As mentioned in the previous posts, which should be taken as a highly recommended if not entirely required prerequisite reading for this post, Sciops can be viewed as a mathematical framework for understanding emergent structures, and their role in consciousness, meta-physics and science more broadly. The primary subject of this post will be to introduce the key ideas of this mathematical framework, especially as it relates to the brain and consciousness, leaving it to future posts to speculate about its broader role in scientific inquiry.

We’ll begin with consciousness, where we seem to have direct and privileged access to certain truths that can be used to establish the key premises for our framework. And given the idea that our conscious representations are a non-arbitrary, but ontologically fundamental entity (an idea that if untrue undermines any epistemological claim), then we can show that these premises, if correct, are key to understanding existence itself and the true nature of reality.

2.1.1 Conscious Structures Eliminate Redundancies

It seems introspectively certain that there are no true redundancies in our conscious experience. Any aspect of our experience either contributes to the quality of the experience in some way (qualia or phenomenon) or it is not experienced at all, but to be entirely redundant doesn’t seem to be plausible. For example, if we were to try to imaging a redundant visual component – if you were to imagine our visual experience as having 3D coordinates, and were said that there were redundant red visual elements at a precise location in this 3D visual space, this idea seems to be incoherent. If there was more red at a specific location in visual space, then either the color would need to change, or the brightness or intensity of light, or perhaps the size of the visual space that was red, or the length of time in which the red persists. But these are all different qualities of the experience that contain new information that can have causal influence. They are not simply redundancies. And the idea that there can be some literal number (eleven redundancies here, two hundred and five there, etc) of redundant reds at some point in space that has no effect on any of these qualities, this idea seems incoherent.

We can imagine the same with sound. If we hear a specific note corresponding to a specific frequency or harmonic, there is no way to imagine experiencing any redundant duplications of this phenomenon. The note could be louder or softer, it could change locations, it could be shorter or longer, it could include other frequencies, but there is no way to experience some number of redundant sounds that play no causal role and add no new information to the experience.

In fact, not only is there not one example in our experience of a redundant structure or quality, but the very idea seems inconsistent with consciousness. If some structure is entirely redundant and has no causal influence, then the experience of it has no meaning.

From this we can conclude that if consciousness can be described or defined by some sort of mathematical structure, that this structure does not contain redundancies.

Defining Redundancies

We can consider two structures to be mathematically redundant when they share the same mathematical structure, including their causal structure, such that the redundant information is not necessary to describe the system. For example if we define a structure with a set of specific elements and relationships, then if two structures are exactly equal, or if one structure is entirely contained already within another structure, and removing one of the structures would make no causal difference to the system we have defined, then one of them is redundant at least with respect to the system we are dealing with.

We can also differentiate total redundancies from partial redundancies. For example two structures are totally redundant only if their structures are exactly identical, and the structure can be fully described while removing all information about the redundant structure. Two structures are partially redundant if part of their structures are identical and some of the information can be compressed into one structure instead of two.

When we are dealing with sets of partially redundant structures, it’s possible to factor out the redundant parts of the structure (the redundant substructure) and isolate it from the non-redundant parts of the structures in terms of our mathematical description of the structure. In fact this is precisely the aim of compression, and concepts such as the Minimum Description Length (MDL) of a structure do approximately this.

We want to be very precise here when describing the implications on complete redundancy elimination for compression. If we are to look at some computational system, for example the brain, such a system could perhaps be explained at some sort of smallest, finest grain level: such as with QFT, or string theory, or whatever theory of everything might emerge. But this description of the system would not be the most compressed description of the system. A system that describes the most important interactions using much larger emergent structures such as neurons would be far more compressed. However that doesn’t mean that non-compressible lower level computations are non-existent, it just means they are not reduced to and absorbed by larger emergent structures.

It’s also worth clarifying that the compression of the entire system (in this example a brain) cannot be lossy – we must retain all information completely since presumably reality actually follows lowest level laws and this cannot just be discarded. Therefore, since we are describing the entire system with loss-less compression, it is mathematically true that only the most compressed structural description can contain zero redundancies, since any other less compressed structure by definition is explaining the same structure with additional unnecessary structural components. Therefore it follows that eliminating redundancies must correspond with the most compressed system.

2.1.2 Defining Sameness with Compression

In order to define any structure that is stable over time we need to be able to rigorously define what structural entities are the “same” structure. For example, if we desire to model the brain using a neural network consisting of neurons, and connections between neurons, with inputs that feed data into the same neural structure across repeated iterations, we need some way of rigorously explaining why Neuron A at time T1 is the same structure as Neuron A at time T2.

This task is trivial if we want to use sameness merely as a semantic tool – as a convenience for creating abstract but non-fundamental structures to make our work easier. But the task is far less trivial if we wish to say that the same neuron is actually metaphysically the same structure across different times / iterations: at a small enough scale it no longer carries all the same properties (in terms of atomic structure, quantum states, etc), so we need some sort of principle for proclaiming that two physically different emergent structures are actually the same structure.

It’s essential in explaining our conscious experience to be able to define sameness. To experience change or time at all, we must represent something as persisting through different states. Kant discusses this in his Critique of Pure Reason: “Without the permanent, then, no relation in time is possible”, “in bare succession existence is always vanishing and recommencing” and “Alteration is a way of existing that succeeds another way of existing of the very same object”.

Thus we need some sort of metaphysical principle that allows us to define two different physical event as ontologically the same structural entity or object. This is where we can use compression and redundancy elimination. If two structures are partially redundant, both in terms of structure and shared causal impact, then the redundant substructure can be defined as the same entity. For example if we have a neuron that persists over time, and the output of that neuron always has the same causal impact, then we can use compression to define the structure such that this neuron is the same entity over time.

In other words, if the most compressed description of the structure models a physical neuron as persisting over time, then we can hypothesize that it actually ontologically exists in that form, and contributes to consciousness in this form. And of course if this neuron always had the same input values and therefore also the same output values, then it would be fully redundant in this sense and could be compressed to just one single experience, but in the real brain this is never the case – neurons have unique inputs over time, and thus we can model the structure with minimal redundancies by modeling it as a single entity that persists over time but can have different input and output values.

In other words, if we have two structures that totally redundant and have no causal difference, then the redundant structure can be entirely eliminated/merged. But if two structures (for example Neuron A at time T1 and Neuron A at time T2) are only partially redundant, then we can describe its structure by factoring out the redundancies and modeling it as one neuron that can have different inputs and outputs over time. And we can speculate that our conscious experience will match the maximally compressed or minimally redundant structure, which can be closely described by the MDL structure or any other method for attempting to maximally compressing a structure. Note that such a syntactical description of a structure translates the structure into a form that can be described by syntax and therefor adds extra information for the purposes of fitting the form to that syntax, but of course the true structure doesn’t need to be translated to a language for humans to understand, it simply exists in the most compressed possible form.

The same principle can be leveraged to compress the countless molecular interactions that together make up an action potential and instead model the meaningfully causal part of the structure as one compressed event: the neuron fired or it did not fire. That is not to say that all the other information has no metaphysical existence or conscious component at all, but that all the other molecular and sub-atomic information is simply not converged into the larger and stable emergent structure of the neuronal firing, and that they play no significant causal role to downstream neural processing which can lead to further convergence.

And of course we can also leverage this principle to model several subsequent neural firing events as a firing rate and neural modulation as weights, such that we can instantiate emergent structures that correspond with these firing rates and are experienced as intensities, degrees and ratios (more on this below).

Perhaps most importantly, convergent (redundancy reduced) emergent events that have downstream effects can lead to further convergence. We will detail this very clearly in the subsequent sections, but just as many molecular interactions can be compressed to a neural firing, and many neural firings to a firing rate, and firing rates over time to the same neuron that exists within a neural network, we can also converge many neural inputs that feed to one neuron into one structure, and many of these neurons which lead to one behavioral output. Thus the convergence of structure via redundancy elimination and compression seems to be a powerful tool for binding. To better understand this however we need to understand the conditionality of these emergent structures.

2.1.3 Binding and Information Loss Across Convergent Structures Using Bayesian Inference

Once we have described any computational system, for example the human brain, as the most compressed structure, we can now consider which reference frames have emerged from this structure. Certainly the reference frames of conscious experience are not equal to the entire structure, which would simply contain all of the information in the entire brain, including at the lowest level, but instead we can define reference frames based on larger emergent logical structures that converge information. For example a single neuron firing could be considered a possible candidate reference frame, if many lower level interactions can be best compressed as a single action potential. Similarly if that neuron firing can be compressed into a firing rate, and a set of neurons firing converge to a single neuron that can be described as a logical gate (for example OR gate, AND gate, etc), then these become possible emergent reference frames. Therefore candidate reference frames emerge when information is converged into a larger unified emergent structure that eliminates redundancies.

It’s also worth clarifying that we cannot simply arbitrarily choose which emergent structures we think are reasonable reference frames for consciousness (ex a neuron firing, a set of neurons, a logical gate, etc). No these reference frames are determined by the computational system as it 1) converges information and 2) loses resolution into this information as things converge. We cannot simply model these emergent structures based on convenience, granting them information that they don’t have. Thus the framework described here is not one of convenient abstraction, but one that creates metaphysically real emergent structures with specific binding and boundaries.

One can imagine that simply compressing multiple fully redundant structures into one (by eliminating all redundancies) can create a more compressed structure, but doesn’t in itself bind together any new information, since in this case we are only binding fully redundant (duplicate) information. Thus any information that is non-reducible and has its own unique causal influence still seems to be a separate entity.

However, consider that we can compress many partially redundant structures such that they have the same internal structure (elements and relationships) and also the same causal influence (produces the same effects) but not necessarily that has the same causes (inputs). Therefore one emergent structure which has been defined based on compression now has the same causal output but contains information about unique non-redundant inputs. Our neuron A above across different times was a good example of this. We can compress the same neuron firing across different iterations as one mathematical structure (a neuron in a neural network with certain inputs, weights and outputs) that can be conditioned on different inputs.

Thus when we have partial redundancies, we can factor out and compress redundancies into one structure, which means that we are now defining multiple physical events as one emergent structure which now can be conditioned on, and merge information from, multiple separate non-redundant structures. And this is what allows multiple non-redundant structures to be bound together into one experience: because an unified emergent structure / reference frame is conditioned on them.

For example we could have many input binary neurons in layer one feeding into one binary output neuron in layer two, such that this output neuron’s effect is solely determined by it’s value (1 or 0) and not based on all of the possible combination of input values. However we can still ask the Bayesian question: based on each output value, what are the probabilities of all the possible input values? These possible input values are now bound into one experience based on our compression of the output into one structure, and the nature of each input value needs to reflect its conditional probability.

Another way of thinking about this idea is to ask the question: based on the value of this emergent reference frame (ex output of Neuron A is 1), what information do we know about all structures that causally influenced this output, and is thus encoded into our reference frame for Neuron A = 1? Crucially, any information that plays no causal role in affecting the output of Neuron A is essentially lost, from the reference frame of Neuron A. This is not to say that all events that are non-causal with respect to Neuron A are metaphysically nonexistent, they are simply not bound into the experience of the emergent reference frame defined by Neuron A.

Similarly, any events that play a small role in influencing the information in Neuron A will be seen as very uncertain with respect to the reference frame defined by Neuron A, since the knowledge of the event corresponding to Neuron A will not tell us much about them. We will discuss this in great detail below when introducing the concept of Resolution.

2.1.4 Structured Information Loss Defines the Quality of Conscious Experience

We have shown so far that:

1) If our conscious experience can be defined by a mathematical structure, it must eliminate redundancies and therefore be equal to the most compressed version of the structure

2) As a consequence, distinct physical events that represent partially redundant structures can define one unified emergent structure that is conditioned on (and contains information from) multiple distinct and non-redundant structures/events, which leads to binding of non-fully-redundant information

3) We can use Bayesian Inferences to calculate the conditional probabilities of all non-redundant causal events/structures, allowing us to understand which information is bound and which is lost and to what degree or resolution/certainty

One of the most profound and fascinating consequences of this is that information loss is rarely random, but tends to follow patterns that highlight certain aspects of mathematical structures while hiding others.

For example consider that many conscious experiences come in degrees of intensity or magnitude: we can hear the same sound become louder or softer, the same color or light can get brighter or dimmer, a smell can become stronger, pain can become weaker or more intense without changing quality. These are all examples, it seems, of a type of experience which increases not in terms of the kind or quality of experience, but the amount of that quality.

Interestingly, this concept only seems possible if we can define multiple physical events as the same event using compression, otherwise how could we have a larger amount or degree of the same thing? The only other solution would be if intensities of conscious experience came from some sort of sub-atomic quantity that also comes in degrees, but we know this is not likely: countless research shows that the experience of greater intensity (brighter light, louder sound, stronger pain, etc) correlates with more neuronal firing, not from some sort of microscopic quantum property. But as mentioned above, to make the metaphysical leap from sub-atomic interactions, to neural firings, to neural firing rates, we need to use compression to define ontologically real emergent structures.

And to be more specific, the experience of magnitude or intensity (ie a larger amount of the same qualia) must come from a convergence of information such that a downstream structure retains information into the number causal inputs, while losing any information that differentiates them. For example, if our neural network is such that five inputs firing corresponds to louder sound than four neurons firing, which are louder than three neurons firing, but all with the same quality of sound, then this tells us that the emergent reference frame that defines our experience has lost any information that differentiates these physically distinct neurons, while retaining information about how many fired. In other words, any differences between the inputneuron fires (their timing, their physical location, their unique connectivity to other neurons) has no causal impact on our reference frame, while the number of input neuron fires does have a causal impact. As a result we experience more of the same thing, rather than a larger number of distinct events.

The same principle would be true if magnitude is encoded not by separate neurons firing but by the same neuron firing faster: these are still physically distinct events that need to be compressed such that the number of events is causal, but the differences between them are not.

This is fascinating because it is our first example of showing how compression and information loss can affect the quality of our experience: instead of experiencing five vs four vs three distinct sounds, we experience one sound that is louder or quieter.

We can now start to discuss more generally how compression can shape different types of structural relationships and consequently phenomenal qualities.

2.1.5 Defining Structures with Elements and Relations

When discussing emergent mathematical structures we need a more broad framework than to deal with any set of numbers (reals, naturals, etc). Instead we will define our emergent structures using Elements and Relations between elements.

Let’s define an element as an empty structure such that information arises not from any intrinsic properties of elements but from relations between elements. As we saw above with magnitude and will see below with more examples, relations are defined based on how information is compressed.

In order for relations to exist between elements, an emergent reference frame must be conditioned on these elements, such that there is some reference frame that binds them together, and the nature of the information loss dictates the type of relations between said elements.

Practically speaking, we typically define elements based on neural activations though of course our framework is more general than this and can be applied to any sort of causal system.

The other terms worth defining immediately are Object and Structure. An object refers to any bound set of elements and relations between them – meaning they are bound together into one experience by some emergent reference frame. A structure refers more broadly to a set of elements and relations (or objects) that correspond to some causal system or subsystem, but are not necessarily all bound together.

The Magnitude Relation

We have already introduced one type of relation in our discussion above: magnitude. If we have an emergent reference frame conditioned on multiple elements such that the number of elements is retained in high resolution (ie is causal and non-compressible) but the differences between each element or object are compressed away (non-causal or minimally causal). The effect of this is that instead of experiencing the elements or objects as distinct events or qualities, we experience them as simply a greater degree of the same event of quality.

As with many other relations, we can think of the magnitude relation in the abstract form: as a relation that can scale indefinitely as we add more entities. When we think of a relation in the abstract, imagining adding more and more elements in succession with the same relation, we will sometimes refer to this as a Dimension. Thus a structure with magnitude 14 for example can be said to have a value of 14 on the magnitude dimension.

The Identity and Non-Identity Relations

We could also imagine compression such that all elements bound together into an emergent reference frame are fully redundant and are simply therefore compressed together as identical to each other. Therefore in this case if we wanted to talk about multiple elements (for example to relate each element to a neural input) then we would say that the elements are simply identical to each other.

The opposite might be true as well: we can imagine a structure such that what is preserved is simply that the elements or objects are not identical. In this case the elements have different effects, but all other information about them is low resolution or non-causal.

The Order Relation

Perhaps the most important and ubiquitous type of relation between elements and objects in our experience is Order. When structures have a well defined order of elements, the implication is that the order itself is causal and potentially high resolution. In other words, if we had for examples four elements bound together: A, B, C and D, then a structure with order A,B,C,D may have a different effect than a structure with order B,A,D,C, which may be different yet from D,A,C,B, etc.

We see order prominently across many conscious experiences: it is crucial to visual experiences where the order of colors, lines and edges is critical to detecting visual objects, but it’s also present in the idea of space itself, as well as time. Spatial points are ordered next to each other, and temporal points are ordered one after another. All of these experiences can be defined by specific structures, but they all share the order relation as a crucial building block.

The easiest order to visualize may simply be a one dimensional linear order. If an emergent reference frame is conditioned on a set of linear ordered elements or objects, then the experience of this reference frame must in some way contain a sense of certain qualities next to other qualities along a linear dimension.

With order alone, we don’t necessarily have any asymmetry. Consequently, experiences that seem to be linearly ordered but asymmetrical (for example pitch) must explain this asymmetry with some other type of relation.

As mentioned above, in order for the order relation to exist it must be instantiated by compression and it can only do so if order is causal. Simply ordering neurons in physical space (shown below) will not create a true structural order. If on the other hand we use something like a Convolutional Neural Network that explicitly applies transformations to a set order of inputs and only that order, then the order of elements can be potentially preserved and causal. Similarly any sort of positional embedding in a neural network gives the model the ability to causally preserve order information, though this is not necessarily guaranteed as certain information about order could be discarded or ignored in deeper layers.

Of course linear orders are just one example of a type of order. A circular order is the same as linear order, but the two ends of the line are connected such that the first and last element are connected. Branching orders can apply the same concept to graph structures where each element could have a varying number of neighbors.

If/Then Relationships

We discussed above how partially redundant structures can create emergent structures that are unified in terms of structure and effects but not in terms of causes. We can also formalize this as a relation between the emergent structure (which we can represent itself as an element or object – often termed the “Object”) and the inputs it is conditioned on (also elements or objects – often termed the “Conditioned Elements”).

With binary inputs we can abstract this relation to an if/then relationship in its simplest form, such that if neuron A fires, then neuron B fires:

If A then B

Interestingly, from the perspective of compression (rather than pure mathematics) if the causation here is bidirectional, then it seems we can compress the relationship tothe identity relation, since if A then always B and if B then always A, therefore (unless the difference is defined some other way) A = B, which makes A and B redundant and can instead just be modeled by one element or object, which we could name AB.

However, if the causation logic is unidirectional, then A is not equal to B, since for example if A then B but not B then A – this implies that B must be able to exist without A. The implication here is that either B must be able to spontaneously exist without cause, or it must be causally influenced by some other object or element in addition to A.

The latter can be modeled in its most simple form as an OR gate: for example we could say that B if A OR X. Therefore it is true that if A then B and if X then B, and therefor if A OR X then B, but not if B then A or X.

Modeling the system as an OR gate comes with a compression tradeoff: from the perspective of the emergent structure of the OR gate (and the relationship it instantiates between inputs and outputs), we lose some information about which input (A or X) actually fired in each iteration, which means less of the information from A and X is merged into Neuron B. However some of the information from A and X is still merged into B, since if B fires we know there is approximately a 50% chance that A fired and a 50% chance that X fired. But what we gain from this tradeoff is that we can now model Neuron B as just one structure, that always has the same causal impact, rather than two separate structures (AB or XB). Ultimately whichever description is a more compressed description of reality that better reduces redundancies will be the one that closer matches what ontologically exists and is experienced consciously.

Stacking Convergent Structures

One can imagine of course that we can stack successive OR gates in our neural network (or logical gate network) such that we may have something like:

if A OR B then C

if C OR D then E

Here it is important to realize that since what is consciously experienced is the most compressed version of the structure, with no redundancies, we would never have any emergent reference frame that experiences both of the above logical expressions in this form, since there is a redundancy present, and both lines can simply be compressed to…

if A OR B OR D then E

This principle ends up being quite consequential: we can imagine a neural network or logical gate network that for example has twenty layers of logical gates, which could be described using twenty successive logical expressions. However the inner conscious experience of the last and most convergent layer, ie the structure as viewed from the emergent reference frame of the last layer, would match the most compressed single logical expression. Consequently, large feed forward computational systems that can be broken down in terms of several computational steps or layers, are often simply experienced as one experience in one moment – but more on this in the next post when we discuss the structure of time in great detail.

2.1.6 Resolution & Binding

Let’s now walk through a few examples to help us calculate precisely how compression binds structures together, and how uncertainty, calculated conditional probabilities, shape the boundaries of those bound experiences.

It’s useful to begin again with the example of an OR gate. Let’s imagine a very simple OR gate such that we have three neurons: Neuron 1A, Neuron 1B, and Neuron 2 where Neuron 2 = Neuron 1A OR Neuron 1B.

Without compressing any neural structures together that are distinct in physical space time, we may have the following computations across time (just as an example):

• Neuron 1A fires > Neuron 2 fires > Downstream effects from Neuron 2 occur
• Neuron 1B fires > Neuron 2 fires > Downstream effects from Neuron 2 occur

Now if we did not yet merge the two instances of Neuron 2 together, we really have what looks like two identity gates where Neuron 1A = Neuron 2 at particular place in spacetime (because it and it alone causes Neuron 2 to fire in this instance) and similarly, Neuron 1B = Neuron 2 in another instance in spacetime. Thus we would compress these together such that Neuron 1A is bound together with Neuron 2 and so is Neuron 1B with a different Neuron 2.

However this is not necessarily the final most compressed result if this logical circuit runs many times, because in order to describe the system with each instance we would need to specify which Neuron 2 fires, despite the fact that the downstream effects are always identical.

Instead we could merge the two instances of Neuron 2 into one structure, an OR gate, that fires when either 1A or 1B fires. But now we need to apply Bayesian Inference, and consider which information is known with what level of certainty from the emergent structure of the now merged Neuron 2 OR gate structure,

If we assume that there are three possible inputs:

• 1A = 0, 1B = 1
• 1A = 1, 1B = 1
• 1A = 1, 1B = 0

Then the most simple way to calculate this certainty is to say that from the event of Neuron 2 firing, both 1A and 1B appear with 67% certainty, and therefore 67% resolution.

We could also consider that perhaps it is very unlikely that both 1A and 1B fire in close enough succession to consider both to be causal to 2, in which case the possibilities are reduced to:

• 1A = 0, 1B = 1
• 1A = 1, 1B = 0

And now the certainty and resolution of each is 50%.

We can now ask ourselves: what does it mean for an element or object to be experienced with 50% resolution?

Actually this concept is very intuitive for us. If you wear glasses or contacts simply take them off and look in the distance. And even if you have perfect vision, simply ask yourself about the visual entities in your periphery. In both cases we have some information about the visual objects we experience, but they appear blurry and uncertain. The same is true of a smell or taste we partly recognize but can’t quite place, or a set of musical notes that sort of blend together.

In fact the concept of resolution, uncertainty, or as Lee calls it “precision” has been discussed in other papers, though not formally connected to Bayesian Inference and conditional probabilities of emergent structures as Sciops has done here (Lee 2021).

Assuming a firing threshold of 1, and that any additional activations once that threshold is met are redundant (and thus multiple inputs = 1 is not a valid input), the resolution of an OR gate as it pertains to each input can be approximated as follows: 

R_I = 1 / T

Where I = a specific input and T = the total number of inputs

So in the example above, the resolution of Neurons 1A and 1B are each 50%. 

It’s also worth noting that since compression will always collapse any structure to reduce redundancies we can have many layers of logical gates or neurons such that the latter layers can have resolution into the earlier layers.

For example if we have

• 2A = 1A OR 1B
• 2B = 1C OR 1D
• 3 = 2A OR 2B

Then it follows that the structure of 3 will have 25% resolution into each 1A, 1B, 1C and 1D:

• 3 = 1A OR 1B OR 1C OR 1D

One can imagine how a few consistent patterns will tend to emerge in complex neural networks. First, our largest emergent reference frames (emergent structures that have converged the most information and contain the most information) will tend to be found downstream towards the latter layers and outputs of our neural network. Second, those large emergent reference frames will tend to lose resolution of earlier and earlier inputs since at each layer we tend to have some information loss and uncertainty. In the example above, 3 would have 50% resolution into both 2A and 2B (if they were their own objects) but only 25% resolution in 1A-1D.

As a consequence we would expect our most significant emergent structures, and therefore most significant conscious structures, to be the ones that 1) bind together the most information and 2) tend to have higher resolution over higher level concepts and objects rather than they do of lower level information and input level information.

AND Gates

An AND gate can be treated in precisely the same way as an OR gate except now there is no loss of information from the reference frame of the output of the AND gate. For example if we have Neuron 2 = Neuron 1A AND Neuron 1B, then if Neuron 2 fires, it has 100% resolution into the values of Neuron 1A and Neuron 1B, and therefore we can fully compress 1A and 1B into the experience of 2.

Of course we can also apply resolution math to structures that have a combination of AND gates and OR gates. Doing the math again assuming that only 1 OR gate input at a time can be causal (the reader can do the math the other way on their own if they wish as an exercise, it’s not challenging):

AND(A, OR(B, C)): 

R_A = 1, R_B = 1/2, R_C = 1/2 

*note in this notation OR(B,C) means B OR C

OR(A, AND(B, C)): 

R_A = 1/2, R_B = 1/2, R_C = 1/2. 

OR(AND(A, B ), AND(A, C) , AND(D, E)):

 R_A = 2/3, R_B,C,D,E = 1/3 

Our framework for calculating uncertainty is extremely precise and its consequences should be made very clear: the uncertainty of an element of object must precisely match the blurriness of that object as it is experienced.

Imagine for example that we have our OR gate where 2 = 1A or 1B. And now imagine that 1A is an object that is experienced as a blue circle and 1B is an object experienced as a blue square. Therefore the experience of the structure defined by 2 is precisely that which is blurred such that it appears as if there is a 50% chance it is a blue circle, and a 50% chance that it is a blue square.

One could imagine a similar example where you are yourself participating in an eye exam and looking at distant letters. Imagine that when you look at the bottom row you see a letter that feels like a dead tie between an F and a P (assuming you are fully confident it could not be any number besides F or P). In this case we could describe your experience of this visual object as having 50% certainty of F and 50% of P.

And of course it’s also true in all of our examples that if the output of the OR gate (ex the output of Neuron 2) has the same downstream effects regardless of which input 1A or 1B fired (as a true OR gate would) then it would be impossible for the downstream effects to have any information about whether our trigger was a blue circle or a blue square (or an F or a P). Thus this uncertainty must be present.

The only caveat here is that our exact resolution math of R_I = 1 / T (and the reason why we described it above as an approximation rather than an infallible rule) assumes that over the course of sufficient time all inputs (ex 1A and 1B) occur with a similar number of times. However this does not always need to be true in reality. For example, imagine the neural network described above lived in a world with 99% blue circles and only 1% red circles, but the strict OR gate connectivity of the neural network remained. In this case the downstream effects may want to assume, despite not having all of the info, that probably a blue circle was seen. Now this downstream assumption in itself can of course not change the resolution of Neuron 2, however compression can. For example it may be most compressed to describe the entire system as Neuron 2 = 99*Neuron 1A OR Neuron 1B. This would be one way of factoring more information into the compressed structure of the OR gate.

2.1.7 Using Uncertainty to Precisely Calculate Boundaries

What may already be obvious to the reader is that we can use statistical uncertainties (and their corresponding experience of resolution) to define precise boundaries to our experience. Specifically, from the perspective of any emergent structure:

1. Any information that is completely non-causal will have zero resolution
2. Any information that is causal will be included in the structure, but only with a certainty / resolution equal to its conditional probability

The resulting boundary is not really a harsh binary cut off, but a probability distribution over all of the information in the computational system defined by conditional probabilities from Bayesian Inference. As a result, much of the information in the neural computations of our brain that seems to allude us is not actually totally excluded from our conscious experience, it is simply very low resolution.

It’s useful to think about neural computations in terms of the following buckets:

Not conscious at all = Any information that has no causal impact at all to any emergent structure when redundancy is removed

Small subconscious experience but not part of our bound primary consciousness = Any information that makes some causal impact to some emergent structure, and cannot be totally compressed away, but makes no causal difference to large emergent reference frames

Low resolution conscious experience = Any information that makes a weak causal difference to large emergent structures such that there is high uncertainty

Experienced with some resolution, but is not verbally reportable = Information that makes a significant causal difference to some large emergent structures (ex influences behaviors) but doesn’t influence verbal structures with a high degree of statistical certainty

Experienced clearly and easy to verbally report ⁃ Information that makes a significant causal impact on large emergent structures and determines verbal activations with high statistical certainty

Here you can see we make a distinction between a structure which is high resolution, and one that causally impacts (selects) words with high certainty that allow for clear verbal report.

The other interesting idea here is the concept of the small subconscious experience. Such an experience survives compression and instantiates a real emergent structure, but may become blurry as we move towards deeper bound structures in the brain. We can walk through such an example with another OR gate scenario.

Let’s say we have an OR gate with three inputs, each which is a visual object which is in high resolution before the OR gate. Input 1A is a circle, input 1B is a square and input 1C is a triangle.

Since the output to the OR gate (let’s call this Neuron 2) is unaware which shape was detected, if will experience a blurry mix between the shapes. In this sense Neuron 2 does not fully compress away each input or make each fully redundant, thereby destroying its experience. Therefore it could be the case that the most compressed structure describing the entire system (approximated by the MLD for example) would instantiate Neurons 1A, 1B and 1C as real emergent structures. If this is the case, then a full resolution experience of a circle, square and triangle really does exist, and lives inside the reference frames defined by neurons 1A, 1B and 1C. Thus they represent small, fleeting, and subconscious (meaning not part of our primary consciousness, but still having some conscious experience) rather than being fully unconscious.

2.1.8 Second Order Relations

We saw when discussing different types of elements and relations that some qualities of our experience can perhaps be explained by relations such as magnitude and order. However these First Order Relations (direct relations between elements) will only take us so far towards describing qualia and attempting to solve The Hard Problem. In order to explain phenomenon like Time, Color, Pitch, Pain and more to any level of satisfaction we will need to introduce Second Order Relations (SORs), which are defined as relations between sets of relations, and Higher Order Relations (HORs: third order relations, fourth order, etc).

Second Order Relations are what allows computational systems, in our framework and Sciops language, to use more complex functions to either transform a structure, or detect a more complex structure.

We will dive into Second Order Relations in great detail in the following posts as we discuss space, time, the experience of causality, etc, but we can imagine a very simple one here to ensure we understand the concept. For example we could imagine a structure such that there is a SOR between a linear order and magnitude: as we move between successive linearly ordered objects, each object corresponds to a magnitude which can be determined by its place in the order.

For example if we wanted this SOR to follow a linear relationship between order and magnitude, we might say that as the place in linear order increases, so does the magnitude of the object by the same amount. Thus it can be modeled as Y = AX where Y represents the magnitude of an object at location X in the order and A represents the constant which determines the size of the increase in magnitude for each step forward in order.

Of course we can imagine more complex SORs where the relationship is not linear but some sort of more interesting curve.

We can also imagine the causality of our system to use SORs in a couple different ways. One utility could be around detecting a specific SOR (such as a specific curved relation between order and magnitude) and then causing some downstream effect only if that specific SOR is detected. Thus we have a logical gate which is conditioned on an SOR. We can call this a Conditional SOR Gate.

We can also imagine in some computational systems the network may actually create the SOR by transforming one set of relations into another. For example we could find a neural network that calculates distances across order and transforms those into magnitudes via neural firing rates. We can call this a Transformation SOR Gate.

Where things get truly fascinating in terms of explaining the qualities of our experience is when the convergence of information is such that in one type of experience we only retain certain structural properties of an SOR or HOR in high resolution, and in other experiences we retain other different properties of an SOR or HOR. Thus we have two new variables that can shape the quality of our experience: 1) what is the nature of the HOR itself (what type of curve, line, etc describes the HOR) and 2) which properties of the HOR are bound in high resolution into our experience and which are lost/low resolution.

These two variables will be absolutely crucial in beginning to understand the difference between the conscious experience of space and that of time for example, which we will discuss in great detail in the next post.

2.1.9 Is Convergence and Resolution Sufficient for Binding and Boundaries?

We have shown above how compressing away redundancies can create unified emergent references frames that bind together different pieces of information, and how the boundaries to these emergent structures can be defined using conditional probabilities.

What is still not entirely clear and open to further research is whether or not these concepts are sufficient for completely solving The Binding Problem and The Boundary Problem. In other words, are these the only mathematical concepts that define binding and boundaries, or do they simply compliment other concepts that are also required (such as recurrence and integrated information). We will leave this as an open question to further researchers who can work on marrying the concepts in Sciops with IIT, RPT, Neurorepresentationalism, etc to see if any new interesting results are found to support or deny either hypothesis.

It’s also an interesting question to ask whether or not the core principles of Sciops outlined in this chapter fully resolve The Grain Problem (Lockwood 1993 ). The heart of that problem is really a question of emergent structures, and how they can arise from the chaotic physical interactions in the brain. At the very least we feel Sciops has helped make tremendous progress on such a problem, and hope to continue to do so in the following posts.

And finally, though in this post we have hinted at the idea of qualities (magnitude, order) Sciops has yet to put forward in this post a fully convincing mathematical description of qualia such that it could be seen as solving The Hard Problem. But we will continue to make progress on this problem in the following posts as we at least partly define space, time, color, pitch, joy/pain and more using emergent structures.

Implications on Recurrent Processing

Many other prominent theories of conscious look to solve The Boundary Problem and Binding Problem based on recurrent processing rather than compression. For example RPT (Lamme 2006) and IIT (Tononi 2004) claim that only bidirectional recursive connections can bind together information, and that only systems of neurons that are recursive can be conscious, defining the boundary between conscious and unconscious.

Sciops shows that at a minimum, the convergence and compression of information, and the statistical certainty of that information, is necessary for binding and boundaries, while leaving open the possibility that this is a requirement but is not sufficient, and perhaps recurrent processing is also required. Thus Sciops can possibly work hand in hand with such theories.

It’s also worth noting that recurrent processing and its two way signaling has significant consequences on the resolutions calculated for any conscious system. For example, top down signaling, in addition to feed forward signaling, could be used to remember which inputs caused which outputs in a neural structure, and essentially retrace the steps back to them. This concept may be practically very important to bind huge amounts of information together into one experience without catastrophic loss of resolution. Thus, Sciops might suspect that recurrent processing may not be so much strictly required for binding as much as it may be practically crucial for high resolution binding in large structures.

Further collaborative research could be done to model recurrent vs feed forward neural networks using the Sciops framework to see if vast differences in binding and high resolution boundaries may result.

2.1.10 Summarizing The Core Argument

Premise 1 – Our conscious experience does not contain redundancies

Our conscious experience appears to contain no completely redundant information. Every distinction within experience must contribute something to its phenomenal structure; otherwise, it is not an additional experienced distinction.

Mathematical Conclusion 1 — Lossless Compression Eliminates Redundancies

Any complete mathematical description of reality that contains removable redundancy cannot be maximally compressed. Therefore, a maximally compressed lossless structure preserves all information about reality while eliminating unnecessary repetition. Lower-level information that cannot be compressed into larger emergent structures must still be preserved elsewhere in the complete description.

Sciops Hypothesis 1 – Conscious Experience Corresponds to Nonredundant Emergent Structures

Sciops proposes that conscious experience corresponds to the nonredundant emergent structures revealed through lossless compression – ie the maximally compressed structure. In particular, conscious reference frames correspond to specific convergent structures within the complete mathematical description, rather than to that entire description.

Premise 2 – Conscious experience as we know it requires sameness / permanence of certain structures

Our experience represents objects and qualities as persisting across changes in their states. To experience something changing, we must in some sense experience it as the same thing across those changes.

Mathematical Conclusion 2 — Compression Can Define Persistent Emergent Structures

When physically distinct events contain partially redundant mathematical structures, these redundancies can be factored into a shared invariant structure. This allows a sequence of distinct physical events to be represented as one persistent emergent entity with changing states, without losing any information about the underlying reality.

Sciops Hypothesis 2 – Such Emergent Structures Can Constitute Persistent Conscious Entities

Sciops proposes that these persistent emergent structures are not merely convenient descriptions but can constitute the objects and reference frames that exist within conscious experience.

Mathematical Principle 3 – Partially Redundant Structures Bind Together Information

When compressed, partially redundant structures can tradeoff full resolution into single causal inputs for a more compressed structure that retains some information into multiple non-redundant causal inputs, thereby binding this information together.

Such convergent structures can only have access to the information that causally impacts them, therefore the resolution of each structure is determined by conditional probabilities over causal inputs. Once an emergent reference frame is defined, Bayesian Inference is used to determine the statistical certainty of each causal input.

Sciops Hypothesis 3 – Convergent Emergent Structures Constitute Phenomenal Binding and Conditional Information Determines Phenomenal Resolution

Sciops proposes that such convergent emergent structures can bind distinct causal information into unified conscious structures, with the conditional information retained about their causal inputs determining their experiential boundaries. Specifically Bayesian Inference can be used to calculate the conditional probabilities of all causal inputs to such reference frame.

Sciops Hypothesis 4 – Resolution Across Structural Entities (First Order Relations, Higher Order Relations) Determines the Quality of Experience

When information loss and therefore resolution is structural, retaining some structural properties in high resolution while losing others, this shapes the quality of our experience and helps define qualia such as order, intensity, time, space and likely even color, pitch, pain, and so on.

2.1.11 Applying Core Sciops Principles To Further Existing Theories

In the last chapter we discussed how Sciops can help significantly further existing theories of consciousness by providing a rigorous mathematical framework for defining emergent structures. Now hopefully the reader has much more context for understanding this claim.

For example, the idea of a Global Workspace (Baars 1988) is almost directly describing the largest and most global possible emergent structure which could be instantiated by compression. GWT essentially hypothesizes that local processing tends to not converge enough information to create a large and stable emergent structure by itself, and only when information is deemed important enough to be broadcast globally do we have enough convergence to create one large bound conscious experience that resembles human consciousness. And now GWT has the mathematical tools to investigate the flow of information globally and calculate the resulting binding and boundaries that may be possible from it, using Bayesian Inference to calculate the certainty or resolution of such information.

Multiple Drafts (Dennett 1991)  rejects the idea of a Cartesian Center for consciousness, preferring instead the idea that conscious experiences emerge more dynamically based on behaviors. This nicely matches everything we discussed above about emergent structures, and Sciops can be applied to specific behavioral outputs, labeling these as hypothesized highly-compressed emergent reference frames. For example if a person speaks a word, one could take that behavior and model it as an emergent reference frame and then ask which neural computations causally influenced this behavior. Then one could apply Bayesian Inference to understand the resolution of all information that influenced that behavior and use this to calculate the information bound into that experience. Each “Draft” could be its own emergent reference frame in this sense, competing to be the best-compressing emergent structure, and therefore the one that defines our primary conscious experience in any given moment.

Recurrent theories (RPT, IIT) hypothesize that bidirectional signaling, or recurrent signaling is necessary for significant conscious experience. In order to make such a claim rigorous, these theories need a mathematical tool for rigorously defining sameness or permanence, otherwise it’s not clear how the idea of recurrence could be ontologically fundamental rather than a more arbitrary semantic distinction. Sciops allows them to use compression to do exactly this as we discussed above, allowing RPT and IIT to rigorously define a single neuron over time for example as the same entity which is capable of signaling in a bidirectional way.

Furthermore, the claims by recurrent theories could be interpreted as either making the strong claim – that only recurrent signaling is conscious at all, or the weaker claim – that non-recurrent processing is minimally conscious.

Further research could be done to attempt to prove either claim mathematically using the framework introduced in this chapter.

To prove the strong claim, one would need to show that recurrent structures in the brain fully compress away all information that resides in feed-forward signaling. In other words, one would need to show mathematically that all such information is either non-causal or redundant and therefor can be essentially absorbed by the structures that partake in top-down as well as bottom-up signaling.

Proving the weaker claim is likely much easier. To do so, further research would simply need to show using Bayesian Inference that information loss is simply too large with feed-forward signaling for any feed-forward information to be retained in high resolution by large (highly convergent) emergent structures. This claim seems plausible given how many neuronal connections there are in the brain and how information uncertainty tends to drop as the number. of inputs increases, approximately according to R_I = 1 / T (where I = a specific input and T = the total number of inputs).

The next few posts will dive into each Sciops principle in more depth shedding even more light on how our framework can be applied to further research.