Reciprocal System of Mathematics - Background
Mathematics and Physics
In his book, New Foundations for Classical Mechanics, David Hestenes observes:
- There is a tendency among physicists to take mathematics for granted, to regard the development of mathematics as the business of mathematicians. However, history shows that most mathematics of use in physics has origins in successful attacks on physical problems. The advance of physics has gone hand in hand with the development of a mathematical language to express and exploit the theory…The task of improving the language of physics…is one of the fundamental tasks of theoretical physics.
One of the most common criticisms of Larson’s works is that they lack a “mathematical language to express and exploit the theory.” Larson developed the system mathematically, but he did not explicitly develop a mathematical language for it. In fact, he was adamant that there was no need for such a development. As he wrote in The Neglected Facts of Science:
- A change in [the definition of motion that is] the base of the [new] system naturally necessitates many modifications of the details of physical theory. However, the amount of change that is required is not nearly as great as might appear on first consideration, because the new development calls for very little change in the mathematics of present-day theory. The changes are mainly in the interpretation of the mathematics, in our understanding of what the mathematics mean. Since the case in favor of the currently accepted theories is primarily–often entirely–mathematical, there is little that can be said, in most cases, in favor of current theory that is not equally applicable to the mathematically equivalent conclusions that I have reached. The substantial advantages of a fully integrated general physical theory are thus attained without any violent disruption of the mathematical fabric of the physics of familiar phenomena. All that is necessary in most instances is some alteration in the significance attributed to the mathematical relations, and a corresponding modification of the language that is utilized. These new interpretations, integral parts of a consistent, fully integrated general theory, can then be extended to a resolution of the problems that are currently being encountered in the far-out regions.
This is especially true, when one realizes that the new system doesn’t supplant the old, but rather enhances it; that is, the base of the current system, vectorial motion, is the motion of objects relative to one another. Of course, this motion is also part of the new system, and, as long as the vectorial motion is not a significant fraction of the speed of light, the new system really has little to add to the current system in its realm of vectorial motion. On the other hand, however, the vectorial motion of the current system cannot be defined without objects. Energy, radiation and matter must be put into the system in order to apply it to the study of natural phenomena.
This is not true under the new system of physical theory. All forms of energy, radiation, and matter emerge from the RST strictly as a consequence of its fundamental postulates. All the theoretical constituents of the universe of motion and their mutual interactions are either motions, combinations of motions, or relations between motions, including the one-dimensional translational, rotational, and vibrational motions, the vectorial motions, of matter.
What the RST brings to the table, so-to-speak, is a new kind of motion, hithertofore unrecognized as motion, the fundamental scalar motion of the universe, manifest in the observation of the continuously forward march of time and space. Clearly, the recognition of this new type of motion doesn’t change the physical laws that science has already discovered that apply to relatively low speeds, and it doesn’t change the usefulness of the mathematical language used to express and exploit these laws, but it’s clearly unreasonable to think that a new mathematical language, suitable for expressing and exploiting the laws of the new scalar motion, would not actually be needed. Not only is it needed, but given the mysterious nature of the relationship between mathematics and physics, what Wigner characterized as the “unreasonable effectiveness of mathematics in physics,” one would expect that it would provide great insight into the new system and its consequences.
What is a Mathematical Language
Larson’s conclusion that “the new development calls for very little change in the mathematics of present-day theory” may have been premature, but deciding that a new bottle is needed for the new wine of reciprocity is one thing, and actually developing it is quite another thing. Just how should one go about developing a mathematical language to express and exploit the physics of scalar motion?
To understand one answer to this question, it’s helpful to begin at the beginning, with the Greek system of mathematics. For the Greeks, mathematics was science, and there were four fields of science:
- 1) Arithmetic - Numbers
- 2) Harmony – Vibration as ratios of numbers
- 3) Physics - The motion of heavenly bodies
- 4) Geometry - Space magnitudes
There was not always consensus on the order of the fields, but it was generally thought that there was a hierarchal relationship between these four fields; that is, that numbers were fundamental, because numbers were required to express ratios, ratios that were not necessarily interpreted in terms of magnitudes of space and time, but as fundamental numbers of harmony. Clearly, geometry too depends upon numbers, but understood in terms of fundamental proportions, not in terms of magnitudes of space and time (motion).
However, centuries after the Greeks, Newton clearly understood that the spaces of geometry are truly dependent upon vectorial motion (mechanics). Unless vectorial motion can describe the “right lines and circles” required by geometry, it would not be able to do its magic. He wrote:
- The descriptions of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn…and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things…
However, this leads us to ask, if harmony depends upon numbers, and if geometry depends upon vectorial motion (mechanics), does this mean then that mechanics (physics) depends on harmony in turn, completing the Greek hierarchy of mathematics (science)?
Harmony was more than the musical vibrations of strings to the Greeks. It was a cosmological concept that is connected with metaphysical meanings of the first four numbers that they called the tetraktys (tetras means four).
Originated by Pythagoras, the tetraktys was so sacred to the Pythagoreans that it formed the basis of their oath:
- By that pure, holy, four lettered name on high, nature’s eternal fountain and supply, the parent of all souls that living be, by him, with faith find oath, I swear to thee.
The dots represent the numbers 1, 2, 3, and 4, and their descent symbolizes the order of creation of the known universe, and the increasing complexity of its manifestation. The four lower dots represent the four elements; the upper, the first principle. However, the mathematical concept of the tetraktys is harmony. It consists of the four fundamental ratios of vibrating strings: the fundamental ratio, or first principle, which is the vibrating frequency of a string of a given length, the second ratio, double the first, which is the vibrating frequency of a second string, half the length of the first, and two more, the fifth and the fourth, which were harmonics contained within the interval of the first and the second ratios (3/2 & 4/3 lengths). This fact was highly mysterious to the ancient Greeks, whose observations founded the vast field of today’s western music.
But how is the harmony of the cosmos related to numbers? Does it depend on numbers, in a way that is similar to the dependence of geometry on physics; that is, on principles “brought from without?” Is there a glory of harmony that enables it to do many wonderful things based on just a few principles of numbers?
Recalling that the dots of the tetraktys represent the numbers 1, 2, 3, and 4, and the increasing complexity and order of the universe, it is clear that these numbers are counting numbers that seem to have little in common with the magnitudes of length, area and volume. Geometry deals with magnitudes of space described by vectorial motion (mechanics), i.e. magnitudes of space and time. Space magnitudes have direction and dimension, and even the ever forward march of time magnitudes has direction in a sense, which is something numbers, as simple quantities, just do not have. Hence, harmony, as ratios of vibrations, is something with space and time magnitude, and these magnitudes of vibration have direction and dimension, but the counting numbers have neither direction nor dimension. For this reason, Euclid was careful to keep them separate from the magnitudes of geometry, something Hestenes believes was a huge stumbling block to the development of modern civilization.
Centuries later, scientists began generalizing the concept of number to include the notion of magnitude, a quantity of length, and extending the counting numbers to rational, and irrational numbers, or to the real numbers of the continuum, which could be represented symbolically. However, while numbers could be adapted to represent the magnitude of length, the idea of the direction of such a length, or magnitude, was not included in the number itself. A number ‘a’, was magnitude in a direction that had to be specified with natural language. According to Hestenes, Descartes was the first to develop algebra systematically into a geometrical language in 1637, but his idea that numbers should be put in a one to one correspondence with the points on a line was so easily accepted at the time that it indicated that the “notion of number underwent a profound evolution” from that held by the Greeks. The idea was rejected by the Greeks because of the “incommensurables” they had discovered, such as the square root of 2, but by giving symbolic names to these magnitudes, they became “real numbers” at last, which “gave a precise symbolic expression to the intuitive notion of a continuous line.”
However, in spite of the huge advance in algebra that the identification of numbers with the magnitude of geometric length made possible, it took another 200 years before Grassmann was able to extend the idea of direction to numbers in a precise and complete manner. Grassmann’s work led to the idea of a “directed line segment,” or vector, and soon algebra contained both real numbers, or scalars, and vectors. The central idea is that the true significance of a number lies not in itself, but “solely in its relation to other numbers.”
Nevertheless, in all of this there was an effectual struggle that yet remained, because the number zero plays a special role in the algebraic rules of both scalars and vectors; that is, like a scalar, if a vector, a, is added to zero the result is equal to the vector. Likewise, if the sum of two vectors, a + b, is equal to zero, then b is a unique vector that is the negative of a.
However, it took centuries to accept the idea of negative magnitudes, because negative quantities were unthinkable, and the square of any scalar is a positive magnitude. According to Sir Michael Atiyah, to bridge the perceived conceptual gulf, mathematicians came up with what is “probably the biggest, single, invention of the human mind in history,” the imaginary number ‘i’, the square of which is declared to be -1. Then, with a legitimate negative scalar, a vector, a, times -1, is negative a, and this, says Hestenes, “justifies interpreting -1 as a representation of the operation of reversing [the] direction” of vector a.
About the same time that Grassmann was working the concept of generalizing number to include direction of magnitude, Hamilton was attempting to generalize number to include three directions, extending the concept of numbers from the complex, used to specify two magnitudes with direction as coordinates in the 2D complex plane, to the concept of quaternions that could be used to specify three magnitudes, as coordinates in a 3D spatial coordinate system.
In short, the concept of numbers with “direction,” or negative and positive numbers with one, two, or three directions, was a long time in coming, but it was the invention of the imaginary number ‘i’, as the square root of -1 and the idea that positive and negative scalar numbers can be identified in a precise and symbolic way with the continuous, directed, line segment, which opened the door for truly exploiting the power of algebra in physics. Soon, with the development of the inner product that recognizes that the difference between scalars and vectors can be treated algebraically to express trigonometric relations in algebraic equations, the algebra of scalars and vectors would become a useful language for describing the magnitudes of the real world.
However, in spite of the powerful advance these innovations provide, according to Hestenes, “this algebra [of scalars and vectors] is still incapable of providing a full expression of geometrical ideas,” and, he writes, “…there is nothing close to a consensus on how to overcome this limitation.” He notes:
- …there is a great proliferation of different mathematical systems designed to express geometrical ideas - tensor algebra, matrix algebra, spinor algebra - to name just a few of the most common. It might be thought that this profusion of systems reveals the richness of mathematics. On the contrary, it reveals a wide-spread confusion - confusion about the aims and principles of geometric algebra.
Hestenes’ position is that the confusion over the “aims and principles of geometric algebra,” referred to in the above quote, has to do with the dimensions of geometrical magnitudes; that is, it is the consequence of a failure to clearly recognize that, lo these many centuries, the objective of mathematicians and scientists has been to completely generalize the concept of number, so that it expresses all the properties of geometric magnitudes, including dimensions. It is Hestenes’ discovery that the world has failed to pursue this most fundamental of all mathematical quests that has led to his life’s work: the development of Geometric Algebra (GA).
GA is founded on the work of Clifford algebras. Clifford, following Hamilton and Grassmann, discovered that the combined ideas of multiple imaginary numbers and geometry, is an evolution from the concept of numbers as quantities to a concept of numbers as magnitudes with direction, and dimension, and it follows a binomial expansion, forming the set of the only normed division algebras known to exist within the first four numbers of Pascal’s triangle. That these first four numbers are precisely the numbers of the Greek tetraktys is obvious. Moreover, the algebra of the fourth line corresponds to the three dimensions of Euclidean geometry. To the Greeks, this is the highest manifestation of physical complexity, and it forms the name of the Hebrew God in the similar Tetragrammaton of the Kabbalists, but modern mathematicians call these numbers “octonions,” the mysterious “fourth dimension,” because they contain a total of eight numbers containing the four dimensions of 3D geometry: the point, the line, the plane, and the volume.
Mathematicians learned to do arithmetic (add, subtract, multiply and divide), with these numbers as algebras of n-dimensional numbers, but they found short cuts too, using matrices that didn’t require them to use the four dimensions of the quaternions, let alone the eight dimensions of the octonions. One of the main difficulties in using numbers of more than two dimensions (2D complex numbers) is that quaternions (4D complex numbers) don’t commute, and octonions (8D complex numbers) do not commute, nor obey the associative law of arithmetic (that is, a x (b x c) is not equal to (b x a) x c in the octonions.)
However, Hestenes’ GA is an associative algebra based on the octonions. It’s still non-commutative, but this is understood as a necessary evil of three-dimensional rotations, as anyone who has ever written flight simulator software can attest. Nevertheless, the key innovation of GA is the definition of the sum of Grassmann’s inner and outer, or wedge, product, as the geometric product. In essence, the geometric product is an expression of the dimensional property of numbers, thereby completing the task of the ages, the generalization of number as geometrical magnitude. Hestenes writes, refering to the basic equations of the outer and inner product:
- [The above equations illustrate] the general rule that outer multiplication by a vector “raises the dimension” of any directed number by one, whereas inner multiplication “lowers” it by one.
Thus, he concludes:
- Clearly, the generalized inner and outer products provide an algebraic vehicle for expressing geometric notions about “increasing or decreasing the dimensions of space.”
However, it should also be noted that in raising and lowering the dimensions of space algebraically in GA, the direction of the number is important. Indeed, this is why the outer product, and thus the geometric product of GA, is non-commuatative, which led Hestenes (following Clifford) to the idea of “orientation,” a new property of geometric magnitudes used to express the direction of n-dimensional vectors in GA. Interestingly, this seems contrary to Newton’s idea that the “right lines and circles” of geometry depend upon “principles brought from without.” Does this mean that Hestenes has managed to combine mechanics (physics) and geometry by eliminating the boundary between them drawn by Newton? We will come back to this question below.
In Search of Harmony
The modern ability to use complex numbers in physics has led to the modern world of technology and science. The power of the imaginary number is thus almost indescribable. Unfortunately, however, its use, while it has led to everything from Maxwell’s equations to quantum mechanics, has actually destroyed the harmony of the ancient Greek tetraktys, which today, in the mind of modern man, is relegated for the most part to the relics of a quaint and naïve people of the past. The role of the harmony of numbers, as the ancient Greeks understood it in the hierarchy of mathematics as science, has been supplanted in the minds of modern mathematicians and physicists by new concepts of symmetry, as laws of conservation. Indeed, even the idea of a harmonious hierarchy of science and mathematics, as the Greeks understood it, is lost on modern man, replaced by a non-hierarchal complex of faintly inter-related concepts.
The modern irrelevance of a mystical meaning found in the harmonious fitting together of numbers, ratios of numbers, vectorial motions, and geometric spaces, is hardly noticed today, even though it was Eugene Wigner, a pillar in establishing a great deal of the modern complexity of science and mathematics, who first articulated the sense that something of the mysterious still lingers in the relation of mathematics and physics in the study of nature. Wigner wrote:
- The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
What’s behind this mystery of mathematics’ relation to nature? Could it somehow be related to the mysteries of numbers, as the ancients understood them? Is the unity and simplicity we seek to find in observations of colliding protons, and galactic explosions, hidden in the concepts of the ancient Greeks after all?
As it turns out, the advent of the RST gives us an opportunity to take a look. The missing key seems to be found in the fact that Hestenes credits Clifford and Grassmann with understanding for the first time the significance that there are two interpretations that can be given to number. Hestenes calls one of these the quantitative interpretation, and the other he calls the operational interpretation of number. Actually, Hestenes uses the operational interpretation of number, which he incorporates in GA, as the meaning of rotations inherent in the imaginary numbers of a 3D spatial coordinate system, which consequently enables him to algebraically represent rotations without coordinates in his algebra. However, here the reciprocal relation of two numbers is of more interest, and the operational interpretation of number that will be employed is that which the ancients used whenever they measured something in a balance.
Rational Numbers Operationally Interpreted
Taking this course, the ratio of two natural numbers (quantities) can be interpreted as a number, and the system of these reciprocal numbers, or ratios, forms the well known rational number system. However, given the reciprocal interpretation of number, the rational number system leads to the system of numbers called the signed integers, with the exception that zero is replaced by the unit ratio. Nevertheless, there are actually three ones in this system: the unit ratio, and two “displaced” ratios of unit magnitude on either side of the unit ratio, corresponding to the negative and positive integers as shown below. Thus, on this basis, the integers become rational numbers with two “directions,” which can be designated positive and negative:
- n/m…,1/2, 1/1, 2/1,…m/n = -x…,-1, 0, 1,…x
Now, returning to the four Clifford algebras referred to earlier, which correspond to the tetraktys, and given the reciprocal system of signed integers, as operationally interpreted numbers with two “directions,” the same binomial expansion of the first four numbers generates the Pascal triangle, but with a difference. Instead of denoting the first algebra, as the algebra of real numbers, the second algebra, as the algebra of complex numbers, the third, as the algebra of quaternions, and the fourth, as the algebra of the octonions, there is simply a collection of n-dimensional scalar numbers:
- 1) 20
- 2) 21
- 3) 22
- 4) 23
Remarkably, when these four powers of 2 are interpreted as scalar numbers with two “directions,” it turns out that the ordinary, commutative, and associative algebra of scalars applies to all of them. In other words, because these numbers have positive and negative “direction,” in multiple dimensions, they correspond in every way to geometric magnitudes with the properties of points, lengths, areas, and volumes. Hence, their very existence predicts an alternate set of principles, “brought from without,” upon which the “right lines and circles” of geometry might be drawn, that do not involve the imaginary numbers of vectorial motion.
This is a significant discovery. Hestenes’ GA, interprets imaginary numbers as rotations that can be used to convert scalar numbers into directed numbers, in which the algebraic inner and outer products become the means for raising and lowering the dimensions of these numbers. In this way, GA provides numbers called “multivectors” that correspond to the points, lines, areas, and volumes of geometry, but, because the non-zero dimensional numbers are vectors, the algebra is necessarily non-commutative, and the idea of “orientation” must be introduced to specify the difference in the “direction” of the vectors; that is, in GA, a^b = -b^a, which means that there are two different ways to “draw right lines and circles,” and depending upon which way the constituent vectors of a line, area, or volume enter into the drawing of a given geometric magnitude, its “orientation” differs.
On the other hand, given the operational interpretation of reciprocal numbers as signed integers, the number of “directions” of n-dimensional scalars becomes a function of their dimension. Thus, there are no “directions” in the 20 = 1 scalar, but there are two in the next higher 21 = 2 scalar, four in the 22 = 4 scalar, and eight in the 23 = 8 scalar. These “directions” correspond to, that is, they are analogous to, the directed magnitudes of geometry, where a point has no direction, while a line has two directions, relative to its center point, a plane has four directions, relative to its center point, and a cube has eight directions, relative to its center point.
Consequently, there exists more than one mathematical way to represent the “right lines and circles” that geometry requires, and if there is more than one mathematical way to represent them, it is highly probable that there is more than one physical way to “draw” them too, given “the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics” that, as has already been pointed out, Wigner calls “a wonderful gift, which we neither understand nor deserve.”
Notice that the fourth line contains 8 “directions” in the four geometric dimensions: 0, 1, 2, and 3. Mathematicians are intrigued with this number 8, because they find it popping up unexpectedly in advanced mathematics. In fact, expanding the binomial expansion to 8 (27) dimensions just seems to create an inverse copy of these first four dimensions, and then the pattern just repeats itself from there, ad infinitum. This is called Bott periodicity, because Raul Bott proved that no new phenomena mathematically appears beyond the eight “directions” of the octonions (eight directions relative to a center point). It is clear to see that it’s related to the initial 3D system and also to the first four numbers of the Greek mystery. Most of the advanced physics research in this connection in modern times has to do with highly complex topics in topology, advanced algebras, and the symmetry and representations of groups.
Nevertheless, it’s fascinating to realize that modern physics is searching for understanding of their cosmological mysteries in the very icon that the ancients revered so much, the Greek tetraktys! To the Greeks, the sum of these four numbers, 10, was profoundly significant. The progression from 1, called the monad, to two, called the dyad, to the triad, to the tetrad, represented much more than a curiously symmetrical progression of numbers. In their minds, the unfolding of these numbers was mystical, because their relationships had an interpretation that seemed divine and represented foundational principles, possessing great power.
For instance, the monad, was seen as the First Principle, the One of unity, the source of all being, constituting the greatest mystery of all - how the One gives birth to the many, without diminishing itself (i.e. n/n = 1/1). While to the Greeks, these ideas were metaphysical, the amazing and inspiring parallel to the real physical concept of the universal Progression, the unit scalar motion, which forms the basis of the theoretical universe of motion, is plainly manifest.
Next is the dyad, which, for the Greeks, represented the power of duality, the idea of positive and negative, order and inverse order, etc. Again, this was interpreted by the Greeks metaphysically, but how striking is the physical parallel that is seen in the operational interpretation of the ratio of two numbers as a number itself. The operational interpretation of one, 1/1 = 1, becomes dual with the number two: 1/2 and 2/1 are inverse numbers; that is, they are equal but opposite numbers on either side of one, just as -1 and +1 are equal, but opposite, relative to zero. Thus, the power of 2 is the creation of duality.
Combining the monad and the dyad leads to the triad. The metaphysical and sexual connotations of this are obvious, but again the impact of the actual physical concept, regardless of metaphysical interpretation, is hard to overestimate in its comprehensiveness. The Greeks saw it as the essence of harmony, because it is the unification of two extremes in the sense that it has the power to bring order out of duality, by representing the mean between the opposites of duality. In the Reciprocal System of Mathematics (RSM), a triad is formed from the combination of a monad, 1/1, and the dual forms of a dyad (1/2 & 2/1), to form a triad (1/2 + 1/1 + 2/1), called a physical reciprocal number (PRN). It turns out that the role of the triad PRN in the RSM is truly fundamental and harmonious, in a most remarkable fashion.
Finally, the tetrad, which for the Greeks represented completion, because it contains all the previous numbers, the 1, 2, 3, and itself, 4, in one number, 10, and forms the basis for the square and the four sides of a pyramid, has exactly this same effect in the RSM in that it represents the completion of a fundamental cycle of numbers that is repeated over and over again as a series of three-dimensional groups that grows in value as it performs its completion role in each group of four numbers. What this means is that the Bott periodicity of eight, which is the subject of so much modern research, is really a period of four, n-dimensional, numbers, the fourth of which is equal to the sum of all from 20 = 1 to 23 = 8: in a sense, an endless cycle of the number 10 in the form of the tetraktys. Moreover, the sum of the triad ratio in the PRN is actually equal to four in one:
- 1/2 + 1/1 + 2/1 = 4/4 = 1/1.
In short, when the perfect symmetry found in the reciprocity of one, operationally interpreted as unity in the form of 1/1, is broken by the duality of two (1/2 & 2/1), it is restored in three, by joining together one and two (1/2 + 1/1 + 2/1), which enables them all to be combined together in four, 4/4. In the next part of this article, this unity is implemented in the form of the RSM and forms the basis of the Reciprocal System of Mathematics (RSM).