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Bearing the view that numbers are only numbers in their practice, mathematicians conceal an aspect that I see as important, though not considered so by them, perhaps because it is taken as superficial or irrelevant. I'm raising the intuitive fact - which I intend to stress here - that these families of numbers have multiple relationships with specific segments of the objective and subjective worlds.

It's obvious, for instance, that whole numbers have a clear connection to the enumeration of collections of discrete objects. It could be said this is their "natural" function in the empirical world. We can state that there are eight children in a room, or that a fleet has fifteen ships, but it's impossible that in a certain room exist 6.43 children, or that a fleet is composed of 15.217 ships. If such numbers were to appear in reference to those collections, they would be the result of statistical calculations, or calculations of some other nature without direct relation to the real world. This occurs because fractions and negative numbers - to mention just these two - are not appropriate for describing a group of children, or the ships comprising a fleet. In abstract procedures of calculation, the use of all numerical forms is legitimate, but only during the intermediate stages. It could never be acceptable as a final conclusion adjusted to the natural world that the number of children in a room were 8.344 or any other fraction, simply because children cannot be divided into pieces that continue to be children.

Nevertheless, negative numbers and fractions can well be adjusted to particular segments of everyday life. A bank account, for instance, can have a positive or negative balance. The inherent properties of negative numbers are appropriate for describing such aspects of reality. This seems to be, as we said before, their "natural" function in the world we live in, although they can be used much more freely in calculation procedures. Fractions are excellent to slice a pizza, or for dividing rural property among heirs, but they can be entirely inadequate to describe other segments of reality, as we have seen. The same is true for vector numbers, transcendental numbers, imaginary numbers and all other families that have been created - or discovered - along mathematical history.

In principle, each of these families of numbers have two different ambits of application in which they can be used: calculations and descriptions. The larger ambit, wider and flexible, encompasses all computational possibilities: the many calculation types and operational possibilities of mathematical analysis. The narrower ambit - and more pragmatic one, being linked to facts of varied natures - is restricted to the segment of totality which can be adequately described by that number family.

A comprehensive study of these two ambits related to the several families of numbers is yet to be undertaken. It seems, at first look, that it would be a barren task devoid of palpable results in terms of scientific advancement. Rather, it can yield pleasant surprises and discoveries if conducted with a breadth of vision capable of perceiving and accepting that totality is not restricted to the physical universe, and that the pathway to knowledge is an endless road whose terminus cannot be seen. If we the men and women of science, philosophy and the arts, who search for a clearer understanding of existence, succeed in breaking down the barriers that we have built around ourselves - barriers represented by specialization, or worse, by our adherence to sterilizing metaphysical assumptions - we will undoubtedly be on a track that leads to the vastest synthesis. Specifically in relation to number families and their respective realms - a subject I am introducing in this lecture - an evident example emerges that something unexpected and valuable may result from the simple and unprejudiced observation of already known facts and relationships under a new light, free of commitments to previously adopted lines of thought.

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Euler has written in his 1770 Algebra:

All expressions like or are impossible, or imaginary, once they represent the roots of negative quantities, and about them we can definitely affirm     they are not zero, nor more than zero, nor less than zero, and this makes them imaginary or impossible.

But history repeats itself, and little by little imaginary numbers became assimilated in spite of being considered absurdities of logic. Later, they became important contributions to mathematical analysis, specially as complex numbers, hybrid entities composed of an imaginary part and a real one, under the general form a + bi. Soon the notation i was adopted to represent the imaginary unit, resulting in i = Imaginary (and complex) numbers became considered another family of numbers, with ontological status equivalent to the others, useful as working tools, without more concern for their intrinsic meaning or what realm of totality they could fit into more properly. Only a slight worry remained, that, when imaginary roots are obtained as the result of an equation, they must be considered as devoid of logic, and as such should be disregarded ⁰³.

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Along with the utilitarian assimilation of imaginary numbers came the notion that there are four distinct units in mathematics. Each one bears its own meaning and properties, all derived from the original One, identified with the unit of whole numbers, the 1 in the series of natural numbers. In a free association of ideas, without any of the prejudices or constraints that imprison those confined to the golden cages of their specialization, I recollect the mythical significance of the Unit in numerous classic cultures, such as the Indian, Chinese and many others of ancient times. I recall the Board of Emerald, on which Hermes Trismegistos left recorded that "as all things are and proceed from One, by the meditation of One, so all things are born proceeding from that only thing, through adaptation".

If I were to not be participating in this event, surrounded by highly skilled scientific minds from diverse cultural backgrounds, searching for the most unexpected interactions that may occur in every field where the human spirit moves, I would feel uncomfortable about leading my discourse towards these considerations. However, in the context of this Symposium, where we look for the unusual outcome, specially if it emerges from new and unexpected relationships between diverse fields of knowledge, I declare that approaches emerging from myth, poetry or even plain speculation, far from being a deviation, are in fact windows which permit completely new insights to our understanding, unattainable by those who limit themselves to the inside of their logic frontiers.

This way, we have at our disposal four different units, each bearing its own properties:

a) the unit of real whole numbers +1, or simply 1;
b) the unit of negative real numbers -1;
c) the imaginary positive unit +i, or simply i, and
d) the imaginary negative unit -i.

We can chart 1, -1, i, -i on two orthogonal axes, thus introducing the Argand-Gauss plane - which allows the plotting of complex numbers, real and imaginary. (see fig. 1)

Fig. 1 - Argand-Gauss plane of complex numbers

At this point of my lecture, I wish to make clear that my thesis is that the segment of totality "naturally" fit by imaginary numbers is the psychic world, the subjective realm, the spiritual side of Being that physicists and cosmologists linked to the old paradigm place outside of their universe, outside their field of research, to which the most radical even refuse to ascribe any "reality" whatsoever.

In regards to this last point, we must concede that those committed to the old paradigm do have some reason, depending on the confines attributed to the term reality. If we are to understand that reality is everything that exists, we glide into another semantic problem of defining the limits of meaning ascribed to the verb to exist. If we don't wish to extend into a philosophical theme that encompasses many interpretations and views, we must adhere to the elementary definition that real is anything complementary to psychic, subjective, abstract, spiritual or imaginary matters. To smooth our way, we can return to the old Cartesian proposition of a res extensa opposed to a res cogitans, a dualistic view of Being which may be contested by many philosophers and scientists that succeeded Descartes, but is invaluable as a tool for understanding totality, which, as we know, can only be grasped when reduced to contrasting opposites.

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Given the limits of this definition, and for the purpose of my lecture, we abandon the wider notion of reality, in spite of its use by Carl Gustav Jung who says that anything that can influence the empirical world should be considered as real . In his conception, that we will not adopt here, thought is real, mathematical abstractions can be real, even our dreams of last night can be taken as real, as long as they influence our lives.

As has already been established, we can only reach knowledge by means of contrasts: the mind proceeds dichotomies to make wholeness understandable. The undifferentiated Oneness, Brahman, the Chinese Tai-Chi, the Innominable, the Essential Being in its omni-embracing totality is unattainable to empirical knowledge. Only through the introduction of duality - the primal notion of polarity - does rational knowledge become possible, and a particular being achieve perception through the elementary contrast between subject and object, shape and background, information and redundancy. This theme in the language of sacred books would read:

In the beginning there was the innominable and undifferentiated Being, omni-embracing in its supreme plenitude, pure eternity devoid of qualities: the primordial Oneness. Its only possible conception is the number One, that in solitude nothing represents, therefore being equal to zero, to nothing, to non-being. In this primal state, Being is equal to non-being and this is not understandable.
But through the meditation of One, duration emerged, precisely in the frontier between existence and non-existence. And the One meditated its duration:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
In its undifferentiated duration, the One simultaneously existed and didn't exist. But in it everything had a potential existence. And the One, while existing, meditated its non-existence, conceived initially as its negative:
-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1...
Afterwards, in a primordial instant, anterior to time itself (hence inconceivable), the One meditated not only its negative but also its intrinsic impossibility to manifest itself as existent, self-immanent in its perfect transcendence, and so appeared the expression of this total impossibility, that later, in the differentiated world, humans will represent as i:
i, i, i, i, i, i, i, i, i, i, i, i, i, i, i, i, i, i...
From this meditation of the One, as a coronation of its early essence in undifferentiated eternity, emerged the denial of this impossibility, the key to the return to itself in plenitude and transcendence:
-i, -i, -i, -i, -i, -i, -i, -i, -i, -i, -i, -i, -i...

Only then was time allowed to begin and, with it, the work of Creation. Meditating on Oneself, the undifferentiated conceived multiplicity when He became aware that within Himself everything existed as a potential possibility. Here we return to the notion of potentia, introduced by Aristotle and, since then, underlying Western thought, without unfolding or ripening until its revival in the context of quantum physics. Heisenberg and Schrödinger used this concept to describe the state of a non-observed particle. This potential existence was the instance that permitted Being to break out of the barrier of eternity, thereby creating information, the quaternion and the rotational motion, essential tools without which existence would never have emerged from the womb of transcendent plenitude.

But... how was this achieved? Through a calculation on the four aspects of the mathematical unit, which men later rediscovered and significantly - and certainly not by blind chance - named power calculation, which implies the concept of potentia. Within these accidents and coincidences, thousands of gods conceal themselves.

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Now I ask for your utmost careful attention, as I intend to demonstrate what I believe to be the first mathematical formulation of the divine proceedings prior to Creation, scrutinized through power sequences calculated on the primordial units. As a result we will see in the symbolic realm of mathematical symbols: the first appearance of the elementary information unit, the bit; the quaternion, which is itself the basis of the intelligible world; the rotation motion, a quintessence of all matter and energy - the two faces of the physicists' universe. Let us begin.

First, let's observe what occurs when we perform a series of power calculations on the elementary unit of positive natural numbers, the primal representation of the undifferentiated and unattainable One, that has given origin to everything:

10 = 1, because any number increased to the power of zero is equal to the positive unit;

11 = 1, because any number increased to the power of 1 is equal to itself;

12 = 1, because the square of the positive unit equals itself, as it happens to any power to be increased, the positive unit always is equal to 1, hence we obtain the series

10 = 1; 11 = 1; 12 = 1; 13 = 1; 14 = 1; 15 = 1; 16 = 1; 17 = 1..., etc.

It's notable that the sequence 1, 1, 1, 1, 1, 1, 1, 1, 1..., the primordial mantra, constitutes the maximum level of redundancy in a system, a complete absence of information, showing us that as long as the One remains identical to itself, it cannot create anything.

Then, let's see what happens to the One when, in its supreme plenitude, it conceives its own denial, represented in the symbolic world of mathematics by the negative unit -1:

-10 = 1, because any number increased to the power of zero is equal to the positive unit;

-11 = -1, because any number increased to the power of 1 is equal to itself;

-12 = 1, because the square of a negative number is always a positive number, hence -1 -1 = +1;

-13 = -1, because this is equivalent to the square of -1, which is 1, as we have seen, multiplied once more by -1, or +1 -1= -1, what leads to the series

-10 = 1; -11 = -1; -12 = 1; -13 = -1; -14 = 1; -15 = -1; -16 = 1; -17 = -1..., etc.

What, then, do we see in the series 1, -1, 1, -1, 1, -1, 1, -1...? Simply the genesis of information, the appearance of the primal duality, the first contrast in the undifferentiated unit, the birth of the bit and of binary language used in the I-Ching (The Book of Change) and computers, fascination of Leibniz, and the most elementary of all possible languages. In the beginning there is the Verb, and the bit of information is the primordial seed of the Verb.

The bit, the elementary unit of information, is recognized as the most basic of all components of the differentiated realm of multiplicity, appearing prior to other elementary units of time, space and energy, which base on it their legitimacy and existential status in the intelligible world.

Now, let's go a step further in the divine process that precedes Creation, examining what will happen when we submit the same calculation of powers to the next unit conceived by the meditation of the One: the imaginary unit i - whose condition of existence is precisely not to exist, a primordial paradox which shows us that divine logic is quite unlike common logic, which so often blocks mankind's path in the search of knowledge:

i0 = 1, because any number increased to the power of zero is equal to 1;

i1 = i, because any number increased to the power of 1 is equal to itself;

i2 = -1, because this was the very genesis of the imaginary unit: i is the notation that made the square root of the negative unit come out of nothingness, so i2 = -1;

i3 = -i, because this is the same as i multiplied by -1, which is done by simply switching the value sign, or i -1 = -i, yielding the series:

i0 = 1; i1 = i; i2 = -1; i3 = -i; i4 = 1; i5 = i; i6 = -1; i7 = -i..., etc.

Clearly, the sequence 1, i, -1, -i, 1, i, -1, -i, 1, i, -1, -i..., witnesses the birth of the quarternion, an archetypal structure of Creation according to the myths of origin of most ancient cultures, and studied in depth by Carl Gustav Jung in its relationship to the foundations of the resident knowledge seated in the abysses of man's unconscious realm.

I don't intend, at this point, to scrutinize the profound significance of quaternion structures, but I will state that they correspond graphically and symbolically to two pair of opposites, as the cross, which is the mathematical representation of the two axis of the Cartesian plane - two axes crossing, forming four right angles. In this primordial case which we are examining, there are two positive-negative oppositions that occur both in the vertical and in the horizontal axis, thus the Argand-Gauss plan. (see fig. 2)

Fig 2 - Quadrants of real-imaginary totality

Certainly, an astute mind would not miss the significance of this structure appearing as result of successively increasing the imaginary unit to powers, a unit defined by its own nonexistence, the unit of something which is more inexistent than a mere absence, an outrage to reason, inadmissible to human understanding and logic. Nevertheless, from this primordial process comes the most basic of all archetypal structures, emerging from something more subtle than nothingness itself, directing with the force of its abstract powers to the correct way to meditate on the imaginary realm, on the subjective side of the universe. The spiritual slope of Being, for short.

Myths of origin found in the most fertile ancient cultures are solidly rooted on this primeval structure of the quaternion. The four directions of the world, the four seasons, the four Vedas, the four truths of Buddhism, the four functions of consciousness, the four elements and the four qualities of substance in medieval metaphysics, the most spread understanding of the cosmology and cosmogony rooted in humanity's collective unconscious, the perfect key to enlighten our awareness of the ineffable Being who Is us.

But our investigation is not yet finished at this point.

In plotting the quaternion on the Argand-Gauss plane, thus creating the primal symbol of the cross, we see the birth of the concept of position, and also of the notion of origin - or center -, the idea of unlimited expansion and that of direction. Most amazingly we witness the genesis of the rotational motion, the spin, the innermost basis of the physical world, whose most ancient graphic form is the immemorial swastika, propelled by the sequence 1, i, -1, -i. (fig. 2)

So, 1, i, -1, -i, the primordial rhythmic pattern, has its origin detected by a most elementary mathematical calculation, in which one doesn't even need to go beyond the unit to begin operating with plural numbers, which emerged well after these proceedings prior to the divine Creation of this particular cosmos where we live.

In closing my lecture, leaving to your imagination the numerous pathways opened by these crystal clear insights, I wish to show all of you who honor my speech with your attention how the primordial cycle closes itself, once again utilizing power calculations, now on the negative imaginary unit -i, whose deepest meaning lies beyond the human capacity of abstract conception. Let's observe:

-i0 = 1, because any number increased to the power of zero is equal to the positive unit, reaffirming definitely the absolute character of the One as the origin of everything and nothingness;

-i1 = -i, because any number increased to the power of 1 is equal to itself;

-i2 = -1, because when we increase the imaginary unit to the square we are transforming it into the real negative unit;

-i3 = i, evidently, thus yielding the series

-i0 = 1; -i1 = -i; -i2 = -1; -i3 = i; -i4 = 1; -i5 = -i; -i6 = -1; -i 7 = i... etc.

In the sequence 1, -i, -1, i, 1, -i, -1, i, 1, -i, -1, i,...., the cycle of primordial rhythms is closed: Ouroboros, the alchemist serpent, with her imaginary ringed body, now runs backwards in the Argand-Gauss plane, ready to bite her own tail and throw us in multiplicity, into the realm of time that, "as all things, is and proceeds from One, by the meditation of One, as all things are born proceeding from that sole thing, through innumerable adaptations".

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1. The early engravings  of a symbol for the number zero appeared in Cambodia and Sumatra in the year 683. (J. Needham, Science and Civilization in China, vol. 3). back to text
2. When Gauss, Riemann and Lobatchevsky created the first non-Euclidean geometries, these were seen by the scientific community - as much as the non-Boolean logics - as mathematicians' playthings devoid of any contact with "reality". The same happens to other mathematical entities, including imaginary numbers and imaginary roots, that come out as solutions to many equations and are discarded because they "do not make sense" or "are not real", making explicit the old confusion between not real and not true. Obviously, the imaginary isn't real, since it's connected to the imaginary slope of wholeness, complementary to the real slope that describes the physical objective universe, to constitute the objective/subjective totality of Being, the unus mundus discerned by Jung.back to text
3. J.C. Polkinghorne,  in his 1984 book The Quantum World, observes that "[Quantum mechanics] also uses complex numbers, worthwhile in the correct occasions yet disastrous as eigen values. Quantum mechanics can be peculiar, but not to the point of yielding the square root of negative one as the result of an experiment".  back to text

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Further readings:

• Asimov, Isaac. Realm of Numbers. © Isaac Asimov, 1959.
• Barker, Stephen F. Philosophy of Mathematics. New Jersey: Prentice Hall, Inc., 1964.
• Boyer, Carl B. A History of Mathematics. John Wiley & Sons, Inc., 1968.
• Dantzig, Tobias. Number - The Language of Science. New York: The Macmillan Company, 4th ed., 1953.
• Davis, Philip and Reuben Hersch. Descartes' Dreams: The World According to Mathematics. New York: Harcourt Brace Jovanovich, 1986.
• Dopp, Joseph. Notions de Logique Formelle. Paris: Éditions Béatrice-Nauwelaerts, 1965.
• Edwards, Elwyn. Information Transmission. London: Chapman & Hall Ltd., 1964.
• Hall, C.S. and  Vernon J. Nordby. A Primer of Jungian Psychology. New York: New American Library, Inc.
• Islam, Jamal Nazrul. An Introduction to Mathematical Cosmology. New York: The Press Syndicate of the University of Cambridge. 1993.
• Jung, C.G. Die Dynamik des Unbewussten. Walter Verlag, AG, Olten, 1971.
• Needham, J. Science and Civilization in China. Cambridge University Press, 1959.
• Peirce, Charles Sanders. Philosophical Writings. New York: Dover Publications, 1955.
• Penrose, Roger. The Emperor's New Mind – Concerning Computers, Minds and Laws of Physics. Oxford University Press, 1989.
• Polkinghorne, J.C. The Quantum World. Longman Group Limited, 1984.
• Salmon, Wesley C. Logic. New Jersey: Prentice-Hall Inc., 1965.
• Skyrms, Brian. Choice and Chance – An Introduction to Inductive Logic. Dickenson Publishing Company, Inc., 1966.
• Wilhelm, Richard. I Ging - Das Buch der Wanlungen. Dusseldorf, Köln: Eugen Diederich Verlag, 1956.

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