Keynote Articles


A Mathematician's View Is there a mathematical argument for the existence of one God?

Reprinted from the peer-reviewed NOETIC JOURNAL, VOLUME 2 NUMBER 2 (April 1999)

Journal-reprint version available here (The God of Abraham-A Mathematician's View.A3lograf.pdf  513K)
©1993, 1996, 1999 Stan Tenen This essay was first published in TORUS, the Journal of the Meru Foundation, Vol. 2 #3 (July 1993).
A version in German was published in Tattva Viveka, Issue #5, October 1996, Frankfurt, Germany.
A shorter version was published in Gnosis #28, Summer 1993

The Noetic Journal published the accompanying piece to this article,  Man Bites Dog,  as an Appendix to The God of Abraham.
This article is usually published with copious illustrations. A few of those illustrations are reproduced here; many more are available in other places on this website. In particular, the reader should refer to the following Meru Foundation Presentation Posters:

Fourier Transform of ALL-THERE-IS  <=>     Fourier Transform of PULSE
The Singular Pulse at "Creation" is the Fourier Transform of an eternity of ALL-THERE-IS;
The Spectrum of ALL Tones and Harmonics, representing ALL-THERE-IS
is the Fourier Transform of the "Creation" Pulse.

If the universe is limited in extent and duration, then its ultimate source must be less than omnipotent. Thus the presumption of ONE exquisitely Infinite source demands that the universe be infinite, open and eternal. The Singular Pulse may be the Kabbalist's "line" (Qav) that extends from the "creation-contraction" (ZimZum) into All-There-Is. Abraham's definition of the One-God and our modern understanding of this universe may well be based on the same fundamental principles.

Here is another view that extends the idea of our closest logical singularity - the sun, the ostensible god of many ancient peoples - into a suitable definition of the Exquisite, Transcendental Unity of the One-God.


From physicist Roger Penrose's, The Emperor's New Mind, page 318-19, italics sic; [square brackets] and bold-face added for emphasis:

"Where indeed does our own low entropy come from? The organization in our bodies comes from the food that we eat and the oxygen that we breathe.
"The question is how we keep ourselves alive throughout out normal (mainly adult) lives. For that, we do not need to add to our energy content.
"...To keep ourselves alive, we need to keep lowering the entropy that is within ourselves.
"Where does this supply of low entropy come from?
"... green plants ... [provide] it by making use of sunlight. The light from the sun brings energy to the earth in a comparatively low-entropy form, namely in the photons of visible light. The earth, including its inhabitants, does not retain this energy, but (after some while) re-radiates it all back into space. However, the re-radiated energy is in a high-entropy form, namely what is called 'radiant heat' - which means infra-red photons. Contrary to a common impression, the earth does not gain [net] energy from the sun! What the earth does is to take energy in low-entropy form, and then spew it all back again into space, but in a high-entropy form. What the sun has done for us is to supply us with a huge source of low entropy. We (via the plant's cleverness), make use of this, ultimately extracting some tiny part of this low entropy and converting it into the remarkable and intricately organized structures that are ourselves.
"The energy the earth spews back into space is spread over many more degrees of freedom than is the energy that it receives from the sun. Since there are so many more degrees of freedom involved when the energy is sent back out again, ... the entropy has gone up enormously. The green plants, by taking in energy in a low-entropy form (comparatively few visible-light photons) and re-radiating it a high-entropy form (comparatively many infra-red photons) have been able to feed on this low entropy and provide us with this oxygen-carbon separation that we need.
"All this is made possible by the fact that the sun is a hot-spot in the sky."5
Professor Penrose is telling us that the "raw" information needed to sustain the self-organization of life on earth is due to the "separation" - the distinction - between oxygen and carbon and this, in turn, is due to the distinction - the contrast - between the hot, bright sun and the cool, dark sky.

This model suggests why in traditional societies (as well as today) those who believe that life is entirely physical and that consciousness is an epiphenomenon of complex physical organization alone believe that the source of life is the Sun. Exclusive materialism is the equivalent of solar worship. In effect, materialists are not atheists. In this context, materialists can truly be said to worship the hot "metal and stone" idol the Greeks called Apollo.

Roger Penrose's TWISTER

We can take Professor Penrose's observations one step further. After all, life on Earth reaches one further step of organization. Human beings (and perhaps a few other creatures such as some primates, elephants and the cetacea) are not only living animals - whose living neg-entropic organization comes from the contrast of the bright physical sun in the dark sky - but we are also conscious and aware of ourselves. Humans have a special hand with an opposable thumb marking our self-reflexive awareness and - some say consequently - we form and use language.

Following Penrose's argument, where does our "hyper" self-aware consciousness get its organizing information? What "hand" "informs" our special, "higher" human consciousness?

Does there exist or can we hypothesize a "higher" "hyper" contrast than between our star and the sky? Conceptually, at least, we can. The contrast between our physical sun and the sky can be idealized and extended by considering, at least in principle, an even higher, brighter source - a source of ALL-THERE-IS.

If the sun is a very bright, yet finite source, very far away, but not infinitely far away, then we could investigate the consequences of an infinite source at an infinite distance from us. We can define a model of an exquisitely singular and omnipotent source.

As with the Fourier Transform - musical harmonics ("music of the spheres") model, our extension of Professor Penrose's High Contrast leads us to an infinite pulse. (It is interesting to note that Kabbalists understand this principle of highest contrast as the paradox of God withdrawing or contracting (tzimtzum) and then projecting.) In this case the pulse is an infinitely energetic Singularity infinitely far away.

This then could be the "hyper" source of our human self-aware consciousness. This conceptually farthest, brightest source is also the conceptually highest information, lowest entropy source. It is the "hyper-neg-entropic" source of universal consciousness that corresponds to the One-God. By (Abraham's) definition this is the most "high-contrast" source.


How are we to model this ultimate, exquisite, unknowable UNITY representing our unique definition of the Singularity of the One-Living-God? Is there an entirely abstract - non-idolatrous - mathematical model that incorporates Singularity, Uniqueness, Self-organization, Universality, Infinitude, Elegance and Simplicity?

In The Laws of Form, mathematician G. Spencer-Brown proposes the "mark of distinction" archetypally distinguishing INSIDE from OUTSIDE as a definition of maximal contrast. Mathematicians have shown that all of formal logic can be derived from G. Spencer-Brown's "mark of distinction." The following is from The Laws of Form, p. xxix (emphasis added):

"The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance.


"Although all forms, and thus all universes, are possible, and any particular form is mutable, it becomes evident that the laws relating such forms are the same in any universe. It is this sameness, the idea that we can find a reality which is independent of how the universe actually appears, that lends such fascination to the study of mathematics."6
BET - "house"
image 3

In Hebrew the letter that most represents this "mark of distinction" between inside and outside is Bet, the first letter of the Hebrew text of Genesis.

It appropriately establishes, by definition, the first logical distinction possible. 

The distinction between archetypal Symmetry and Asymmetry is also primary and of absolute contrast. Each co-defines the other. Without a representative standard of asymmetry how could we unambiguously define symmetry?

Archetypal symmetry can be represented by the most compact structural forms (in any given dimension). The five Platonic solids (Tetrahedron, Cube, Octahedron, Icosahedron, and Dodecahedron) and the Archimedian semi-regular solids can define fundamental symmetries in 3-dimensions. (Mathematicians and physicists derive the formal symmetry groups that they use from these polyhedral archetypes.)

image 4
Sample Platonic polyhedra based on drawings from Keith Critchlow's Order in Space, p. 19.7

Archetypal asymmetry can be represented by a dynamic form that continuously breaks symmetries as it unfurls. Meru Foundation research suggests that this form is a particular, explicit vortex, which we call "Naked Recursion" ("naked" in the mathematical sense - unadorned, without any other quality) and which has been traditionally associated with "the flame of consciousness", the archetypal living process of a "fruit tree yielding fruit whose seed is inside itself" (Genesis I.11.), and with its highest human embodiment - our hands.

As we will see later, besides its traditional association with the heart Chakra, what is most astonishing about the vortex-shaped model HAND is its direct relationship to both our personal consciousness and to our cosmological model of Unity, Singularity and Wholeness.

hand and flame

Traditionally mystics and Kabbalists have referred to the asymmetry vs. symmetry model of highest contrast as "Flame and Coal" (Absolute Unity can only exist when the "flame" is wedded to the "coal"),8 the "Light in the Meeting Tent," or, perhaps, in Plato as the "same (symmetric) and the different (asymmetric)" (although many other interpretations have been given for this phrase).

flame ten


Many representations of ultimate contrast are variations of the Inside and Outside model. These include the hypercube and hypersphere which are modeled by placing a small cube (or sphere) inside of a larger one. The In-Out direction models the fourth dimension.

image 12

Following Eddington, mathematical philosopher Arthur M. Young in his The Reflexive Universe, points out that the surface of a 4-dimensional Hypersphere is the same as the volume of a solid Torus in 3-dimensions.9

nasa hol

In living things the Seed is INside and the Fruit is OUTside, the Sperm is INside and the Ovum is OUTside:

Flame and Coal
Light and Vessel
Seed and Fruit
Wave and Particle
Female and Male
Spirit and Matter
Mind and Body
Transcendance and Immanence

In these embodiments the asymmetrical dynamic element is conceptually INside (Seed, Mind, Spirit) with its symmetrical projective complement OUTside (Fruit, Body, Matter). See Note #2. There are very many other examples.

The modern physical concept of the complementarity of Wave and Particle as two mutually exclusive (highest contrast) aspects of all fundamental entities also models the Inside / Outside relationship.

knot squ

Mathematicians suggest that Singularities - formal mathematical definitions of Unity - can be modeled by Torus knots. Torus knots are simple "basket-weaving" or wreathes wound on rings. [For more complete illustration, see The 3,10 Torus Knot, Ring, Sphere, Tetrahelix and Hand.] The simplest knot is also the simplest Torus Knot. There are explicit vortex models based on particular Torus knots. The torus itself also defines the primary distinction between Inside and Outside.


Break TORUS Ring and pull apart A, B, and C


Continuous CreationThe illustration of CONTINUOUS CREATION is an attempt to show how a range of spiritual metaphors from different traditions can be reconciled in one geometric model of the High Contrast of Wholeness and Singularity. It shows some of the names by which the DIMPLED SPHERE Torus is known in different cultures and spiritual traditions. In this model, Continuous Creation refers to the unbroken chain of life: ...Seed-Tree-Fruit / Seed-Tree-Fruit / Seed-Tree... It represents the reflexive self-organizing process that is the natural transformation and unfoldment of every "SEED" (Singularity, Tao, Sun) via its "TREE" (Unfoldment, Flame, World Mountain) into new "FRUIT" (Wholeness, Hand, World) of its kind. It represents the general principle of the projection of life from life, endlessly. It is a model of one cycle of Singularity, Unfoldment, and Wholeness.

The IDEALIZED meditational "Fruit tree yielding
fruit whose seed is inside itself," in
the form of a DIMPLED-SPHERE Torus, is
defined by pairing off the letters in the first
verse of the Hebrew text of Genesis.

It is the minimal, most compact and most elegant representation
of the sequence of letters in the first verse.

CONTINUOUS CREATION represents the verse from Genesis (I,11) that is quoted in the introduction to the Sefer Zohar: "Fruit tree yielding fruit whose seed is inside itself."

The first verse of the Hebrew text of Genesis represents Naked (mathematically unadorned) Recursion (...acorn-oak-acorn-oak-acorn-oak....) in the form of a Human Hand.


The historical samples of the Hebrew Letters reproduced in our poster, The Hebrew Alphabet - FIRST HAND (below) are from Reuben Leaf, Hebrew Alphabets, 400 B.C.E. to Our Days. For each letter, the single right column (next to the printed letter name) shows the form of the letter from the Elephantine Papyrus, circa 300 BCE. The four samples in the middle are Merubah ("square-form") rabbinic script (often referred to as "Rashi" script) from Islamic Spain, circa. 1000-1300 CE.11

The far left column of Hebrew letters shows photocopy reductions of actual photographs of various views (shadowgrams) of the same physical model hand. [For a full-size rendition of this poster, see The Hebrew Alphabet - FIRST HAND.]

Because we can feel the model HAND on our hand and because we can always "see" our own hands in our mind's eye, we can also see the model HAND in our mind's eye. Close your eyes and point to the ceiling. You will find that you can tell how your hand is oriented and where your fingers are pointing even with your eyes closed. Now open your eyes and confirm that your inner vision of your hand was correct. This means that each gesture that displays a particular letter in our hand also displays that letter in our minds.

Since letters generated by hand gestures can represent directions of conscious focus in our minds, a sequence of these letters can be used to specify and reconstruct a meditation. This may explain why certain Scriptural texts were canonized and why these letters are rightly called "sacred." Thus the same letter-signs are elements of both cosmology and consciousness at the same time.

Meru Foundation research has also shown that because the natural meaning of hand gestures is universal for all humans,12 each letter displayed (and seen in the mind's eye) by each gesture has explicit meaning. For example, the name of the Hebrew letter Pe (Greek Pi, Arabic Fe, English "P") traditionally means "mouth" (or swallow, puff, or engulf). It can only be seen in the model hand when it is worn on our hand - within the limits of motion of our arms and wrists - when our hand points to our mouth, thumbs in, fingers flared, in a puffing gesture. Every other letter is similarly explicitly defined by the gesture that displays it to us. It is usually possible to "decipher" the operational meaning (not the idiomatic meaning) of root words in all languages (that can be phonetically related to Hebrew, Greek and Arabic) by examining the sequence of letters alone.

There are many other examples we could suggest from every culture and field of study.

Finally, there is the Hebrew word for Unity ("one"): E-cHoD. It means "archetypal acuteness or sharpness." (Aleph - "archetype of"; cHet - "encompassing"; Dalet - "dividing.") By its exquisite singularity, E-cHoD represents the essence of "highest contrast." The so-called "credo of Judaism," the Sh'Ma (in the daily prayer book) states that "..., God is E-cHoD (One, exquisitely Singular)."

This ancient model of the Unity of physics and consciousness is also compelling and potent in a modern sense. As physicist Nick Herbert notes in his discussion at the end of his book Quantum Reality, Beyond the New Physics:

The perspective presented above suggests that we should not lightly dismiss the spiritual claims of our traditional religions. If we dismiss these teachings because they appear to be self-serving apologia presented by interested partisans, as has been suggested to me by friendly scholars, then we are explicitly excluding the possibility that they have real meaning. Whether or not spiritual discussions are often "damned by the faint praise" of faith, they must still be explored honestly. Those who have preserved and protected these traditions may not be the same persons who can make use of their content.

Although it is bound to be considered insulting (and thus to have no validity) by those who do not understand geometry and who therefore adhere to exclusively literal literary understandings of traditional spiritual traditions, there may in fact be no possible way to describe Exquisite Singularity without mathematics.

Although some sources insist that it is anecdotal, it is generally believed that over the entrance to the Platonic Academy it was written that:


"Only he who is familiar with geometry shall be admitted here."

Our spiritual traditions are the vessels of our faiths. The function of these vessels, like nut-shells or seed-husks, is to protect the kernel with its germ of life, the flame, the hand of the transcendent, inside. If these vessels had not survived, there would be no hope for rebirth of the germ - the "light" - they carry.

Perhaps our presumption that our sages of the past were not really knowledgeable or wise has blinded us from seeing their knowledge and wisdom. The Abrahamic claim of an exclusive, Singular and Unique One-God only appears to be prideful puffery when our sight is overly narrow and literal. We misinterpret our partial view of the "vessel." Once we are aware of the distinction between Inside and Outside, between "vessel" and "light," we can appreciate the true significance of the use of a real model of Singularity for the definition of the Oneness of God. 

Note #1:

Kaplan's footnotes, from page 42 of The Handbook of Jewish Thought (Brooklyn, N.Y, Maznaim Publishing Co., ©1979):

32. Some say from the age of three; Nedarim 32a; Bereshith Rabbah 30:8, 64:4, 95:2; BaMidbar Rabbah 18:17; Shir HaShirim Rabbah 6:1; Esther Rabbah 6:5; Tanchuma, Lekh Lekha 3; Zohar 3:302a; Raavad. Avodath Kokhavim 1:3; Kesef Mishneh; Migdal Oz; Hagahoth Maimonioth 1, ibid.

33. Such as by ridiculing and destroying his father's idols: Bereshith Rabbah 38:19; Tanna DeBei Eliahu Zuta 25 (57a).

34. Bereshith Rabbah 39:1; BaMidbar Rabbah 14:7; Yad Avodath Kokhavim 1:3; Beth Elokim, Shaar HaYesodoth 48.

35. Abraham was thus the world's greatest astronomer; Bava Bathra 15a. He is also the leading personage of his time; Kiddushim 32b; Rambam on Genesis 26:29, 40:14.

Note #2:

It is interesting to note that in most of these examples the positions of the complements could be reversed. For example, in our models the geometric structure (Coal, Vessel, Particle, Tetrahedron, etc.) is considered to be "male" because these structural elements, represented by polyhedra, are rigid, discrete, and logically precise, while the complementary process (Flame, Light, Wave, Spirit, etc.) is considered to be female because the vortex forms which represent the process are cyclic, continuous, and dynamic (or "emotional"). But it could be the other way around:. The polyhedra could be seen as vessels and outside (like a womb) and thus "female" and the vortex forms could be seen as penetrating and inside (like a penis) and thus "male." This is similar to how Yin and Yang continuously flow into and mutually define each other as each becomes the other.

This article and its illustrations (unless otherwise noted), and all other material on this page, are
©1993, 1996, 1999 Stan Tenen, and licensed to Meru Foundation,
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