale to an observer.

And of course this organic hierarchy has holistic order. There is downward causation – top-down constraint as well as bottom-up construction. The global scale is as fundamental as the local or atomistic scale.

So the organic hierarchy is quite different from the conventional mechanical hierarchy. But before we get into describing it in detail, let’s further sharpen our impression of what the mechanical idea of a hierarchy is all about.

mechanical hierarchies

Hierarchy comes from the Greek words, sacred and rule. The term was in fact coined in medieval times by theologians to describe the three ascending ranks of angels. But the idea itself had long been a familiar one.

It could hardly be otherwise in a world where since ancient times societies were composed of hierarchies of slaves, citizens, noblemen and rulers. Being complex systems themselves, human civilisations quite naturally fell into this fundamentally logical mode of organisation!

So the general notion of a hierarchy is familiar enough. Things are grouped according to size or importance. In a network, every element is the same size and evenly connected. More like a democracy perhaps. But in a hierarchy, things form an ascending order of scale and connectedness. Generally speaking, small simple stuff gets constructed to make things that are increasingly larger and more complex, with new properties or authorities.

It is standard in some traditions of hierarchy theorising to divide hierarchies into two types – nested and non-nested. But actually they don’t seem that different.

A nested hierarchy is where there is a grouping of stuff within stuff. A process of interpolation. So cells are used to make organs and organs to make bodies (and bodies perhaps to make populations or societies). Each level of organisation is nested within a larger level of organisation.

A non-nested hierarchy is more like a chain of command. The usual example is an army where each of the components is a human individual, and what changes is the rank – the extent of their authority and responsibilities. So we have privates, corporals, captains, majors and generals. The general clearly does not include all the other ranks inside himself as parts of his own body. The system seems non-nested,

However this is a weak kind of distinction for reasons which should become increasingly clear. But already we can say that an army consists of squads, platoons and regiments, so the general is in some sense made up of his men. As a set of power relations – a realm of semiotics – an army is a nested hierarchical system.

There are some other issues that bedevil the usual mechanical view of a hierarchy. A big question is whether the causality of a hierarchy is bottom-up, with each new level being emergent upon the last, or whether there is also a top-down causal drive?

A bottom-up causality is the one usually favoured by reductionist thinkers. For example, there is the hierarchy of science itself. Physics constructs chemistry which constructs biology which constructs psychology. Or in set theory terms, we could say [physics [chemistry [biology [psychology]]]].

In this view, all the causality resides in the bottom level, the realm of the fundamental. The smallest and simplest stuff. So physics is the set of causes and chemistry is an emergent collection of effects. And even within chemistry, there is the same bottom-up causality. H₂O is the chemical structure and liquidity is what happens when you get a bunch of water molecules together. Enough quantity creates a new quality, enough substance produces a novel form.

This view of causality of course creates all manner of paradox. It leads to age old problems such as how mind could emerge from matter. Or having emerged, how it could then serve any purpose. If consciousness is the result of neurological complexity, how then can consciousness have any causal impact on the shape of those neural processes?

It is as if the mind is the tune of a player piano. A roll of slotted paper mechanically determines the striking of each note. A beautiful melody is heard. But the tune has no causal role in the process. It does not play the piano in any sense. It is merely “output”.

However even the ordinary mechanical conception of a hierarchy can overcome this problem by allowing for holistic, cybernetic, heterarchical or otherwise top-down flows of causation - a downwards pressure of constraints to match the bottom-up tide of construction. In the holistic view, the whole shapes the parts even as the parts are combining to make the whole. There is an interaction across scales of organisation so that higher and lower create the organisation of the system between them – synergistically or mutually.

A general and his army is an example of this kind of holistic order. The general commands and his army acts. There we clearly have top-down causality. But if we look closer, we can see that the situation is complex. There is a hierarchical decomposition of his commands – they are broken down across a succession of scales from broad general instructions to detailed local ones. A command from on high to hold the left flank becomes eventually a host of particular commands such as dig that foxhole, or patrol that wood.

And for an army to act, the causality also has to build up from the parts to the whole. Many soldiers will try to do many things and eventually a general will have to respond to what he learns of a developing situation. The left flank may or may not be held.

The story for conscious brains is much the same. The brain as a whole can form its broad aims. These feed down to shape local actions. A conscious (or rather attentional) desire to cross a room becomes broken down into the set of subconscious (or habitual) steps needed to make it happen. Meanwhile information is also feeding upwards to shape those broad plans. Our foot may land on an unseen banana skin and new global priorities will capture our attention.

Holism is about a dichotomistic interaction across scales of order. In a complex system, the local and the global, the broad aims and the fine detail, get pushed further and further apart. And yet remain smoothly in balance, able to move together to achieve cohesive outcomes.

An army of course is in practice usually only weakly cohesive. Certainly in the heat of battle its internal communications are quite poor. A brain operates with seamless efficiency by comparison, blending attention and habit, general intentions and routinised actions, with practised ease (though the existence of a “machinery” is exposed by psychological effects such as change blindness).

Anyway, the point I want to make here is that there is a fair amount of conceptual confusion about hierarchies and the causality they represent. Obviously a hierarchy is a way to get complexity out of collections of simpler components. But the deeper principles of hierarchy theory – a complex model of causality – are not clearly understood even among cyberneticians and system scientists.

the story of water

Right, we now have a pretty clear idea that the organic hierarchy involves three components – an upper and lower boundary and then a flat middle ground sandwiched in-between the two developing extremes of scale. Let’s continue by considering the story of the boundaries in more detail. And we can do this in two steps – first by describing boundaries in static, mechanical terms, and then by taking the proper dynamic and organic view.

Think about a glass of water as a system. Every system must be constructed of something – some substance. There will be some finest grain of stuff. Water for example is built of water molecules – two hydrogen atoms glued to an oxygen atom.

A water molecule is of course a form as well as a substance. An H₂O molecule has a structure or organisation that gives it its particular properties. As school chemistry textbooks tell us, the key to H₂O is the way the negative oxygen atom binds to the two positive hydrogen atoms in a V with an angle of 109.47°. This creates a dipole molecule, slightly polarised and thus able to form weak attractive bonds. The exposed negative rump of the oxygen molecule can become the fleeting attachment point for the hydrogen atoms of one or two other H₂O molecules.

In fact the idea of a rigid V-form molecule is an exaggeration. Water molecules are always breaking up and reforming. They are continually splitting into H+ and OH- fragments, which is why water readily forms acid or base mixes. Even pure water is dissociating constantly on a sub-millisecond time scale.

The V-form shape is also dynamic – an average state rather than a fixed alignment. If we could see water molecules on their own local scale, they would appear more as a blur of activity. (And that is with kinetic movement, before we get into the issue of quantum fuzziness.)

But anyway, for the moment we are talking mechanically here. So for the sake of modelling, we will treat an H₂O molecule as a fixed, cookie-cutter, component, a rigid Lego building block. Any two molecules will be identical and interchangeable. They will be granted existence – once made, they need no further causal underpinnings. They are not a feature like a whirlpool that needs continuing forces to persist. And the form of the molecules is sufficient to explain their causal properties.

There is the dipole V-shape as we have said. There are also some other important properties such as the fact that water molecules spin easily. This is because the oxygen atoms are 16 times heavier than the pair of hydrogen atoms attached to them. A combination of free spinning and the drag that the molecules exert on each other through their fleeting bonding naturally leads to vortex-like behaviour.

Of course while all water molecules are exactly alike, they also differ. The mechanical – or atomistic - view says they have fixed identities, but can vary individually in certain qualities like their speed and their direction. One molecule could be going slowly north, another rapidly south-east. So their local properties – the ones inherent in the shape or form of a molecule – are fixed, making them cookie-cutter components. But their relations to the wider world – their global dynamics - are free to vary. Local and global properties have this critical (need I says dichotomous?) difference.

Anyway, back to our glass of water. We can start with the idea that "water" is made of some local stuff. There is a smallest grain of causality. We can call this the local limit or lower boundary of the system because it does not make sense to break things down any further. Break an H₂O molecule into its individual atoms and we will lose sight of the essential properties that come from its form. So there is crisp lower limit to the causal story. Then from this lower boundary of stuff we can begin to construct. As we put together H₂O molecules, we start to get emergent behaviour – the appearance of a system with global properties.

Now how many H₂O molecules are enough to make “water”? This is a version of the Sorites paradox (or what today we would call the domain of nanothermodynamics). The Sorites paradox asks how many grains of wheat does it take to count as a heap?

But a heap of wheat is really always just a heap. The question of how many water molecules is rather more meaningful because at some point a sufficient number of H₂O molecules will undergo a phase transition to form a body of water. There will be enough interactions to create a wetness, viscosity, surface tension. boiling and freezing point, pH balance, temperature, density, pressure and the many other global properties that water may have.

We can see here that we thus have both a lower limit – an atomistic grain – and also an upper limit – that which marks the appearance of a global order. It does not make sense past a certain point – a certain scale! – to seek the local causes of a system, But equally, past a certain scale we can safely say that all the global properties have also kicked into place.

A droplet or a bucket or an ocean of water all have the same essential liquid character. You cannot make their water more watery by adding yet more local substance. Once there are enough H₂O molecules assembled to effect a phase transition (and nanothermodynamics will tell us how many one day), then the global form exists. There will be the steady, prevailing, persistent, equilibrium, state of order that we call water.

So we can see that the upper bound, the global limit, is a concrete kind of thing. There can always be more water, a larger body of substance. But there is a limit to watery-ness, a limit to the global form. The next question is whether this global limit exerts downward causation? Does it seem concrete because it is the whole that also shapes the parts?

downward causation

In the mechanical view, all causality is bottom-up. The H₂O molecules are the fundamental scale of causality in the system, the global watery state is merely an emergent outcome – an effect. Wholes are merely supervenient – the sum of their many parts. Yet still we can see how there is also top-down constraint acting here.

Let’s help matters by imagining a glass worth of water floating freely in space. And then being a little more unrealistic, let’s grant that outer space is warm enough for this blob of water to stay liquid rather than to first boil away with its own internal heat and then freeze into a scattered mist of ice crystals. So perhaps this experiment is taking place inside an orbiting space shuttle.

Surface tension – an emergent effect due to the weak attraction of the water molecules for each other – will curl the water up into a sphere. The sphere will impose an internal pressure and density. The molecules inside will become spread out with an average motion and location.

And importantly this sphere is a global form that is “large” both in space and time. We can easily see how it extends to either side of any individual H₂O molecule in spatial terms. But the globally prevailing state also extends “to either side” in terms of the passage of time. Just like relativity (and relativity is a global scale view!), we always have to treat scale as a spatiotemporal phenomenon, not just a spatial one. The mechanical world is just a succession of instants, but the organic world has "thick" time.

Anyway, once the blob of water has self-organised into a steady sphere, it will have developed an equilibrium, an ambience. It will have a rolling weight that carries it forward as a system. We can say that the system has a memory (it is marked by where it has been) and it shows anticipation (its current state now also predicts its future state).

We can test this by injecting some new H₂O molecule into the sphere, travelling with arbitrary speed and direction. Quite quickly, as the new molecule bumps and collides with the other molecules, it will also fall (or rise) to the sphere’s average level of behaviour. Energy exchanges will on the whole tend to cancel out the extremes. So the system’s ambience – its constraining effect - surrounds the individual molecule. The ambience exists before any particular localised action and also persists further into the future. The general state of a system is larger in scale both in terms of space and time, in both its extent and its duration.

Of course, an ambience can be disrupted – rewritten – with enough local change. Add enough hot or cold molecules and we might blast the sphere apart, or tip it into freezing. This is no problem because we have already agreed that the local substances can and do construct the global form. There is this kind of bottom-up causality in an organic system. But what we want to now acknowledge is that there is also a concrete global limit that exerts a downward constraint. The whole does also organise its parts.

So the causality of any system is not simple but complex. Or dichotomised at least. The shift from a mechanical conception of such a hierarchy to an organic one lies in recognising that the global limit is as fundamental as the local. And that global is a spatiotemporal or diachronic idea. Too often global is treated as merely “large” – large at some particular instant in time. Or the synchronic view.

Once we begin to understand the nature of local and global limits – here so far described using a pretty mechanical model, a sphere of water – then we can move on to a fully organic model of a hierarchically-organised system.

But first we should say something about the middle ground that forms between the boundary limits of a hierarchy, the realm sandwiched between the upwards causal thrust of local construction and the downwards causal pressure of global constraint. And to talk about middles, we need to branch off into the realms of thermodynamics. And thus to statistics.

The mechanical world view is based on the idea of closed and static systems. Hence they have middles best described by Gaussian statistics. Organicism, by contrast, assumes that we are dealing with fundamentally open or dynamic systems. And their middle grounds are characterised by a different "natural" statistics – one that we can call fractal, power-law or scale-free.

So this thread on hierarchies must take a break first to spend a page on an organic approach to thermodynamics and statistical mechanics, then a second page on fractals and power-law distributions. After that we can return to the specifics of organic hierarchies.

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