**Big Emergences :**

- The emergence of life from an ambient "soup" of inanimate molecules.
- The emergence of consciousness from a cranial soup of neural networks and other brain parts that are not conscious
- Computational emergences - Cellular Automata :

Examples :*Life*by John Horton Conway ; many examples in Stephen Wolfram's*A new kind of science*. The general idea is a simple algorithm, repeated application of which leads to totally unexpected "emergent" behaviours and patterns. There are well known examples from (computational) biology - along the lines of, simple postulated individual interactions from which emerge coherent behaviours of schools of fish, flocks of birds etc ; simple rules of growth and development which lead to emergent morphologies of coat patterns etc on animals. - The emergence of a coherent regulated society from the (more or less) free participation of millions of individuals.

Both the big and computational emergences have a kind of implicit payload of rabbit-out-of-the-hat magic and rare privilege - events that you are unlikely to see very often and only in special staged situations, such as at the primordial birth of life, or of consciousness, or on running a computer program like Conway's Life or on occasional witness to the impressive synchrony of a flock of birds or school of fish. But they are (in my view) no different to everyday emergences...

**Little Emergences :**

- 8 people sitting at a table at morning tea discussing a single topic, one speaker is speaking. One topic and thread emerging from a complex of 8 different people.
- an iron atom : 56 protons and neutrons sitting at a table at morning tea discussing a single topic - how to be an iron atom. One coherent and very useful metal atom emerging from 56 unruly nuclear personalities.
- a well designed computer program interface, marshaling millions of pixels and bits of information into a simple coherent metaphorical world with which the user interacts.

**Unifying Small, Large (and Computational) Emergences : Dimensional Reduction**

My idea for a metaphysics of emergence, is a factorisation into

(1) an input factor space (I) of relatively high dimension

(2) an output product space (O) of relatively low dimension

(3) mediation of the transform from input space to output space by a complex (i.e. complicated) intermediate factor space (N) of negative dimension.

So that we may write formally as :

I x N = O

with

dim(I) + dim(N) = dim(O)

In words - an input space of very high dimension is formally composed with a negative dimensional space to yield an output space of low dimension - the dimension of the product space is the sum of the dimensions of the two factor spaces.

For a visual metaphor - we might think of the intermediate space as being like a lens, which transmits and transforms a coherent input scene into a transformed coherent output image. All three of the elements of this metaphor have themselves internally a coherent space-like structure, and with a formal composition of the lens space with the input-scene-space generating a coherent but transformed output product space.

Taking as an example a small emergence - conversation at morning tea - the input space has dimension 8, which is the dimension of the conversation space if all 8 individuals talked independently and simultaneously ; the output space has dimension 1 - the single coherent conversation that occurs. The dimensional reduction is mediated by a relatively intricate set of social mores, hierarchies, and behaviours - but which in this metaphysics we will notate and size as a negative dimensional space of a certain dimension - in this case -7.

A living creature enjoys a coherent high-level existence within a space that is quite distinct from the melee of chemical reactions that it is supported by. Furthermore by comparison with the lower substrate - it is a much *simpler*, lower dimensional space. High level laws of behaviour, nutrition, reproduction operate at this level. Just as in conscious life, our brains build for us an experience of a seamless space of streets, trees, flowers, colours, selfhood....that is both quite distinct from the substrate of our brain and the actual physical activity of the external world, and also vastly simpler.

But what is the point of introducing the mathematical fictions - the negative dimensional space , and the abstract composition of spaces ?

The main idea is that this analysis is philosophically useful - but also with a hope for empirical potential in that it suggests a way to predict or measure the dimensional size of these internal engines of emergence. As a philosophical guide, the analysis recommends that, where we see simple ("low dimensional") behaviour or structures emerging from a complex ("high dimensional") input substrate, then we should expect that this emergence will always be mediated by a *complex* intermediate structure,with a (negative) dimensional "size" almost as high as that of the input substrate.

And as an engineering guide, the analysis recommends that (for example), where we wish to engineer software interfaces or societies that are simple and coherent, we should in general expect that the internal engines of organisation that generate these interfaces and societies will be complex and messy. And vice versa we should perhaps be wary of simple and elegant internal engineering data and object models and political ideologies - according to the metaphysics-of-emergence formula, this internal elegance will be achieved at the cost of emergent societies and software interfaces that are *less* coherent and *more* complex.

So what does the complicated intermediate negative dimensional space N look like ? It is no more possible to visualise a negative dimensional space than it is to visualise other mathematical inventions such as negative numbers and potential energies : but we can say that it encapsulates a forest of dimension-reduction-engine-room internals that actually get the job of dimensional reduction done...constraints, relationships, surfaces, volumes, intersections, knottings and braiding of dimensions, dynamic censoring and suppression of dimensions...that our new metaphysics parlays into a space-like structure.

Philosophically this analysis is in contradiction to the cellular automata view - it encourages* *us to look for *complex* machinery underlying simple emergent phenomena, rather than *simple* machinery. (Obviously this particular antagonism will require some explication in view of the long history and many examples and proponents of cellular automata explanations of emergent phenomena).

Freeman Dyson wrote an essay entitled "Why is Life So Complicated". He meant, why is the machinery of life so complicated. He writes *"It seems to be true, both in the world of cellular chemistry and in the world of ecology, that homeostatic mechanisms have a tendency to become complicated rather than simple"*

My answer would be - the machinery of life is complicated because

(1) the non-living world (I) with which it interacts is complicated : dim(I) is large

(2) life itself (O) is - almost tautologically - simple : dim(O) is low

(3) the machinery of life (N) is therefore mathematically required by the metaphysical emergence formula to be complex since

dim(N) = dim(O) - dim (I)

Life is complicated because there is a higher level metaphysics governing emergence, and for the same reasons that societies are complicated, the laws of nuclear physics that make iron atoms possible are complicated etc.

The simplicity of life itself - as opposed to the machinery that supports life - is perhaps the key part of this equation. One of the striking things about a living creature is its coherence and unity of purpose - how so many intricate parts mesh into a coherent whole that....swims, runs, searches for food, mates....has a being. But this is another way of saying that life itself - the end product of all of the machinery - exists in a rather *low* dimensional space. Indeed one way of explicating "being" is that it is a space of dimension 1 - this is the dimension of the thread of existence that we envisage for ourselves, stretching linearly back into the past and into the future.

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