The recursively generational brain

Large brain projects

At least two large brain projects are currently proposed. Of these, the Human Brain Project (HBP) is at a more advanced stage of implementation. It is a 10-year venture into building a working model of the brain, simulated on a super computer. The HBP is the most ambitious attempt of its kind ever, and a flagship research programme of the European Union. A similar project called the Brain Initiative was announced by President Obama in the US, but the aim of the American project is different in focusing on the development of technologies for understanding the brain’s structure and function rather than the simulation of these structures and processes.

Much hype surrounds these projects. Henry Markram, co-director of the HBP, sees the project as “the Higgs boson of the brain,” and in the US, Thomas Insel, the director of the National Institute of Mental Health compared the Brain Initiative to the Apollo space programme. Indeed these projects are quite major, especially in terms of the funding they require. The European Union recently awarded €1bn to the HBP, and $3bn may be required for the Brain Initiative. But according to HBP scientists, understanding the human brain is one of the greatest challenges facing 21st-century science (HBP, 2013a). Still, there are questions about how this is to be realised, especially in light of it being a case of the recursively generational brain.

Much of the answer is provided in the HBP’s response to frequently asked questions (HBP, 2013b). The answers to these carefully listed questions display a clear logic. The brain is a complex entity consisting of many parts that interact. It has to be understood at different levels, from the molecular to the cognitive, using both bottom-up and top-down approaches. Each level produces properties that can be contained in models that are constructed to explain these properties. In bottom-up approaches level substrates are explored experimentally, whereas top-down approaches are governed by model-driven theory. This circular method is an economic way to fill knowledge gaps. If a model successfully explains a function at a particular level, further detail of the model’s substrate need not be explored experimentally because understanding of the structure and function of the substrate’s components can be expanded on the basis of the model. On the other hand, when a model fails to describe the emergent properties of a substrate, more experimentation is needed to inform the model. The guiding principles are integration and interaction. In the end the working model of the brain will be a hierarchically structured gigantic integration of numerous integrated, interacting models. It will also be a biological object ultimately based in a physical reality that will allow its form and function to be implemented in a silicon substrate, in the form of neuromorphic computing.

But perhaps the most telling response concerning the nature of the project comes from Henry Markram himself. In the concluding moments of a television interview the presenter, Neelie Kroes, asks a final question, namely: What is your dream? Answering this question, Professor Markram captures the human essence of the project with his characteristic passion:

We want to understand how the brain represents reality. This is the most magical, it is really fundamental science. There is all kind of electrical activity, chemical processes. But it all translates into this world, and we all build our own reality to some extent. We have a lot of subjective influences, and ultimately as scientists our fundamental question is to understand how the brain builds, represents reality and allows us to manoeuvre in that reality. … We are strangers to ourselves because we do not understand our brain. We do not understand that we are on a wild horse that is just running around commanding us. Deeper knowledge of the brain I believe just on a fundamental level will have a profound impact on society. (Kroes, 2013)

A deconstructed dream

A brief analysis of Markram’s response reveals how scientists attempt to skirt the obvious difficulty of a recursively generational brain, namely that an understanding of the brain is an understanding by the brain. The dream Markram describes is not a fleeting dream, not the kind that disappears with the first light of day. It is a dream anchored in reality, or at least two realities: the first, a reality that can be represented in the brain; and the second, the reality of ourselves—the fact that we will not be strangers to ourselves anymore once we get to know the brain. Markram foregrounds the idea that the brain represents reality. But he modifies this approach, adding to representation the notion of subjective contribution. However, when he gets to the point where he states that the brain builds reality he quickly replaces “builds” with “represents” again. Of course this manoeuvre cannot be held against him because, to begin with, he did not fully commit to the brain as builder. According to him the building of our own reality happens to some extent only, and ultimately it is the various kinds of electrical activity and chemical processes that translate into this world. Still, Markram’s manoeuvre is more complicated than a straightforward replacement. He first uncovers an undecidable, the undecidable between the brain that represents and the brain that builds. But just as he is about to recognise this undecidable as a stranger within himself he executes a second move. When he replaces how the brain builds with how the brain represents, he also replaces the stranger within himself with the stranger to himself, the stranger before him. The undecidable moment of a brain that builds and a brain that represents, of a reality that is created and a reality that already exists to be represented, of a representation that is always already a presentation, this undecidable is replaced with a knowable scientific object, a brain that can be known in depth. For Markram the recursively generational brain does not pose a problem because the generating brain is subsumed and replaced by the generated brain. When Markram dreams of taming the wild horse that is just running around commanding us he does not see that the real wild horse, the undecidable of the recursively generational brain, has already bolted. There is no way of reigning in what remains undecidable between the brain that dreams and the dreamt brain.

An arena for the tamed horse

The undecidable (the wild horse) does not suggest the impossibility of large brain projects because an undecidable does not forbid exploration. It may disrupt conventional approaches, but it normally does so only when pushed to ultimate conclusions, which still leaves much space for manoeuvring. In terms of Markram’s metaphor: there are always arenas where tamed horses can be put through their paces. For example, ever since Bertrand Russell pointed out that the set that refers to itself establishes a paradox, and Gödel formulised incompleteness, we have known that no axiomatic system can be self-sustained. But this knowledge did not stifle mathematics, a complex and advanced system thriving on axiomatic deduction. Cybernetics is another example. We know that cybernetic control is subject to infinite regress and that second order cybernetics creates paradoxes, but such difficulties do not distract from the usefulness of these approaches, be that as frameworks of understanding or models in practical application. Thus, within limits, much can be learned from large brain projects, the only difficulty being that these limits tend to present themselves sooner and more forcefully when the object of exploration is also the condition for exploration. We saw above how quickly Markram encountered this limit and how powerfully the limit repelled his attempt to avoid it.

Markram’s arena for the HBP has been staked out clearly. This is an arena in which the brain represents reality. If the brain also builds reality it is only to some extent, and it happens within the arena of representation. The building brain can shape what appears as the represented, but it cannot change the fact of representation. In Markram’s arena what presents itself for representation is not representation itself, but a reality that awaits its representation, a reality that is already in place and ready to be duplicated (modified to some extent) inside a brain. The representation that Markram is interested in is a duplication that presupposes a reality that awaits representation and the image of this reality. Markram’s representation is bounded by and cannot exceed the reality–image structure.

Being the most basic form of a reality that can be presupposed, physical reality is the reality awaiting representation. Markram’s representation is bounded by, and cannot exceed, the physical and the image of the physical. It is a representation that happens within space–time, a representation that is always coordinated by space and time. Therefore studies of spatial-temporal representation in the brain will advance the HBP. However, work that relies on a notion of physical reality that incorporates qualities that would not normally be understood as traditionally physical would not have a certain place in the arena of the HBP.

Studies concerning spatial-temporal representation focus on neurons, groups of neurons, or brain structures like the hippocampus that react to particular spatial and temporal aspects of the environment with the aim to identify how organisms orientate themselves in space and time, or how they organise events in space and time (e.g., MacDonald, Lepage, Eden, & Eichenbaum, 2011; O’Keefe & Nadel, 1978). But there are also studies pointing to the possibility that the brain creates a spatial-temporal framework that it then uses to structure the world (Chadwick, Hassabis, & Maguire, 2011; Maguire & Hassabis, 2011; Moser, Kropff, & Moser, 2008).

Studies concerning spatial-temporal representation in the brain are comfortable with a monistic ontology in which cognitive processes are emergent properties of a brain anchored in physical reality. But the nature of physical reality has been troubled by developments in the discipline of physics, leading to understandings of physical reality that would not fit well in the HBP arena. Examples of these are Barad’s (2007) notion of the intra-activity of physical reality and Penrose’s (2001, 2005, p. 809) suggestion that a theory of perception is required to understand the collapse of the wave function (a fundamental process in physical reality), and that the brain is ultimately beyond computational simulation. Hameroff (2001) implemented Penrose’s ideas as a brain model. Although the model is physical, in its ultimate reduction it still harbours Penrose’s reservations about the collapse of the wave function and the computability of the brain. Markram’s arena for the tamed horse may therefore not have been penned as securely as he hoped for.

Catching up with the horse that bolted

In the end the wild horse that Markram allows to get away (the undecidable of the brain representing the brain) may never be tamed but it has to be saddled. Ultimately the recursively generational brain is a complex entity. It is not a simple solid, well-bounded structure that calls upon itself in an uncomplicated manner within space and time. It does not simply repeat itself in time. Self-reference, the ability to call upon itself, to represent itself is fundamental to its structure. The recursively generational brain is not simply an already existing generation that is repeated and that is modified in the repetition to constitute a next generation. There is a sense in which the fact of repetition, of representation precedes structure. The structure (the brain) is constructed in and through the process of representing itself. However it is not easy to comprehend this idea. It is quite difficult to consider representation without first imagining the structure, the brain that represents itself, as a structure, a generation that exists before representing itself as the next generation. Hence the complications in the process that Derrida (1972/1982) called différance. Technically speaking différance is the play of signification, the signifier becoming the signified, becoming the signifier in the production of a text. Similarly, the generating brain (the current generation) becomes the generated brain (the next generation), which in turn becomes a generating brain in the production of the recursively generational brain. This brain is the outcome of the interplay between being generating and being generated. But there is no ultimate foundation in a being that is simultaneously and equally the source and the product of its generation.

The following discussion takes différance as a point of departure, but it does so in a format that serves the current analysis. Although much of the subtleness of différance is lost in this approach it still enriches the notion of recursive generation.

Consider the brain calling upon itself in its own generation. Figure 1A depicts this as a process in which the source of generation and the product thereof constitute a single entity. But this is an untenable over-simplification. In terms of différance, signification has two irreducible moments, namely a moment of deferring and a moment of differing. In signification (the most fundamental meaning - making process) the signified is the signifier deferred and differed from itself. The signifier becomes the signified when the self-representation (the smooth extending of the signifier as itself) is interrupted by difference. Similarly the generating brain is extended (deferred) from itself (depicted in Figure 1B) and becomes the generated brain when difference interrupts the smooth extension of itself (Figure 1C). The differentiation occurs within sameness. The generated brain is the generating brain deferred from itself, but this similarity is interrupted and the generating brain is differentiated as the generated brain. This means similarity is interrupted by an exteriority, a difference inserted inside similarity. This exteriority should not be misunderstood as a space that already exists, surrounding the recursively generational brain. There is no presupposed existing external space. The exteriority within sameness relates the sameness to its outside, that is to an outside that belongs to the sameness. The difference that interrupts the self-deferring of the generating brain relates the generating brain to its outside. Figure 1C depicts this process. In the process of the generating brain slipping away from itself in the deferring of itself, in its similarity, being interrupted by difference, the generating brain becomes related to its outside. The generating brain is deferred and differed from itself as the generated brain. It “sees,” “experiences,” “comprehends” the outside of itself. The arrow in Figure 1C depicts this fundamentally temporalising and spatialising process. An entire space–time comes into being in the deferring and differing of the generating brain.

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Figure 1.

Différance and the recursively generational brain.

Here the recursively generational brain is viewed in terms of a brain that defers and differs itself from itself. As such the brain already exists. But différance is more subtle. It does not assume or presuppose the existence of a brain that precedes the processes it engages in. Nor does it presuppose such processes. Différance is the interplay between, on the one hand, a brain that differs and defers, and on the other, a differing and deferring that precedes the brain. As such the exact meaning of différance is beyond one’s grasp. But différance nevertheless offers a particular prescription, namely that neither the generating brain nor the generated brain should be considered onto-epistemologically superior to the other. Différance does not allow the generating brain to be claimed by the generated brain (which is the ultimate dream of the HBP), but rather recognises and defends the onto-epistemological undecidability between a brain that generates and one that is generated, the ontological abyss (Markram’s wild horse) that prohibits a final claim of the recursively generational brain.

Riding the horse that bolted

Différance’s recognition and defence of the onto-epistemological undecidability constituted by the recursively generational brain is not a call to surrender, but a challenge rather to mount Markram’s wild horse. In essence the challenge is to maintain the onto-epistemological balance between the generating brain and the generated brain. Technically this requires an analysis of the symmetries of the transformations that constitute the recursively generational brain.

Symmetry is generally known as a pleasing property of objects that share some form of similarity, for example being mirror images of each other. However, in its technical form the notion of symmetry is more abstract (see, e.g., Ash & Gross, 2006, pp. 13–29; Penrose, 2005, pp. 247–254). It refers to symmetrical transformations in which particular elements or features are preserved. Davis and Gribbin (1992, p. 239) provide a clear and uncomplicated explanation of the fundamental importance of symmetry in physical processes. That the laws of physics are preserved in mirrored and time-reversed transformations of the universe is an indication of exactly how foundational symmetry is in understanding our world. For the purpose of the present article a symmetrical transformation may be understood simply as a transformation that brings an entity back to itself, or more precisely a transformation that plots an entity onto itself.

The recursively generational brain has several symmetries to consider. These symmetries guarantee the existence of the generating brain and the generated brain. More precisely, the generating brain exists if for a transformation of itself another transformation can be found that transforms it back into itself. Such transformations are symmetrical. The same argument holds for the generated brain. The existence of the recursively generational brain requires the existence of the generating brain and the generated brain. Therefore it requires the symmetry of the symmetries of the generating brain and the generated brain.

Consider the symmetries of the generating brain. Suppose the generating brain transforms into the generated brain and from the generated brain back into the generating brain (see Figure 2). These transformations are symmetrical if the generating brain is preserved in the process, in other words if it is the same before and after the transformations. The symmetry of these transformations also establishes the generated brain as the generated brain that belongs to (is produced by) the generating brain.

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Figure 2.

The generating brain deferred and differed from itself as the generated brain.

However, not all transformations are symmetrical. Transformations are non-symmetrical when the generating brain is not preserved in the process of transformation. In other words, the generating brain is transformed into the generated brain, but the transformation from the generated brain into the generating brain does not bring the generating brain back to itself. The non-symmetrical transformations leave a surplus, which is the difference between the generating brain before and after its transformations. In Figure 2 this surplus is indicated as “a.” The surplus needs to be extracted in order to allow the transformations to be symmetrical. Thus the transformations are symmetrical provided the surplus. The surplus is a temporal-spatial phenomenon. It is a surplus in the deferring (temporalising) and differing (spatialising) of the generating brain.

A similar logic prevails in the analysis of the symmetries of the generated brain. Figure 3 depicts the transformations that establish the generating brain as the generating brain that belongs to the generated brain. These transformations have to be symmetrical for the generated brain to be preserved in the process. Non-symmetrical transformations produce a surplus (indicated as “b”) that needs to be extracted in order to allow for symmetrical transformation and the preservation of the generated brain.

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Figure 3.

The generated brain deferred and differed from itself as the generating brain.

The recursively generational brain requires a symmetrical relationship of the symmetries of the generating brain and those of the generated brain. In other words it requires symmetry between the generated brain belonging to the generating brain and generating brain belonging to the generated brain. This means symmetry between the outcome of the inductive processes involved in the “production” of the generated brain and the outcome of the deductive processes involved in the “production” of the generating brain. In this case the surplus created by non-symmetrical transformations (“c” in Figure 4) is the difference between the surplus of the transformations of the generating brain (“a”) and the surplus of the transformations of the generated brain (“b”).

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Figure 4.

The recursively generational brain as the generating brain and the generated brain deferred and differed from each other.

This abstract model of the recursively generational brain describes how the brain relates to itself in its attempt to describe itself. As such it indicates the basic conditions for the objectification of the brain, namely the symmetrical relationships between a generating brain and a generated brain. But in this model the brain is not constituted as a physical object of observation. The only physical correlates are the surpluses “a,” “b,” and “c” that appear as space–time phenomena, and these phenomena are not fixed, independently existing physical phenomena. The condition for their existence is the symmetrical transformations of the recursively generational brain. Yet they are the only hope for physical observation of the brain, and the exact nature of their physicality is a matter of speculation rather than fact. The phenomena associated with “a” and “b” are likely to be a matter of physical forces rather than physical objects, because these surpluses are the correlates of one-sided transformations, that is the transformations of either the generating brain or the generated brain. Physical force (carried by bosons, a fundamental particle group) is the only objective element in a one-sided perspective of reality. Surplus “c” may be object-like because it is a surplus generated by both the generating and the generated brain. Unlike forces, objects (represented by the fundamental particle group, called fermions) rely on a double (720 degree) perspective to be constituted as an independent, objective reality. But regardless of the exact physical nature of these surpluses, observing and probing them does not offer direct views of the brain. They provide information about the symmetrical transformations intrinsic to the recursively generational brain.

The following example offers a concrete illustration of the physical manifestation of a recursively generational brain.

An example of a very simple brain

Imagine the brain as an equilateral triangle (Figure 5). Although a considerable oversimplification, this triangular brain nevertheless offers a detailed illustration of the processes described above.

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Figure 5.

The brain as an equilateral triangle.

The property of the equilateral triangle that is of interest here is the symmetry it displays when rotated in a two-dimensional plane. The triangle is symmetrical at 120 degree intervals. To illustrate, consider the triangle rotated clockwise through 120 degrees (see Figure 6). The triangle looks exactly like before. The figure of the brain is the only indication of the rotation. Without this drawing inside the triangle one would not be able to distinguish the rotated triangle from the original one. A further 120 degrees’ rotation brings the triangle to its next symmetrical position at 240 degrees, and a third symmetrical rotation positions the triangle at 360 degrees, which is similar to the original position.

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Figure 6.

The equilateral triangle rotated clockwise through 120 degrees.

Suppose these symmetries offer a complete description of the triangular brain and suppose rotation captures the processes of recursive generation. These presuppositions enable one to describe the events that occur when the triangular brain engages in transformations that preserve itself as generating brain. The preservation of the generating brain requires these transformations to be symmetrical. In other words, the transformation that changes the generating brain (the broken-line triangle in Figure 7) into the generated brain (the solid-line triangle in Figure 7) and the transformation from the generated brain back into the generating brain should restore the original position of the generating brain. The symmetry of these transformations also establishes the generated brain as the generated brain of the generating brain.

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Figure 7.

The equilateral triangle at a rotated position (solid outline) relative to its original position (broken outline). The rotation defers and differs the triangle from itself.

Let the transformations in question be a clockwise rotation (from the broken-line triangle to the solid-line triangle) followed by a counter clockwise rotation (from the solid-line triangle to the broken-line triangle). Rotation (transformation) involves deferring and differing. The triangle is deferred in maintaining its shape. In other words, in its rotation the triangle remains what it is, namely an equilateral triangle of a particular size. But the triangle is differed from itself in being repositioned at an angle to itself (see Figure 7).

Let the clockwise rotation that transforms (that defers and differs) the generating triangle into the generated triangle be g_θ1, with g the function of clockwise rotation and θ₁ the degrees of rotation. Let h_θ2 be the counter clockwise rotation of θ₂ degrees, transforming the generated triangle into the generating triangle. The recursive generation is given by h_θ2g_θ1. Reading this formulation from right to left, as convention dictates, means the counter clockwise rotation, h, is executed on the result of the clockwise rotation, g. The transformation, h_θ2g_θ1, would be symmetrical if it restores the original position of the generating triangle. For example, let both θ₁ and θ₂ equal 40 degrees. Then h_θ2g_θ1 = h₄₀g₄₀. After the transformation, g₄₀, the generated triangle is at an angle of 40 degrees clockwise to the generating triangle, and after h₄₀ the generating triangle is at an angle of 40 degrees counter clockwise to the generated triangle, which means the generating triangle is restored in its original position, and hg is a symmetrical transformation.

The notion of symmetrical transformation can be used to index the complexity of the recursively generated object (in this case the triangular brain). A higher degree of possible symmetrical transformations means a more complex object. For the triangular brain the symmetrical transformation described above (in which θ₁ = θ₂) is not the only symmetry. The transformation hg is symmetrical whenever it produces a result that is symmetrical to the original position of the generating triangle. As the generating triangle has symmetries at 120, 240, and 360 degrees of rotation the transformation hg is symmetrical whenever it produces one of these results. For example, h₆₀g₁₈₀ constitutes a symmetrical transformation because a 180 degree clockwise rotation followed by a 60 degree counter clockwise rotation positions the generating triangle at 120 degrees. A slightly more complicated example of a symmetrical transformation is h₁₆₀g₄₀, which brings the generating triangle back to 240 degrees.

The possibility of symmetrical transformation is a presupposition and also the existential condition of a recursively generational system. For example, the existence of the triangular brain is guaranteed by the fact that it is preserved in symmetrical transformations (rotations). However, this existence is self-absorbing. The generating triangle and the generated triangle are superimposed on one another, meaning that the generating brain induces the generated brain as much as it is deduced from the generated brain. A close analysis of the symmetrical transformations shows why. The symmetrical transformation h_θ2g_θ1 does not require g_θ1 and h_θ2 to be individually symmetrical. For example, neither g₁₈₀ nor h₆₀ is a symmetrical transformation of the triangular brain, but h₆₀g₁₈₀ is. Consider the transformation g_θ1 with the original broken line triangle (generating brain) deferred and differed as the solid line triangle (generated brain). The deferring of the generating brain (the maintaining of its sameness) is interrupted by differing, relating the generating brain to its outside. This is shown in Figure 8 where “a” marks the area of the generated brain that lies outside the generating brain. However, this outside is not a separate outside, a space that exists independently of the generating brain. It is part of the deferred brain and therefore related to the generating brain. It is the exterior of the generating brain. The line segment “A” differentiates the generating brain from its exterior. When g_θ1 is followed by h_θ2 and h_θ2g_θ1 constitutes a symmetrical transformation, the generated brain is transformed into the generating brain. The surplus “a” and the line segment “A” shrink to 0, and the relationship between the generating brain and its exterior disappears. Under symmetrical transformation the triangular brain has an existence of self-absorbing inductions and deductions.

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Figure 8.

The rotated triangle (solid outline) contains a surplus, “a,” that constitutes an exteriority of the original triangle (broken outline) and a line segment A (broken line) that differentiates the exteriority of the generating brain from itself.

There are more non-symmetrical transformations than symmetrical transformations. Symmetrical transformations do not leave any surpluses, but non-symmetrical transformations do. When h_θ2g_θ1 is not symmetrical the generated brain is not returned to a symmetrical position of the generating brain. For example the non-symmetrical transformation h₆₀g₉₀ does not fully return the generated brain to the generating brain. It leaves the generated brain at an angle of 30 degrees relative to the generating brain, with a corresponding surplus of “a” (see Figure 8). The size of this surplus, the extent of the exteriority, shows the degree to which the non-symmetrical transformation deviates from being a symmetrical transformation. The surplus, the exteriority, is produced in the deferring (temporalising) and differing (spatialising) of the generating brain. As such, the surplus is an exteriority that is fundamentally temporal and spatial, and the extent of this exteriority (the size of “a”) reflects the magnitude and complexity of space–time features.

There are numerous configurations of θ₁ and θ₂ that render non-symmetrical transformations of h_θ2g_θ1. Each configuration produces a particular surplus “a” and a corresponding segment “A”. Therefore, a series of transformations produces a series of surpluses. When summed, these surpluses constitute the exterior of the generating brain, as illustrated in Figure 9. When strung together the line segments (“A”) corresponding to each surplus (“a”) trace the broken line that differentiates the generating brain from its exterior. Through its transformations, the repeated deferring and differing from itself, the generating brain demarcates itself in relation to its exterior.

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Figure 9.

The exteriority of the generating brain is the sum of the surpluses of non-symmetrical transformations. The exteriority is limited in being undefined at moments of symmetry (120, 180, and 360 degrees of rotation) and being undecided at moments of maximum extent (60, 180, and 300 degrees of rotation).

The exterior of the generating brain is not without limits. It is not unbounded or infinite. It has two kinds of limit, namely moments at which it is undefinable and moments at which it is undecidable. The exteriority of the generating brain is undefined at a moment of symmetry, and it is undecided at the midpoint between symmetries. For the triangular brain the undefined moments are at 120, 240, and 360 degrees of rotation, and the undecided moments occur at 60, 180, and 300 degrees (see Figure 9).The exteriority is undefined at moments of symmetry because at these moments there is no surplus “a”. At its moments of symmetry the triangular brain is not deferred and differed from itself. There is no temporality and no spatiality, and thus no features in space and time. The lack of definition is fundamental. Technically speaking exteriority is punctuated by discontinuity.

The undecidable is not a discontinuity. It is a well-defined moment within exteriority. It occurs at moments of maximum magnitude and complexity of space–time features. Between the moment when “a” stops expanding and the moment it starts shrinking is the moment at which it is neither expanding nor shrinking. These are moments at which equally important but opposing space–time features are pitched against each other with no way to decide.

Thus far the discussion has focused largely on the generating brain being deferred and differed from itself as the generated brain. The process was illustrated in terms of the transformation h_θ2g_θ1 that constitutes the existence of the generating brain and establishes the generated brain as the generated brain of the generating brain, when symmetrical. However, the discussion would not be complete without consideration of the counter transformation, g_θ4h_θ3. When symmetrical, this transformation constitutes the existence of the generated brain and establishes the generating brain as the generating brain of the generated brain. Although this addition may seem superfluous, it is not. It is required to counter the ontological bias of a generating brain that defers and differs itself with the ontological bias of a generated brain that defers and differs itself. In other words, what needs to be discussed is the symmetry between h_θ2g_θ1 and g_θ4h_θ3.

The processes involved in the transformation, g_θ4h_θ3, are similar to those of h_θ2g_θ1 (see Figure 10). The surplus, “b,” depicts interiority and the line segment “B” differentiates the generated brain from its interior. The extent of interiority (size of “b”) reflects the magnitude and complexity of temporalising and spatialising features. The surpluses of repeated non-symmetrical transformations add up to an interior that is constituted as the interior of the generated brain (consider Figure 9 with ∑a replaced by ∑b), an interiority limited by the undefined and the undecided.

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Figure 10.

The original triangle (broken outline) contains a surplus, “b,” that constitutes an interiority of the rotated triangle (solid outline) and a line segment B (solid line) that differentiates the interiority of the generated brain from itself.

Symmetry between h_θ2g_θ1 and g_θ4h_θ3 means surplus “a” and surplus “b” cancel each other, leaving no resultant surplus. However, in most instances this is not the case. The transformations h_θ2g_θ1 and g_θ4h_θ3 are more likely to be non-symmetrical than symmetrical and, therefore, to leave a resultant surplus “c.” The logic of surplus “c” is somewhat complicated. The surplus follows from an ontological undecidable. The ontological primacy of the generating brain versus the generated brain is undecided. Recall that, assuming an ontological bias towards the generating brain, the exteriority of the generating brain is found in the generated brain. In other words, the generated brain captures or claims the exteriority of the generating brain. Similarly, assuming an ontological bias towards the generated brain, the interiority of the generated brain is captured in and claimed by the generating brain. However, there is no guarantee that the exteriority of the generating brain is captured fully by the ontologically assumed generated brain, and likewise no guarantee that the ontologically assumed generating brain fully captures the interiority of the generated brain. Thus part of the exteriority of the generating brain, and part of the interiority of the generated brain, may be left unclaimed. Surplus “c” is this unclaimed territory, an exteriority that is not claimed as the generated brain and an interiority that is not claimed as the generating brain. It is the difference between the broken line that differentiates the generating brain from its exterior and the solid line that differentiates the generated brain from its interior (see Figure 11).

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Figure 11.

Surplus “c” as the difference between the broken line (A) that differentiates the generating brain from its exterior and the solid line (B) that differentiates the generated brain from its interior.

Considered jointly, Figure 8 and Figure 10 offer a simplified illustration of surplus “c.” Hiding the details of the underlying logic, the difference between line segments “A” and “B” can be read simply as the difference in their lengths. Under repeated rotations (being repeatedly deferred and differed) the line segments “A” trace the outline of the generating brain, the line segments “B” the outline of the generated brain. The differences between the line segments “A” and “B” outline a more complicated object, namely a regular hexagon, with six symmetries (see Figure 12). This object manifests when the broken line triangle and the solid line triangle are angled at 60, 180, and 300 degrees. In other words, it manifests at moments of maximum extent and complexity and in the limits of the undecidable.

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Figure 12.

The differences between line segments “A” and line segments “B” trace the outline of the regular hexagon at the intersection of the generating brain (broken outline) and the generated brain (solid outline).

In the world of triangular brains that recursively generate themselves, the regular hexagon is the recursively generational brain. It is the object that manifests in the triangular brain’s search for its own reality. If there were to be a large Triangular Brain Project, the outcome of the project would not be a generating brain claimed by an ontologically assumed generated brain. No matter how complicated the systems that claim to capture the generating brain, its simulation will always be interrupted by matters that are undefined and issues that are undecided, matters and issues that arise from a symmetry not captured fully by a generated brain.

Revisiting Markram’s dream

Markram’s dream is governed by representation. He wants to understand how the brain represents reality, and he wants to use this understanding to represent himself. To realise this dream, to live his dream, Markram has to ignore the fundamental difference between the dreamer and the dream. He wishes to reduce himself to a brain that represents reality. Therefore, when Markram identifies the brain as the wild horse that needs to be tamed, he has already manoeuvred the escape of the real wild horse, namely the fundamental difference between the brain that represents and the brain that builds. The brain that builds, the generating brain, is replaced by the representing brain. This is a representation without generation. Markram’s dream does not allow for a generating brain that calls upon itself in its own generation. The large Human Brain Project cannot capture the complex object that represents the recursively generational brain. The physical correlates of this object are the outcomes of non-symmetrical transformations between the generating and the generated brain. It is not something that can be simulated without participating in the simulation. No matter how complicated the computer systems that claim to capture the generating brain, its simulation will always be interrupted by matters that are undefined and issues that are undecided. In the world of computer simulation the recursively generational brain can only be a cyborg, a brain that participates in the simulation of itself instead of merely watching itself being simulated. In the land of triangular brains this cyborg would be the regular hexagon.