Consciousness: The Last Frontier of Geometry - Maharishi ...
Consciousness: The Last Frontier of Geometry - Maharishi ...
Consciousness: The Last Frontier of Geometry - Maharishi ...
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<strong>Consciousness</strong>: <strong>The</strong> <strong>Last</strong> <strong>Frontier</strong> <strong>of</strong> <strong>Geometry</strong><br />
Catherine A. Gorini<br />
Introduction. Each discipline <strong>of</strong> knowledge studies different aspects <strong>of</strong> life and uses<br />
different methodologies and procedures. Many disciplines, however, find that their<br />
distinct paths reach a common end: questions about consciousness. Thus, consciousness<br />
is regarded as the last frontier in humanity’s quest for knowledge. Here, we will describe<br />
how the discipline <strong>of</strong> geometry, that part <strong>of</strong> mathematics that investigates patterns <strong>of</strong><br />
space, shape, and form. We will see that geometry can also lead, in a very natural way, to<br />
concerns relevant to the study <strong>of</strong> consciousness and can give unique insights into the<br />
ultimate nature <strong>of</strong> consciousness.<br />
Throughout its history, mathematics has been highly regarded as a discipline that can<br />
<strong>of</strong>fer valuable insights outside <strong>of</strong> its designated domain. Thousands <strong>of</strong> years ago, Plato [1,<br />
page 69] affirmed that “geometry will draw the soul towards truth.” Descartes, several<br />
hundred years ago, modeled his comprehensive search for truth on the methodology <strong>of</strong><br />
geometry. Today, Keith Devlin [2, page 9] recognizes that “as an entirely human<br />
creation, the study <strong>of</strong> mathematics is ultimately a study <strong>of</strong> humanity itself.”<br />
In this paper, we will first look at how other disciplines have encountered this last<br />
frontier <strong>of</strong> consciousness and then focus on several themes <strong>of</strong> geometry—continuity,<br />
higher dimensions, infinity, symmetry, homogeneity <strong>of</strong> space, and non-Euclidean<br />
geometries—to see what connections they have with consciousness. Finally, we will<br />
discuss the significance <strong>of</strong> this view <strong>of</strong> geometry.<br />
<strong>Last</strong> <strong>Frontier</strong>s. Physics, the study <strong>of</strong> matter, force, and energy, has encountered<br />
Heisenberg’s uncertainty principle, which guarantees that merely observing a system<br />
changes the system in a way that cannot be minimized or avoided. This means that the<br />
role <strong>of</strong> the observer is an integral part <strong>of</strong> any system being observed. This has two<br />
important implications. <strong>The</strong> first is that understanding the process <strong>of</strong> observation,<br />
whereby an observer is connected to an object <strong>of</strong> observation, is necessary for any<br />
complete theory <strong>of</strong> physics. <strong>The</strong> second is that the observer and the observed,<br />
consciousness and matter, mind and body, are not separate but are intimately connected<br />
at their very basis. Thus, a unified field theory <strong>of</strong> physics must ultimately account for the<br />
role <strong>of</strong> consciousness.<br />
Biology, the study <strong>of</strong> living organisms, has located one fundamental structure at the<br />
basis <strong>of</strong> life: DNA. Encoded in an organism’s DNA is all <strong>of</strong> the information about the<br />
growth, development, functioning, and behavior <strong>of</strong> the organism. Biologists know quite<br />
precisely how certain sequences <strong>of</strong> DNA codons produce specific proteins, but they know<br />
nothing about how the structure <strong>of</strong> DNA and the biological systems it controls are<br />
connected to the phenomenon <strong>of</strong> human awareness or consciousness. Perhaps a complete<br />
and correct understanding <strong>of</strong> this “blueprint <strong>of</strong> life” could lead to a deep understanding <strong>of</strong><br />
the nature <strong>of</strong> consciousness.
continuity <strong>of</strong> the straight line and <strong>of</strong> angular measure [6, Appendix 5].<br />
Until very recently, the most appropriate model for space, time, matter, and energy<br />
has been the continuum. Calculus was developed to handle changing relationships<br />
between continuous quantities. Beginning with the discovery <strong>of</strong> molecules and atoms,<br />
however, each continuous model used by physics has been shown to be inaccurate at<br />
sufficiently small time and distance scales. <strong>The</strong> most recent theories <strong>of</strong> physics, which<br />
model space and time with space-time foam, show that even space and time cannot, at<br />
their finest level, be modeled by a continuum.<br />
It must be the case, then, that continuity is characteristic <strong>of</strong> how the observer, as a<br />
mathematician or physicist, interprets or understands space, time, and matter. Continuity<br />
must be more closely associated with the consciousness <strong>of</strong> the observer than with the<br />
object <strong>of</strong> observation. <strong>The</strong> continuum is an infinite, expanding structure, flowing evenly.<br />
It is the basis for measuring or quantifying. We can conclude that these qualities are<br />
necessarily attributes <strong>of</strong> consciousness.<br />
Higher Dimensions. <strong>The</strong> three dimensions <strong>of</strong> space that we live in appear to be<br />
inviolable, yet they can be extended mathematically in a very natural way to higher- and<br />
even infinite-dimensional abstract spaces. Higher-dimensional objects or spaces are<br />
typically used to provide a convenient geometric framework in which to consolidate all<br />
possible members <strong>of</strong> a certain set. For example, the infinite-dimensional phase spaces <strong>of</strong><br />
physics consist <strong>of</strong> every possible state a specific particle or system could have; the time<br />
evolution <strong>of</strong> the particle or system is then a trajectory through the phase space. Similarly,<br />
the feasibility region <strong>of</strong> a linear program is the many-dimensional set <strong>of</strong> all possible<br />
points satisfying a given collection <strong>of</strong> inequalities. <strong>The</strong>se higher-dimensional spaces are<br />
used because they organize information in an effective way that allows for further<br />
computation. <strong>The</strong> physical bounds <strong>of</strong> three dimensions are not boundaries to our<br />
awareness; it is natural and useful for the mind to operate in an all-inclusive way,<br />
unrestricted by the boundaries <strong>of</strong> the space we live in. Our consciousness more naturally<br />
functions from a unified level, creating a totality or wholeness out <strong>of</strong> all possible distinct<br />
parts.<br />
Infinity. <strong>The</strong> infinite is at the heart <strong>of</strong> mathematics. Considerations <strong>of</strong> continuity and<br />
higher-dimensional spaces lead us inevitably to the broader concept <strong>of</strong> infinity in<br />
geometry. <strong>The</strong> straight line is infinite in extent and has infinitely many points. <strong>The</strong>re are<br />
infinitely many geometric shapes, classified according to the various attributes (number<br />
<strong>of</strong> edges, measure <strong>of</strong> angles, and so on). Fractals show the intricate and beautiful<br />
structure that the infinite can support. What can be said <strong>of</strong> the human awareness that has<br />
created, studied, and classified the infinities <strong>of</strong> geometry Certainly the faculty that can<br />
comprehend and manipulate those infinities must itself be even more infinite.<br />
<strong>Consciousness</strong> is infinite, it is capable <strong>of</strong> comprehending infinity, and it must be the<br />
holistic value <strong>of</strong> which all other infinities are only a part.<br />
Symmetry. At the basis <strong>of</strong> the concept <strong>of</strong> symmetry is the recognition that two<br />
different objects can have essentially the same structure. For example, the symmetries <strong>of</strong>
a square depend on the congruence <strong>of</strong> each side with the other three sides and each angle<br />
with the other three angles. Our innate appreciation <strong>of</strong> symmetry depends on our ability<br />
to simultaneously discriminate (between the different parts <strong>of</strong> an object or between<br />
different objects) and to locate sameness or equivalency (<strong>of</strong> those parts or objects).<br />
<strong>Consciousness</strong> can locate differences, but more importantly, it can use differences as<br />
the basis for harmonizing or balancing those differences through the location <strong>of</strong><br />
symmetries. <strong>The</strong> natural enjoyment <strong>of</strong> unity in the presence <strong>of</strong> diversity explains the<br />
universal use <strong>of</strong> symmetry in the decorative arts. Thus, we see that consciousness is at<br />
once discriminating and harmonizing, synthesizing parts into ever greater wholes.<br />
Symmetry also depends on the comparing <strong>of</strong> an object to itself, using an object itself<br />
as its own measure. This is reflective <strong>of</strong> the self-referral quality <strong>of</strong> consciousness, the<br />
ability <strong>of</strong> consciousness to know, observe, and interact with itself.<br />
Homogeneity <strong>of</strong> Space. <strong>The</strong> fourth postulate <strong>of</strong> Euclid says that all right angles are<br />
equal; this means that space is everywhere the same in Euclidean geometry. This is<br />
certainly not how the eye perceives the space around us, since a right angle looks “right”<br />
only when viewed straight on. It is also not the best model <strong>of</strong> the space we live in, as we<br />
see from Einstein’s General <strong>The</strong>ory <strong>of</strong> Relativity. This suggests that homogeneity is more<br />
naturally a characteristic <strong>of</strong> the consciousness <strong>of</strong> the mathematician than <strong>of</strong> the space we<br />
live in.<br />
Different Geometries. For hundreds <strong>of</strong> years, Euclidean geometry was regarded as<br />
“true,” the only possible geometry and the actual description <strong>of</strong> the space we live in. <strong>The</strong><br />
discovery <strong>of</strong> non-Euclidean geometries that are provably as consistent as Euclidean<br />
geometry inspired a reexamination <strong>of</strong> the nature <strong>of</strong> mathematical truth, forcing<br />
mathematicians to accept mathematical truth as relative truth only. <strong>Consciousness</strong> is<br />
capable <strong>of</strong> integrating opposites, harmonizing them into a single viewpoint that<br />
transcends the diversity created by the intellect.<br />
Conclusions. <strong>The</strong> discussion here <strong>of</strong> these few geometric topics affirm that the study<br />
<strong>of</strong> geometry is an important and appropriate study for someone interested in<br />
consciousness and the meaning <strong>of</strong> life. In geometry, one is studying patterns <strong>of</strong><br />
consciousness and gaining knowledge about how consciousness functions within itself.<br />
<strong>The</strong> unique perspective <strong>of</strong> geometry, focusing on patterns <strong>of</strong> shape and form, is<br />
complementary to other approaches to the study <strong>of</strong> consciousness.<br />
<strong>The</strong> study <strong>of</strong> geometry suggests that consciousness is an infinite, unbounded,<br />
homogeneous continuum that can support great diversity. It is more natural and effective<br />
for the mind to construct a whole out <strong>of</strong> infinitely many possibilities than to deal with<br />
finitely many cases. <strong>The</strong> intellect is harmonizing and unifying, capable <strong>of</strong> locating the<br />
common features <strong>of</strong> different structures. Finally, human consciousness, even when<br />
restricted by the precision and logical correctness <strong>of</strong> mathematics, is capable <strong>of</strong><br />
encompassing different, even contradictory, truths.
<strong>The</strong>se conclusions about connections between geometry and consciousness have<br />
implications for the student <strong>of</strong> geometry. Students <strong>of</strong> geometry should consider<br />
themselves to be also students <strong>of</strong> consciousness. <strong>The</strong> development and refinement <strong>of</strong> the<br />
intellect has always been considered to be important for mathematicians, but we see now<br />
that an intellectual understanding <strong>of</strong> the nature <strong>of</strong> consciousness can support and enrich<br />
our understanding <strong>of</strong> geometric concepts.<br />
<strong>The</strong> integration <strong>of</strong> the study <strong>of</strong> consciousness with the study <strong>of</strong> geometry is part <strong>of</strong><br />
the curriculum at <strong>Maharishi</strong> University <strong>of</strong> Management. At the beginning <strong>of</strong> their studies,<br />
all students take a course covering the nature <strong>of</strong> consciousness, intelligence, and<br />
knowledge led by <strong>Maharishi</strong> Mahesh Yogi. This course serves as a foundation for their<br />
study <strong>of</strong> all other disciplines, including geometry. All students also practice the<br />
Transcendental Meditation and TM-Sidhi program as an experiential method for research<br />
into the nature <strong>of</strong> their own consciousness. This foundation has proved to be successful<br />
and rewarding in the study <strong>of</strong> geometry.<br />
Finally, I would like to note that <strong>Maharishi</strong> Mahesh Yogi, the world’s foremost<br />
scientist in the area <strong>of</strong> consciousness, views mathematics as really nothing other than the<br />
study <strong>of</strong> consciousness:<br />
Mathematical knowledge deals directly with the functioning <strong>of</strong><br />
the field <strong>of</strong> intelligence—consciousness. <strong>The</strong> principles <strong>of</strong><br />
Mathematics are universally valid principles <strong>of</strong> knowledge that<br />
describe the dynamics <strong>of</strong> the field <strong>of</strong> intelligence—the functioning <strong>of</strong><br />
the mathematician’s own consciousness. [5, page 160<br />
From this perspective, geometry is necessarily one aspect <strong>of</strong> the study <strong>of</strong> consciousness.<br />
<strong>The</strong> real fulfillment <strong>of</strong> mathematics, however, is in what it can achieve for humanity:<br />
Vedic Mathematics provides the steps <strong>of</strong> invincibility, enlivens<br />
the total potential <strong>of</strong> Natural Law, establishes mastery over Natural<br />
Law and thereby places life on the invincible level <strong>of</strong> ‘Victory before<br />
War’—life without problems—invincible defence against anything<br />
that is not useful to life. [4, page 351]<br />
This vision <strong>of</strong> what mathematics can achieve embodies the goals mathematicians have<br />
held throughout history and is an inspiration to all mathematicians.<br />
References<br />
[1] Ronald Calinger. Classics <strong>of</strong> Mathematics. Moore Publishing Company, Oak<br />
Park, IL 1982.<br />
[2] Keith Devlin. <strong>The</strong> Language <strong>of</strong> Mathematics: Making the Invisible Visible.<br />
Freeman, New York, 1998.
[3] David Hilbert. Foundations <strong>of</strong> <strong>Geometry</strong>. Open Court, La Salle, IL, 1971.<br />
[4] <strong>Maharishi</strong> Mahesh Yogi. <strong>Maharishi</strong>’s Absolute <strong>The</strong>ory <strong>of</strong> Defence. Age <strong>of</strong><br />
Enlightenment Publications, India, 1996.<br />
[5] <strong>Maharishi</strong> Mahesh Yogi. <strong>Maharishi</strong> Speaks to Students: Mastery Over Natural<br />
Law (Vol. 3). Age <strong>of</strong> Enlightenment Publications, India, 1997.<br />
[6] James Smart. Modern Geometries. Brooks/Cole, Pacific Grove, CA, 1998.