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A Psychological Critique of Mathematicians

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In this paper, we'll examine engrained thinking habits in even the most powerful human intellects that contributed to existential risks now confronting life on Earth. They also relate to limitations of mathematical logic and algorithmic computation. Finally, we'll consider unfreezing deeper natural abilities to help avert the danger.
A Psychological Critique of Mathematicians
RW Boyer, PhD
In this paper, we’ll examine engrained thinking habits in even the most powerful
human intellects that contributed to existential risks now confronting life on
Earth. They also relate to limitations of mathematical logic and algorithmic
computation. Finally, we’ll consider unfreezing deeper natural abilities to help
avert the danger.
Key terms: Gödel’s theorem, Turing test, Omega theorem, the I’, transcending, consciousness
In my scientific writing, I usually minimize the use of personal pronouns. Although a bit
awkward, it seems consistent with attempts to convey at least an impression of sober impersonal
‘objectivity’. After all, isn’t science about ‘objective knowledge’ uncontaminated by personal
‘subjectivity’? I’m comfortable with an impersonal style because it reads as less egoistic less
‘me’ as the story, directing attention toward the content. But this paper applies a more personal
style, for a reason. Part of the reason is a trend to ‘real world’ narratives away from impersonal
‘just the facts’ styles. Though this may seem superficial, it reflects major developments toward a
new science of mind recognized, for example, in physicist Bernard Kastrup’s1 recent article in
Scientific American, “Physics is Inexorably Pointing to Mind.” I mention this not just to explain
the more personal style of this paper, but for the reason that it is the key issue: the overlooked
direct experience of ‘I’.
This key issue also relates to long-time frustrations I’ve had communicating with
mathematicians. I honor the contributions of mathematicians and consider them to have the most
powerful intellects. But as we’ll consider, the need is to gain more practical knowledge – an issue
reverberating through the history of philosophy, science, and mathematics. Things have gotten so
far out of hand in applications of mathematical science, due to engrained thinking habits, that
ominous existential risks to human life on Earth are immediately at handsuch as chemical and
electromagnetic pollution of our external and internal environments; nuclear, bio, and cyber
weaponry; and potentially uncontrollable genetic reengineering and AI robots. Though ‘hard
science’ has contributed to many practical solutions, it also has led to even harder problems now
imminent. This is the consequence of not utilizing all the natural levels of our minds as human
My background is cognitive science and counseling psychology (considered ‘soft science’ at
best). I’ve carefully listened to and observed behavioral expressions of inner feelings and
thoughts to explore systematic means for higher human development. Doctoral training in
psychology and writing psychological critiques were recommended to me by a teacher I consider
the most important in my life, leading educator in consciousness studies Maharishi Mahesh
Yogi. Pursuing his recommendations, this ‘psychological critique’ is to get to the essential
element missing in Western and overlooked in Eastern intellectual traditions directly relevant
for mathematicians.
To begin with an example, a few years ago while discussing infinity with a brilliant
mathematician I was again frustrated by another instance of the habit of mathematicians to
objectify discussions and avoid consideration of the inner dynamics of their own thinking. So,
somewhat jokingly, perhaps imprudently, I suggested that mathematics needs an “I” axiom. This
didn’t seem to go over well coming from a ‘fringe scientist’ illiterate in the language of
mathematics. But maybe, this ‘outside the box’ approach helped my own understanding – to zero
in on what’s been missing.
Setting the stage to examine ‘the overlooked direct experience of ‘I
Toward deeper understanding of how to unfreeze engrained thinking and deepen our knowledge,
it is widely held that reasoning and experience are the two primary means to gain valid
knowledge. Basically, reasoning involves rational thinking, logic, and understanding; and
experience involves experiencing the world through the organs of sensory perception and action.
There is a core distinction in this between subject as inner ‘observer’ and observed as outer
‘object’. It relates to subject/object independence (once held essential to science, but challenged
by recent advances). Also associated with mind/body duality, it is a core phenomenal feature of
the ordinary waking state of consciousness in which modern science and mathematics largely
have been practiced.
One basic point long overlooked is that reasoning and experience are subjective. This concerns
the contrast between the ‘first-person perspective’(1st-pp) held to be subjective, unreliable and
unscientific; and the ‘third- person perspective’ (3rd-pp) held to be objective, reliable, and
‘scientific’. But there is no observing ‘outside’ of an observer. 3rd-pp entails a 1st-pp; there is no
3rd-pp that is not also more fundamentally a 1st-pp. So, what distinguishes 1st and 3rd-pps? The
3rd-pp is what an observer attends to that is considered ‘separate’ and ‘outside’ of her/his self.
3rd-pp ‘objectivity’ is based on what ‘outside’ observers agree on together. If each individual
observer’s intra-subjectivity agrees with other individual observer’s intra-subjectivity
agreement across individual observers then there is inter-subjective agreement. This supports
the efficacy of empirical experiences as ‘objective’, through broad consensus not just private
perspectives alone. Use of measurement and calculation tools already agreed-on as reliable
strengthen the consensus. (Of course, some 1st-pp could be more accurate than a widely accepted
3rd-pp consensus; and also of course, science is not a democratic majority consensus to validate
scientific facts and laws.)
As to its relevance to this ‘psychological critique’, I’ll describe a thinking pattern I’ve noticed to
be fairly common among mathematicians. But first, I should note that the pattern isn’t unique to
them, and doesn’t apply to all of them maybe you also if you are a mathematician. Here are
two anecdotes to exemplify this particular point. When a new brochure on bipolar disorder was
placed in the lobby of the mental health center where I worked, every client who came in to talk
with me that morning read the brochure, and each one ‘self-diagnosed’ as bipolar none were.
Also, in teaching abnormal psychology I explain the ‘student syndrome’ that in first learning of a
disorder, some students begin to think they have it (you can sense the increased anxiety in the
The pattern I’m referring to includes tacit assumptions that subjective processes of reasoning
somehow become ‘objective’ in mathematics; and that empirical experience somehow is
essentially ‘objective’ science uncontaminated by subjectivity. But neither reason nor experience
are outside subjective mind. Isn’t the inconsistency in this obvious something fundamental
missing? Let’s consider behavioral aspects of the pattern, then deeper psychological aspects with
profound implications about the current fragmented state of human knowledge and of life on
Here's an anecdote to illustrate this next point. While a physicist/math teacher and I were
traveling cross-country, we detoured to visit a national park. As we entered the park and he first
glimpsed its grandeur, he spontaneously reflected, “I’m having a very unscientific feeling right
now”. Delighted by this uncharacteristic self-discloser, I’ve much enjoyed his progress opening
to deeper feelings, and to our deepening friendship. I bring this up to identify further the pattern
as including emotional distancing less attentive to or expressive of feelings, more an
‘intellectualizing’ mode. This can involve missing cues that could aid communication
sometimes interpreted by others as being insensitive, or preoccupied. The pattern also frequently
involves deflecting discussions that challenge engrained thinking habits. (Perhaps just
circumstantial, or perhaps indicative, in my counseling work with thousands of people, I don’t
recall a single mathematician as being a client.)
I would characterize the pattern as focusing on discriminative rather than unifying processes.
This isn’t at all to take away from the crucial value of intellectual discrimination. It’s to
emphasize integrating discriminative and unifying aspects of the intellect very important and
useful for each one of us. Let’s now examine deeper aspects of the pattern, and the principle of
‘going outside’ the intellect to levels of mind deeper ‘inside’ that are quite relevant for
Gödel’s Incompleteness Theorem
Kurt Gödel’s profound insight into the inherent limitations of mathematical systems is relevant
to understanding the key issue in this paper. The Incompleteness Theorem holds that
mathematical systems complex enough to include simple arithmetic can be consistent or
complete, but not both at the same time. Implications of this limitation are explored, followed by
limitations of mechanistic systems related to Alan Turing’s and recent other work, and later to
the practical significance of unfreezing engrained thinking habits and accessing more
fundamental human abilities. We will consider that the sense of consistency and self-certainty
fundamental to mathematics come from ‘outside’ the rules of mathematical systems. To address
this foundational issue, we’ll start with a simple but subtle framework described by philosophers
Jonathan Shear and Neil Sim2 (pps. 1-3), who state:
Gödel himself…felt strongly that his theorem, combined with Turing’s theoretical work on
computers, should yield the result that
Either there exist infinitely many number theoretic questions which the human mind is unable to
answer or the human mind contains an element totally different from a finite combinatorial
mechanism (such as a nerve net acting like an electronic computer). I hope to be able to prove on
mathematical, philosophical, and psychological grounds that the second alternative holds.
In recent years Roger Penrose… [argued]… that Gödel’s and Turing’s theorems do in fact imply that
the human mind can think in mathematically useful ways that transcend algorithmic computation in
general, and those made by algorithmic mechanical computers in particular…[and] that the
implications of Gödel's theorem…establish that there must be a noncomputational ingredient in
human conscious thinking… [T]his non-computational step of going outside algorithmic rule-based
combinatorial operations, and evaluating something semantically rather than merely syntactically…
leads Penrose to say that
human minds can reflect on the meanings of the symbols in a formal system, and move from that
system to a new system outside of it that has new rules capable of doing useful things the original
system cannot do. But a computational system cannot…it just has the rules it already follows.
Shear and Sim 2 (pps.7-15) examine this basic distinction using the simple framework of the
“I-it” structure of intentionality” [involving] (i) one’s bare awareness at the “center” or “here”…with
(ii) attention oriented outward towards…objects that might appear “there” in it. The “I- pole”…is…
the locus of one’s point of view…[and] objects…in this space…[are] “its”… The “eye” (or “I”) of the
mind, is always outside the field of its objects, and thus doesn’t need to go anywhere to be outside…
What is shifting from inside the system to outside is not any mathematical entity…only one’s
attention. And shifting attention in this way is…how the mind naturally operates…
Shear and Sim summarize implications of this “I-it” duality associated with ordinary experience:
Gödel’s proof was revolutionary in showing that mathematics cannot be completed deductively from
finitistic systems alone. But it was also revolutionary both (a) in relying on the “going outside” step
and (b) including explicit reference to it in the proof itself… With this, reference to the
mathematician’s mind became part of the objective structure of mathematics itself… The shift of
attention involved in the “going outside the system” move of Gödel’s proof, in other words, is simply
an instance of a general, pre-mathematical, non-verbal mental ability to shift attention in all sorts of
ways within one’s awareness… Gödel’s incompleteness proof thus appears to imply two very
different, complimentary kinds of incompleteness with respect to algorithmic systems. On the one
hand, moving “outward” from the proof to the theorem and its consequences, it shows that no
algorithmic system can generate all the truths of arithmetic, much less mathematics in general. And
moving “inward” from the proof to its source, it implies that no such system can model the range of
mathematical thought necessary to apply, much less create and understand, the proof’s crucial “going
outside the system” step. In other words, no such system can properly model the “I-it” structure and
the full range of its potential content.
Elaborating on these points, Gödel’s Theorem implies that the bottom line of the human ability
to gain knowledge is not mathematical logic and intellectual reasoning. There are pre-
suppositions and assumptions not completely verifiable via the senses, rational thinking, or
intellectual reasoning, from deeper ‘inside,’ about how the world is. The fundamental inner felt
sense of consistency, rightness, and self-certainty go ‘outside’ the logic of the discriminative
There are levels of mathematical structures whole numbers, integers, rational numbers, real
numbers including irrational numbers, imaginary numbers, complex numbers. But mathematics
also relates to such things as axioms, undefined quantities; principles of simplicity, elegance,
symmetry, balance in nature, and belief in the comprehensibility and inevitability of laws of
nature. None of these are verifiable alone by the senses or by logical proofs – frequently they’re
attributed to be self-evident. These deep inner feelings are further underlain by a sense of
knowingness that I am the knower, that I am, that I exist; and that I am conscious. 3
The bottom line of our ability to gain knowledge is not the outer-directed senses they cannot
reveal deeper levels of mind, including intellect. In turn, mind and intellect cannot reveal
consciousness itself that is needed for sensing, thinking, reasoning, and feeling. The deeper
levels are inner aspects of the ordinary functioning human mind. They are natural phenomenal
experiences underlying the intellect, associated with being conscious of being the inner “I”
(subject) that is aware of outerits” (objects). The “I” or individual self naturally can direct
attention to go outside mathematical systemsto other outside sensory “its(objects), and also
to go insideto thoughts, feelings, memories, and to the inner sense of self. We’ll soon extend
this discussion to the crucial issue of how to unfreeze engrained thinking patterns.
Turing Test
Alan Turing4 also sought to clarify limitations of algorithmic processes, and suggested that
computational functions themselves cannot completely replicate the full range of human abilities.
The concept of the ‘Turing test’ is frequently applied to compare AI capabilities to natural
human capabilities. If behavioral outputs of AI systems cannot be distinguished from natural
human behavior, then is this not prima facie evidence that there are no fundamental differences?
Further, can they not be equipped via AI technology with even superior abilities (after all, nature
built us, so it has been done)? Sophisticated robotic machinery with more complex algorithmic
tools out-perform natural human information processes, with potential for much more dramatic
progress. In the face of this progress, questions are arising whether humans have sufficient
ability to handle the rapidly increasing complexities of life on planet Earth. Some claim that AI
systems have already advanced to the ‘point of no return’ (the ‘Singularity’) when humans can
no longer stop AI systems from dominating human civilization. Some even propose that more
advanced AI should control human civilization to avoid crises on Earth caused by human
This is an extremely important concern about the materialistic, physicalist worldview. 
 objectify           
                  
        matter
doesn’t have a material basis. Scientific realism is now extending beyond the physical. However,
there’s great reluctance to accept a non-physical basis of matter, though quantum and unified
field theories certainly can be interpreted as implying it, and are progressing toward it.3
This brings up deep questions about what constitutes life, mind, and consciousness now
prominent in modern science. Will AI machines develop the ability to think, feel, and be
conscious? Or. are there features of human life not accountable for within a closed mechanistic
physical world? What are the appropriate criteria to pass the Turing test? How far down the
physical time and distance scale is required to match human functions well enough to conclude
robots actually do think, feel, and have consciousness and thus should be afforded the same
legal rights as humans? Wouldn’t AI need to go much deeper than macroscopic neural nets in the
brain/body to cellular, molecular, chemical, elementary particle, atomic, sub-atomic, and
quantum field levels of nature? But deeper layers are held to be inert, insentient, random and
indeterminate, whereas human thinking, feeling, and consciousness are certainly not. Even all the
way down at the bottom line of physical spacetime at the Planck scale, the ground state of the
physical universe, nature is held to be random quantum fluctuations; and conscious mind has not
been found on any physical level only its correlates. It further is now being argued that
randomness is a conditional view from within the physical system, and there are causal
influences on the purported closed physical system from outside it a deeper creative aspect
appearing to be random but coming from some transcendent source in a much larger orderly
universe comparable to ‘going outside the system of mathematical logic’ as in Gödel’s
Theorem. This relates to the next point about limitations of physical systems.
Omega Theorem
The Omega Theorem is a significant step in distinguishing artificial computational systems and
natural human systems. Proposed by Cathy Reason and Kushal Shah,5 6 it is a simple approach to
a mathematical proof that computational systems built of physical parts cannot have the ability of
‘self-certainty,’ or feeling of rightness 7 which humans do have, as naturally validated in our
own ordinary empirical experience. ‘Self-certainty’ refers to “being able to assert its own
conscious experience” 5 (p. 105). Reason and Shah state that
Conscious macrostates are usually assumed to be emergent from the underlying physical microstates
comprising the brain and nervous system of biological organisms…[W]e show that conscious
macrostates are not emergent from physical systems and they also do not supervene on physical
microstates 5 (p. 103).
To summarize for brevity here, after defining key terms, Reason and Shah state 5 (pps. 111-112):
The proof of this theorem can also be expressed in terms of a couplet of statements:
1. S(t) [a physical system] cannot determine that the function [self-certainty] has been performed
without first determining that the process which performs it is physically real;
2. S(t) cannot show that the process which performs (self-certainly] is physically real without first
performing the function…
These two statements together generate an infinite regress… Therefore, if a physical system is unable
to perform self-certainty according to some model, it will be unable to perform self-certainty
according to any model supervening on that model.
Theorem 1b: No theoretical model of a system which supervenes on any physical system S(t), which
is capable of human-like reasoning (assumed to be sound in principle), can perform the self-certainty
function… and neither can any such system be recorded by any external device as having asserted ‘I
am self-certain’.
The theorem is consistent with the need for deeper views of nature to address the human mind
and consciousness. Gödel showed that human experience goes beyond mathematical logic,
Turing asserted that computational systems themselves cannot fully replicate human abilities,
and the Omega Theorem shows that physical systems cannot include the human ability of ‘self-
certainty’. These clearly imply there are aspects of human nature ‘outside’ mathematical
systems, ‘outside’ the discriminative intellect, ‘outside’ ordinary reason and experience,
‘outside’ the brain/body, and ‘outside’ the ‘physical’. They point to levels of nature that
transcend algorithmic and physical computational systems. It is instructive that Plato considered
mathematical objects to be ‘Universal Forms’ that exist as objectively real in nature. He held that
they were not just in individual minds, but exist in a more abstract level of reality beyond our
ordinary familiar physical world. Where would such forms be, and how could we access such
deeper aspects of nature to open up more fundamental natural abilities to gain more universal
practical knowledge?
‘Going Outside Mathematical Logic’ and Transcending the Intellect
Referring back to the “I-it” framework of ordinary experience, Shear and Sim2 (p. 5) explain:
The seeming impossibility of giving clear empirical sense to the “I,” generally taken for granted by
Western philosophers since Hume and Kant, might well thus seem to render the experiential sense of
the “I-it” structure utterly problematic. Nevertheless, despite this conceptual problem, the structure
remains obvious to ordinary experience, and basic to Western phenomenology… Eastern
phenomenological investigations however suggest that we can address this problem
directly… to facilitate development of various mental states and abilities…in meditation
procedures… Put simply, a major idea underlying these meditation procedures is that
removing all the phenomenological objects that ordinarily absorb our waking attention can
allow basic, but otherwise “overshadowed” features of consciousness to emerge… Thus, for
example, Maharishi Mahesh Yogi, founder of the Transcendental Meditation program,
describes it in the language of Yoga and Vedanta as follows:
When the object of experience has diminished to the point where it has disappeared, the mind
ceases to be the experiencing mind… But during this process of transformation, it first gains the
pure state of its own individuality… When the mind gains this state…It holds its individuality in
the void—the abstract fullness around it—because there is nothing there for it to experience. It
remains undisturbed, awake in itself…
Now we’ve arrived at the crux of the subject: how to unfreeze levels of mind underlying
intellect. In the ancient Vedic tradition of Yoga, there is a correspondence between ontological
levels of reality and epistemic experiences of them. The macroscopic sensing system matches
macroscopic physical layers. When such experiences predominate and there are no or only
fleeting deeper experiences, the ordinary belief is that the local physical domain is the only real
domain of nature. This is the result of lack of deeper refinement and development of mind, which
Yoga practice is designed to alleviate. The inner “I”, self, or ego is not eliminated or annihilated
with higher development, but rather refined and developed to its universal value, through
transcending to its basis in consciousness itself. Systematic, reliable means to transcend develop
their clarity. Without regular transcending to refine mind and body, there may be only marginal
progress throughout an entire life-span, with few clear experiences of “I” and the unified state of
pure consciousness itself. This underlying, transcendent level, deeper ‘inside’ the logical
discriminating intellect and the individual self or ‘I’, is easily open to direct experiential
validation. 3
The inner intuitive sense relates to the sense of consistency and self-certainty fundamental to
logic and mathematics. This deep conviction is strengthened by the consensual validation of
teachers, colleagues, and other historical authorities important in knowledge training, as well as
consistent experience in successfully applying the principles empirically in our daily life. But at
a deep experiential level, it is a subtle inner feeling of almost unquestionable clarity and certainty
that something is self-evident, self-consistent, inherently reasonable, sensible, even necessarily
true. This core inner feeling goes beyond the discriminating intellect and the consistent
principles of logic the intellect applies. However, again, it is underlain by deeper levels including
a sense of knowingness, direct apprehension that I am the knower; again, more fundamentally, a
sense that ‘I am’, that ‘I exist’; and that ‘I am conscious’. The human mind has the natural ability
to direct attention ‘outside’ toward the vast diversity of phenomenally ‘separate’ objects of
experience (“its’). And it also has the natural ability to attend to and settle deeper ‘inside’ to the
“inner “I” – from the field of diversity to a more integrated, unified sense of self, and to ‘bare’
The “I-it’, subject-object duality is so engrained in ordinary waking experience that deeper
underlying levels of mind and pure consciousness itself have been almost completely
overlooked. The concern here is not with mathematics, but rather habitual attention that entrains
an externalizing ‘outside’ focus, not balanced with integrating experiences that are deeper
‘inside’. Vedic proponent Sri Nisargadatta8 zeroes in on what’s been most fundamentally missed:
You are taking duality so much for granted, that you do not even notice it…. Truth…is nearer than
the mind and the body, nearer than the sense ‘I am’… You have objectified truth and insist on your
standard proofs and tests, which apply only to things and thoughts… Your true being is…completely
free from all self-identification with whatever it may be, gross, subtle, or transcendental (pps. 368-
Essentially, what has been overlooked is the direct experience of ‘I’ and the natural means to
experience the inner basis of mind and self actual transcending the ordinary “I–it” subject-
object duality to the underlying unified state of consciousness, which is described extensively in
Yoga. This natural process of transcending deeper ‘inside’ involves the phenomenal experience
of softer and softer thinking and feeling to the pure, silent, inner unbounded wakeful ground state
of mind (like a wave settling into the ocean). That silent inner wakefulness is naturally accessible
as the ever-present background to all intuition, feeling, thinking, perceiving, sensing, and
behaving. It is so simple, effortless, and subtle as to be long overlooked. Referred to throughout
history by the wise, but its technology forgotten, seekers concluded that it must require arduous
effort and a reclusive lifestyle. Concentration, contemplation, introspection, or being more
‘mind-ful’ tend to increase rather than decrease mental activity. Though these practices can have
useful benefits, they are not designed for transcending to unbounded inner silence (samadhi). 9
Psychophysiological research on the Transcendental Meditation technique supports this
distinction. Maharishi explains:
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Those whose hearts and minds are not cultured, whose vision concentrates on the gross, only see the
surface value of life. They only find qualities of matter and energy… They do not enjoy almighty
Being in Its innocent, never changing status of fullness and abundance of everything that lies beyond
the obvious phase of forms and phenomena of matter and energy, and of mind and individual... Pure
Being is of transcendental nature because of Its status as the essential constituent of the universe. It is
finer than the finest in creation; because of Its nature, It is not exposed to the senses…
So, what’s the practical significance of transcending in addressing the imminent existential
threats? The deep rest and rejuvenation resulting from regular transcending activates healing
mechanisms that unfreeze engrained thinking habits for improved mental and physical health.
Extensive experimental research supports this technology as natural means for stress reduction,
increased mental acuity and intelligence, and more coherent and peaceful behavior even
throughout society. It has the power to unfold much deeper coherence and harmony with nature.
As Maharishi 11 states: “Transcending thought is infinitely more valuable than thinking” (p. 444).
In the ancient Vedic Yoga tradition, there is “Nothing so purifying as knowledge” 12 (p. 311).
Expanding one’s state of consciousness naturally facilitates deeper, more unified knowledge.
The diversity and complexity of today’s world can be so stressful that thinking and behavior
become fragmented. When diversity dominates and the underlying unity is overshadowed, our
families, homes, neighborhoods, cities, states, nations, and world family become more chaotic.
Substantive practical advances to make our world family more unified and peaceful won’t come
from dismantling and reengineering nature as if it were just insentient random bits of physical
matter. “Diversity is destructive only in the absence of unity.” 13 Attuning with nature by settling
deeper to more refined unifying levels of the mind results in more coherent thought, behavior,
and relationships with each other and the natural ecology of our home on Earth. Coherent
processes deeper in each of us can naturally counteract engrained thinking patterns that have
reduced understanding of nature to a meaningless, valueless random nothing. Mathematics is of
great value in the toolbox of modern science, and mathematical logic has been a core training
objective in modern education in this Age of Reason and Science. But we are now seeing the
cumulative effects of looking ‘outside’ to objectify experience, while missing more powerful
levels of mind deeper ‘inside’. Knowledge is for living the full value of nature in a unified
wholeness of understanding and experience. That knowledge within each of us, long overlooked,
is easily accessible, and can ‘avert the danger’ we’ve created by not utilizing it. It’s as
profoundly simple as That.
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13 Maharishi Mahesh Yogi, (2003). Maharishi’s Global News Conference, June 25.
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