Quantity and Quality

Prime numbers, are the fundamental building blocks of the natural number system. A prime number has no factors other than itself and 1. So, for example, 7 is a prime number, as it has no other factors other than 7 and 1.

Now, bearing in mind that in dynamic terms numbers have dual aspects both as quantities and qualities, we will define prime numbers in a likewise manner.

Thus we can list on a horizontal scale the set of prime numbers 2, 3, 5, 7, 11 ..... ,

and also on a vertical scale the same set 2, 3, 5, 7, 11 ..... .

The first set represents numbers as quantities, all defined within the same dimension (i.e. number as quality). This is always taken - where not otherwise stated - as the 1st dimension.

Strictly speaking this first (quantitative) set should be listed as 2¹, 3¹, 5¹, 7¹, 11¹.....

Here the focus is on prime numbers as quantities (within the given number quality or dimension "1").
 
 

The second set represents numbers as qualities, with the same number (as quantity), raised to different prime powers or dimensions. Again - unless otherwise stated - the given number is again 1.

Again, strictly speaking this second (qualitative) set should be listed as 1², 1³, 1⁵, 1⁷, 1¹¹ ..... .

Here the focus is on the prime numbers as qualities or dimensions (to which the given number quantity "1" is raised).

In the former case, the inclusion of the dimensional characteristic (i.e. the 1st) does not affect the quantitative interpretation of the number. For example 2¹ = 2. Therefore in quantitative terms, reducing the combined quantitative-qualitative expression of 2¹ to the merely quantitative expression of 2 apparently makes no difference.

In the latter case, the inclusion of the quantitative characteristic (i.e. the number "1"), this time does not affect the qualitative interpretation of the number. 1² = 2 (i.e. the dimensional characteristic here "2" remains unchanged regardless of the number quantity used).

Therefore in qualitative terms, again reducing the combined qualitative-quantitative expression of 1² to 2, apparently makes no difference.

However, because both sets - in reduced terms - are now formally identical, the misleading step is then made of reducing the second qualitative set of numbers in turn to the first quantitative set.

Thus conventionally, numbers in mathematics instead of signifying dynamic two way relationships (with complementary aspects), are treated one way as static entities.
 
 

This might all seem far removed from psychological development, but in fact there is a fascinating direct connection.

In my "Transforming Voyage", I defined the first two stages of physical understanding (at the linear level), as the primitive stages. The very word "primitive" bears direct comparison with "prime" where the basic building blocks of experience emerge.

The first of these stages emerges from a state, where both conscious and unconscious minds are totally undifferentiated. This leads - and I am deliberating using language that is suggestive of the mathematical connections - to complete confusion as between objects and dimensions.

The child is not yet able to place objects in an environment of space and time, but rather confuses both in an immediate and fleeting experience. Indeed, this always remains the basis of instinctive response, where the quantitative experience (i.e. what is projected objectively) is so spontaneous and immediate, that it eludes separate qualitative control. Subjective emotion directly attaches to the object so that both literally are confused.

Thus the emerging infant, initially greatly confuses quantitative impersonal with qualitative personal experience. Constancy of phenomena is not possible, because this requires placing them in a dimensional environment of space and time, whereas for now, phenomena are directly experienced as dimensions so that there is the confusion of whole with part.
 
 

Now when we look at prime numbers, we can see an exact complementary pattern.

As we have seen, by definition a prime number has no factors (other than 1 and itself).

In mathematics, when we multiply numbers together, we are introducing a qualitative (or dimensional) characteristic to what is interpreted in solely quantitative terms.

This is most easily appreciated, when we multiply the same number by itself.

Thus 2 * 2 = 2². Thus through this operation a number - which was defined in the 1st dimension - is now through multiplication defined in the 2nd dimension. In fact a qualitative - as well as quantitative - transformation has taken place. By and large this qualitative aspect is ignored in mathematics with the quantitative result once more expressed in the 1st dimension..

Even when different numbers are multiplied together, the same dimensional transformation is involved.

Thus 2 * 3 is also (qualitatively) a two dimensional number. Geometrically this can be appreciated by representing the numbers as the sides of a two dimensional rectangle. However, once more in quantitative terms, the result (i.e. 6) is expressed in a reduced form in the 1st dimension.
 
 

The fascinating thing about prime numbers, is that they cannot be represented as the result of combined quantitative and qualitative operations because prime numbers have no factors.

Thus prime numbers can be expressed alternatively as horizontal quantities or vertical qualities, but not as the expression of a combined operation.

Composite numbers - using prime components - on the other hand always involve both a quantitative and qualitative transformation. Thus a quantitative value and a qualitative value are involved. The quantitative value is the (reduced) value of the operation (i.e. expressed in the 1st dimension). As we have seen 2 * 3 is a composite number. Its quantitative value is 6 (i.e. reduced value in 1st dimension). Its qualitative value - simply the number of prime factors involved - is 2.

Incidentally 6 is a perfect number with a prime number of factors (i.e. 2). In other words it is vertically prime. All perfect numbers, for example - as they have a prime number of factors - are therefore vertically prime.
 
 

When we link up the primitive structures (psychological), with the prime structures (mathematical), we can see clearly the complementary relationship.

With the primitive structures, there is total confusion as between quantitative and qualitative experience. Phenomena in experience cannot be yet separated from the dimensions of space and time.

With prime numbers - in complementary fashion - there is total separation of quantitative (i.e. reduced numerical value) and qualitative (i.e. dimensional) characteristics.

In future, therefore - bearing in mind this complementarity - I will refer to the primitive structures as prime structures.
 
 

Masculine and Feminine Numbers
 

There are other fascinating aspects to prime numbers, with a complementary psychological significance.

With the exception of the 1st prime number "2", which is an even number, all other primes of course are odd numbers.

In most cultures odd numbers carry a "masculine"", and even numbers a "feminine" connotation. The masculine principle relates to the one-way cognitive (rational) approach, culturally considered more typical of men, whereas the feminine principle relates to the two-way affective (intuitive) approach which is considered more typical of women. Thus the masculine principle can be directly related to the first odd number (i.e. 1), whereas the feminine principle can be related to the first even number (2). Thus by extension all odd numbers are "masculine" and all even numbers "feminine".

Prime numbers - with the exception of 2 - are particularly "masculine", in that as well as being odd, they are highly independent (i.e. have no factors).

Psychologically, the prime structures (i.e. primitive impulses), are particularly "feminine" (i.e. pure instinctive response rather than reason). Thus the "masculine" prime numbers complement the "feminine" prime structures.

The first prime number "2" which is even and "feminine", has a special significance. I shall refer to this as the root prime number. Psychologically, it is related very closely to the fundamental unconscious symbol of 0. As we have seen 0 arises from the dynamic (negative) counterbalancing of the (positive) 1. Mathematically this can be simply expressed as 1 - 1 = 0. Thus the psychological interpretation of nothingness (i.e. zero) is dynamic, involving complementary opposite poles.

Now 2 is just a mathematically reduced interpretation of this dynamic relationship. When each pole is experienced separately it is positive. (It is only in complementary dynamic relationship that the positive-negative polarity arises). Thus when the conscious linear approach is used, in terms of the fundamental unconscious relationship, it appears as 2. Thus there are - in static terms - two poles involved in terms of this relationship.

The dynamic 0 (unconsciously), then in terms of conscious interpretation is 2.

Therefore the conscious process itself is fundamentally one dimensional, and the unconscious is fundamentally two dimensional (in reduced static terms).
 
 

2 which is the reduced interpretation of 0, carries the same unconscious (intuitive) connotations. Not surprisingly, as intuition is a defining characteristic of the feminine principle, 2 in most cultures is seen as a "feminine" number. By extension all even numbers (which are divisible by 2) are similarly treated as "feminine" numbers. By the same reasoning all odd numbers (which are not divisible by 2), are treated as "masculine" numbers. Indeed the prime numbers (excluding 2) which are all odd comprise a special subset of the odd "masculine" numbers. They carry the psychological connotation of being especially masculine.
 
 

Mersenne Primes
 

Biologically, we have the observation that every male as an infant foetus is initially inseparable from the mother - in physical terms - and therefore enjoys a feminine identity. What may not be quite so obvious is that this parallels the very nature of the prime number system, which starts with an even (feminine) number.

Indeed, this would suggest that every prime number can ultimately be derived from 2. A famous example of this approach is the set of Mersenne primes. (Another is the set of Fermat primes).

Now Mersenne primes are especially interesting in that they also generate another fascinating class of numbers - with considerable psycho-mathematical significance i.e. perfect numbers.
 
 

A Mersenne prime is always of the form 2ⁿ - 1 (n is a positive integer). For example,

2⁵ - 1 = 31 is a Mersenne prime.

Now, there are two points which I wish to point out which illustrate the transrational approach.
 
 

1) The power of 2 (i.e. the qualitative vertical number) must itself be prime, if the resulting number (i.e. the reduced quantitative horizontal number) is to be prime. In our example, the qualitative number 5 is prime, and the (reduced) quantitative number 31 is prime.
 
 

2) The resulting prime number is closely associated with a highly composite number.

32 (i.e. 2⁵) is the most composite possible for its size. It has five factors all involving the minimum number of 2. Then by reducing by 1, we move to the opposite extreme of a prime number with no factors. Furthermore this seems to be a natural feature of the number system. In other words on either side of the most highly composite numbers (i.e. numbers that are powers of 2), can be found numbers at the opposite extreme with very few factors (frequently being prime). This is just another example of the principle of the complementarity of opposites naturally occurring in the number system. By the addition or subtraction of 1, a highly composite number tends to collapse to its opposite prime or near prime state.
 
 

Indeed, we can push this principle to the extreme, by starting from 2, attempting to continually generate prime numbers, through a special recursive process.

Thus 2² - 1 = 3. Therefore, with 2 as the base "qualitative" prime number (i.e. power), through our operation, we generate a new reduced "quantitative" prime.

Now we keep on substituting this derived "quantitative" prime as the new exponent or power. So we keep exchanging "quantitative" for "qualitative" numbers.

Thus 2³ - 1 = 7, which is the new (reduced) "quantitative" prime. This in turn becomes the new "qualitative" prime, so that,

2⁷ - 1 = 127. This is also prime, and this "quantitative" number, becomes the new "qualitative" number,

with 2¹²⁷ - 1 = M₁₂₇. This again is a "quantitative" prime number (with 39 digits).
 
 

This automatically leads to the (surely) interesting hypothesis that the next term, which is 2 ^M₁₂₇ - 1, is in turn a prime number (the exponent of 2 having 39 digits), which would be incomparably larger than any yet discovered, ultimately rooted in the 1st root prime "2". Of course we do not have to stop here. We can substitute this "new prime" as the exponent of 2 to generate a further prime and so on indefinitely.
 
 

Furthermore, when we represent the L.H.S in unary form, and the R.H.S. in binary form, the above terms can always be represented by the use of just one digit (i.e. 1).

Thus the Ist line can be written

1111 - 1 or 111(unary form) = 11 (binary form) with the second term,

11111111 - 1 or 1111111(unary form) = 111 (binary form) etc.,

Thus what is involved in the above is a continual (ordered) mathematical switching as between binary and unary format generating prime numbers, which can be represented as a system of (differentiated) ones.

As we have seen, in psychological terms, the conscious (horizontal) process is a unary system, while the unconscious (vertical) process is a binary system.

Therefore in complementary manner, the continual (disordered and confused) psychological switching as between binary (unconscious) and unary (conscious) format generates the (purely) instinctive behaviour of the prime structures, which can be represented as a system of undifferentiated oneness.
 
 

Twin Primes

Another example of how creative use of the transrational approach is in the of twin primes (i.e. primes differing by 2).

Twin primes are closely related to what I term diagonal primes. There are many forms in which these can take place. One way is to set up a two dimensional grid, with horizontal and vertical scales (representing "quantities" and "qualities").Then one orders two dimensional numbers (i.e. numbers with two factors), - in ascending magnitude - on the second row, numbers with three dimensions (three factors) on the third row etc. (Strictly speaking one should leave out, initially, non-prime rows such as the 4th). Now a diagonal number will have the same co-ordinates on both axes. Thus, for example, 6 which is the 2nd number in 2nd row is a diagonal number. Likewise 18, which is the 3rd number in 3rd row is a diagonal number. It is striking how often these numbers lie between twin primes (in this case 5, 7 and 17, 19 respectively).

It also leads to a view of composite numbers as simply prime numbers in dimensions other than 1. For example 6 can be defined as a two dimensional prime.

So by the addition and subtraction of 1 from these diagonal numbers, we very often generate twin primes.
 
 

This would also strongly suggest that the famous twin prime hypothesis (unproven) that the set of twin primes is infinite, is directly tied up with the twin nature of the prime number system itself, which is both horizontal (quantitative) and vertical (qualitative). As these systems are infinite - due to their complementary nature - then the set of twin primes which is connected with their diagonal coincidence - is also infinite. It would also strongly suggest that the frequency of twin primes (as a fraction of the natural numbers) is closely related to the probability of double prime incidence (i.e. the frequency of a horizontal prime multiplied by the frequency of a vertical prime).

The psychological counterpart of twin primes could be outlined as follows. Here we are referring to twin prime structures. At the prime stage of infant development, there is in fact a very close coincidence as between (conscious) quantitative and (unconscious) qualitative experience. In directional terms, the experience then splits down into a twin left hand (subjective) and a right hand (objective) component, representing primitive experience of the self and of the world. However, because of the undifferentiated nature of the experience both directions are confused.

Psychologically, it is common to refer to the intuitive and rational processes as involving right hand side and left hand side brain activity respectively. Twin primes, lying on either side of diagonal numbers, complement this situation.
 
 

Infinitude of Primes

There is a theorem by Euclid which proves the infinitude of primes. It is justly famous. However there are other means of inferring the same result.

The prime structures are very closely related to the unconscious mind. The unconscious is the direct source of infinite perception, whereas the conscious is the direct source of the finite. Therefore the prime number system - with this close unconscious link - can be readily appreciated as infinite.

There is another more mathematical way of deriving the same conclusion based on the dynamic principle of the complementarity of opposites.

Here one assumes that the natural number system is infinite, and that the horizontal and vertical prime number systems are complementary. One then initially formulates the hypothesis that the horizontal number system (as quantities) is finite. This then implies that the vertical prime number system (as qualities) is infinite, for an infinite natural number made up from a finite set of prime quantities, would require an infinite number of prime factors. However this would imply that the horizontal and vertical prime number systems are not formally identical which contradicts our basic assumption.
 
 

Physical Reality
 

This unconscious nature of prime numbers is also evident in another way also. Despite so many attempts no one has ever been able to devise a formula which would generate all the prime numbers.

In other words there is a certain random behaviour to prime numbers, rather reminiscent of quantum theory. In quantum mechanics, even though the behaviour of a particular particle cannot be exactly predicted, yet where large groups of particles are involved, predictive accuracy greatly increases, though still bound by laws of probability. It is very similar with prime numbers. Even though the existence of any particular prime number cannot be guaranteed yet where large groups are involved, predictive accuracy greatly increases so that the bigger the set of natural numbers involved, the more accurate can be the predictions regarding the amount of prime numbers contained therein.

This connection should not be too surprising when one realises that quantum behaviour in the physical world relates particularly to elementary particles. The most elementary particles in turn can be considered as primitive or prime particles emerging out of the unconscious ground of the material world. Now simultaneous with the emergence of prime particles is the emergence of prime dimensions. Thus we cannot yet separate (quantitative) phenomena from the (qualitative) dimensions of experience.
 
 

Superstrings

Indeed there is a remarkable similarity here with some of the basic assumptions of Superstring Theory. Here the most fundamental particles are literally viewed as tiny one dimensional objects. So just as prime numbers - the basic constituents of the number system - are inherently one dimensional (i.e. no constituent factors), so it is likewise with these "prime" particles. Furthermore it is understood that such particles are subtly intertwined with their dimensions. This complements prime numbers where quantities and dimensions (i.e. factors) are separated. These strings "vibrate" (i.e. interaction of prime particles with prime dimensions) leading to the generation of more composite observable particles of natural phenomena. This resembles the means by which the interaction of prime numbers with prime dimensions leads to the generation of the natural number system. So we have here a simple - yet compelling - mathematical way of providing an understanding of some of the basic insights of Superstring Theory. However it cannot successfully pose as the Theory of Everything. Prime numbers are not the most fundamental mathematical quantities. In like manner therefore, these prime particles (i.e. superstrings) cannot represent the fundamental structure of reality.

If, in psychological terms, we admit two interconnected processes in experience, of conscious and unconscious, and the complementarity of physical and psychological reality, then nature has its own equivalent to the unconscious, where behaviour takes place in the same non-linear fashion.

This also implies that the fundamental nature of primitive instinctive behaviour at the psychological level, is very much complementary to that of elementary particles at the physical quantum level. In other words the same basic processes can be identified in both cases.

While referring to prime numbers and elementary particles, it also strongly suggests that the search for the definitive "fundamental particles" is in fact an illusion. Because of the inevitable confusion of phenomena and dimensions at the most primitive levels, it becomes impossible ultimately to identify prime particles (i.e. fundamental building blocks of matter). Indeed, what have already been discovered - in phenomena that can be directly or indirectly detected - are really in the way of composite particles. Even if science proceeds further (e.g. experimentally identify the existence of all quarks). this will still represent a composite level of particle identity.

The fundamental particles are infinite - like prime numbers - and by definition (since the phenomena are inseparable from the dimensions mutually created) not experimentally verifiable. Thus as we approach the potential for existence, particles become increasingly unstable and ephemeral, ultimately eluding detection (in terms of what can be experimentally validated).
 
 
 
 

Mathematical Axioms

I will deal here with one more fascinating application of the "prime" concept.

As is well known, mathematics can be represented as an axiomatic system. Axioms represent basic assumptions which are necessary to proceed but not derived from within the system itself. Thus though mathematics prides itself on its rational approach, it starts from a basis which is decidedly pre-rational. Axioms are unconscious assumptions which are impossible to rationally verify. In fact they represent - just like prime numbers - the basic building blocks of any system.
 

In the past, it was quite common - even when acknowledging the non-rational nature of axioms - to at least limit them to a few which would be universally acceptable.

Euclidean geometry is perhaps the best example of this approach. An impressive edifice - based on just five axioms - was built gaining such a level of acceptance, that for approximately 2,000 years it was understood as the only valid geometrical system. However in the 19th century, by simply changing one of these axioms, other geometries with great relevance to the modern world emerged.

The point is that there is no limit to the number of possible axioms which can be adopted. The number of prime axioms - like prime numbers - is infinite. It is true that most possible axioms will be dismissed intuitively as irrelevant, but this process is - as the Euclidean history demonstrates - highly arbitrary.

Seen in this light, conventional mathematical "truth" itself is highly dubious. Its validity greatly depends on a blind faith in assumptions at so many different levels which are themselves non-rational. Much of this faith in fact masks deep unconscious prejudice which is the legacy of the attempt to reduce mathematical understanding to (mere) rational format.
 
 

Direction of Experience

The second stage of the prime structures, relates to the direction of understanding, where the ability to distinguish internal (subjective) from external (objective) experience emerges. Thus initially, in psychological terms, these two directions of experience are completely undifferentiated and confused.

The existence of the world cannot be separated from existence of the self. Thus what is positive (the phenomenal world) and negative (the knowing self) - in relative terms - have as yet no separate existence. Because existence is still so confused with non-existence, experience is highly ephemeral, with phenomena - like sub-atomic particles - being created and instantly dissolved with each moment.
 
 

Now again we have the a complementary process underlying prime numbers.

We have in dynamic terms, complementary number systems in horizontal and vertical terms reflecting conscious (rational) and unconscious (intuitive) poles of understanding, We also have in dynamic terms complementary number systems in positive and negative terms reflecting external (objective) and (subjective) poles of understanding.

The root psychological basis of positing and negating in experience relates to this dynamic interaction as between object and subject.

To posit psychologically, is simply to become conscious of a phenomenon. This always involves momentary separation of subject and object. When I am conscious of an object and posit it, I give it a separate existence, independent of self. Equally when I am subjectively conscious, I posit the self, in turn giving it a separate existence independent of phenomena. I am therefore in the moment of self awareness, conscious of being separate from the world. Thus to move from positive (i.e. conscious) experience of the world, to positive experience of self, requires cancelling out to some extent the former experience. Likewise, moving from positive experience of self to positive experience of world , in turn involves to some extent cancelling out the self. Thus negation in psychological terms, is simply this cancelling out activity, which switches the direction of experience. This enables dynamic interaction between self and the world to take place.
 
 

Of course, in mathematics positive and negative numbers are used. But once again, they are interpreted in a reduced static fashion.

In dynamic terms, every number is two-directional. From one direction we have the objective number entity (in relation to the subjective mind). From the other direction we have the subjective self (in relation to the objective number entity).

The positive number system - in dynamic terms - involves taking one of these directions (i.e. the former), where we concentrate explicitly on objective number entities (and implicitly on the subjective self).

The negative number system - in relative terms - is the other direction, where we explicitly concentrate on the subjective self (and implicitly on the objective number entities).

The two poles are complementary aspects of what is a dynamic system.

Now, the conventional approach, is to reduce understanding to one direction only seeing numbers as static objective independent entities. Indeed, for so many, the attraction of mathematical activity is in this perceived absolute objective nature of the pursuit. In truth, numbers represent dynamic interactions, rendering all knowledge of them relative.
 
 

One of the fascinating aspects of prime numbers is that they have no directional significance. Prime numbers - preceding the natural number system - are initially defined as positive numbers.

Now, both subjective and objective poles in isolation are positive. It is the interaction between them that creates, in relative terms, the positive-negative polarity.

Prime numbers therefore, represent a stage where both directions (mathematically) are completely separated. This complements the corresponding prime structures, where both directions (psychologically) are still unified (i.e. undifferentiated and confused).