Mirror Neurons, Mirrorhouses, and the Algebraic Structure of the Self

Mirror Neurons, Mirrorhouses,
and the Algebraic Structure of the Self

Ben Goertzel1, Onar Aam, F. Tony Smith, Kent Palmer
1Novamente LLC

November 18, 2007


Abstract. Recent psychological research suggests that the individual human mind may
be effectively modeled as involving a group of interacting social actors: both various
subselves representing coherent aspects of personality; and virtual actors embodying
“internalizations of others.” Recent neuroscience research suggests the further
hypothesis that these internal actors may in many cases be neurologically associated with
collections of mirror neurons. Taking up this theme, we study the mathematical and
conceptual structure of sets of inter-observing actors, noting that this structure is
mathematically isomorphic to the structure of physical entities called “mirrorhouses.”
Mirrorhouses are naturally modeled in terms of abstract algebras such as quaternions and
octonions (which also play a central role in physics), which leads to the conclusion that
the presence within a single human mind of multiple inter-observing actors naturally
gives rise to a mirrorhouse-type cognitive structure and hence to a quaternionic and
octonionic algebraic structure as a significant aspect of human intelligence. Similar
conclusions would apply to nonhuman intelligences such as AI’s, we suggest, so long as
these intelligences included empathic social modeling (and/or other cognitive dynamics
leading to the creation of simultaneously active subselves or other internal autonomous
actors) as a significant component.


Introduction

The thesis of this paper is that there are certain abstract algebraic structures that typify the
self-structure of human beings and any other intelligent systems relying on empathy for
social intelligence. These algebraic structures, called quaternions and octonions, are
familiar to mathematicians, and also play a critical role in modern theoretical physics
(Dixon, 1994).

The argument presented in favor of this thesis has two steps. First, it is argued that much
of human psychodynamics consists of “internal dialogue” between separate internal
actors – some of which may be conceived as subselves a la (Rowan, 1990), some of
which may be “virtual others” intended to explicitly mirror other humans (or potentially
other entities like animals or software programs). Second, it is argued that the structure
of inter-observation among multiple inter-observing actors naturally leads to quaternionic
and octonionic algebras. Specifically, the structure of inter-observation among three
inter-observers is quaternionic; and the structure of inter-observation among four inter-
observers is octonionic. This mapping between inter-observation and abstract algebra is
made particularly vivid by the realization that the quaternions model the physical

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situation of three mirrors facing each other in a triangle; whereas the octonions model the
physical situation of four mirrors facing each other in a tetrahedron, or more complex
packing structures related to tetrahedra. Using these facts, we may phrase the main thesis
of the current paper in a simple form: The structure of the self of an empathic social
intelligence is that of a quaternionic or octonionic mirrorhouse.

There are obvious echoes here of Buckminster Fuller’s (1982) philosophy, which viewed
the tetrahedron as an essential structure for internal and external reality. And there is a
next step that is even more Fulleresque in nature: the structure of a group of socially
interacting individu als is that of a tiling of part of space using adjacent quaternionic or
octonionic mirrorhouses.

There is also an intriguing potential tie-in with recent developments in neurobiology,
which suggest that empathic modeling of other minds may be carried out in part via a
“mirror neuron system” that enables a mind to experience another’s actions, in a sense,
“as if they were its own” (Ramachandran, 2006). Building on existing data and theories
regarding mirror neurons, we hypothesize a “neural mirrorhouse system” supporting a
mirrorhouse-structured self.

In the remainder of the paper, we will explore these issues from the perspective of
psychology, biology and mathematics, finally returning at the end to a discussion of what
it means phenomenologically, from the internal perspective of the experiencing mind.

The Intrinsic Sociality of the Self

In what sense may it be said that the self of an individual human being is a “social”
system?

A special issue of "Journal of Consciousness Studies" (Thompson, 2001) provides an
excellent summary of recent research and thinking on this topic. A basic theme of
several of the papers is as follows:

? The human brain contains structures specifically configured to respond to
other humans' behaviors (these appear to involve “mirror neurons” and
associated “mirror neuron systems,” on which we will elaborate below).
? these structures are also used internally when no other people (or other agents)
are present, because human self is founded on a process of continual
interaction between "phenomenal self" and "virtual other(s)", where the
virtual others are reflected by the same neural processes used to mirror actual
others
? so, the iteration between phenomenal self and actual others is highly wrapped
up with the interaction between phenomenal self and virtual others

This line of research focuses on exploration of the dynamics by which self is
fundamentally grounded in sociality and social interactions. The social interactions that
structure the self are in part grounded in the interactions between the brain structures

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generating the phenomenal self and the brain structures generating the virtual others. That
is, they are part of the dynamics of the self as well as part of the interactions between self
and actual others. The key point is that human self is intrinsically not autonomous and
independent, but rather is intrinsically dialogic and intersubjective.

Another way to phrase this is in terms of “empathy.” That is, one can imagine an
intelligence that attempted to understand other minds in a purely impersonal way, simply
by reasoning about their behavior. But that doesn’t seem to be the whole story of how
humans do it. Rather, we do it, in part, by running simulations of the other minds
internally – by spawning virtual actors, virtual selves within our own minds that emulate
these other actors (according our own understanding). This is why we have the feeling of
empathy – of feeling what another mind is feeling. It’s because we actually are feeling
what the other mind is feeling – in an approximation, because we’re feeling what our
internal simulation of the other mind is feeling. Thus, one way to define “empathy” is as
the understanding of other minds via internal simulation of them. Clearly, internal
simulation is not the only strategy the human mind takes to studying other minds – the
patterns of errors we make in predicting others’ behaviors indicates that there is also an
inferential, analytical component to a human’s understanding of others (Carruthers and
Smith, 1996). But empathic simulation is a key component, and we suggest that, in
normal humans (autistic humans may be a counterexample; see Oberman et al, 2005), it
is the most central aspect of other-modeling, the framework upon which other sorts of
other-modeling such as inferencing are layered.

This perspective has some overlap with John Rowan’s (1990) theory of human
subpersonalities, according to which each person is analyzed as possessing multiple
subselves representing different aspects of their nature appropriate to different situations.
Subselves may possess different capabilities, sometimes different memories, and
commonly differently biased views of the common memory store. Numerous references
to this sort of “internal community” of the mind exist in literature, e.g. Proust’s reference
to “the several gentlemen of whom I consist.”

Putting these various insights together, we arrive at a view of the interior of the human
mind as consisting of not a single self but a handful of actors representing subselves and
virtual others. In other words, we arrive at a perspective of human mind as social mind,
not only in the sense that humans define themselves largely in terms of their interactions
with others, but also in the sense that humans are substantially internally constituted by
collections of interacting actors each with some level of self-understanding and
autonomy. The primary contribution of this paper is a specific hypothesis regarding the
structure of this internal social mind: that is corresponds to the structures of certain
physical constructs (mirrorhouses) and certain abstract algebras (quaternions, octonions
and Clifford algebras).

Mirror Neurons and Associated Neural Systems

A number of thinkers have tied together the above ideas regarding self-as-social-system,
with recent neurobiological results regarding the role of mirror neurons and associated

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neural systems in allowing human and animal minds to interpret, predict and empathize
with other human and animal minds with which they interact. The biology of mirror
neuron systems is still only partially understood, so that the tie-in between mirror neurons
and psychological structures posited here must be viewed as subject to revision based on
further refinement of our understanding in the biology of mirror neurons. Ultimately, the
core ideas of this paper would remain equally valid if one replaced “mirror neurons” and
associated systems with some other, functionally similar neural mechanism. However,
given that we do have some reasonably solid biological data -- and some additional,
associated detailed biological hypotheses – regarding the role of mirror neurons in
supporting the functions of empathy and self, it is interesting to investigate what these
data and hypotheses suggest.

In simplest terms, a mirror neuron is a neuron which fires both when an animal acts and
when the animal observes the same action performed by another animal, especially one of
the same species. Thus, the neuron is said to "mirror" the behavior of another animal –
creating a similar neuronal activation patterns as if the observer itself were acting. Mirror
neurons have been directly observed in primates, and are believed to exist in humans as
well as in some other mammals and birds (Blakeslee, 2006). Evidence suggestive of
mirror neuron activity has been found in human premotor cortex and inferior parietal
cortex. V.S. Ramachandran (2006) has been among the more vocal advocates of the
important of mirror neurons, arguing that they may be one of the most important findings
of neuroscience in the last decade, based on the likelihood of their playing a strong role in
language acquisition via imitative learning.

The specific conditions under which mirror neuron activity occurs are still being
investigated and are not fully understood. Among the classic examples probed in lab
experiments are grasping behavior, and facial expressions indicating emotions such as
disgust. When an ape sees another ape grasp something, or make a face indicating
disgust, mirror neurons fire in the observing ape’s brain, similar to what would happen if
the observing ape were the one doing the grabbing or experiencing the disgust. This is a
pretty powerful set of facts – what it says is that shared experience among differently
embodied minds is not a purely cultural or psychological phenomenon, it’s something
that is wired into our physiology. We really can feel each others’ feelings as if they were
our own; to an extent, we may even be able to will each others’ actions as if they were
our own (Lohmar, 2006).

Equally interesting is that mirror neuron response often has to do with the perceived
intention or goal of an action, rather than the specific physical action observed. If
another animal is observed carrying out an action that is expected to lead to a certain
goal, the observing animal may experience neural activity that it would experience if it
had achieved this goal. Furthermore, mere visual observation of actions doesn’t
necessarily elicit mirror neuron activity. Recent studies (Buccino et al, 2001, 2004)
involved scanning the brains of various human subjects while they were observing
various events, such another person speaking or biting something, a monkey lip-
smacking or a dog barking. The mirror neurons were not activated by the sight of the
barking dog – presumably because this was understood visually and not empathically

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(since people don’t bark), but were activated by the sight of other people as well as of
monkeys.

There is also evidence that mirror neurons may come to be associated with learned rather
than just inherited capabilities. For instance, monkeys have mirror neurons
corresponding to specific activities such as tearing paper, which are learned in the lab and
have no close correlate in the wild (Rizzolatti, 2004).

Perhaps the most ambitious hypothesis regarding the role of mirror neurons in cognition
is Rizzolatti and Arbib’s (1998) Mirror System Hypothesis, which conjectures that neural
assemblies reliant on mirror neurons played a key role in the evolution of language.
These authors suggest that Broca’s area (associated with speech production) evolved on
top of a mirror system specialized for grasping, and inherited from this mirror system a
robust capacity for pattern recognition and generation, which was then used to enable
imitation of vocalizations, and to encourage “parity” in which associations involving
vocalizations are roughly the same for the speaker as for the hearer. According to the
MSH, the evolution of language proceeded according to the following series of steps
(Arbib et al, 2006):

? S1: Grasping.
? S2: A mirror system for grasping, shared with the common ancestor of human and
monkey.
? S3: A system for simple imitation of grasping shared with the common ancestor
of human and chimpanzee. The next 3 stages distinguish the hominid line from
that of the great apes:
? S4: A complex imitation system for grasping.
? S5: Protosign, a manual-based communication system that involves the
breakthrough from employing manual actions for praxis to using them for
pantomime (not just of manual actions), and then going beyond pantomime to add
conventionalized gestures that can disambiguate pantomimes.
? S6: Protospeech, resulting from linking the mechanisms for mediating the
semantics of protosign to a vocal apparatus of increasing flexibility. The
hypothesis is not that S5 was completed before the inception of S6, but rather that
protosign and protospeech evolved together in an expanding spiral.
? S7: Language: the change from action-object frames to verb-argument structures
to syntax and semantics.

As will be discussed below, one may correlate this series of stages with a series of
mirrorhouses involving an increasing number of mirrors. This leads to an elaboration of
the MSH, which posits that evolutionarily, as the mirrorhouse of self and attention gained
more mirrors, the capability for linguistic interaction became progressively more
complex.


Quaternions and Octonions

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In this section, as a preparation for our mathematical treatment of mirrorhouses and the
self, we review the basics of the quaternion and octonion algebras. This is not original
material, but it is repeated here because it is not well known outside the mathematics and
physics community. Readers who want to learn more should follow the references.

Most readers will be aware of the real numbers and the complex numbers. The complex
numbers are formed by positing an “imaginary number” i so that i*i=-1, and then looking
at “complex numbers” of the form a+bi, where a and b are real numbers. What is less
well known is that this approach to extending the real number system may be generalized
further. The quaternions are formed by positing three imaginary numbers i, j and k with
i*i=j*j=k*k=-1, and then looking at “quaternionic numbers” of the form a + bi + cj + dk.
The octonions are formed similarly, by positing 7 imaginary numbers i,j,k,E,I,J,K and
looking at “octonionic numbers” defined as linear combinations thereof.

Why 3 and 7? This is where the math gets interesting. The trick is that only for these
dimensionalities can one define a multiplication table for the multiple imaginaries so that
unique division and length measurement (norming) will work. For quaternions, the
“magic multiplication table” looks like

i*j = k
j*i = -k

j*k = i
k*j = -i

k*i = j
i*k = -j

Using this multiplication table, for any two quaternionic numbers A and B, the equation

x * A = B

has a unique solution when solved for x. Quaternions are not commutative under
multiplication, unlike real and complex numbers: this can be seen from the above
multiplication table in which e.g. i*j is not equal to j*i. However, quaternions are
normed: one can define ||A|| for a quaternion A, in the familiar root-mean-square manner,
and get a valid measure of length fulfilling the mathematical axioms for a norm.

Note that you can also define an opposite multiplication for quaternions: from i*j = k
you can reverse to get j*i = k, which is an opposite multiplication, that still works, and
basically just constitutes a relabeling of the quaternions. This is different from the
complex numbers, where there is only one workable way to define multiplication.

The quaternion algebra is fairly well known due to its uses in classical physics and
computer graphics (Hanson, 2006); the octonion algebra, also known as Cayley’s
octaves, is less well known but is adeptly reviewed by John Baez (2002).

The magic multiplication table for 7 imaginaries that leads to the properties of unique
division and normed-ness is as follows:

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1
i
j
k
E
I
J
K
i
-1
k
-j
I
-E
-K
J
j
-k
-1
i
J
K
-E
-I
k
j
-i
-1
K
-J
I
-E
E
-I
-J
-K
-1
i
j
k
I
E
-K
J
-i
-1
-k
j
J
K
E
-I
-j
k
-1
-i
K
-J
I
E
-k
-j
i
-1


Actually this is just one of 480 basically equivalent (and equally “magical”) forms of the
octonionic multiplication table (as opposed to the 2 varieties for quaternions, mentioned
above). Note that, according to this or any of the other 479 tables, octonionic
multiplication is neither commutative nor associative; but octonions do satisfy a weaker
form of associativity called alternativity, which means that the subalgebra generated by
any two elements is associative.

As it happens, the only normed division algebras over the reals are the real, complex,
quaternionic and octonionic number systems. These four algebras also form the only
alternative, finite-dimensional division algebras over the reals. These theorems are
nontrivial to prove, and fascinating to contemplate.

Modeling Mirrorhouses Using Quaternions and Octonions

Now let’s move from algebras to mirrors – houses of mirrors, to be precise. Interestingly
shaped houses of mirrors!

Mirrorhouses are structures built up from mutually facing mirrors which reflect each
others’ reflections. The simplest mirrorhouse possible to construct is made of two
facing mirrors, X and Y. X reflects Y and Y reflects X.
One convenient way to describe mirrorhouses is to introduce hypersets, as described e.g.
in (Barwise and Etchemendy, 1989). Hypersets are mathematical sets that are freed from
the Axiom of Foundation, so that unlike an ordinary mathematical set, a hyperset A may
contain A as an element, or may contain a set that contains A as an element, etc. The
general utility of hypersets for modeling complex systems is discussed in (Goertzel,
1994).
In terms of hypersets, a simple 2-mirror mirrorhouse may be crudely described as:
X = {Y}

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Y = {X}
(ignoring the inversion effect of mirroring).
Note that if we try to unravel this hyperset by inserting one element into the other we
arrive at an infinite regress:
Y = {X = {Y = {X = {Y = {X = {Y = {{X = {Y = {...} } } } } } } } }
This corresponds to the illusory infinite tube which interpenetrates both mirrors.
Suppose now that we constructed a mirrorhouse from three mirrors instead of two. What
hyper-structure would this have? Amazingly it turns out that it has precisely the structure
of the quaternion imaginaries.
Let i, j and k be hypersets representing three facing mirrors. We then have that
i = {j,k}
j = {k,i}
and
k = {i,j}
where the notation i={j,k} means, e.g. that mirror i reflects mirrors j and k in that order.
With three mirrors ordering now starts playing a vital role because mirroring inverts
left/right-handedness. If we denote the mirror inversion operation by "-" we have that
i = {j,k} = -{k,j}
j = {k,i} = -{i,k}
and

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k = {i,j} = -{j,i}
But the above is exactly the structure of the quaternion triple of imaginaries:
i = j*k = -k*j
j = k*i = -i*k
k = i*j = -j*i
The quaternion algebra therefore is the precise model of the holographic hyper-structure
of three facing mirrors, where we see mirror inversion as the quaternionic anti-
commutation. The two versions of the quaternion multiplication table correspond to the
two possible ways of arranging three mirrors into a triangular mirrorhouse.

When we move on to octonions, things get considerably subtler – though no less elegant,
and no less conceptually satisfying. While there are 2 possible quaternionic
mirrorhouses, there are 480 possible octonionic mirrorhouses, corresponding to the 480
possible variant octonion multiplication tables!

Recall that the octonions have 7 imaginaries i,j,k,E,I,J,K, which have 3 algebraic
generators i,j,E (meaning that combining these three imaginaries can give rise to all the
others). The third generator E is distinguished from the others, and we can vary it to get
the 480 multiplications/mirrorhouses.

The simplest octonionic mirrorhouse is simply the tetrahedron:



More complex octonionic mirrorhouses correspond to tetrahedra with extra mirrors
placed over their internal corners. This gives rise to very interesting geometric structures,
which have been explored by Buckminster Fuller and also by various others throughout

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history.

Start with a 3-dimensional tetrahedron of 4 facing mirrors. Let the floor be the
distinguished third generator E and the 3 walls be I,J,K (with a specific assignment of
walls to imaginaries, of course). Then, by reflection through the E floor, the reflected I J
K become i j k, and we now have all 7 imaginary octonions. This relatively simple
tetrahedral mirrorhouse corresponds to one of the 480 different multiplications; the one
given in the table above.

To get another we truncate the tetrahedron. Truncation puts a mirror parallel to the floor,
making a mirror roof. Then, when you look up at the mirror roof, you see the triangle
roof parallel to the floor E. The triangle roof parallel to the floor E represents the
octonion -E, and reflection in the roof -E gives 7 imaginary octonions with the
multiplication rule in which -E is the distinguished third generator.

Looking up from the floor, you will also see 3 new triangles having a common side with
the triangle roof -E, and 6 new triangles having a common vertex with the triangle roof -
E.

The triangle roof + 9 triangles = 10 triangles form half of the faces (one hemisphere) of a
20-face quasi-icosahedron. The quasi-icosahedron is only qualitatively an icosahedron,
and is not exact, since the internal angle of the pentagonal vertex figure of the reflected
quasi-icosahedron is not 108 degrees, but is 109.47 degrees (the octahedral dihedral
angle), and the vertex angle is not 72 degrees, but is 70.53 degrees (the tetrahedral
dihedral angle). (To get an exact icosahedral kaleidoscope, three of the triangles of the
tetrahedron should be golden isosceles triangles.)

Each of the 9 new triangles is a "reflection roof" defining another multiplication. Now,
look down at the floor E to see 9 new triangles reflected from the 9 triangles adjoining
the roof -E. Each of these 9 new triangles is a "reflection floor" defining another
multiplication. We have now 1 + 1 + 9 + 9 = 20 of the 480 multiplications.

Just as we put a roof parallel to the floor E by truncating the top of the tetrahedral
pyramid, we can put in 3 walls parallel to each of the 3 walls I, J, K by truncating the
other 3 points of the tetrahedron, thus getting 3x20 = 60 more multiplications. That gives
us 20 + 60 = 80 of the 480 multiplications.

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