True Nature: Notes on Spirit and Science

Symmetry is intimately related to the idea of invariance, the persistence of something amidst change (e.g., changes in time, space, or perspective). This is also one meaning of objectivity. Thus, symmetry is a key to understanding the objective world, as is well known in physics. The basic idea here is not confined to the esoteric realms of theoretical physics. It can be seen in our everyday experience, if we just look at it right.

Consider a child’s visual experience of a wooden block, i.e., a cube. As the child turns it around and looks at it from different perspectives, the visual appearance changes. The shape of the image changes depending on which side is facing the child. Never does the child see the entire block at once (one side is always hidden, facing away). Yet, the child learns to correlate all these different 2D visual images, abstracting from them a 3D object that persists. This object, however, is not itself experienced. It is conceptually constructed and posited as existing “behind” the variety of 2D visual appearances in experience. At the heart of this connection between the multitude of appearances and the invariant reality behind them is the principle of symmetry.

Put simply, symmetry is an invariance amidst change. More precisely, consider something X (e.g., a set of two points in the plane) and a transformation R that changes X to X’ (e.g., a rotation of the plane). Now suppose we have abstracted a feature D that can be defined for both X and X’ (e.g, the Euclidean distance between points). Then we say that D is a symmetry of R if D is invariant with respect to the transformation R (e.g., distance is a symmetry of rotation if distance is unchanged when the plane is rotated). So, although rotations of the plane change the positions of points, the distances between those points are not changed. We can also express this in terms of a rotation of the coordinate system: although the coordinates of points change when the frame of reference (or perspective) is rotated, the distance between two points remains the same. Thus, the distance is an “objective feature” that persists amidst changes in perspective, while the coordinates are “mere appearances” that depend on the perspective and are not objective.

In physics, Noether’s theorem establishes a correspondence between symmetries in the dynamical laws of a system and conserved quantities (i.e., invariant features of the system). For example, if the laws are rotationally symmetric, then angular momentum is conserved. Simply put, Noether’s theorem relates the objective dynamical features D of a system to the transformations R of its dynamical laws that leave the laws unchanged, i.e., that represent mere changes in perspective and not real changes.

In mathematics, Felix Klein’s Erlangen program characterizes different geometries by their  corresponding symmetry groups. Euclidean geometry, for example, corresponds to the group of rotations, reflections, and translations because the geometric features that characterize Euclidean geometry (e.g., distances and angles) are invariant under those transformations. Different geometries can then be related to each other by relating their corresponding symmetry groups. For example, projective geometry has a larger group of symmetries, and it has fewer invariant features (e.g, cross-ratio, incidence, tangency, colinearity, but not distances or angles). The symmetry transformations correspond to changes in appearance due to change in perspective rather than change in something objectively real. This is why the larger symmetry group corresponds to fewer objective features. As physics moves towards higher symmetry, objectivity dissolves until it vanishes in the limit of perfect symmetry where everything is seen as a hierarchy of symmetry groups corresponding to successively deeper levels of perspectives.

Consider a set X and the group G of all possible transformations of X to itself (i.e., the group of automorphisms of X). One feature of X that is symmetric under all the transformations of G is the size, or cardinality, of X. Most often, though, the more interesting features of X are not symmetric under G, but are symmetric under some subgroup of G. For example, if X is the real number line then the Euclidean distance between two points is invariant under the Euclidean group E(1) of translations and reflections, but is not invariant under scalar multiplication. E(1) is a subgroup of G that corresponds to the merely apparent transformations of perspective, the transformations that do not change anything real. The factor group G/E(1) represents the transformations that are regarded as objectively real, factoring out all the merely apparent transformations of the symmetry group E(1). These real transformations effectively define what it means for objects to undergo objective changes, i.e., to change from one object to a different object. The identity element of G/E(1) corresponds to the transformations of perspective that do not change the identity of objects.

Let’s reconsider now the example of the child and the block. Although the child does not know abstract mathematics, the different perspectives on the block are understood as transformations of a symmetry group E(3) with corresponding invariants of distances and angles. Because no other transformations are experienced except for those in the group E(3), the child posits a real (i.e., invariant) object existing in a 3D world with a constant identity amidst the changes in perspective. This is how symmetry and invariance is implicitly built into the very roots of our experience of an objective world.

A keystone of modern science is the view that everything physical exists within time and space. Of course, these notions have themselves gone through radical revision, but they still form the basis for formulations of modern physics, and only recently have some theoretical physicists begun to consider the possibility that time and space might emerge from some more fundamental objective reality.

If we carefully examine the process of measurement, however, we see that time and space actually emerge empirically. To illustrate, let’s first consider just time. When multiple measurements of position are performed, we assign the measurements a linear ordering to form an ordered sequence of measurements. We then correlate these sequential measurements to construct something called time that is a useful parameterization of position (and perhaps other constructed quantities). So, to do all this we need to presuppose some ability to define a linear order of measurements, and to define the position measurements in some systematic way so that they can be correlated with each other in some coherent way. You might say, we need to superimpose some primitive structures of protospace and prototime in order to even start studying the world in a systematic way by relating measurements to each other. And when we do this, various kinds of time and space might emerge, such as Newton’s space and time and Einstein’s space and time.

To elaborate a bit on these ideas of proto-time and proto-space, if we start from a purely empirical basis, then you might say we have a set of measurements, which by nature are discrete. Then proto-time would be the linear order of this set, giving it the structure of a linear sequence of measurements. The continuous time parameter would then emerge as a kind of interpolated continuum constructed as a layer that emerges when we correlate the discrete sequence of measurements.

The proto-space is a kind of presupposed unit of measurement and method for systematic measurement of position using that unit. Its the very least we need to measure something like space or distance or length. Once this is defined, then if it is consistently used in all the measurements, then it provides a coherent way to make the correlations among the measurements at different times in the sequence. Without any such standard for position measurements, there is no basis for making sensible correlations between different measurements in the sequence. Then, just as continuous time emerges as a derivative layer from the discrete sequence of measurements, continuous space emerges as a derivative layer expressing the coherence between position measurements.

Thus, time and space emerge not objectively, but are constructed through the very process of measurement itself. This provides a new conceptual foundation for the formulation of physics that does not presuppose objective time and space.

The so-called “collapse” of the wave function in quantum theory is often illustrated by the wave/particle duality. When a photo propagates through a double-slit apparatus, it behaves like a wave. Yet, if it is observed, the non-local wave is collapsed into a single localized particle. However, both theory and experiment show that this is not a clear-cut either/or  distinction, as it is misleadingly presented in traditional discussions of the double slit experiment. The interference pattern is not simply there or not, but is gradually deteriorated as more information about which slit the particle went through can be extracted from the photon measurement. This suggests that, in general, there is never any discontinuous or sudden collapse of the wavefunction. All that is ever happening is that we’re pushing information around with measurement interactions in a completely continuous (unitary) way.

Not only is collapse of the wave function totally unverifiable and nonphysical, but another big problem with collapse is that it is in blatant violation of the Schrödinger equation! Any other scientific hypothesis that both violates known laws of physics and is not verifiable would normally be immediately rejected as pseudo-science. Why, then, has the notion of collapse stuck? Perhaps because one consequence of rejecting collapse would seem to be that it would lead us inevitably to the many worlds interpretation. Strange as the many worlds interpretation may be, however, it does have the virtue of being consistent with the laws of physics, at least as we know them so far.

The many worlds interpretation is often rejected as outrageous because it seems to imply that all the separate “worlds” have some actual existence, just like ours. But, it’s more like none of the “worlds” have actual existence, including ours. To make an analogy with the theory of relativity, it’s not like there are many actual velocities of the earth in space, each existing as its own separate actualized “world.” Rather, it’s that the earth has no actual objectively existing velocity at all. Velocity only has meaning relative to a reference frame, and reality does not have any privileged reference frame. We happen to observe things in the reference frame of the Earth where that velocity is zero. If we were on the Moon, things would be different. Is there really some mystery here? How is this so different from quantum theory? The original “relative state” formulation of quantum theory seems to be in line with this view, and calling it a “many worlds” theory is just as misleading as calling relativity theory a “many worlds” theory. It’s just “many reference frames” and one world. One might complain that the “one world” is a strange one, but that’s no less true in relativity theory where nothing has any objective mass, length, time, etc. The only objective realities are the four-dimensional invariants. These are almost as weird as coherent superpositions.

It is good to remember that physical theories in general are abstractions, describing a reality that is beyond our direct experience. We experience our immediate sensations of sight, sound, etc., and never directly experience the abstractions of “atoms” or “fields” which are only indirectly inferred from experience. (The same is actually true of a “chair” or “rock” as well.) These may be useful abstractions, but we never actually experience them directly, and can never know if they really exist the way we think. In fact, we don’t really know that they exist at all. We could be a brain in a vat or having a lucid dream right now. Science tries to balance the belief in some objective reality with the fact that we can never know the thing in itself. As Heisenberg wrote,

We have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.

It is actually more radical than Heisenberg suggests. Consider again the double-slit experiment. A simple photon which “measures” which slit the particle went through does not actually collapse the wave function to be localized in just one region of space. It merely entangles itself with the system.  Provided no decoherence has taken place so that the coherence of the original system is not washed out in many degrees of freedom of the measurement system, then there is no sense in which an irreversible measurement interaction has taken place. So one is still free to decide what will ultimately be measured. Because there has not been any interaction with a particular well-defined measurement apparatus (by which I mean a device that involves decoherence) the attributes of the system are likewise still undefined.

The above situation with regard to a quantum system is analogous to not having defined any particular well-defined reference frame in relativity. If I do not specify a reference frame for an observation of a monolith floating in space, then it has no definite well-defined value for various properties such as velocity, mass and length. Once the reference frame is specified, however, then one can meaningfully talk about definite values for these quantities. Similarly, once one specifies a particular measurement apparatus (that involves decoherence), then one can say there is a well-defined meaning to talking about certain properties. The coherence is lost and there is no practical possibility to erase that measurement choice after the interaction with the measurement apparatus and choose instead to measure a complementary observable. And all observers will agree on what is measured.

In connection with this, Pauli has this interesting statement:

Just as in the theory of relativity a group of mathematical transformations connects all possible coordinate systems, so in quantum mechanics a group of mathematical transformations connects the possible experimental arrangements.

And Bohr writes:

In neither case [of quantum theory or relativity theory] does the appropriate widening of our conceptual framework imply any appeal to the observing subject, which would hinder unambiguous communication of experience. In relativistic argumentation, such objectivity is secured by due regard to the dependence of the phenomena on the reference frame of the observer, while in complementary description all subjectivity is avoided by proper attention to the circumstances required for the well-defined use of elementary physical concepts.

Admittedly, the analogy with relativity only goes so far. In the case of relativity, the choice of reference frame is sufficient to provide a unique and definite value for physical attributes. In quantum systems, on the other hand, although the interaction with a particular decohering measurement apparatus gives a particular observable well-defined meaning, it still does not result in a definite value (i.e., the wavefunction is not collapsed). The analogy with relativity, it seems, is a similarity between the choice of reference frame and the choice of a particular decohering measurement apparatus. These choices are sufficient to give well-defined meaning to certain physical quantities. The difference seems to be that in quantum theory, even though the quantities may have well-defined meaning, they still have not been actualized. For example, once the atom has interacted with the Geiger counter and poison bottle, it makes sense to say that Schrödiner’s cat is either alive or dead (there is no longer any coherence that would allow one to perform a measurement of a complementary observable to the alive/dead observable).

The actualization of a particular value could be described in terms of the many worlds interpretation as the choice of which world “you” get identified with. In relativity, though, one can actually imagine something analogous, but we don’t regard it as a mystery for some reason: The description of the world according to relativity does not specify which moment in spacetime we should be experiencing as “here and now”. So, what determines which point in Minkowski space is “actualized” in our experience as here and now? Why should we experience this here and now rather than some other? This question seems quite similar to the question of why we experience ourselves in one of the many worlds as opposed to some other. What “collapses” us into a particular here and now? Clearly, there is no such collapse, just as there is no collapse in quantum theory. The theory is an abstraction from the here and now. If we get confused and think that we really live in the abstraction, then we become perplexed at how the specific here and now is mysteriously “collapsed” from all the possibilities in the general, abstract world we’ve dreamed up.

There is also an interesting similarity between the role of decoherence, which effectively cuts us off from ever detecting any of the worlds that have decohered from ours, and space-like separation in relativity. There are spacelike separated regions of spacetime that can not have any interaction or communication with us. So, what justification is there for saying that they exist at all? They can never be observed or verified to exist. Is this really any different than the other branches of the universal wave function that we can no longer detect because of decoherence?

The heavens have long symbolized the eternal, changeless perfection of the divine, while the earth has symbolized the ephemeral, changing flux of existence. Since ancient times, this division of heaven and earth has been a profound symbol of the separation humans feel from their divine source. We experience ourselves as ephemeral beings exiled from the eternal source from which we came, and we long for a return and reunification with this source. We long to reconnect heaven and earth, to experience the deep connection between the two.  Our desire to know the nature of the heavens is inherently religious.

Although heaven and earth have, at first sight, very different natures, they are also interconnected. The radiance of the Sun shines upon the Earth from above and gives life to everything here below.  The Sun evaporates waters from the oceans, drives the winds that carries that moisture to the mountains, where in falls, creating powerful rivers; the Sun feeds all vegetation with the energy they need to grow.  When animals eat these fruits and vegetables, they partake of the Sun’s power, releasing its energy in every beat of their hearts.  All life is deeply and profoundly connected with the Sun. This undeniable link between heaven and earth is a symbol of the bridge between eternity and time.

The connection between heaven and earth is manifested in the synchrony of heavenly and earthly cycles. Although the stars appear essentially eternal, a perfect image of timelessness, the Sun, Moon, and planets move through regular cycles in what Plato called a “moving image of eternity.” And these cycles of the Sun are synchronized with the daily rhythms of life. In the cycle of day and night we experience the duality of light and dark, life and death, warm and cold.  Our bodies pass through cycles of activity and rest, while our minds pass through cycles of consciousness and unconsciousness.  Our inner lives, as well as our outer lives, are powerfully attuned to this cycle. But while the passing days on earth result in aging and death, the cycles of the Sun repeat perfectly with mathematical precision. Although the heavens move, there is no imperfection, decay or death in their movement.

Because the cycles on earth are reflections of those in the heavens, we can come to know the heavens through knowledge of ourselves, and we can come to know ourselves through knowledge of the heavens. This revelation, expressed by the ancient insight, “as above, so below,” was certainly one of the most profound ever experience by the human species, and provided the root metaphor for many ancient cosmologies. Ancient astronomy was seen as a revelation of the profound connection between the rhythms of heaven and earth, and of the harmony of the entire cosmos.  The mathematical understanding of the observed astronomical cycles was thus a sacred science.

Of course, the daily cycle of the Sun is but one of the many astronomical cycles that can be observed. Another obvious but longer cycle involving the Sun is the annual cycle of the seasons. Each year, the length of the day gradually increases to a maximum at the height of summer and decreases to a minimum in the depths of winter. Like the daily cycle of day and night, all life is organized around this annual cycle. The trees drop their leaves in the fall and blossom in the spring.  Winters are dark and cold, while summers are sunny and warm. Spring is the time of planting seeds, while autumn is the time of harvest.

The Moon has its monthly cycle of four phases, which naturally divide the month into four weeks. Like the Sun, the Moon also influences the patterns of life on Earth.  The entire oceans of Earth rise and fall in the ebb and flow of the tides under the direct influence of the Moon.  These tides take sea creatures onto land, and take land creatures out to sea, providing an impetus for life to transition between land and water.  Even as land creatures, our physiology still remains influenced by the lunar cycle.

Because these the cycles of the Sun and Moon have clear connection with cycles of life on earth, it was natural to assume that the cycles of the other planets were similarly connected with life in some mysterious and subtle way. The basic premise of ancient cosmology, that heaven and earth are interconnected, implied that all heavenly cycles have some kind of imperfect reflection in the patterns of life. Thus, the study of the planets and their relationships with each other was viewed as a key to understanding ourselves and life in general.

Each celestial sphere has its own temporal period corresponding to the duration of its cyclic movement in the heavens. These rhythms correspond to frequencies having harmonies with each other.  The combined movement of all the celestial objects is thus a grand symphony of cosmic proportions. It is said that Pythagoras was so spiritually developed that he could “hear” this music, presumably because the heavenly spheres are simply the outer reflections of our own inner divinity.  Thus, insofar as we are conscious of this inner divinity, we are conscious of these qualitative aspects of the outer divinity as well.

The coherent and precise understanding of the cycles of the Sun, Moon, and planets requires a mathematical treatment. It is here, though, that mysterious problems emerge, for it was found that the various cycles do not harmonize with each other. The month is not equal to an integral number of days, nor is the year equal to an integral number of days or months. It was not possible to construct a single calendar that perfectly combines all these cycles into one coherent framework. For example, if the month is defined to be exactly 28 days long, then after several months the new moon will no longer begin at the start of the month. Similarly, if a year is defined as exactly 365 days, then the calendar will gradually drift out of sync from the seasons of nature. Driven by a faith in the comprehensibility of the cosmos, the ancients struggled with these patterns, looking deeper into the relationships between the cycles. This led to the discovery of even more subtle patterns, such as the precession of the equinoxes. Gradually, over thousands of years, the sphere of the stars gradually shifts ever so slightly. It is as if the cosmos has different asynchronous clocks to measure cycles having different periods of time. But their relationship to each other remained obscure.

The temporal cycles of the heavens were also evidently spatial cycles: the Sun is seen to follow a circular path around the Earth, as does the Moon and the sphere of the stars. The geometric circle is the perfect spatial image of temporal recurrence: Just as a cycle in time exhibits change yet repeatedly returns to an identical time again, so movement around a circle undergoes change yet repeatedly returns to an identical point. The different temporal cycles thus naturally suggested different circles in space, with the Sun, Moon, and stars pictured as concentric spheres. However, while the movement of the Sun and Moon was uniform, the movement of some planets was not: sometimes they went one direction, then other times they would reverse and go backward for a while, only to reverse again and continue forward. This retrograde motion posed a significant challenge to the ancient astronomers. What could explain this non-uniform motion?

For centuries, the retrograde motion was explained with complicated epicycles. Copernicus simplified the model by placing the Sun at the center of the solar system. The reversals were then explained to be illusions of perspective, and the daily motion of the sun around the earth only an appearance and not real. Although this retained the classical circular motion, this was a significant step away from the ancient worldview. Not only was the Earth no longer fixed at the center of the cosmos, but the reality of the cosmos shifted radically. No longer did the apparent motions of the Sun and Moon correspond to their real motion. What motivated and justified such a sacrifice? Although both the Sun-centered and Earth-centered systems both explained the appearances, the Sun-centered system introduced by Copernicus so much more simple and elegant than the complicated system of epicycles. It is remarkable that this intellectual elegance was sufficiently powerful to the human mind to usurp the obviousness of sensory appearances as well as the centuries-old dogma of the Aristotelian worldview.

Copernicus set the stage for another radical departure from the classical worldview: the dropping of uniform circular motion. As empirical observations of the planets became more precise, even the Copernican model was unable to fit the data without awkward, ad hoc modifications. After extensive effort to make a circular path fit the data, Kepler concluded that the planets must, in reality, follow elliptical orbits with the Sun at one focus. This bold step usurped the circle from its centuries-old place as the fundamental shape of heavenly motion. As with Copernicus, this revolution took place because the elliptical orbit provided a much more simple and elegant match with the appearances. Even though the circle itself is simpler than the ellipse, it did not have any simple correspondence with empirical observations of the planets. The ellipse, on the other hand, provided a perfect fit. Thus, the aesthetic quest for intellectual coherence drove astronomy to deeper and more subtle understanding of the patterns of the heavens, revealing truths that were not obvious in the appearances.

Not long after Kepler, Newton made a profound unification of the terrestrial and heavenly in his discovery of the universal laws of motion and gravitation. These same laws governed both motion of terrestrial objects and those in the heavens, providing an unprecedented unification of heaven and earth, guided again by the quest for mathematical coherence and unity. The ancient cosmology, which had been based upon the specifics of our particular solar system, was now seen as just one of many possible solutions to Newton’s general laws of motion. With Newton, a giant leap in abstraction was taken, grounding thought in universal mathematical laws rather than specific geometric models. The dichotomy of time and timelessness still exist, but are no longer associated with the obsolete distinction between heaven and earth. Instead, timelessness is a property of the mathematical laws that govern the entire cosmos, both heaven and earth alike, while time is experienced as a property of our specific cosmos, which is one solution to those universal laws. In short, the distinction between time and eternity shifted from a spatial distinction between earth and heaven to a distinction in the levels of manifestation that are universally omnipresent. At every point in space there is contact with timelessness insofar as the general laws are universal. And at every point in space there is contact with time insofar as this universe is a particular instance of those general laws. In this sense, modern science has accomplished the ancient religious quest to unite the realms of eternity and time.

The creation of the cosmos, or cosmogenesis, may be symbolized as an emanation from a nondual reality (1) into duality (2), trinity (3), and further multiplicity. This archetypal pattern is represented in the universal symbol of the tree of life. At its base, the tree is a single trunk, representing the cosmic axis, or axis mundi. The branching of the tree represents the emanation of multiplicity from the unity. Despite this branching, however, the tree remains a single, living organism. Although apparently divided, the tree is and always remains whole.

The Tree of Eternity has its roots in heaven above and its branches reach down to earth. …The whole universe comes from him [Brahman] and his life burns through the whole universe.
-The Upanishads. Mascaro, Juan, tr. (New York:  Penguin, 1965). p. 65.

The Pythagoreans used a more abstract symbol, the tetraktys, which is an arrangement of ten dots in triangular form:

This symbolizes the development from the single unity (*) into duality (**) then trinity (***) and four-fold multiplicity (****). Although the tetraktys symbol does not have the organic qualities of the tree of life, this mathematical symbol brings out more explicitly the features of harmony and order. For example, the numerical ratio 1/2 between the first and second level correspond to the musical octave. The next two levels give the ratio 2/3, which is the musical interval of the fifth, and the last two levels give the ratio 3/4, which is the musical intervals of the fourth. The levels of cosmic creation thus correspond to musical harmonies. This is the seminal insight at the basis of the “music of the spheres” connecting the structure of the cosmos with music through mathematics.

In Timeaus, Plato describes the cosmos as being built from mathematical archetypes. With only one dot, there is not much structure, but with two dots there is now a line, and with three dots a triangle. With four dots a solid object can be specified. Thus, the basic physical elements are viewed as constructed from the non-physical mathematical patterns. From the simple One, structure emerges first as subtle archetypal forms, but then reaches the point where, mysteriously, it becomes physical. 

ONE

Common to most of the most influential worldviews of humanity is the insight that reality is, in its deepest essence, unified. To illustrate:

The universe, therefore, is nothing but Brahman.  It is superimposed upon Him.  It has no separate existence apart from its ground.
-Shankara. Shankara’s Crest-Jewel of Discrimination (Viveka-Chudamani). Isherwood, Christopher, tr. (Hollywood:  Vedanta Press, ). p. 70.

In essence things are not two but one. …All duality is falsely imagined.
-The Lankavatara Sutra 

There is in reality neither truth nor error, neither yes nor no, nor any distinction whatsoever, since all—including contraries—is One.
-Chuang Tzu (A Treasury of Traditional Wisdom, p.979)

The One … is there before every oneness amid multiplicity, before every part and whole, before the definite and indefinite, before the limited and the unlimited. It is there defining all things that have being, defining being itself … . It is there beyond the one itself, defining this one.
-Pseudo Dionysius (Paul Rorem, Pseudo Dionysius, Paulist Press, 1987, p.129)

Like the symbol of the tree, it is a One that is not exclusively a single trunk or a multiplicity of branches, but somehow both.

TWO

The root of all things is difference.
-Ibn Arabi (William C. Chittick, The Sufi Path of Knowledge, SUNY, 1989, p.67)

The first departure from this original unity is a single distinction, giving rise to a basic duality expressed variously as one/many, heaven/earth, infinite/finite, subject/object, transcendence/immanence, ultimate/relative, emptiness/form, eternity/time, being/becoming. This distinction, however, is not ultimately real. The different branches of the tree are only apparently separate. But if we forget this, then we fall into delusion and suffering:

“In the beginning God created heaven and earth,” that is, the first fall of all is from the One into two, from unity into number, from what is perfect, undivided and indistinct into imperfection, division and distinction, and from the whole into parts.
-Eckhart, Meister.  Meister Eckhart: The Essential Sermons, Commentaries, Treatises, and Defense. Colledge, Edmund, tr. (Ramsey, N.J.:  Paulist Press, 1981). p. 100.

And, as the Bhagavad Gita reminds us:

There are two spirits in this universe,
The perishable and the unperishable.
The perishable is all things in creation.
The unperishable is that which moves not.
But the highest spirit is another:
It is called the Spirit Supreme.
He is the God of Eternity
Who pervading all sustains all.
-The Bhagavad Gita. Mascaro, Juan, tr. (New York:  Penguin, 1962). p. 107.

THREE

When another distinction is made, duality splits into the tree-fold structure of the trinity. This is expressed variously as Father/Son/Holy Spirit, Body/Mind/Spirit, Sat/Chit/Ananda, Dharmakaya/Sambhogakaya/Nirmanakaya, Gross/Subtle/Causal. This more refined structure provides a more explicit expression of the implicit aspects of the original One. And the process continues indefinitely to increasing multiplicity.

Not only is this pattern of cosmogenesis reflected in the traditional metaphysical systems of the world, but it also manifests in modern physics. In the Big Bang cosmological theory, symmetry breaking leads to the manifestations of distinctions between the four fundamental forces of nature. Prior to 10^-43 seconds all the physical forces of nature were unified in perfect symmetry. After 10^-43 seconds, the force of gravity emerged as a distinct interaction. Then, after 10^-35 seconds, another symmetry broke and the strong nuclear force was distinguished. And at 10^-10 seconds, the weak nuclear interaction was distinguished. Although much more sophisticated than the Pythagorean tetraktys, the essential pattern of multiplicity unfolding from unity is the same.

Most problems in physics are genuine problems with interesting solutions. There are a few problems, though, that are described as problems of physics, but are actually pseudo-problems. They arise from a fundamental misunderstanding about the nature of theories. One such problem arises as the explanatory gap between the abstract description of the world by a theory and the concrete and particular present moment, actualized here and now. This can take several forms. For example, there is the problem of explaining how it is that, of all the possible universes consistent with the laws of physics, we happen to live in this particular one, with the fundamental constants of physics having the particular values that they do. Another is the measurement problem of quantum physics, which is the problem of relating the probabilistic description of the world given by the theory to the particular world that is actually experienced. We know from our immediate experience that this particular world exists here and now. The problem is that we don’t have an explanation of how such a specific actual world is selected from the general description of reality that the theory provides. We expect the theory to dictate not only the general laws of many possible worlds, but to explain as well why it is that this particular world is actual rather than various other possibilities.

To see how these problems are pseudo-problems, we need only realize that physical theories are abstractions from the here and now. They are conceptual systems that posit and describe a general reality, much of which is forever inaccessible to experiment (e.g., decohered regions of a many worlds theory, or space-like separated regions of space-time). We always begin and end with the present moment. In fact, we never escape it. With our theories, though, we imagine a vast world extended into the future and past, reaching far into the depths of intergalactic space, and including various branches of possibility in a reality that includes many worlds of possibility. The problem begins when we forget that this is all imagined in the present moment, and take these theories to be describing a primary reality.

For example, if we think that the quantum theoretical description of the world is real, we can become perplexed trying to explain how to relate it to our present experience, e.g., how is it that the present experience is “actualized” from the many possibilities? Why is this possible outcome actualized and not another? Or consider the four dimensional space-time of special relativity. Why is the present here-and-now located at this particular point in space-time, and not another? This problem only arises because we have forgotten that we never really left the present here-and-now in the first place. The measurement problem, the actualization of a particular “here-now” in spacetime, is looking at it backwards. The key is to recognize that we do not live in the abstraction but in the here and now, and have never left it. The key is to become aware of how we are abstracting from the present here-now, and what presuppositions of time and space we are making when we do so. This approach to fundamental physics removes the pseudo-problems from physics that have their root in our own misconception of the nature of scientific theory. And it opens up a more fruitful approach to understanding the basis of physical theory and its relationship to the inescapable present moment of experience.

What is the nature of time? In our contemporary culture, this is considered a question for physics to answer. In Einstein’s general relativity, time (as well as space) are relative and dynamic, depending on gravity and mass. Time is part of the nature of the physical world. In the Timaeus, Plato also expresses a view in which the time is created along with the physical world as an image of a deeper eternal unity:

Wherefore he resolved to have a moving image of eternity, and when he set in order the heaven, he made this image eternal but moving according to number, while eternity itself rests in unity; and this image we call time. For there were no days and nights and months and years before the heaven was created, but when he constructed the heaven he created them also. They are all parts of time, and the past and future are created species of time, which we unconsciously but wrongly transfer to the eternal essence.

In other words, time is not fundamental but a derivative image. And the movement of this image is mathematical. The manifest multiplicity in time is the mathematical movement of an underlying eternal unity. This Platonic vision is remarkably similar to the notion of symmetry. Put simply, a symmetry is an imagined change that produces no change. Where there is an invariant amidst variation, there is a symmetry. For example, a rotation of a circle by 90 degrees changes the circle but leaves it unchanged in a deeper, more fundamental sense. The symmetry transformation is a moving image of the eternal invariant structure. And in modern physics, fundamental constants of motion (conserved quantities) are directly related to symmetries through Noether’s theorem. Mathematical symmetry expresses the relationship of time and eternity, the sense in which time is a moving image of eternity, change within the unchanging, variation within an invariant.

A physical law itself is also an invariant that expresses the variations in the physical world. A law of physics provides a mathematical form that describes the invariant relationships between variable quantities. For example, F=ma is (within classical physics) a universal formula that is invariant across all times and places. While the values for m, a, and F can change, their relationship expressed by this equation is an invariant amidst this change. Thus, physical law itself is a symmetry that reveals the physical world as a moving image of eternity.

A paradox of sorts emerges at this point. On the one hand, the very notion of symmetry, of invariance within variation, presupposes a very basic form of change and time. On the other hand, physical time itself is part of the physical world that manifests as a moving image of eternity. Thus, time seems to be at once both cause and effect of manifestation. It is here that we are prompted to distinguish between layers of time. The most basic “proto-time,” the bare possibility of imagined change of the unchanging, is without any elaborate structure. Physical time, on the other hand, is rich with structure. It is entangled with space in a four-dimensional curved space-time whose dynamics depend on gravity and mass. The nature of physical time is part of the solution to equations of physics. But these equations themselves presuppose a more basic proto-time that is embedded in the very notion of law itself. The origin of this proto-time is presupposed by physical law, and is therefore not within the domain of physics per se. It is beyond or outside of physics. One might say it is part of meta-physics. Deeply understanding the foundations of physics necessarily takes one into this realm beyond physics, just as studying the foundations of a building takes one into the Earth. Structural engineering ultimately opens up to geology.

The concepts of linear and circular are root metaphors in many worldviews. For example, linear and circular provide root metaphors for the conception of time. The linear and circular are fundamentally different in several respects. The circular, for example, has the quality of recurrence while the linear has the quality of infinite extension without repetition. The presence of these contradictory root metaphors in different worldviews raises the question of which one is correct, or, if both are somehow correct, how they can be reconciled.

What appears paradoxical from limited perspectives often has a straightforward reconciliation from a more comprehensive perspective. The linear and circular, I will argue, can be integrated through the mathematics of projective geometry. The circle can be seen, in a precise mathematical way, to be identical to the line. This model provides a way to see, without contradiction, how a circular conception of time (eternal recurrence) is equivalent to a linear conception of time.

For the following discussion, the reader is referred to the figure to the right which shows a blue circle horizontally bisected by a red line. A point on the bottom of the circle serves as a pivot point for a black line that freely rotates through the pivot point. At any particular non-horizontal angle of orientation, the black line intersects exactly one point on the horizontal red line and exactly one other point on the circle. This creates a one-to-one correspondence between points on the circle and points on the line. The one exceptional case occurs when the black line is exactly horizontal. It then intersects neither the circle nor the horizontal red line. The pivot point, therefore, corresponds to no point on the horizontal red line. The pivot point is the singular point of projection that is not itself projected.

In mathematical terms, this model illustrates how, despite their apparent differences, the line and circle are fundamentally isomorphic to each other. Thus, this isomorphism provides a reconciliation of circular and linear conceptions of time by showing how they are merely two ways of viewing the same fundamental structure.

As a metaphor, this model also provides insight into the relationship between circular and linear perspectives. For example, instead of drawing the red line horizontally through the blue circle, instead draw the red line vertically through the circle and select the rightmost point on the circle as the point of projection (in other words, rotate the red line and pivot point in the original figure 90 degrees counter-clockwise). The same points on the circle now correspond to points projected along the vertical red line. Similarly, any other choice of pivot point will map the points of the circle to points along a different red line. These different red lines can be interpreted as symbols of different worlds projected from the same circle from the perspective of a particular pivot point on the circle. This provides a metaphor for how different worlds may be viewed as projections from a single reality, while each world is at the same time isomorphic to that reality, and hence to all the other worlds. If the circle and line are given a temporal interpretation, the model provides a way of reconciling time conceived as infinitely extended in a line with time conceived as cyclic and recurring. The two are different ways of representing a single common fundamental structure, which we might call with Plato a moving image of eternity.

Finally, we may also observe that the circle has an underlying rotational symmetry. That is, a circle can be rotated without changing any essential property of the circle. To put it in somewhat paradoxical terms, a rotation of a circle is a change that results in no change. We can imagine rotating a circle, but in doing so we do not in fact do anything. Thus, in a sense, each point on the circle is equivalent and indistinguishable from every other point. The symmetry of the circle thus provides a symbol for the identity-in-difference of all the points on the circle. Each pivot point is both the same and different from all other pivot points.

Note: The ideas in this post are largely based upon a 2004 article by the author entitled “The Integral Sphere: A Mathematical Mandala of Reality” located at http://www.integralscience.org/sphere.html. The article provides a more detailed discussion of the model and generalizes it to higher dimensions.