Friday, 2 December 2016

Irrreducibly Complex Society, Biology & All That Mathematical Jazz


In response to my recent blog on evolutionary emergence and the fecundity of bottom up localised decisions, a friend, and excellent blogger, Tim Reeves, shared his thoughts in the shape of his mathematical spongeom sketch of evolution, and in the shape of a comment in which he claimed that:

"There is no way to distinguish between right and wrong in the trial and error computation which ultimately must make the appropriate selection, except to say that top-down transcendent constraints skew the statistics in favour of certain classes of outcomes."

As Tim rightly says, evolution requires a physical substrate of active information on which to run, and a constraint of the physical laws to do so (this is something I've written about in more depth before). However, I don't think the comparison is the best objection to the social elements I covered in the evolutionary emergence blog. Let me explain.

What’s happening in biological evolution underneath that layer is that there is a huge biochemical morphospace that has a connected structure through which evolution’s reducible complexity can traverse. Take, for example, irreducible complexity and reducible complexity - they refer to the arrangement of stable organic structures in evolution’s ‘morphospace’, but they cannot most primarily be understood at the level of the organism, because morphospace is not an adaptive landscape where we visualise the relationship between genotypes (or phenotypes) and reproductive success, and model fitness on the height of the landscape.

Morphospace is a mathematical model of the form of connectivity between patterns – so a reducibly complex morphospace means that the biological structures that populate the evolutionary landscape form a connected group. The notion of a gigantic sponge made up of very tiny fibrils that connect the evolutionary structure together sits fine with me. If the connection has no broken off parts then the random walk of evolution can move across the whole structure.

In fact, this is a particularly good illustration because sponges are composed entirely of mobile cells which can move about between different layers of tissue and reallocate themselves to take on different tasks. Sponges have totipotency, which as you may know, is the ability of a single cell to divide and produce all the differentiated cells in an organism. This allows any fragment of a sponge to regenerate into a self-sustaining organism.

A good analogy for markets?
I'm afraid though, the analogy of every tiny fibril representing the complete structure of the entire world’s biochemistry causes me problems when we get onto markets. You have to remember, no one is saying markets don't evolve without a top structure. What is being said is that activity at a local level is, just like natural selection, producing complex and sophisticated outcomes that look too well cultivated to have occurred by Smithian localities, and there we find a good analogy to the 'naturalism' of biological evolution.

You’ve probably heard of the term irreducible complexity in relation to the debate about intelligent design. The ID debate uses a fairly rudimentary definition of irreducibly complexity in relation to the evolvability of organisms (as per my definition above). At a deeper level, at the level of computation, if a system can only exhibit the full extent of its output by running it then we can say that system is computationally irreducible. Obviously we humans cannot understand the full implications of this when it comes to the entire universe, but the universe is a nexus of activity which has an underlying story that we may define as being computationally irreducibly complex.

What this means is that to simulate a like for like model of the universe would involve computation of universe-size proportions, just as to simulate the whole history of biological evolution would involve computation of the whole 4.5 billion years of biological evolution.

Extending the analogy to society, consider human history (let's call it H) as the sum total of everything that has happened in the vast search space, just as morphospace is the totality of the search space in the biochemistry that facilitates evolution. If we imagine H to be a computational model representing every piece of data and information linked to humans, then we can say that H is computationally irreducible, because H is locked together by every contingency (that is, all the constituent facts contained within H) so the removal of one fact would change H to something that isn’t H.

I realise that if you're unfamiliar with this kind of talk, notions like computational irreducible complexity are going to be quite conceptually opaque to the mind, so hopefully a simpler illustration will help, in the shape of what's called a magic square. A magic square, if you not familiar with it, is a configuration of numbers that give a beautifully succinct illustration. If you look at the square of numbers below, you'll see that the removal of one number from a global size magic square would change the whole structure. 

 1   35   34   3   32   6
30  8    28   27 11   7
24  23  15   16 14   19
13  17  21  22  20   18
12  26   9   10  29   25
31  2     4   33   5    36

What’s significant about the magic square is that its numerical structure totals 111 when summing columns, rows and the diagonals. When added together each sequence of six will obtain 111 each time. Change any one of the lines and the magic 111 will be thrown out. In other words, altering the configuration will disrupt the overall connectivity of the square. Mathematics has conceptual forms that entail magnificent symmetry, and the removal of any of those proprietary parts means it falls down like a house of cards in a gale force wind. 

Returning to human history (what we called 'H'), obviously H by definition can only be one computational set because the only way to change that set would be to produce an alternative history, and we can’t do that, because we are then talking about some other kind of H. Altering the configuration of the magic square will disrupt the overall connectivity of the structure – and that must be what the human story is like – you cannot change one constituent part without changing the whole. 

Obviously, constituent parts could be changed with seemingly no effect on the whole, but that’s not really happening – it’s only because we see reality through the tiny lens of our first person perspective that we don’t see the change to the whole.  Pretty much every word you speak to others, every decision you make, and every course of action you take has implications far wider than you can imagine. 

Let’s take a fairly extreme example to illustrate. Say you go back to April 20th 1889 and kill baby Adolf Hitler; it is obvious that the whole human story will be inexorably altered from that point onwards.  The 20th century would look unrecognisably different, even if at the time your act didn't appear to have global consequences. Using Hitler is only an extreme way of describing what would apply at smaller levels too.  Suppose you just want to go back in time and stop the little girl who lives down the lane getting run over.  Even that small intervention would have vast ramifications beyond your scope – you’d start a social butterfly effect that impinges on the rest of the human story. 

The H we call human history is irreducible, because the removal one component makes it something other than what it is. History tends to be viewed first off at a local level (one’s own history perhaps) and those local connections are woven into the standard cause and effect relations that make up the bigger picture. 

Just recently a young cyclist was knocked down by a car on the roundabout near where I live. With local introspection it is hard to imagine that this event had any real bearing in China or Brazil, but it would do – not immediately directly of course, but given H it must have an effect somewhere down the line.  Because things are not immediately obviously connected to us, each event in the global structure of H appears local and disconnected, whereas all events are actually woven together in an interconnected whole.  Clearly though not every event has global relevance – if I suddenly scratch my nose for 3 seconds that won’t have any bearing on China or South America will it (butterfly effect notwithstanding). 

I said that reality is seen through its of conceptual layers – well just as overall human history is incompressible, at the individualistic level the history of a person's life is the same – change one bit of it and it is no longer that unique system.  So in that sense even the rubbing of my nose is part of a unique personal history for me.  I am in my 40th year of being alive on earth, so my life's history amounts to around 14,500 days, or 350,000 hours, or 21 million minutes, or just over 1.2 billion seconds and counting. 

Of course we can work with a compression based on how much time I've been asleep in that time, or eating, or driving, but just like with evolution, the history of my life can only be fully analysed with the same 1.2 billion seconds computation time.  In other words, even if such a project were possible, to run a program of my life to see how my thoughts, feelings and emotions had changed and developed from birth to now would involve factoring in every single experience and influence and neuronal processing, and that's what I mean at a wider level by history being incompressible. 

To change one element of it means changing the totality of the whole thing. On a grander scale the world's history of the socio-personal is the same - to set up a computation mapping every event, the influence of those events on people, and the cognita processing those influences would take a timescale of the same length as the world's socio-personal history.

The perceived format of the magic square and the irreducibly complex nature of mathematics, the universe, evolution, human history, and whichever lens we choose to gaze through, provides us with a great metaphor for life – reality appears to us locally, and it can be extended into a much broader multilayered picture of interconnected stories – be they social, historical, biological, physical or mathematical. And what appear at the local level as a collection of separate and somewhat disconnected mini narratives actually weave a global pattern, which is itself embedded in an even bigger mathematical object that we only sparsely sample through this brilliant narrative we call physical reality. 

Now we get to the nitty gritty of Tim's question in relation to the freedom of markets and where it's beneficial for them to be artificially interfered with. As a reminder:

"There is no way to distinguish between right and wrong in the trial and error computation which ultimately must make the appropriate selection, except to say that top-down transcendent constraints skew the statistics in favour of certain classes of outcomes."

What you have to really consider is what in societal terms is the overall transcendental physical regime that constrains possible behaviours/outcomes? Society is constrained by all sorts of overarching authority figures (rule of law, regulations, mandatory exchanges of money, etc) but the question at hand here is not to deny their existence, it is to locate areas where they are excessive and counterproductively interfering in our beneficial transactions and our liberties (as per my recent paper on trade).

So to put it in algorithmic terms, the debate is about whether the algorithmic means by which the seek, find, reject and select computation is carried out in human behaviour better at a local level, or whether it is better imposed by authority figures. Given that local behaviour is concomitant with local knowledge and local incentives, it is fairly evident that there are many areas of society in which the authority figures are constraining the societal value (the consumer surpluses + the producer surpluses) at a level lower than would be the case without their interference.

Society is, of course, subject to physical constraints – most notably, energy, labour and knowledge – that limit the rate of progress and the directions society can take. But as we've covered before, many decisions made locally have knowledge-based and incentive-based advantages that top-down command decisions do not have (see here for example)

Appendix
As a bonus, if you don't mind ending on a tangent, the whole sponge is computationally irreducible notion also demonstrates what is it that the Intelligent Design school have got wrong about irreducible complexity about irreducible complexity being unevolable in biological morphospace. Using the fairly standard scaffolding example that is often posited as an illustration – once you've removed the scaffolding and the cranes from, say, a Gothic cathedral, and destroyed any knowledge of cranes, cathedral construction becomes irreducibly complex. If we look for examples of this in biology and use a comparison between heavy stones in cathedrals and cells in organs, whereas with the Gothic cathedral the stones are simply too heavy to have been hoisted there, there are similar occurrences in biology where if subsequent mutations remove the antecedents of symbiotic systems there is no way to logically regress the path of a complex evolutionary adaptation.

If one component was removed then it could not function, and this seems to be due to two possibilities: The proteins (or protein complexes) have been modified further since the addition of the hypothetically removed part. If such modification creates dependence on this more recent part then its removal would be detrimental. Secondly, as we’ve said, the structure evolved with 'scaffolding'. It is hypothetically possible that a structure could only be stable when complete, but that it could also be stable if another structure is present.  Remove this second structure, just as scaffolding is removed when building work has finished, and the first structure remains stable. However, remove a single part of the structure and the scaffolding is needed for support again, if it is not there the first structure collapses.  Both of these are ways in which irreducibly complex structures can evolve – but crucially they describe irreducible complexity in the biological patterns of morphospace, not in the computational ‘sponge’ whole. 

The problem with the IDists idea of irreducibly complex systems is that if morphospace does contain lengthy leaps that refute reducible complexity such large leaps taken would never be stumbled upon because they would be too computationally complex. You see, the Intelligent Design school is thinking of Irreducible Complexity in terms of subtraction of functional elements which leads to “instability” in the organism. But if we think back to our pattern of morphospace with the sponge, those concepts of “scaffolding” (and everything else in the biological patterning) will be embedded in such a complex configurational nexus that biological irreducible complexity could only be found in morphospace with a computation of the same length. So the IDists are barking up the wrong tree with their version of irreducible complexity.

N.B: Point of clarity: logical incompressibility is different from morphospace because it is dealing with a different heuristic. Logical incompressibility has to do with equations and algorithms used for data compression, although in mathematical terms it is true that physics effectively defines a set of stable structures in morphospace. The subject we have dealt with is tricky because we are looking at how mathematical patterns are configurations of laws over physical systems, and it is easy to confuse mathematical patterns with the physical systems they support. 
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