Open Journal of Philosophy
2013. Vol.3, No.4, 443-450
Published Online November 2013 in SciRes (http://www.scirp.org/journal/ojpp) http://dx.doi.org/10.4236/ojpp.2013.34065
Open Access 443
A Unified Theory of Mind-Brain Relationship: Is It Possible?
Shashidhar Belbase
College of Education, University of Wyoming, Laramie, Wyoming, USA
Received May 11th, 2013; revised June 11th, 2013; accepted June 18th, 2013
Copyright © 2013 Shashidhar Belbase. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
The mind-body relationship has vexed philosophers of mind for quite a long time. Different theories of
mind have offered different points of view about the interaction between the two, but none of them seem
free of ambiguities and questions. This paper attempts to use a mathematical model for mind-body rela-
tionship. The model may generate some questions to think about this relationship from the viewpoint of
operator theory.
Keywords: Philosophy of Mind; Unified Theory of Mind; Operator Theory
Introduction
This paper argues for possibility of a mathematical model of
operator theory of mind-body relationship as a unified theory of
mind. At first, I would like to briefly review the existing theo-
ries of mind-body interrelation. Different theories of mind (e.g.,
materialism, dualism, and idealism) have their points of views
about correspondence between mind and body. There have
been different points of views offered regarding the mind-body
relationship. Then, I would like to argue in favor of a mathe-
matical model of a functional operator between mental states
and brain states. Finally, I would like to discuss a quantum
operator which is possible between mind-brain functional in-
terfaces.
Philosophers have attempted to give a coherent view of the
mind-body relationship; however, it seems that none of these
endeavors has succeeded in offering a unified explication of
this thorny relationship. We need to look no further than a
broad outline of the four main theories so far expounded under
the categories of monism and dualism to recognize the failure
to provide a unified view (Cf. Anderson, 2003). First, Arm-
strong (1993) has argued for a materialistic theory of the mind,
which maintains that the mind can be explained in terms of a
universal material law. Thinking, reasoning, believing, remem-
bering, perceiving, and understanding, general capacities of the
mind, are nothing but the workings of universal material law.
Some philosophers (e.g., Ward, 1894) have organized this law
in two distinct forms of organization: chemical organization
and biological organization, which are fundamentally based on
interactions and organizations among different material sub-
stances. The materialistic theory of mind assumes that mental
states are reducible to physical states (Robinson, 2012), i.e.,
monism, and concludes that “primitive reality is material”
(Marchal, 2013: p. 2).
Second, some philosophers, perhaps Rene Descartes most
famously among them, have argued that the mind and body are
two separate substances, i.e., dualism. Cartesian dualism as-
sumes that mind and body cannot be explained in terms of a
universal material law because the mind is non-corporeal and
non-physical. According to this theory, the mind is non-physi-
cal, mental substance and the body is physical, corporeal sub-
stance. The body and mind exist separately and have distinct
features with distinct properties (Robinson, 2012). Therefore,
“dualism assumes two fundamental, non-reducible, realities:
mind and bodies, together with some possible associations be-
tween them” (Marchal, 2013: p. 2).
There are basically two kinds of dualism: substance dualism
and property dualism. Substance dualism considers mental and
physical substances as two distinct substances, mental and phy-
sical. Property dualism considers mental and physical proper-
ties as two distinct kinds of properties in relation to mind and
body. Substance dualists claim that mind is non-physical sub-
stance that does not have space extension whereas body con-
sists of physical substance that has space extension. Rene Des-
cartes defended this form of dualism and hence it is also known
as Cartesian dualism. Property dualists claim that there is only
one kind of substance, that is physical or material substance but
these substances have two kinds of properties, one is physical
properties and the other is mental properties (Howard, 2012).
Third, other philosophers, such as Kant, Berkeley, and Hegel,
have emphasized the supremacy of mind over material sub-
stance. Call this, broadly speaking: “idealism”. According to
this view, there is no other realm except the mental realm. On
some naive accounts, there is no mind-independent reality.
Everything in the world can be viewed as one’s mental con-
struct. Proponents of idealism believe in the sovereignty of
mind over physical objects (Heil, 2000). Idealism concludes
that “primitive reality is mental” (Marchal, 2013: p. 2).
Fourth and finally, there is the view that suggests neither the
mind nor the body exists. If the two do not exist, then there is
no need to worry about whether the two interact. What we con-
strue to be either mental or physical is mere illusion. It con-
cludes that “the primitive reality is neither material nor mental”
(Marchal, 2013: p. 2). Call this: “nihilism”.
Hence, the four points of view concern the ontology of mind
and the mind-body relationship. Whereas the four predominant
theories of mind have attempted to provide an argument show-

S. BELBASE
ing how the mind and body interact, very few people have fo-
cused upon the relationship itself. The four views have not
given us any reason to believe that we ought to be in the busi-
ness of discussing the interaction if we have not understood
what the status of this so-called “relationship” is.
In this context, I would like to briefly discuss the correspon-
dence of mind-body relationship in different philosophical do-
mains. Then discuss eliminative materialism and why this the-
ory failed to explicate bidirectional correspondence of mind-
body interaction. As alternative to this view, I propose a mathe-
matical model of mind-brain function space and operators in
the space. I argue that such mathematical model at a basic level
can help us to understand mind-body interaction. The model is
not an effort to solve mind-body problem, but to provide an
alternate approach to look at the problem.
Correspondence of Mental and Brain States
Now I would like to discuss how major philosophical schools
view the correspondence between mental and physical states.
This can be interpreted in different ways in dualism and mo-
nism. Substance dualism claims that mental and physical states
are independent of each other and hence there is no direct cor-
respondence between these two states. Property dualists view
that there is bidirectional correspondence between mind and
body. However, epiphenomenalism rejects the bidirectional
correspondence. This view claims that mental states are causal
effect of brain states whereas there is no effect of mental states
on brain states. Leibniz’s view about mind-body relationship is
not based on causal relationship, but it is a non-causal corre-
spondence between the two (Kulstad & Carlin, 2008).
Functionalism, a version of materialism, claims that mental
and brain states have causal correspondence. This view com-
pares the mind-body interaction with functioning of a computer.
Within this view mind-body relationship can be understood in
terms of their functional interaction as input, process, and out-
come (Block, 1980). The recent version of materialism, elimi-
native materialism, does not accept the bidirectional corre-
spondence between mental and brain states (Feyerabend, 1963).
This view eliminates the mental state. According to this view,
mental state is nothing more than a psychological state and it
has no separate existence from the brain state (Churchland,
1981). I would like to discuss eliminative materialism in a se-
parate subheading.
Eliminative Materialism
Churchland’s (1989) eliminative materialism is an extreme
kind of monistic material reductionism, which recommends that
all folk-psychological talk about the “mind” be eliminated in
favor of a kind of neuro-speak. Paul Churchland (1988) has
clarified the notion of eliminative materialism in the following
way:
[W]hen neuroscience has matured to the point where the
poverty of our current conceptions is apparent to everyone, and
the superiority of the new framework is established, we shall
then be able to set about reconceiving our internal states and
activities, within a truly adequate conceptual framework at last.
Our explanations of one another’s behavior will appeal to such
things as our neuropharmacological states, the neural activity
in specialized anatomical areas, and whatever other states are
deemed relevant by the new theory. Our private introspection
will also be transformed, and may be profoundly enhanced by
reason of the more accurate and penetrating framework it will
have to work with—just as the astronomer’s perception of the
night sky is much enhanced by the detailed knowledge of mod-
ern astronomical theory that he or she possesses (Churchland,
1988: pp. 44-45).
Churchland has claimed that our conception of mental states,
such as beliefs, desires, fears, pains, joys, emotions, and attitudes,
must be re-conceived. We must eliminate such folk-psycho-
logical talk in favor of a more precise alternative derived from a
discussion of neuro-physiological changes. There is no such thing
as private introspection or separate mental phenomena beyond
brain function (Churchland, 1989; Lycan & Pappas, 1972).
The Churchlands’ eliminative materialism may be considered
hyper-reductive, in that, atoms, molecules, and neurons play a
significant part in the theory of mind (Cf. Bickle, 1997). It is to
recognize that “Humans, like everything else in the universe,
were at bottom just organized assemblies of atoms, although the
organization was neither understood nor assumed to be simple”
(p. 240). According to this view, our body is simply an organi-
zation of atoms. This kind of materialistic theory of mind as-
sumes that a living system can be continuously reduced into
smaller sections (e.g., bodies into systems, systems into organs,
organs into tissues, tissues into cells, cells into organelles, or-
ganelles into complex molecules such as DNA and RNA). Ul-
timately, on this view, a theory of mind is reduced to a theory
of brain and the molecules or neurons that make it up.
Eliminative materialism could not explicate the mind-body
interaction in the sense that it failed to provide logic to how
mind is reduced to body or how bodies (material substances)
organized to form mental states or properties. Eliminating a
mental state as a part of folk psychology does not resolve the
issue. If we bring consciousness in relation to mental state,
eliminative materialism has not clarified how the neurons give
rise to consciousness and how the consciousness give rise to
mental states with beliefs and desires. It is not yet clear why
other theories of mind be eliminated before neuroscience itself
provides a convincing explanation for sophisticated mind-body
interaction in a subtle way. Its attempt to reduce theory of mind
into theory of brain, grossly, fails to establish such interactional
correspondence. Therefore, a mathematical model of corre-
spondence between mental and physical states may provide an
alternative to this theory for a broader understanding of mind-
body interaction. For this, I would like to introduce a very sim-
ple mathematical analytical tool to look at correspondence be-
tween mental and physical states.
A Mathema tic al Fu nc ti o n
We can assume mathematical logic as a product of mind and
it may represent a mental state. That means when we have the
logic of interrelation of two things (either physical or non-
physical) we are in a state of mind in which we experience
mental stimulation. The degree of stimulation may depend on
the gravity of the logic in terms of its value and consequence in
our decision within the logic. The logical interpretation of such
and such (e.g., x is bigger than y) is purely a mental. There is
no such thing either relational or un-relational (e.g., x bigger
than y) beyond our perception and mental construction. Our
scheme of “bigger than or smaller than or equal” is a represen-
tational and relational one. When we say x is bigger than y, we
observe them or measure them or perceive them and make a
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conclusion. It also depends on relative states of x and y during
observation or measurement. Our construction of the relation-
ship is subjective and mental even when we claim an objective
measurement. Likewise, we can assume different biological,
chemical, and physical changes in the brain corresponding to
each state of logic. We may have experienced such physiologi-
cal stress in the brain based on our logic of whether we did
something good or bad. The perception of doing something
good or bad is our mental state. We may continue sweating.
Even we may continue feeling of a headache and a stress on
brain. These are real experiences. Therefore, we can see and
feel a correspondence between mental state and brain state.
Sometimes we can think of this kind of correspondence in
terms of mathematical functions.
Now, I can think of a set of all of my mental states constitut-
ing a space Ls (i.e., a space of mental states) and a set of all
brain states constituting a space Bs (i.e., a space of brain states).
Now the question is: Is there a strict correspondence between
Ls and Bs? Whether there exists such correspondence may de-
pend on how the two spaces Ls and Bs interact each other. Cor-
respondence theorists (e.g., Benedetti et al., 2010) accept that
there is such a relationship between mental and brain states.
However, this correspondence can be of two kinds: one is a
strict and strong correspondence and the other is a loose and
weak correspondence. Therefore, for each element of Ls there
is a either strong or weak corresponding element in Bs and vice-
versa. A strong correspondence is strict one-on-one correspon-
dence where a weak correspondence is a correspondence of one
element in a domain (initial space) with more than one element
in the range (the final space). This correspondence can be in-
terpreted as both interactional cause and effect relationship or
without any causal interaction, which is a parallel correspon-
dence. The correspondence of mental state Ls with brain state
Bs shows can be described by function (Figure 1) relations. All
such possible functions constitute a space of functions Fs.
The set of all mental states (or logical states) in the space Ls
and set of all brain states in the space Bs can be interrelated
with the function F either as onto or into. These onto and into
functions can be one-to-one and onto, one-to-one and into,
many-to-one and onto, and many-to-one and into.
One-to-One an d o nt o F un ction
If the function F is one-on-one and onto, then for each brain
state there is a distinct mental state and vice-versa (Figure 2).
We can express this function as F(L1) = B1, F(L2) = B2, F(L3)
= B3, and so on. The function F describes each brain state as a
unique feature of corresponding mental state or each mental
state as a unique feature of corresponding brain state. This kind
of relationship between mental and brain states can be consid-
ered as a consistent mind-body relationship. Each mental state
(e.g., L1, L2, L3, …) is paired with a corresponding brain state
(e.g., B1, B2, B3, …). The order may not be the same but the
pairing is distinct as in the Figure 2. When F is one-on-one and
onto, then F−1 (inverse of F) exists and that maps Bs to Ls. We
can predict one’s mental state once we know his or her brain
state and vice-versa. However, the correspondence of L1 to B1
or B1 to L1 does not mean that they are identical. It is not that
L1 is reduced to B1 or B1 is reduced to L1. Correspondence of
L1 to B1 or B1 to L1 is a non-reductive transformation. In this
sense, this correspondence is different from Armstrong’s (1993)
identity theory.
Figure 1.
Functional model of mind-brain correspondence relationship.
Figure 2.
One-to-one and onto function of mind-brain relationship.
Let’s assume that a person has a mental state of pain on his
sciatic nerve. For each pain state (i.e., a mental state of being in
the pain), he or she has a corresponding nerve problem at the
sciatica and then it is responded by his or her brain with a neu-
ral firing. Every moment he or she has this kind of pain at sci-
atic region, there is one-to-one correspondence between mental
state (with extreme feeling of the pain) and the severe nerve
pressure and corresponding brain state (i.e. a physical state).
The mental states and brain states both have this correspon-
dence through a function F.
One-to-One an d into F un ction
If the correspondence of mental states and brain states is
one-on-one into, then for each mental state in Ls there is a
unique element in brain state within the space Bs. That means
for mental states L1, L2, L3, … there are brain states B1, B3,
B5, … but also there are brain states B2, B4, B6, … not associ-
ated with any mental states. Likewise, there may be the rela-
tions of brain states B1, B2, B3, … to mental states L2, L4,
L6, … but also there are mental states L1, L3, L5, … not asso-
ciated with any brain states (Figure 3). From the view point of
strict correspondence of mind-brain, that is token identity the-
ory in relation to cause and effect, and then there should not be
such state which breaks the one-on-one relationship. However,
physicalism has not yet proved strict correspondence between
mind and brain (Earley Sr., 2008; Hendry, 2006). Neuroscience
has speculation of strict one-on-one correspondence, but it has
yet far from the proof. This shows that there are cases which do
not obey strict one-on-one relationships (Figure 3). It is like
there is a software program (analogous to mind) in a computer
but it has no corresponding hardware (analogous to body) or it
may have some hardware without software to operate it within
the computer. In one-to-one and into case between Ls and Bs
there is no strict correspondence between mental and brain
states. In such a case, either some brain states or mental states
are idle (without pairing) or they do not have active roles in
such correspondence.
Let’s assume that one has a certain body part like a tail (i.e.,
a physical state) by mutation, but this does not have any rela-
tionship to his or her mental state. When he or she is in a men-
tal state of fear and wants to use that part for safety, he or she
has no idea about how to move it and how to use it for safety
purposes. Now the physical state of being with a tail-like part is
not corresponded to a mental state. He or she even cannot think
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of using it. However, similar organ in some other animals (let’s
say a dolphin) can be used for navigation. That means a dolphin
may make sense of direction within water with this part how-
ever a person with such a body part can not.
Likewise, we can suppose that there is a physical entity
named “zeta” which we cannot perceive because we lack the
appropriate sense organ to know about it. A dolphin, however,
can perceive “zeta” because it has, e.g., a “zetan organ”. We
cannot comprehend “zeta” because we lack the “zetan organ”
the dolphin possesses. Just as we cannot experience the zeta,
there may be certain physical states that we do not comprehend
because we lack the ability to sense it.
Many-to-One and onto Function
If a function is many-on-one and onto, then for each brain state
in the domain there are more than one elements in the mental
states, and vice-versa (Figure 4). We can express this function
as F(L1, L2) = B1, F(L3, L4) = B2, F(L5, L6, L7) = B3, F(L8,
L9) = B4, and so on. The function F describes each brain state
as a unique feature of corresponding mental states or each
mental state as a unique feature of corresponding brain states.
This kind of relationship between mental and brain states can
be considered as an inconsistent mind-body relationship. Mul-
tiple mental states are paired with the same brain states. When
F is many-to-one and onto, then F−1 (inverse of F) does not
exist (Figure 4). Most possibly, we cannot predict one’s mental
state once we know his or her brain state and vice-versa.
Let’s assume that one’s mental state is very unstable. He or
she is constantly changing thoughts, beliefs, and values. He or
she imagines different kinds of things. These imaginations are
not consistent with his or her beliefs and values. Then the per-
son may have multiple mental states that correspond to a certain
brain state. This places his or her brain into an immense cogni-
tive pressure. He or she may feel extreme mental pressure and
then the brain simply cannot process all those mental events at
a time. The inverse brain state to mental state function stops
and the brain may stop further working. This may lead to a
serious psychological problem leading to unbalanced thinking.
The same brain state has now many mental states. The person
may be mentally ill.
Many-to-One and into Function
If a function is many-on-one and into, then for each brain
Figure 3.
One-to-one and Into Function of mind-brain relationship.
Figure 4.
Many-to-one and onto function of mind-brain relationship.
state in the domain there are more than one elements in the
mental states, and vice-versa (Figure 5). We can express this
function as F(L1, L2) = B1, F(L3, L4) = B3, F(L5, L6, L7) =
B5, F(L8, L9) = B7, and so on. Although all the elements of the
domain (Ls) seem to be paired with an element in the range
(Bs), but there are some elements (e.g., B2, B4, B6, …) re-
mained unpaired. The function F describes each brain state
either a feature of corresponding mental states or it just does
not involve some elements in Bs in the relationship. This kind
of relationship between mental and brain states can be consid-
ered as an inconsistent mind-body relationship. Multiple mental
states are paired with the same brain states. When F is many-
to-one and into, then F−1 (inverse of F) does not exist (Figure
5). Most possibly, we cannot predict one’s mental state once we
know his or her brain state and vice-versa.
Let’s assume that one’s mind is in very active state. He or
she has several mental states. Despite his or her multiple mental
states at a time he or she may be using the brain partially. The
multiple thought processes in the mind may not have corre-
sponding brain processes as distinct physical phenomena. Many
intelligence powers within the mind are working at the same
time, however the brain (the physical part) may not be suppor-
tive to all the mental functions. Then many mental states may
have correspondence to a few brain states leaving some other
brain states even without relating to any mental events.
Above discussion shows that a specific mental state even
may have corresponding multiple brain states or vice versa.
Then either one mental state may have multiple brain states or
one brain state may have multiple corresponding mental states.
It is not yet known how the function space Fs actually behaves
in the interfaces of Ls and Bs. The function between the mental
and brain state interface should be a very sophisticated one due
to myriad of interactions going on in both directions—from
mind to brain and from brain to the mind. The mental states
within Ls and brain states within Bs are not static. Every mo-
ment they are changing. The dynamic relationships of Ls and
Bs can modify Fs. The space of functions Fs may not be a func-
tion in classical mechanics but it also may represent an instan-
taneous function in continuous flux. Given that it may represent
a quantum state of Ls and Bs, the function becomes a quantum
function.
Any theory of mind in the past has not succeeded to describe
whether such function is specific and certain. In a broader sense,
without limiting Fs either in classical or relativistic or quantum
state it may represent a holistic concept of mind-body corre-
spondence when extended with a set of operators of infinite
dimensions. Because such a condition might exist, we can de-
rive a unified theory of mind in terms of a unified mind-brain
function; however, the boundary of the spaces Ls and Bs might
be an issue in such a model.
Figure 5.
Many-to-one and into function of mind-brain relationship.
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Complexity with Boundary
Let’s assume that one-dimensional space constitutes length-
wise all functions of the mind and body. The perception of a
length or distance between two fixed points through eyes creates
a brain state B1 (with neuron firings) and corresponding mental
state L1 with the consciousness of the length of an object or
distance between two points in the space. The observation of the
length creates a schema of distance between two points in the
brain. This schema then serves as a basis for mental state of
being conscious to the distance. That is why one does not jump
down from a height of ten fit thinking that it would harm his or
her body. This sense of distance in mind as a mental state and
corresponding schema in brain state create such functional rela-
tionship of one dimensional space. Hence, the associated func-
tions could be F1 with one dimension (i.e. distance).
We can assume that two-dimensional space constitutes all
functions of mind and body associated with the area. The visual
perception of an area through eyes creates area schema and a
corresponding brain state B2 (with neuron firings). The visual
schema of area as an extension of two dimensional space gen-
erates an awareness and corresponding mental states L2. This
mental state is exhibited with the consciousness of surface area.
The function that corresponds B2 to L2 or vice versa could be F2
with two dimensions (i.e. length and width). Again, we can
assume that three-dimensional space constitutes all functions of
mind and body associated with volume. The schema of volume
through visual perception to the brain creates a brain state B3
(with neuron firings). The three dimensional spatial distribution
of points in space or three dimensional extension of an object
creates an awareness and corresponding mental states L3 with
the consciousness of volume. The correspondence of B3 to L3 or
vice versa creates a space of functions F3 with three dimensions
(i.e. length, width, and height).
We can continue increasing the dimensions of a space.
However, we have no idea if our brain states have such stimuli to
make schema of dimensions greater than three. Our awareness of
such higher dimensions can continue forming mental schema of
such dimensions. That is what we do in mathematics of higher
dimensions that physically do not exist. The addition of new
dimensions can continue to infinite number (at least theoreti-
cally). Some physical dimensions beyond length, width, and
height can have brain states in terms of other attributes of objects.
However, mentally, we can think of and conceive a space with
infinite dimensions.
The mathematically universal set Ls and Bs have their
boundaries. Is it possible that the sets Ls and Bs have their cor-
responding boundaries? In terms of the continuous flux of
mental and brain states, it seems that these spaces do not have
specific boundaries (i.e. in terms of elements within the spaces).
A logic space including all Ls’s does not have any upper and
lower bounds (i.e. There is no such upper-ness or lower-ness in
mental states). Mathematically, we cannot enlist all the ele-
ments of Ls in a sequence. Even at a certain point of time, we
are unsure of what mental states we are in. It is possible that
there are multiple mental states we are in at any given time.
Enumerating mental states seems to be an impossible task one
ought not to undertake. For example, when we studying for an
exam and trying to memorize or to critically analyze important
concepts, our mental states cannot be individuated and counted.
Although my brain may have a finite number of neurons at a
time, these finite numbers of neurons may have uncountable
number of brain states. The brain states are not simply the states
of individual neurons, but these states can be considered as
ongoing physiological, chemical, and physical processes or
changes. It may not be possible to segregate different brain
states at the same time. Our brain functions in a very complex
way. It can process all perceptual and intuitive information at a
time. That means it can process visual, auditory, olfactory, and
other retrospective and prospective information at the same
time. It may come up with a complex brain state. Hence we
cannot expect a clear-cut boundary of sets Ls and Bs, which
means these states are fuzzy. The fuzziness of Ls and Bs gives
impetus to interpret the mind-body relationship in alternative
ways. However, a mathematical model of mental and brain
states seems a less fuzzy and, therefore, more palatable inter-
pretation of the mind-body interaction.
Reconceptualizing the Functional Model
I discussed possibilities of different mental states and brain
states in the forms of correspondence relationships. A mathe-
matical functional model of the mind-body relationship is a
plausible model because it rests on how Fs is characterized and
positioned within different philosophical schools. Idealists,
dualists, and materialists all accept some tenable position com-
mon in discussions of the mind-body relationship. The debate,
however, if my argument is sound, lies in how they understand
functions in Fs.
For an idealist, Fs is nothing more than a function in domain
Ls with the range Bs where any brain activity is a function of
mental activity. The supremacy of Ls over Bs through Fs could
be established by characterizing the function space Fs. Like
wise, for a materialist, Fs is nothing more than a function in
domain Bs within a range Ls and any mental activity is a func-
tion of brain activity. A predominant theme in the work of neu-
roscience is understanding how the brain functions in a way
that generates certain mental states (if any).However, they do
not seem to recognize the importance of a function when they
discuss the body and mind. In my understanding, they have
failed to characterize function space Fs or even acknowledged
that such a function space Fs exists as a model.
For a dualist (both substance and property dualists), Fs is
nothing more than a function space in domain Bs with the range
Ls or vice versa. In both cases, Fs may not be equivalent on two
independent theories of mind because they are quite different
functions due to nature of direction of fit from Ls to Bs or Bs to
Ls. For each and every traditional position in the theory of mind,
idealism, materialism, and dualism, the main problem lies with-
in the understanding nature of functional space Fs.
Despite the complexity of Fs, the functional space of Fs may
yield fruitful results for the mind-body relationship. The one-to-
one onto, one-to-one into, many-to-one onto, and many-to-one
into functions constitute basic mathematical relational proper-
ties of domains and ranges in terms of Bs and Ls within the
function space Fs, which may provide a basis to generate a
mathematical functional model of the mind-body relationship.
Modern neuroscience uses brain-scanning technology (e.g.,
fMRI) to study brain states. This technology is still not devel-
oped well enough for us to say that these scans may form an
accurate prediction of such states. For example, we cannot gen-
eralize from the results of a brain scan to what Ls and Bs are in
ordinary conditions. We may conclude that Ls and Bs merely
seem to be just one aspect of the mind-body relationship.
We can purport that both Ls and Bs are real or complex do-
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mains and ranges of functions within a logical space Fs; we can
come up with a complex set of all possible functions in Fs
(space of mind-brain functions) that complicates the model. F1
is a brain function that maps L1 to B1, F2 is another brain func-
tion that maps L2 to B2, and we can continue this way to form
an innumerable number of brain functions in the space Fs.
There is no simple model of function F of mind-brain corre-
spondence that can represent a holistic concept of the mind-
body relationship. The non-reductive transformation of brain
functions in terms of a mathematical model with a set of func-
tions Fs mapping from Ls to Bs or vice versa is never a com-
plete one. Hence, the unified theory of mind in terms of func-
tion F of mind-brain may not help us understand the mind and
body relationship in a complete sense. Then we need to con-
ceptualize a new unified theory of mind in conjunction with a
broader operation on the functions in Fs.
Conceptualizing an Oper a t o r for Functions
The mathematical functional model of mental states in Ls
and brain states in Bs with functions in Fs shows that a unified
theory of mind is possible if all the functions are brought to-
gether with an operator or operators. The mind-brain corre-
spondence function model is a mathematical model that may
include all possible functions Fs of brain states in Bs and men-
tal states in Ls within a set of operators U. Here, an operator,
such as U, is a mathematical device for a change in the function,
in Fs, to which it is applied. We can apply different kinds of
operators in relation to mathematical functions. The simple
operators are addition, subtraction, multiplication, and division.
More complex operators are differentiation, integration, and
transformations. In logic, operators are and, or, negation, etc.
Each operator employed alters the behavior of the function and
gives end results different from the original function. The set of
operators U is a model that operates on the functions Fs. Sup-
pose an operator “P” in the set U can transform a function in Fs
with a brain state in Bs and a mental state in Ls into a third state,
a “deton” state Ds (a form of life-state). The operator P in U (as
a unified function operator for all Fs) can be used to translate
both brain state and mental state into D. D is a complex life
state. The set of operators, U, has, both Bs and Ls as input to
produce D as output. Accordingly the functions with the opera-
tors can be considered a holistic theory of mind because U in-
cludes all kinds of operators within the space of functions Fs.
Mathematically, the set of operators U is a modulator of all
functions, either one-to-one or many-to-one or onto or into
functions. We can discuss the model with an analogy to the
Hilbert space.
An Analogy to the Hilbert Space
Analogizing the mathematical functional model set out above
to Hilbert space would suffice to describe the operator model of
mind-brain functions. A Hilbert space (in mathematics and
physics) is an infinite dimensional space of inner products of
vectors. Hilbert space is useful to express the status of mind-
body system or relationship digitally (Kapteina & Zhang, 2008),
since this space has been used in physics to solve body-body
interaction problems. However, there may be a possibility to
apply this space to solve mind-body problems. We can assume
that the mental events that compose mental space is a vector
space including mental states. The mental states as a vector state
can be realized in terms of differentials of mental states with a
parameter of time. We can imagine an infinite dimensional
operator P in U within a functional space Fs. The set U is a set of
operators in Hilbert space. The functional space can be an inner
product space of Bs(t) and Ls(t) where t is a reference point in
time. The vector space connotes a physical space but in
mathematics it is an ideal space where we can assume both
mental and physical vector spaces, then Fs: <Ls(t)|Bs(t) > →
Ds(t). The inner product space of mind and brain corresponds to
the space of all functions of mental states Ls to functions of brain
states Bs.
There are different linear and nonlinear operators in Hilbert
space to interpret the nature of different kinds of inner product
space. In the mind-body relational space of functions Fs, we can
work out analogous operators. These operations have not yet
been discussed in the literatures of philosophy of mind and even
neuroscience. Now the research in these fields can come up with
such operators that can represent interrelation of elements of an
inner product space of mind and brain to deton states. Hence, we
can imagine a multidimensional (even infinite dimensional)
operator P in U for a holistic understanding of mind-body rela-
tionship. This kind of model opens the possibility that there can
be different operators to describe unique combinations of men-
tal and brain states. This multidimensional operator theory
seems to present a unified theory of mind, an alternative to
Churchland’s eliminative materialism (1989) and Stern’s matrix
theory of mind and brain (1992).
Quantum Operator in Mind-Body Function
Our mental states and brain states function within the world
of uncertainties. These uncertainties arise due to changes in the
time, space, and environment with which both mind and brain
interact. This change is continuous. Therefore, any two mental
or brain states are not identical. Our mind changes every time
and we have never ever experienced two identical thoughts.
Likewise, our brain states are changing within every moment
due to change in the supply of oxygen and nutrients and also
different perceptual and introspective phenomena. Such state of
flux of mental state and brain state relates to quantum phe-
nomenon of mind-body relationship. Some scholars, for exam-
ple Goertzel (1993), tried to relate mental states with quantum
physics. He stated that “according to quantum physics, no
physical entity is ever in a definite state; the most one can ever
say about a given entity is that it has certain probabilities of
being in certain states” (p. 133). When we consider differential
states of mind and brain with time, Heisenberg’s famous un-
certainty principle might be applicable to describe such rela-
tionship.
The relative change of mental states


Ls and brain states


Bs can be used as analogous to two components of
Heisenberg’s uncertainty principle. The degree of change in
mental states and brain states can be very small but their prod-
uct might be greater than a very small possible differential
value such as half the Plank’s constant divided by pi. Mathe-
matically,




LsBs M

 and M might be equivalent to
h/(2π) where h is Plank’s constant. This relationship is analo-
gous to Heisenberg’s uncertainty principle for the momentum
state and position state of a particle. A question may come in
relation to what does this product really represent. Can it be
applied to mental states and brain states? To me, mind as such
has no separate existence out of our body. Then, it should be
something associated with the brain or body. When we take it
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as a part of the body that continues to exist until this body ex-
ists, then it should have an existence in the form of a physical
entity either as matter or energy or some form of synergy
(combination of matter and energy). This synergy may be the
actual function Fs that transforms mental states to brain states
or brain states to mental states. There is always change in this
synergy with


SLs where M is analogues
to h/(2π) in Heigenberg’ principle. When the change in this
synergy is zero, then the mind and brain both may possibly
collapse. All the functions of mind-brain states with the space
Fs can operate in a different way at different situation relative
to time and space. The operator P within the set of operators U
may carry different kinds of transformations in Fs and result
into different deton states. We can assume this operator as a
quantum operator because it operates on mind-brain functions
based on probabilistic modulation of Bs and Ls to Ds. Here, F
is mind-brain function of inner product of mental state and
brain states. After modulation by an operator P on F, the system
within the synergy produces a life state (i.e., deton state) D.
Hence the life state of a person is the result of different quan-
tum operations on mind-brain functions in Fs. This model may
have some empirical implications for future research in mind-
brain relations.

Bs M
Some Implications
The operator P in U can be thought of having multiple di-
mensions and hence it can modulate any kind of functions in
space Fs into deton state D. We ought to consider some impli-
cations of the unified theory of mind-brain functional model.
For example, we can consider deton states from one-to-one into
and onto functions when operated through any element P of U.
Since the input function converts all brain states into unique
mental states and vice versa, the modulated life state with an
operator P might be a normal life with a consistent mind-brain
relation.
The case of many-to-one onto and into functions when
modulated with the operator P in U might result in quite differ-
ent deton states than the one discussed earlier. Here the one
mental state may have many corresponding brain states or vice
versa. In such a case the modulation of functions in Fs with the
operator P in U can produce deton states either seriously dis-
turbing mental states or brain states. The modulation of func-
tions that correspond one mental state into multiple brain states
can result in a life state that is full of diverse behaviors, e.g., the
person may act akratically. Hence, knowledge of different func-
tional interactions between mind and brain and influence of
operator P on these functions might help in understanding the
complex mind-body problem.
Challenges
There are some issues in relation to the model with a set of
operators in U as a unified theory of mind-brain relationship.
Ontologically, if Ls is purely non-physical and Bs is purely
physical, then the functions Fs and its space that relates Ls and
Bs might be challenged. Epistemologically, a mathematical
function in Fs and its space must be interpreted as a fuzzy space.
The non-physical nature of Ls makes the functions Fs as an
inner product space of Ls and Bs fuzzy. The operator theory
with operators in U might have fuzzy nature due to very subtle
correspondence of mental states in Ls and brain states in Bs.
Alternately, if both Ls and Bs are physical, then the model with
functions Fs as inner product of Ls and Bs vectors and the op-
erators in U seem more viable. Since both Ls and Bs are physi-
cal states and can be modulated with operators in U on func-
tions in Fs, then it can be seen as parallel to infinite dimen-
sional Hilbert spaces. There will be a direct analogy of space of
functions Fs with the Hilbert space. Maybe, the function space
Fs may have better interpretability and applicability analogous
to other spaces beyond Hilbert space (e.g., Probability space).
Philosophers can work with mathematicians, neuroscientists,
and physicist to uncover more about such spaces within which
all mental states and brain states can be better understood and
predicted in terms of mathematical models of observables and
state variables. More properties and interrelations analogous to
Hilbert space and others have to be studied further in order to
develop such a unified theory of mind and brain. This model
has many implicit and explicit challenges of identifying the
appropriate mode of functional correspondence between mental
states in Ls and brain states in Bs through an operator in U.
New Direction
The fluctuation of mental states in Ls and brain states in Bs
may or may not be predicted by a model depending on the na-
ture of the model and associated variables within it. The pre-
dictability of mental or brain states depends on the strength of
the functional model and associated operator. The functional
model of mind-brain relationship has to go through this exami-
nation. If a model functions well with a wide range of possibili-
ties, then it could be a viable model, though it may not repre-
sent the reality as it is. A model is not a reality. It is only a way
to describe the relationship between state of mind and brain in
this case. Operator model on functions within a space of mental
and brain states analogous to Hilbert space may open a possi-
bility of new interpretation of mind-body relationship. None-
theless, this model needs further attention from philosophical
and scientific community either in the similar form or a differ-
ent takeover.
Acknowledgements
I would like to acknowledge Joseph Ulatowski for providing
constructive feedback and comments on an earlier draft of this
paper, which was written for his class, Topics in the Philosophy
of Mind. I would like to thank anonymous reviewers for en-
couraging comments/feedback to the manuscript.
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