Four Dimensions

Picking up where we last left off, our dilemma of the three and four continues into the realm of mathematics. Preston Harold explains:

In the mathematician’s view, the physicist deals with four continua, or – more precisely – with four dimensions. In geometry, four dimensions would mean that one has four independent directions. For example, this could be seen by drawing four lines through a given point, all perpendicular to the others. In this narrow sense the universe has only three dimensions. The mathematicians extended the concept of dimensions to any situation where events can be described by independent coordinates, and where certain simple laws hold. In this broader sense Einstein found it convenient to use four independent coordinates, with time playing the role of a fourth dimension. In pure mathematics, as well as in its applications to physics, it is often convenient to use many more dimensions, even infinitely many.

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But the idea of wholeness, or of continua itself, as one dimension greater than any number of dimensions has not broken through the tetragrammaton – through the confines of four. For example, Einstein speaks of the “bold” interpretation of the modern quantum theory associated with de Broglie, Schrodinger, Dirac, and Born – he says their interpretation “is logically unobjectionable and has important successes to its credit. Unfortunately, however, it compels one to use a continuum the number of whose dimensions is not that ascribed to space by physics hitherto (four) but rises indefinitely with the number of the particles constituting the system under consideration.”

For how long does the number not ascribed to space rise? Is it infinite? Jesus will have something to teach us here but before we get to him, we will detour into the world of music.

Until next time, peace.

Triad or Tetrad?

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Difference emphases on either the three or the four can be found within varying cultures, the beginnings of which are lost in the sands of time.

Although scientists move away from the quantitative view toward the qualitative view and acknowledge the validity of both positions, the dilemma of three and four is by no means resolved – its beginning is lost in antiquity and its end is not yet in sight. As to its beginning, Jung says that number helps more than anything else to bring order into “the chaos of appearances…primitive patterns of order are mostly triads or tetrads,” and he points to I Ching, Book of Changes:

“…the experimental basis of classical Chinese philosophy…one of the oldest known methods for grasping a situation as a whole and thus placing the details against a cosmic background – the interplay of Yin and Yang… there is also a Western method of very ancient origin which is based on the same general principle as the I Ching, the only difference being that in the West this principle is not triadic but, significantly enough, tetradic…”

He refers also to the alchemists’ tackling of the problem of three and four, seeing the dilemma stated in the story that serves as a setting for the Timeaus and extending all the way to the “Cabiri scene in Faust, Part II…recognized by a sixteenth-century alchemist, Gerhard Dorn, as the decision between the Christian Trinity and the serpens quadricornutus, the four-horned serpent who is the Devil.”

Of course western religion and culture has been based on the tension between the three and the four, both being primary factors in the Holy Scriptures. The four is stated outright: YHVH, even translated into English as a four-letter word, LORD. The three is implied in the three visitors to Abrahm, the Christian Trinity, etc. Returning to alchemy’s approach of the problem, Preston Harold says:

Wolfgang Pauli discusses the controversy between Johannes Kepler, discoverer of the three famous laws of planetary motion, and Robert Fludd, in his day a famous alchemist and Rosicrucian. Pauli says that Kepler’s ideas “represent a remarkable intermediary stage between the earlier, magical-symbolical and the modern, quantitive-mathematical descriptions of nature,” indicating a way of thinking that produced the natural science which today is called classical. Kepler, a devotee of Euclid’s geometry, insisted upon strict mathematical methods of proof. His premise was that “Mathematical reasoning is ‘inborn in the human soul’…” His is a trinity-concept, his symbol “contains no hint of the number four or quaternity.” Fludd, however, was a mystic with great aversion to all quantitative mensuration: “It is significant for the psychological contrast between Kepler and Fludd that for Fludd the number four has a special symbolical character, which, as we have seen, is not true of Kepler.” Fludd drew his inspiration from Moses, and he brilliantly defends his stand on the nature of the soul. Kepler, however, appears to best him in all scientific argument until one realizes that Kepler considered the quantitative relations of the parts to be essential while Fludd considered the qualitative indivisibility of the whole. Pauli says, “modern quantum physics again stresses the factor of the disturbance of phenomena through measurement,” as Fludd (and Goethe) insisted upon. He concludes that the only acceptable point of view appears to be one that recognizes both the quantitative and the qualitative, “the physical and the psychical” as compatible, embracing them simultaneously.

This attitude eases the argument, but it does not resolve the dilemma of three and four, as may be seen in a mathematician’s explanation of continua.

We will explore this mathematical explanation in our next post. Until then, peace.

The Dilemma and the Pearl

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Chapter 8, “The Dilemma and the Pearl,” begins with Preston Harold asking us what type of outlook we bring to the world around us – are we a “three” or a “four?”

Wolfgang Pauli says that two types of minds have battled through history: first, the thinking type that considers the quantitative relations of the parts to be essential – and secondly, the intuitive type that senses the qualitative indivisibility of the whole.

The first type mind is posed on the side of three. This type took its stand with Euclid, resting upon his well-known axiom: the whole is equal to the sum of the parts. This axiom, along with the rest of Euclidian geometry, dominated Western thought until the late 19th century. One might say that Euclidian geometry still dominates, for the revolution in mathematics that tumbled it from sacred pre-eminence has not yet seeped down to the layman’s level, and many students will learn first, by rote, Euclid’s axioms, imbedding in the subconscious mind these fallible statements which have been presented as unquestionable truth…

One might say that the three represents Rene Guenon’s “reign of quantity,” the historical manifestation of the descent from form (quality) toward matter (quantity), and the “nothing but-ness” of stark materialism. Tradition calls this period the Kali Yuga, the age of the demon, Kali, or the iron age.

Today, the second type of mind, posed on the side of four, insisting upon the qualitative indivisibility of the whole, regains much of the standing lost in recent centuries. As regards the sum of the parts in relation to whole being, scientists, dealing with one whole atom and the sum of its parts, have found that in the formation of a nucleus from protons and neutrons some of the mass of the particles apparently is converted to energy. The chemist sees that the combined action of several elements taken together is greater than the sum of them taken separately. Mathematicians working with transfinite number theory confront the concept that the whole can equal one of its parts. In short, one is forced to alter his concept that a discrete whole within the universe can be divided and its parts regathered to equal the sum of the whole…

Anyone wishing to look further into the “qualitative indivisibility of the whole” would do well to search out the works of the Goethean scientist Henri Bortoft. You can thank me later ;-).

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Of course both the three and the four have their place in our world but how do we go about regaining the balance between the two? This is what we will continue to explore in Chapter 8. Until next time, peace.