This section is about multiplication. In the large sense, multiplication brings two things together to make a third. In the case of numbers, this leads to simple arithmetic. In the case of two inebriated men at a bar, it can lead to a bar room brawl. The ancients, both in the West and the East, were interested in bringing two principles together, one masculine and one feminine. The multiplication of these two principles created the Cosmos. We will visit the ontological and epistemological roles of gender later. For the moment, we are interested in multiplying together two different ways of thinking, two different takes on reality. In mathematics, there are so many different kinds of multiplication that it can be very overwhelming. We are particularly interested in the role of multiplication in geometry. There is one kind of geometry that is pertinent.. Continue reading “The Shape of Mind”
In soon to appear book and its appendixes, we have mapped out the foundations of a new kind of geometry based on the right side scientific paradigm. When talking about the shape of knowledge, we must also talk about the shape of geometry. Traditional left side spatiality, like Hilbert space for example, is notable for its lack of shape.. Continue reading “The Shape of Space”
In any ground breaking project there is a polemical streak and this work is no exception. Topics covered in this blog have raged across the axis of traditional left side science and our proposed right side science. The arena for this epic tussle has been the nature and structure of scientific knowledge. What we have failed to do is clarify exactly what we mean by scientific knowledge. We have argued that there is another kind of scientific knowledge than left side conventional science. This was the right side science. Now we must ask the question as to whether there are any other kinds of knowledge, knowledge that escapes the scientific tag. Embarrassingly, there is another axis of knowledge that is dramatically distinct from the scientific. In this blog, we have ignored this other axis, an axis of equal importance as the left-right science axis.
To begin with, our topic here is universal knowledge, knowledge that includes scientific knowledge as a special case. Universal knowledge is composed of two, and only two, fundamental ingredients. These ingredients are semantics and logic, in the large sense. In order to understand the universal shape of knowledge, one must understand that there are two orders of logic and two orders of semantics: both have a first and a second order form. The notion of first and second order logic is a quite familiar to present day logic, and has been even formalised from an axiomatic perspective. However, this is not the case for semantics. Even the very prevailing notion of semantics is hazy, let alone any notion of first and second order semantics. We must rectify that situation as we proceed.
Attempting to explain the shape of knowledge can take up reams and reams of pages and still not get anywhere very fast. Our best recourse is simply to illustrate the shape of knowledge with the semantic square. As usual, everything finds its place and we get the fit shown below.
Semiotic square of the four kinds of knowledge
From the diagram we identify logic with the masculine gender and semantics with the feminine. As we have seen, the pure feminine F typed entity has extent but no presence. On the other hand, the pure masculine M typed entity enjoys presence (it is presence) but has no extent. Logic plays in the masculine register and so becomes the logic of presence. The masculine becomes the ultimate determiner of what is and what is not. Semantic plays in the feminine register involving the interplay of that which can claim some substantiality. The minimal requirement for substantiality is extent. The pure feminine has extent but no presence.
Now we come to the question of scientific knowledge. We start with a rough-hewn definition of science in the context of the semiotic square . The parts of the square that qualify as science are the boxes where both M and F are present. M provides the logic and F provides stuff with extent. Science thus fits into the slot of being the “logic of stuff”. This would be the MF version corresponding to traditional left side science. Reversing the order, we get the FM science corresponding to the “stuff of logic.” This corresponds to the right side science that we have been developing. One could surreptitiously slide in the comment that the MF science studies dead stuff whilst the FM science studies living stuff.
Note that we are using the semiotic square as our compass in our attempt to avoid Kant’s curse of the “fine spun argument.” Keep in mind that one does not need to write a doctoral dissertation each time one consults a compass.
Thus, it appears that the main dialectical opposition of this blog has raged along the MF-FM diagonal of the semiotic square. There we find two kinds of science, the traditional left side and the right side science that we are promoting. The semiotic square nicely characterises these two kinds of science. Left side science is based on the paradigm of second order logic and first order semantics. In other words, all the traditional science, including mathematics are based on abstraction provided by a second order logic and the shallow first order semantics. On the other hand there is the generic, and universal oriented right side paradigm, which is totally devoid of abstraction and its higher order generalisations. To remove abstraction from the pudding, the paradigm only allows first order logic. Where it shines, is that it can handle non-trivial second order semantics.
It appears that if you want a science with non-trivial semantics, you have to throw away abstraction and its higher order logic. Vice versa, if you want the generalisation power of abstraction, you have to throw away higher order semantics and use the rather trivial default version based on first order semantics.
Just before going on to explain what is meant by these different orders of logic and semantics, we cast one more glance at the semiotic square of knowledge. Apparently, scientific knowledge works along the MF-FM diagonal of the square. This leaves two other kinds of knowledge left out of our science equation. The other diagonal consists of FF and MM knowledge. This is a topic that we will have to come back to later. For the moment, whatever this kind of knowledge may involve, we will refrain from characterising as “non-scientific.” Rather, we will call these potential science, the subtle sciences. Such sciences have a distinctly Eastern flavour. We shall briefly discuss these subtle sciences later.
Our immediate task is to clarify what is meant by first and second order logic and semantics.
We start with first order logic. From a traditional left side perspective, first order logic comes down to the logic of propositions, the propositional calculus. The propositional calculus involves well-formed formula, called propositions. Each proposition has a truth value of either true or false and is made up logical conjunctions, disjunctions, and negations. In brief, propositions are mathematical logical expressions made up of abstract symbols combined with AND, OR and NOT primitives.
Second order logic is an extension of the propositional calculus and is called the predicate calculus, In addition to the propositional calculus structure, the predicate calculus allows the abstract symbols to be treated as variables with values ranging over sets. Each variable x can take on a range of value restricted to a particular set of values A. This simple construct provides the necessary equipment for abstract logical reasoning. The reasoning is formalised by the addition of two logical primitives called universal and existential operators. The so called universal operator is used to mean that a predicate is valid “for all x,” The existential operator means that “there exists an x” for the logical expression to be true.
Second order logic is the basic construct that enables abstract reasoning. An essential characteristic of abstract reasoning is that the objects of reason are not required to exist. Whether something exists or does not exist may be true or false, depending on the assumptions. The reasoning is based on generalisations and, despite the “universal quantifier” terminology, has no concept of universals. The universals belong to right side science.
Generalisations and general laws apply to everything in a closed world. For example, the Second Law of Thermodynamics is a general law and so applies to everything in a closed system. The law states that everything in in the confines of such a system drifts to a state of maximum entropy, that is to say, to a state of thermic death. One can say that general laws apply to everything but not everywhere. They only apply to within the closed system. The general law is only valid in the confines of a sealed bottle. The bottle may be made of glass and contain a mixture of gases. Glass is a favourite material for making sealed systems for the left side sciences. Another favourite material for building a closed system is axioms. Axioms make very fine watertight bottles and ensure that everything enclosed within is sure to be headed towards thermic death.
On the other hand, universal laws apply everywhere but not to everything. Instead of applying to the closed system, the universal law applies to the open system, the system that, instead of living in a bottle, lives within itself. Instead of drifting to thermic death, the universal system will tend to proliferate and diversify, producing life.
In addition to second order logic, left side science must have recourse to a first order semantics. We associate logic with the masculine and its punctual nature. We associate semantics with the feminine and its extensive, non-punctual nature. Semantic expresses itself in the form of oppositions between contraries. In the case of first order semantics, there is only a single opposition involved. On the other hand, with second order semantics there are two oppositions, often involving one opposition applied to itself. The traditional left side sciences, including mathematics, only use first order semantics.
Traditional left side mathematics only uses semantics of the first order. Mathematics constructs its semantics from the fundamental opposition between a collection on one side of the opposition, and the objects making up the collection, on the other. This leads to Set Theory, the general expression of first order semantics for practically all of mathematics. The Set is on one side of the opposition and the Elements of the set are on the other. The elements of the set provide some kind of primitive notion of extent. For example, the set of points making up an interval of the real line, is such an example of extent. It is quite remarkable that this is the only semantics that axiomatic mathematics really needs. Feed it Set Theory and off it goes. No fuss.
It is equally remarkable that all of the traditional sciences of our day are based on first order semantics and second order logic. These sciences operate under the heading of the neon light, flashing MF typing of knowledge. It would appear that this kind of knowledge is favoured by those ethnicities that belong to the Christian tradition, cultures with an MF disposition.
We now turn to knowledge of the FM type. According to our analysis, FM type knowledge is based on second order semantics and first order logic. From a linguistic-cultural point of view, we have associated the FM disposition the Islamic tradition supported by the Semitic language Arabic. However, from a philosophical point of view we are lead to the Stoics. The Stoicism was the “least Greek” of the ancient Greek philosophies. Moreover, all the early founders such as Zeno, Cleanthes and Chrysippus, were all of Semitic origin. Thus the cultural typing of Stoicism might justifiably be classed as having “Semitic tendencies.” Be that as it may, we type the knowledge speciality of the Stoics as being of type FM.
The Stoics had their brand of first order logic and they consistently expressed an aversion to employing second order logic and its attendant preoccupation with abstraction. The Stoics only reasoned in particulars arguing that generalisations do not exist. Socrates can exist but Man and mortals do not. There is no such thing as Man. There is no such thing as mortals. Abstract generalisations do not exist. They rejected the species and genus of Aristotle saying that they had no need for them. In modern mathematical terms, they rejected sets. All of modern mathematics is based on sets in the form of Set Theory. Without Set Theory, there can be no traditional mathematics. If a Stoic were alive today, he would still reject Set Theory. The Stoic has no need for such abstractions. The Stoic is content with the logic of Chrysippus, which faithfully avoids anything but the particular. After all, only particulars can exist and that is what concerns the Stoic.
Of course, traditional mathematics goes the other way and reasons over the elements of an abstract set of objects, the set of green apples, the set of prime numbers, for example. First order logic avoids such abstract thinking and only talks about qualities relating to the existence of a particular entity. In their purest form, the qualities involved have nothing to do with the greenness of apples or even the primeness of a number. The qualities are the generic qualities of the generic entities. What matters is whether one has or possesses the quality or not. “if you have the first and the second quality …” is the premise of Chrysippus’ first of the five undemonstratables. The logic does not say what the quality is, but rather whether it is or is not. Relative to you, the quality is if and only if you happen to have it possession at the time. This is an ontological logic. Despite avoiding abstraction, the first order logic reasoning of the Stoics becomes surprisingly profound, as explored in the appendices.
We now turn to semantics. Before moving on to second order semantics, we take another look at semantics of the first order.
Firstly, who uses first order semantics? We know that modern mathematics uses first order semantics and only first order semantics. We notice that this statement did not make the reader suddenly sit bolt upright, which is the reaction we wanted. In fact, the reader’s eyes seemed to have even started to glaze over. In search of a more engaging means of explanation, we come back to earth where people and things actually exist, and not just in the imagination.
We remark that if one looks around us hard enough, one will surely discover an acquaintance, a relative even, who only uses first order semantics in their everyday life. Such people are easy to spot. Moreover, not all of them are mathematicians. The key giveaway is that the person concerned is totally incapable of putting themself in someone else’s shoes. For example, such a person is incapable of putting themself in your shoes. In order to accomplish such a feat, one needs second order semantics. In brief, first order semantics implies a total lack of empathy.
The inability to put yourself in someone else’s shoes leads to the worldview that you are the centre of the universe. This is an inevitable consequence of a first order semantics view of the world. The most famous exponent of this worldview was Ptolemy, of the first century AD. Ptolemy was a gifted mathematician that wrote on many scientific topics. The most famous was his geocentric model of the world based on a set of nested spheres. This incredibly complicated system held sway for over a thousand years until finally replaced by the much simpler heliocentric model.
One wonders whether there are any extremely over complex Ptolemaic scientific abominations around in modern times. One does not need much prodding to come up with a likely candidate. String Theory. Perhaps we should express our admiration for the String Theorists. Their achievements are even more laudable when you realise that they have accomplished so much, and only using first order semantics.
The above explanation of first order semantics is probably as clear as mud. Perhaps we will have to turn back to mathematics itself to bring some sort of rigour to bear on the question. We must turn to the empathy free zone of modern mathematics.
Without going into details, we can say that the kind of mathematical geometry possible with first order semantics is rather trivial compared to the geometry possible with higher order semantics. This is very important as we rely on mathematicians to describe to us the shape of the universe we live in. However, no mathematicians or mathematical physicists to our knowledge have ever pointed out the fine print in their deliberations. They simply inform us that, as a consequence of applying their mathematical theories, it turns out that the world is shaped in this or that particular way. Nowhere in the description is the caveat that, by the way, the expressed views herein have all been based on first order semantics and only on first order semantics. Sadly, there are no labelling laws for modern mathematical products. This must change.
So what kind of geometry do you get when you only use first order semantics? The answer is surprisingly simple. Some mathematicians even boast about how simple it is. They see it as a triumph of applying abstraction. To begin with, they claim that all spaces are n dimensional. Mathematicians cannot stop themselves from generalising. The letter n is a very general number. That way you cover all bases and so it is hard to be wrong. Then comes the decisive factor. All the various mainstream versions of space mathematics have exactly the same geometry! Technically, they all have the same affine geometry. This is truly remarkable. Lines behave like lines and points behave like points in all these vastly different mathematical spaces. The only difference from one mathematical version of spatiality to another is the distance between points. Mathematicians handle this detail by ascribing a different metric artifice, called a metric tensor, to each space. In this way, for example, an ordinary Euclidean space can become Minkowski spacetime geometry by simply swapping the metric tensor.
Practically all these mainstream mathematical spaces are special cases of a Hilbert space, and so the construct goes back to David Hilbert. A ferocious critic of Hilbert was the great Henri Poincaré. Curiously, as an aside, Poincaré was ambidextrous. We could certainly say that about his mathematics too, but he was both genuinely left and right handed with the pen and, it appears, also with the mind. The ambidextrous Poincaré goes head to head against the (presumably) right handed, left paradigm dominant Hilbert: it is a nice image albeit without any grand significance..
Anyhow, history has it that the abstract axiomatic geometry of Hilbert eventually prevailed over the objections of Poincaré. However, the battle is not over. Armed with the realisation that the Hilbert kind of geometry is only based on first order semantics and that there is our second order semantic alternative, the picture may indeed rapidly change. However, this next time round, there will be no conqueror nor conquered. The only thing to settle will be as to which side of the semantic equation is the Master and under what circumstances.
In brief then, mathematics relying on first order semantics results in a very simple, abstract kind of geometry. Simplicity is always an admirable quality when it comes to scientific explanations; according to Ockham’s razor the simpler the better, However, the simple always runs the risk of falling into the abyss of being simplistic. Ptolemy’s thesis that the earth was the centre of the universe was also simple, but looks at the headaches that gave him, and all the poor astronomers that followed him for a thousand years. Modern day String Theory theorists utilise the simplicity of a geometry based on first order semantics and seem to get the same kind of headaches. It is out of our expertise to criticise the details of their work, but looking from afar, it might be that things could be simplified by a paradigm shift or two.
In the appendices of our book, we look at geometry based on second order semantics. In the process we are lead to alternative interpretations of imaginary numbers, the basis for any fundamental geometry. In fact, we are lead to back to our starting point. The imaginary numbers interpreted as MF, FF, and FM typed entities! These same typed entities can be interpreted from a spacetime geometric perspective as cones and heaves of lightlike, timelike and spacelike arrows. In addition, we investigated the enigmatic MM typed entity and intuitively started to understand it as a flip-flopping Figure-Ground, “is” and “is not” kind of geometrical dimension.
When it comes to logic, the best place to start is with Aristotle. As well as being the greatest philosopher of all time, Aristotle was also the greatest fence sitter of all time. With him, our neat dichotomy between left side and right side thinking meets a blank. This man has a foot firmly placed on both sides. Nowhere is this more apparent than with his categorical logic and in particular his square of oppositions. In this section, without going into too much detail, we summarise the aspects that immediately concern our project.
We end this chapter by presenting an alternative approach of breath taking simplicity and elegance, that pioneered by Chrysippus, considered by many to be Aristotle’s equal. However, to understand the full significance of Chrysippus’ approach, we will need to add some important clarifying innovations on our part that will not be found anywhere else. This is quite an important chapter. If the reader has progressed this far into our exposition, it will be well worth the effort to understand what follows, even for logic specialists. Great beauty lies ahead.
What follows is not rocket science. Each step is easy to understand. The hard part, as always, is to grasp the full significance of the matter.
Aristotelian logic occupies a central place in what is nowadays called classical logic. This was the logic studied by the learned peoples, mainly monks, of medieval Europe for a thousand years. During this time, the notation was refined and elaborated, but the essence barely changed. Even by the Enlightenment, Kant was known to exclaim that the only logic that one needed to know was that of Aristotle.
Aristotelian logic was a central component of what he called the Organon, Greek for tool or organ. Syllogisms are logical arguments made up of three parts, a major premise, a minor, and a conclusion. The most famous is the very familiar:
Major: All men are mortal
Minor: Socrates is a man
Conclusion: Socrates is a mortal.
Aristotelian logic is sometimes referred to as term logic where each proposition of a syllogism is made up of two terms. What interests us is how many kinds of term are necessary for such a logic. This also interested Aristotle. He argued that there were four distinct kinds of term. During the Middle Ages the Scholastics gave each a letter as shown below.
Figure 26 The four kind of terms. The Scholastics later labelled them with four letters.
The Four Terms and the Left Side
Aristotelian logic was half modern and half ancient. We will suspend judgment on which was the better half. The modern half is exhibited in two ways: it relies on abstraction and its logic is static. The abstraction can be seen in the use of the existential qualifier “All”. “All men” for example, means every man. By referring to “all men” or every man, one is referring to an abstraction, a generalisation. As the Stoics pointed out, abstractions and generalisations do not exist. In addition to abstraction, there is the fact that the logical representation of these syllogisms can be covered by Venn diagrams as shown below. The terms can be said to have “Venn Diagram” semantics. This characterises the logic as a static, synchronic mechanism.
Both of these aspects, the abstract and synchronic nature of the logic, are characteristics of left side thinking. By default, left side thinking has become synonymous with the modern.
Figure 27 Venn diagrams for the four terms of Aristotle
The Four Terms and the Right Side
However, what is not modern in Aristotle’s logic is that his infrastructure of the four kinds of terms is not determined by a set of axioms, but rather by a pair of oppositions and the opposition between these oppositions. This is exactly the approach we have been using to construct our semiotic squares. Firstly, obtain a pair of oppositions. Employ one opposition to define a left right dichotomy and the other opposition for the front back structure.
In Aristotle’s case, the left right dichotomy is a strict logical opposition between the affirmative form and the negative. The second opposition is between the universal and the particular. Both these oppositions must be true dichotomies in order to construct a non-trivial semiotic square. This is a technical point, but a very important one and will be discussed later when considering Aristotle’s square of oppositions. It turns out that there can be certain cases where an opposition is not a true dichotomy. This can occur when the subject of a term has no existential import. In other words, when dealing with empty sets such as “All centaurs”.
Figure 28 The semiotic square for the four terms of Aristotle’s Syllogistic logic. The square is formed from two oppositions, the negative/affirmative, and the universal/particular.
Superimposing the two oppositions in the one structure, we get the semiotic square as shown in Figure 28 The semiotic square for the four terms of Aristotelian logic. The square is formed from two oppositions, the negative/affirmative, and the universal/particular.
During the middle ages, the scholastics labelled the four kinds of terms with the four letters A, I, O, and E. Syllogisms consist of three propositions, a major, a minor, and a conclusion. Each syllogism could thus be labelled by a triplet of letters taken from the four-letter AIOE alphabet. This fascinated the Scholastics and, many years ago, entertained the author’s curiosity for some time. The reason for the author’s interest was that such a system did have some resemblance to the triadic structure of codons in the genetic code. With a bit of effort, one can make some kind of rapprochement between the AIOE alphabet of the scholastics and the genetic-cum-generic AUGC alphabet, but the effort is probably not justified, as there are richer pickings elsewhere, notably in Stoic logic.
The genetic codon structure only has 64 combinations. What we have ignored for the syllogism is the detail of how the three propositions in each syllogism hook together. We have ignored the fact that there are four different figures of the syllogism. Thus taking into account the four figures, instead of 64 possible syllogisms there will be 256. Only nineteen of these syllogisms are regarded as leading to a valid conclusion.
Aristotelian logic provides a logical tool that is applicable to the contingent world. Unlike modern logic, it also brings with it some nontrivial semiotic infrastructure, the square of oppositions.
(see my online syllogism machine for exploring Aristotelian )logic.
The Square of Oppositions
Aristotle described how the four kinds of terms could be placed in a square illustrating the various oppositions between them. He then went about characterising each kind of opposition, although the subalterns were not mentioned explicitly.
The oppositions between universal statements are contraries. Contraries have the property that both cannot be true together. One may be true and the other false. It is also possible that both can be false together. On the other hand, subcontraries involve oppositions between particulars. In this case, both cannot be false together.
Figure 29 (a) The modern logic version of the oppositions. (b) Aristotle’s square of oppositions.
The Modern Square of Oppositions
Of great interest to us is an opposition at a higher level altogether, the opposition between Aristotle’s syllogistic structures and modern logic. The dramatic difference between the two approaches was clearly illustrated by George Boole, in what has become the modern version of the Square of Oppositions.
Modern logic differs from the ancient logic by simply replacing the universal with the general, in other words with the abstract. This can be achieved by using labels and the logic becomes symbolic logic. Thus, the term ‘All men’ is replaced by the abstract version ‘All X’. The thing gets replaced by a label and introduces different semantics. The label becomes simply a placeholder and as such, like any placeholder, may be empty. The logicians explain this as relaxing the requirement of existential import. From a classical mathematics perspective, the generalisation introduced by modern logic is to allow sets to be empty.
Once the reasoning becomes abstract, the logical difference between yellow centaurs and canaries evaporates. Not only that, but all the oppositions except the contradictories have also evaporated. For example, both sides of the contraries opposition ‘All centaurs are yellow’ and ‘No centaur is yellow’ are true. The contraries opposition has evaporated.
Figure 29 (a) shows the resulting modern logic version of the square of oppositions. The square has virtually collapsed and only the contradictories and the subcontraries survive. We have deliberately drawn the modern version on the left side relative to Aristotle’s square to illustrate that this is the left side variant of logic. The other variant is Aristotle’s seed for the right side version. The left side involves abstract, symbolic logic. The right side in the diagram represents Aristotle’s version of elementary generic logical structure. In practice, the modern symbolic logic approach boils down to a simple bipolar nominalism where the basic opposition is between two particulars, I and O. The letters A and E act as pure label signifiers for the I and O respectively, acting as the signified. The contradictory oppositions A-O and E-I model the relationships between signifier and signified. In essence, the system becomes a simple two letter system labelled by A and E. Thus, although we have not shown that modern day logicians only use half a brain, we are starting to see that they reason using only half an alphabet.
This is our first exploit into the differences between abstract, symbolic logic and generic logic. We can do better. This will be our task in the next section where we investigate Stoic logic and discover great beauty in the land of Chrysippus.
Generic Logic and the Stoics
In this section, we are going to look at the Stoic version of Square of Oppositions. This will undoubtedly upset the scholars. There is no explicit record that the Stoics ever proposed an alternative to Aristotle’s Square of Oppositions. However, we are not constrained by historic Stoicism. If something is missing from the puzzle then we must endeavour to reverse engineer it. We attempt to follow in the tradition of Chrysippus himself, of whom it was said:
… in many points he dissented from Zeno, and also from Cleanthes, to whom he often used to say that he only wanted to be instructed in the dogmas of the school, and that he would discover the demonstrations for himself. (Laërtius)
Stoicism, particularly the Early Stoa, is a very tightly integrated body of thought, much tighter than what might be imagined, especially after Chrysippus had a hand in the matter. Traditionally Stoic philosophy involves a tight integration of physics, ethics and logic. Likened to an egg, the yolk was physics, the white ethics, and the shell was logic. Logic protects and holds it all together.
Stoic logic differs dramatically from that of Aristotle. There is no static classificatory apparatus. There are no species and no genera. There is no extension or comprehension of terms. The figures and modes of the syllogistic evaporate into thin air. To the Stoics, Aristotle’s syllogistic logic was “useless.” (Chénique, 1974) In contrast, Stoic logic is starkly oriented to the individual. As such, it incorporates one aspect that might entitle the logic to be considered a generic logic, a logic free from abstraction. It is this kind of logic one needs to construct and deconstruct a real world, not an abstract world. However, what precisely is a generic logic? Our immediate task is to answer this question. Now proficiency in logic demands a certain dexterity and agility of the mind. In this respect, the author has been blessed with a mind as nimble as the Titanic and just as infallible. This helps explain his reaction when thirty years ago he first came across Stoic logic. Almost nothing is left of Stoic texts in modern times. Nevertheless, a rough sketch that can be coagulated into less than a page or so has survived. It was Chrysippus’ Ground Zero, viewed from a logician’s point of view of course, a great logician’s point of view. The author was quite excited: it is not every day that one comes across an explanation of the structure of the Cosmos spelt out in hard-core logic, all on one piece of paper to boot. There they were, Chrysippus’ five logical undemonstratables. According to Chrysippus, all reasonment stems from these five logical gems. The author stared at the five gems like a stunned plover. If this is it, he could not see it.
Over the years, the author came back numerous times to the five undemonstratables in an attempt to really ‘get it’. A favourite reference was Éléments de Logique Classique (Chénique, 1974). The pages were getting quite dog-eared. Each time brought about the same stunned plover reaction. He still did not get it. However, Chrysippus’ third undemonstrable stuck out a bit from the other four. It looked very much like the modern logic operation called the Sheffer stroke, named after Henry M. Sheffer. It is called a stroke because that is the way it is symbolically written, as a vertical stroke. The author would look stony eyed at the third undemonstrable and ask himself: “What does it mean?” He knew that if he asked someone trained in logic they would patiently, and perhaps condescendingly, explain that it means “NOT this AND that”. In plain English, it means not both. It is sometimes called the NAND operation. It is important in logical networks because any network can be built uniquely using NAND gates. In other words, any other logical operation can be built up uniquely using NAND. They would then go on to explain that Charles Sanders Peirce had earlier in an unpublished work (Peirce,
1880) come up with the mathematical dual which is now called the Peirce Arrow, written as a vertical arrow. This is the logical NOR operation. Such thoughts would make the author’s eyes glaze over. He would ask himself, “But,what does it really mean? What did it mean to Chrysippus?” Superficially, it simply looked as if Chrysippus was the first to discover the propositional calculus. Granted he had made the discovery several thousand years in advance of the moderns, but this is the kind of thing one would expect from a master logician. If that was all there was to it, then Chrysippus would have nothing much more to offer the moderns. The undemonstratables could be simply seen as an early attempt to systemise and even axiomatise the propositional calculus. Chrysippus could be brought into the modern camp and branded as one of them. Surely, there must be a deeper message here.
For a long time, Stoic logic remained as a kind of lurking nemesis in the author’s mind. The five undemonstratables, where did they come from? What is the underlying principle? For a philosophical system as tight and unified as the Stoic’s, the logic must have the same basic epistemological and ontological signature as their physics and ethics.
Our approach so far has been based on an intuitive interpretation of how the physics and ethics could be constructed from a fundamental ontological dichotomy. The dichotomy can be understood linguistically as the difference between the verbs to have and to be. Two fundamental entities were proposed that expressed primary difference free of any accidental, empirical attributes. In this scenario, one entity had the attribute of being devoid of any specificity whatsoever whilst the entity was this attribute. One entity has an attribute; the other entity is this attribute. The difference between these two entities was said to be a difference in gender. Using this gender construct, the primary attributers of reality can be constructed from first principles. The attributes are not harvested empirically, but synthesised, calculated.
We note in passing that our gender terminology is more reminiscent of Indian and Chinese philosophy traditions. Rather than explaining the beginning of a creation cycle in terms of a union between the feminine and masculine, the Stoics tended to restrict their vocabulary to the masculine register, where only Zeus and his seed seem to feature. For the Stoics, the two principles translate to the active and the passive principles. We prefer reading this as the masculine and feminine principles because, in this relativistic domain, the gender concept is much more generically neutral. What is active can be passive and what is passive can in turn be active.It is much easier to talk in terms of gender where the masculine can play the role of feminine to produce the MF type for example, and vice versa. At any rate, all of this is just a debate about terminology. Using Ockham’s razor to cut through the debris, we will stick to the simple and clean gender terminology as the basis for our generic science. Maybe some Stoics did too.
The approach leads to the four elementary letters of the ontological alphabet based on the binary gender typing MF, FF, FM and MM. For reasons that will later become apparent, we allocated the letters A, U, G, and C respectively from the genetic code for this four-letter ontological alphabet. The detailed algebra of this ontological code based on this fourletter alphabet has yet to be determined. This code is capable of describing and proscribing any being whatsoever, including the universe, itself a being. Any being must have its own ontological DNA, so to speak. It is in this way that a being can be sure of what it is.
Using this gender construct, the theory of the four ontological elements can be explained in terms of four kinds of substance typed by the binary gender typing MF, FF, FM and MM. This corresponds to the ancient terminology of air, earth, water and fire respectively. If it starts with M it’s light stuff, if it starts with F it’s heavy stuff and so on, according to the ancients.
The four elements have multiple instances, are mobile, and mix. The four binary types also apply to something that is not mobile and is located at the centre of the universe; or rather, at the centre of its universe. This is the generic subject, what can be thought of as any being whatsoever. This is the generic template of mind. The whole universe gyrates around this entity. Not only are the four typed bodies fixed at this location, they are fixed in relation to each other. The two bodies with a binary gender starting with M are located on the right and the F on the left. The two bodies with gender typing ending in M are located in the front lobes, the F in the back. The question regarding telling the difference between left, right, front and back can be resolved by looking at the gender typing. This might seem a bit tautological, but that is the way things work in this world. Here relativity is not only endemic it is generic.
At this point in the game, there is also the question of whether the four kinds of mobile elements circulate to the exterior of this generic mind or in the interior. This question cannot be answered at this stage of the ontological development, as what is interior to subject and what is exterior is still unqualified.
Finally, we come back to the main question in hand, the Stoic logic question. What is the ontological interpretation of Chrysippus’ five undemonstratables?
The Fifth Element
In order to answer this question, we have to start thinking in terms of quintuplets rather than just quadruples. Before we tackle the logic quintuplet, it is worthwhile looking at Stoic physics. What is the fifth element?
Aristotle argued for a fifth element in his physics, which he called aether. A fifth element was necessary to fill the heavens above the terrestrial world and to explain the constant, unchanging rotation of the stars.
The Stoics also added a fifth element to their system, calling it pneuma, an ancient Greek word meaning ‘breath’. In this perspective, the four elements air, earth, water and fire were considered passive, whilst the pneuma expressed the active principle. Unlike Aristotle’s aether, the pneuma permeates everything and expresses the Logos for both the Cosmos and the body.
Some accounts say that pneuma is created from the fire and air elements. From our previous analysis, we know that the fire and air elements of antiquity have the gender coding MM and MF respectively, which indicates a primary gender of masculine for both. For the Stoics, the masculine gender was interpreted as embodying the active principle, which would explain fire and air being associated with the active principle. The other two elements water and earth are gendered as FM and FF respectively and so are primarily feminine and hence considered as embodying the passive principle.
So far, we have provided the fundamental ontological justification for the ancient four element based physics that was adopted by the Stoics. However, there was no trace of any fifth element in our development. A clue to the missing fifth element can be found in Chrysippus’ five undemonstratables, in particular, the third undemonstratable. As for the other four, we will use them to resurrect a square of oppositions for the Stoic logic of Chrysippus that is comparable to that of Aristotle. Just as the medieval scholars gave four letters to the four terms of the Aristotelian square, we will do the same for the logic of Chrysippus. However, instead of the medieval AIOE lettering we will use the AUGC lettering of the generic-cumgenetic code that we are developing. As for the fifth term, it has no letter. There is no fifth letter in the genetic code. There is no fifth letter in the generic code.
In the biological genetic code, the AUGC lettering based on the codon triplets of letters, code amino acids that go to making up protein. There is no sign of any fifth player in the scheme of things. Moreover, there is no need to try to find a fifth player as it has been there right from the beginning of our development. It forms the core of the very essence of gender, the generic building construct of anything that aspires to be.
To be what one is does not come easily. Being is not something handed out on a plate. To be requires Oneness of the being in question and it is up to that being, and that being alone, to maintain and express its own Oneness.
Generically, this is expressed by the generic entity characterised as totally devoid of any specificity whatsoever. This total lack of specificity must be the case as the generic entity can be anything whatsoever. Such an entity acquires Oneness by interaction with a subject. The subject may imagine it, think about it, touch it, measure it, claim it, or whatever the interaction, the result is that of a collapse to Oneness relative to the subject. Relative to the generic entity itself, the Oneness arises from the one single characterising attribute that the generic entity possesses, that of the absence of any qualifying specificity. The irony of the situation is that this total lack of qualifying specificity plays the role of attribute. The generic entity has an attribute. This attribute that the generic entity has, if it is not to violate First Classness, must be an entity in its own right. This entity is the attribute. Thus, there are two different entities, which share the one single attribute. One entity has this attribute; the other entity is this attribute. The difference between these two entities can be thought of as a difference in ontological gender. The entity that has the attribute is feminine, whilst the entity which is the attribute is masculine.
The masculine entity endows Oneness. As attribute it is pure singularity, pure Oneness. The remarkable thing with the resulting science that can be built upon this initial union of the feminine with the masculine is that entities that are more qualified can be constructed, based on compound gender typing. The first compound types are the four binary combinations of the feminine and masculine and can be considered as the four fundamental “letters” of the generic code. Thus, the four compound genders MF, FF, FM, and MM are allocated the letters A, U, G. and C respectively. The arguments for this particularly allocation will be advanced later.
Ontological gender appears, albeit informally, in the cosmologies of the many different civilisations from the West to the East and Far East and beyond. In this work, we present a formalised version of the construct. The formalisation is not an abstract axiomatic, but a formalisation of a different kind, the generic formalisation. The generic formalisation leads to a new kind of science that we could call generic science.
The essential ingredient in such a science is the rapport between the feminine and the masculine. These two gendered entities are different, but indistinguishable. They are indistinguishable because there is no way to compare them: they both share the one single attribute between them. The attribute that they share is that of Oneness: one has it, the other is it. There is a tension between these two entities. If ever the bond between them were broken, then that would spell the end of the world, at least for it.
The tension between the two genders expresses itself at the microscopic level between all the individual masculine-feminine compounded gender typing of a complex organism. Threats to Oneness of the Organism abound at all levels. This complex compounding of tensions throughout the organism is probably what the Stoics referred to as the pneuma.
We will continue our discussion of the pneuma, this mysterious “fifth element” further on. In the meantime we take another interlude to hammer home the intuitive understanding of the simplest but most profound concept of all, that of ontological gender.
A Light Interlude
Gender is not an abstraction. Any being is it and has it. Any individual being is gender typed. Once again, this is not an abstraction. For example, every cell in an animal’s body contains a copy of its chromosomes consisting of long strings spelling out the genetic coding of the individual. The coding is built up of words consisting of triplets of the four letters AUGC (using the RNA convention). From a gender perspective, the genetic coding has a deeper structure than that determined by biochemistry. The four letters can be represented in terms of binary gender typing MF, FF, FM, and MM respectively. Thus if the reader wants to know his or her gender typing then all they need to do is to translate their AUGC based genome into the gender equivalent.
Gender can be illustrated more concretely than even this biological version. Consider the following two rather tongue in cheek examples of gendering. The examples may help to overcome the bad habit of always thinking with a left brained mental disposition. Each example aims to prove once and for all which gender is superior to the other. The reasoning is only vaguely inspired by the Hindu Naya five-step syllogism and so lacks some of the rigor.
The masculine gender is superior to the feminine gender.
Take the case of a bird in the hand and the birds in the bush.
The bird in the hand is of masculine gender as it is in possession of the subject, a fact that is absolutely and even tautologically true because the bird is in the subject’s hand. Conversely, the birds in the bush are of the wildcard species, not in possession, and undoubtedly hard to catch. These birds are obviously of feminine gender.
In order to prove the proposition we invoke the age-old proverb:
A bird in the hand is worth two in the bush.
This goes to show that the masculine gender is superior to the feminine (by at least two times).
Not to be out done, there is another argument, which demonstrates the converse.
The feminine gender is superior to the masculine gender.
Take the grass on your side of the fence and the grass on the other side of the fence.
The grass on your side of the fence is of masculine gender as it is in possession of the subject (which is you). This fact is true for tautological reasons. However, the grass on the other side is not in the subject’s possession, and remains tantalising out of reach, a real wildcard. That grass is obviously of feminine gender.
In order to prove the proposition we invoke the age-old proverb:
The grass is always greener on the other side of the fence.
This goes to show that the feminine gender is superior to the masculine gender (and much more desirable).
There is another variant of proposition 2 that employs the notion that the spouse on the other side of the fence is more desirable than the spouse on this side of the fence. We will not go into a detailed analysis of this case, but the variant has some value as it goes to show that gender is only obliquely related to sex. A spouse of one sex can be gender typed masculine or feminine, depending on context.
The Gender Algebra of Oneness
As we have said often, left side sciences are based on abstraction, dualism, empirical attribute harvesting, and employ a labelling, categorising, taxonomic epistemological technology. Right side science replaces the abstract with the generic, dualism becomes a monism and the attribute harvesting from the field and laboratory is replaced by attribute construction from one single attribute based on the generic gender construct. As for epistemological technology, labelling, the linear descriptions and essentially rhetorical approach of left side science give way to the dialectical where concepts and semantics are expressed in purely relative terms. Concepts and constructs are expressed in the form of oppositions. Moreover, whilst left side science specialises in the tunnel view of reality, right side science must always address the whole.
Right side science must always take the holistic view. This is the essence of monism; nothing can be left out of the picture. This means that right side science always deals in wholes. A whole is totality viewed from a particular point of view, the point of view of the present subject. The subject is always present in right side science. In the traditional left side sciences, including mathematics, the subject is always absent. As such, left side science specialises and only recognises one-half of reality: it is half-world science.
Right side science must be a monism. There can only be one such science. The Stoics were pioneers in this area with their unified version of the monism, and we follow in their tradition. However, we make no attempt at doctrinal orthodoxy.
The incredible thing about right side science is that it is independent of scale. One still sees the universe as a whole from even the most apparently microscopic point of view. The science is in fact starting point invariant. It does not matter where you start; you always get the same science. There is one and only one science with this unique property. As a theory, the theory is its own invariant.
The science is independent of scale and independent of starting point. It does not matter where you start; you always get the same theory: That is the theory.
Left side sciences are dominated, even swamped by an ever-increasing avalanche of attributes. Contrast that with the attributes in Generic Science, the right side science. There is only one attribute for the whole science! This single attribute is the attribute that the pure feminine F has. It is the pure masculine M. The feminine has an attribute. The masculine is that attribute. That defines what gender is all about. One entity has it; the other is it. All other attributes are simply built up from different combinations of the masculine and the feminine.
It might be thought that the feminine F is also playing the role of attribute, which of course it is, but only by role-playing it, not being it. And so here is the difference. We can be absolutely sure of what the masculine attribute means. We understand the masculine as it means one and only one thing. It means pure Oneness. It represents pure certainty, because we know what Oneness means. The situation with regard to the feminine is the opposite. It represents the opposite of certitude. It represents total ignorance. We do not have a clue of what the feminine actually is, not a clue. This is the absolute expression of the Uncertainty Principle. It is also the secret of Generic Science. Generic Science is the only science that can talk about something that it knows absolutely nothing about, and talk about in absolutely certain terms. Moreover, it can express its ideas in algebraic form.
Now the reader may have had a similar experience to the author as in the following. The author remembers vividly, many years ago, when his Technical School mathematics teacher started the day’s lesson with great drama, something he was prone to do quite often. His name was Harry Sermon. Harry wrote up on the blackboard one letter. It was the letter x. Now apparently x was just like a number. Apparently, you could add it, subtract it, multiply by it and so on; remarkable. Even more remarkable was that you could do all this without having a clue of what the actual value of the number x was! Remember that? The lesson of course was an introduction to elementary algebra. Now having gone through what we now realise was not a complete waste of time, we find ourselves faced with the prospect of learning the algebra of the Cosmos, a worthwhile enterprise surely. Instead of the letter x, we have the ultimate unknown of all time and possibly even outside time, the letter F. This is the wildcard of the cosmic algebra. Its value can be any entity whatsoever. In the case of x in our first algebra lesson, at least we knew it corresponded to a number, even if we did not know its value. In the case of F, we do not have a clue about anything, whether it is a number or god knows what.
Our project is to develop a way of describing our pure ignorance of the world. Now some people might say that if you do not know what you are talking about then you should shut up. However, we take the high road, the road of Socrates’ confession of ignorance. He is reputed to have said that the only thing he knows with absolute certainty is that he knows nothing with absolute certainty. In addition, this is our position; and our task is to develop algebra up to the task of expressing such wisdom.
One of the main points made in this work, is that we have apparently been beaten to the gun. Instances of the algebra, the algebra of Socrates’ confession of ignorance, are everywhere. In your body, repeated in every cell is the same description over and over again, of what you are and are to become. All that has to be done is to “solve for F”, based on the situation on hand. The question, ‘Why bother?” may be raised by those not motivated to solve things, particularly algebraic things. It is a good idea to solve these Socratic gems of wisdom, as a failure to do so may challenge your very existence. If your body gets its Oneness equations tangled up, you could be in real strife.
Only two letters are needed for this code, the letter M and the wildcard F. Combined in pairs they make up the four letters of the alphabet of the generic, pardon, the genetic code. These are the letters A, G, U and C, all binary compounds of F and M. The macroscopic organism is organised as an immense compound articulated by these four letters, an immense compound of F and M gendered entities. M expresses the Oneness of the organism, F the unqualified, the unknown, “Solve for F”, and perhaps you have your life in a nutshell. However, solving for F may take a lifetime and could depend a bit on what crops up along the way.
is sans sujet. We will sketch out here a more fundamental approach to semiotics
and the semiotic square that does include the subject.
, one associated with Ferdinand de Saussure (dyadic, arbitrariness of the sign etc.) and one associated with Charles Sanders Peirce (triadic). In our view, the approach of de Saussure is not semiotics
, but General Linguistics. Like Greimas, the approach of de Saussure is sans sujet. If there is a subject, it is part of the Spectacle, not the Spectator. It is merely what Hegel referred to as the empirical ego. In this perspective, the de Saussure approach is like that of the traditional sciences and mathematics. All of these sciences are sans sujet. We call all of these traditional science left side sciences. Left side sciences claim to be objective, which is another way of saying that they only concerned with a reality of objects where any reference to the subject has been excluded. They are all sans sujet. As such these sciences look at the world from a very specific point of view. This point of view has been described as the “view from nowhere” or the “God’s eye view”. This is a general characteristic of science sans sujet. It is a general characteristic of all the sciences and mathematics of today.
Figure 1 The generic semiotic square is constructed from the feminine masculine opposition applied to itself.
Pure Gender Algebra
, a vertical axis. The square becomes the “Chrysippus cube”! We have used the convention of the implication arrows in the diagram going left to right to signal the upwards direction, and the downwards for the right to left. Talking intuitively, this indicates that the top two entities have an “upward flow” and the bottom two entries have a “downward flow”.
|Chrysyppus Logical Semiotic Square|
The technical core of the paper will not be presented in the blog.
The process of formalising knowledge in the left side science is a relatively straightforward affair. The basic technology is already in place: It is called mathematics. Mathematics provides the tool for formalising the traditional left side science knowledge. When it comes to formalising right side science, one immediately comes up against a brick wall. None of the mathematics works. The obstacle is the draconian constraint of FC. FC must not be violated. The problem is that traditional axiomatic mathematics violates FC right down to its very core.
However, all is not lost. Because axiomatic mathematics is a formal system, it can be exploited to formalise the obstacle to formalising an FC compliant system. Mathematic formalises the way the problem must not be tackled. Axiomatic mathematics formalises the wrong way to go, that is to say, the wrong way to tackle the Kantian problem. Having a formal statement of the obstacle to progress, all we have to do is to find the way around the obstacle. If we cannot do it with mathematics as it stands, we will need something else.
Looking down at the very foundation of mathematics, we come to Set Theory, the elementary mathematic of collections. Without a formal notion of collections of things, there can be no formal mathematics. There are many axiomatic systems that claim to formalise Set Theory. Each system has a different set of axioms, but all systems contain one pivotal axiom, the Axiom of Choice. Faced with a Set of elements, which may even be infinitely denumerable, how can you distinguish one element from the other? How do you choose? The Axiom of Choice imposes sufficient structure on the system to solve the problem. Equivalent to the Axiom of Choice is Zorn’s Lemma, which is easier to understand. The lemma effectively states that the elements of any set can be uniquely labelled with real numbers. Thus, using real numbers as labels, there always exists a unique labelling of elements such that one element can be distinguished from the other.
The very reliance on an axiom, any axiom, violates FC, as no such a priori constructs are permissible in a First Class system. What is of interest with the Axiom of Choice is that it situates the way that mathematics resolves the distinguishing problem. Firstly, it has to resort to a construct at the axiom level. Secondly is equivalent to using an ad hoc labelling technology, a characteristic of all left side sciences. The Axiom of Choice, and its fundamental lemma, thus articulates quite clearly, the way not to proceed: Don’t use labels.
Structure is in the mind of the beholder. For the left side sciences the beholder is the impersonal subject providing the much sought after ‘mind independent’ point of view. This primary opposition between the impersonal subject and its object is ignored by left side science and replaced with an opposition of its own making, that of the rigid dichotomy between abstract theory and its object. In left side mathematics, the primary dichotomy becomes that between a set of axioms and a world of deductively explorable mathematical objects so predetermined, either explicitly or implicitly.
For right side science, the mind of the beholder is of primordial importance and is always present. Not only is the impersonal subject present, but also the personal. There are many ways of interpreting these two kinds of subject. As mentioned previously, the subject as placeholder and the subject as value is one possibility. A more mathematical flavour might be to call them the “covariant” and “contravariant” subjects, but one must be on guard not to slip into abstraction ways of thought. Both these two kinds of subject are simultaneously present in any whole considered by right side science, The science of wholes is the speciality of the right side of the epistemological brain. What matters is the generic subject formed by a highly primitive, primordial Clifford-Grassmann style “geometric product” of these two subjects (together with their respective worlds). The end result is a the generic subject in the form of a “quaternion” kind of Three-Plus-One structure, a semiotic square which can be more formally understood in terms of the ontological gender typing construct. In the right side science paradigm, this artifice occupies centre stage at all times. One could even say that it is centre stage.
One way of understanding the generic subject is to realise that it suffers from an incurable disease. The disease is called monism. Patients suffering from monism exhibit the pathological symptoms of being totally incapable of distinguishing the difference between the real world and their conception of it. Both appear to be the one and the same thing. Curiously, most human subjects, at least when not on hallucinogenic drugs or suffering from a deep schizophrenic episode, also seem to exhibit these symptoms .
Right side science not only must articulate the basic architecture of the generic subject but also of the generic objects. There are four types of generic object, four bases distinguished one from the other by binary gender typing. The typing of bases is determined relative to each other and ultimately compatible with the polarity conventions established by the subject, the ultimate arbitrator of type. These four bases can be represented by four binary gendered typed arrows. The problem now is to establish how these arrows can be combined to form elementary structures, without violating FC.
From a left side science perspective, if a right side science were at all possible it would present as some kind of meta science, metaphysics or meta mathematics equipped with its own metalanguage. Such a science is not possible under the ambit of left side paradigm dominated, as it must be, by its atomistic and dualistic worldview. However, even though fundamentally incompatible with FC, some accommodations can be made to achieve a kind of Partial First Classness (PFC). The resulting science will not be a true metaphysics but at least pass as a poor man’s cousin.
The Sad Story of Mereology
One such accommodation is the rather obscure quasi-mathematical discipline called mereology, a left side attempt at a science of wholes and parts. Mereology is an exercise in mathematical logic. It achieves PFC by removing the rigid set theoretic dichotomy between sets and the elements that they contain. This is achieved by ignoring any explicit reference to the elements of a set and only considering containment relations between sets. Sets do not contain elements, they contain other sets. Contained sets are parts of the containing set. Different axiomatic schemes are set up to formalise this kind of structure where wholes contain parts and PFC is achieved by both parts and wholes being sets.
Mereology is of interest because it is essentially an attempt to formalise has-a relations between entities. Such structure finds echoes in the class inheritance structure of Object Oriented programming systems, for example. There are also echoes with our initial development of right side science where the has-a relation is paramount. However, right side science grants comparable prominence to is-a relations. In fact, the basic building block involved the gender construct where the feminine ontological gender corresponds to the has-a relation and the corresponding masculine gender to the is-a relation. The core of right side science, with its ontological vocation, consists of the dialectic of the has-a and is-a relationship. In mereology the has-a relation is axiomatised in terms of some kind of partially ordered structure such as set inclusion. As for any ontological is-a structure, that is hard wired into the axioms. Being a left side science mereology does not entertain any kind of is-a has-a dialectic.
A. N. Whitehead, in his philosophical quest for a holistic rationalist science, extended mereology concepts to geometry and achieved a geometric PFC (Whitehead, 1919). In this case, the rigid dichotomy between geometric objects with extension and geometric objects with no extension (points) was avoided to produce a pointless geometry. A pointless geometry is a right side kind of geometry. However, the geometry was caste in a left side, abstract, dualistic, atomist framework. In the final count, the system inevitably violates FC on practically every other front. Nevertheless, mereology is worth mentioning here as it expresses many of the aspirations of right side science even though it fundamentally lacks the necessary equipment to deliver the goods. In this respect, the mereology-based paper “Steps Toward a Constructive Nominalism” (Goodman, et al., 1947) is notable. In espousing constructionism and nominalism, the paper articulates important hallmarks of right side science. In addition, the authors start the paper with the doctrinal declaration: “We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether.” This rejection of abstraction is yet another fundamental tenant of right side science. However, declared within the confines of left side abstract axiomatic technology this anti-abstract belief becomes a bit of an oxymoron. It is like the Christmas turkey that struts into the kitchen valiantly declaring that it does not believe that turkeys are food.
From our perspective, mereology is interesting more for its aspirations than its achievements. What we want is a left side discipline that can properly formalise the very essence of mathematics and that comes from within mathematics itself. What we need is an abstract theory of abstract mathematics, and that naturally leads to Category Theory, the meta-mathematics of mathematics. It is with Category Theory that we can find a formal specification of the kind of structure that is anathema to our right side science. We will use Category Theory as a formal negative indication of what we are up against in trying to resolve the Kantian problem.
Category Theory Structure Violates FC
Category Theory provides abstract representations of mathematical structure in terms of a collection of objects and a collection arrows or morphisms between the objects. The specificity of mathematical structure is represented by the arrows and in no way by any explicit internal structure of the objects. The approach is thus structuralist in nature. Representation of the most elementary mathematical structure starts with placing two arrows end to end. This represents the composition of two arrows. Composition of arrows must satisfy two axioms, identity and associativity relying on the structures illustrated in Figure B 3 . Both of these structures violate FC.
Figure B 3(b) represents the composition of two arrows f and g to determine a third arrow h thus satisfying associativity. This violate FC because the arrow g is in an absolute ordering relationship with the arrow f. In a system satisfying FC, no entity can be absolutely before or after any other. Thus, even the two arrows shown in Figure B 3(c) is an FC violation. Thus not only is associativity prohibited but any kind of composition. We could call this disallowing of any absolute ordering relationships, the Parmenidean condition. For FC, the only thing that is must be immediate, not anterior, nor posterior.
A mathematical category requires the notion of composition identity defined for each object. This requires arrows that close back on themselves to form a loop as shown in Figure B 3(a). This structure also violates FC as it infers than the same entity can be different to itself. We will call it the Heraclitus principle expressed by the saying that “You can’t put your foot into the same river twice”. It is a special case of the Parmenidean condition. This prohibition is a subtle one but suffice to say that it can be represented by a prohibition on circular arrows.
Without getting into messy details, it suffices to say that the formal axiomatic mathematical Category abstractly states the minimal structural characteristics that a system must possess in order to qualify as mathematics. What interests us is not mathematics, but its opposite, anti-mathematics. We informally define anti-mathematics, as being everything that mathematics is not. At the abstract pinnacle of mathematics, we find Category Theory. The anti-mathematical counterpart will be the Anti-Category. The only thing in common between the Category and the Anti-Category will be that they both exploit an arrow theoretic methodology.in one way or another.
The Anti-Category and the Kantian Conditions
The conditions on the Anti-Category can be summed up as:
- Unlike the Category, the Anti-Category cannot be abstract. This can be achieved by prohibiting dualistic structures, the essence of abstraction.
- Thus, unlike the Category, the Anti-Category cannot tolerate a duality between a collection of objects and a collection of arrows. For the Anti-Category not to violate FC, the mantra is that all entities are arrows. In this way, any entity will possess extent. From an ontological point of view, we reiterate the Stoic mantra that only bodies exists. Point like entities do not exist.
- Unlike the Category, there can be no identity, no associativity and not even composition of arrows.
- There can be no axioms as any such predetermining structure violates FC.
We will call these conditions, the Kantian conditions for determining a formal structure that is totally devoid of any predetermining considerations. Realise an apparatus that satisfies the Kantian conditions and one has resolved the Kantian problem. In other words, one would have provided a formal basis for right sides science, the monistic counterpart of the dualistic left side sciences. Bot easy, but it can be done.
The axiomatic formalisation of mathematical categories is quite precise. Taking these conditions in the negative provides draconian requirements on the right side counterpart to the Category, the Anti-Category. Briefly, arrows determining anti-categories cannot form loops or be concatenated end to end. This leaves plenty of slack for finding a solution to the riddle. At least the Kantian problem is starting to look tractable.
Arrow Theoretic Methodology
Category Theory is based on an arrow theoretic methodology. It expresses its fundamentals in terms of arrow diagrams. Our task is to develop the right side counterpart of the Category in terms of the Anti-Category. If we can achieve this objective then we will have made a breakthrough is resolving the Kantian problem, the fundamental thrust of this paper. Thus, we claim that, in addition to the known left side arrow theoretic methodology of Category Theory, there must be a complementary right side arrow theoretic methodology. Our task is to bring this right side version of arrow theoretic methodology into the light of day. In the process, we will see that the traditional left side version specialises uniquely in the syntaxical aspects of structure and is virtually devoid of fundamental semantic considerations. On the right side of the equation, we will demonstrate that right side arrow theoretic structures are virtually syntax free, concentrating uniquely on semantics. One could say that traditional left side abstract approach to semantics leads to a syntax only account: Abstract semantics distils down to syntaxical expression. On the other hand, rhea right side paradigm approach to semantics leads to generic, non-abstract semantics. This kind of semantics is ultimately expressed in the gender calculus in the form of a syntax free generic code, a code capable of coding any semantics whatsoever that is compatible with FC.
The previous post outlined the overall structure of the world religions with the diagram shown above (Figure 3) This structure can be studied in more detail.
The Theological Square of Squares
The theological semiotic square provides a way of understanding the four world religions and how they relate to each other. Every religion has its own semiotic square and so there are semiotic squares “within” the semiotic square. Take Hinduism for example. The One in the One-is-One doctrine is the Brahman, the impersonal god that cannot be worshipped directly. The Brahman can be comprehended in the form of a triad of personal gods consisting of Shiva, Vishnu, and Brahma. This leads to the semiotic square in Figure where Vishnu occupies the same front top slot as did the Christ god and Shiva occupying the same back left slot as Allah. Brahma, who is rarely worshiped takes up the Buddhist slot. Theologians warn that this is an error. The Christ god and Vishnu are quite different as are Shiva and Allah.
Shankara’s Quintuple Dosage
Taken together with the unqualified semiotic square in Figure 3, the overall structure feature a configuration of five squares. Back in medieval times, The Indian Shankara put forward a non-theological version of this structure in his Theory of the Quintuple Dosage based on a theory of five intertwining elements. As a matter of passing interest the diagram can be found in the author’s previous book (Moore, 1992) and elsewhere.
The Real and the Imaginary
Science without subject
Science with Subject
Real and Imaginary Numbers
In a later post, the right side slant will be presented, free of any metaphysics. In this case, you get hyper-complex numberswith one real and three kinds of imaginary number.
Both the left side and the right side version of real and imaginary numbers should be included in a balanced education.
Key Phrases: Semiotic square, genetic code, generic code, DNA, start codon, left right hemispheres, the divided brain, epistemology, anti-mathematics, masculine, feminine, gender differentiation, Generic Science, Semiotic structure
As can be seen from the diagram, we end up with four different kinds of science. Kant homed into the front right side of the diagram, that of synthetic a priori judgments that, in theory, should synthesise new knowledge that was necessarily true. This is the domain where the Kantian question, addressed by this book, is located. How do we produce right hand, front side knowledge?
If we admit polyanalytic philosophy into the fray then, to be fair, we should also admit its totally opposite number, polysynthetic philosophy, yet to be born. Whilst analytic philosophy delights itself by listening to the tinkle of meanings flowing from natural language, usually English, on the other polysynthetic side, a different language is spoken where single words can be so large and the chords struck so vibrant that the music can last for a lifetime.
Key Phrases: semiotics square, genetic code, generic code, DNA, start codon, left right hemispheres, the divided brain, epistemology, anti-mathematics, masculine, feminine, gender differentiation, Generic Science, Semiotics structure
Philosopher David Chalmers remarks that the confidence in the traditional scientific method “comes from the progress on the easy problems.” Over the past decade or so, Chalmers has argued that it is time to tackle what he famously calls the “Hard Problem”, notably to develop a rigorous, scientific theory of consciousness. Chalmers’ Hard Problem is Hard to tackle because its requirements are antithetical to the very essence of the scientific method. The objectivity of the scientific method demands that only the object data be under consideration. All reference and interactions with the knowing subject must be eliminated. Thus to turn the tables and make the knowing subject the object of scientific enquiry means that all data has disappeared. And thus the problem of knowing the subject, this entity without data, becomes indeed a very “Hard Problem”.
As we have sketched in this book, this kind of problem has a long history, going back to Aristotle who saw it as the problem of developing the First Science, which he called the First Philosophy. Kant raised the ante in his time, calling for a science that didn’t rely on any a priori experience. Kant called such a science, metaphysics. In modern times, we now see it presented as the challenge of understanding consciousness badged as the “Hard Problem”. Nothing has changed over the past few millennia; whether it is called metaphysics or the Hard Problem, the problem still remains distinctly difficult.
Chalmers’ Hard Problem nomenclature raises possible objections. The emphasis on consciousness, as the last man standing, implies that traditional science has victoriously swept all before it, conquering practically everything in its way and has finally come to the final and last frontier to be conquered. Charmers offers no critique of the scientific method except that when it comes to consciousness it doesn’t work. This ignores the many foundational present day crises that riddle present day traditional science. What is needed is not just a science of consciousness but the noble, unifying science that Aristotle and so many others since have called for.
Having said this, we have no fundamental disagreement with what Chalmers has been saying. He is just presenting the scenario in terms of the measured language of Analytic Philosophy. He has considered all of the armaments and munitions at our disposal, inspected the terrain and has reported back to base. Despite all the equipment we possess and may develop in the future, it appears starkly apparent that there is absolutely no way we can win the battle. Game over.
Chalmers’ message is clear. If you want to win the war, you will have to start from scratch. You need an entirely different scientific methodology.
This, of course, is precisely our message. In order to start getting traction we have illustrated our thinking by using the biological brain as a metaphor for the required epistemological framework to do the job, the epistemological brain. The traditional sciences are what we call the left side sciences and correspond to the left hemisphere of our epistemological brain: reductionist, analytic, abstract, and obsessed with raw data. To resolve the Hard Problem we need another kind of science, the unifying right side science, the one that mysteriously operates out of the right hemisphere of our conveniently confected epistemological brain. In employing this pseudo-biological terminology, we take the same convenient path as Chalmers and effectively rebadge the ancient metaphysics problem as an organisational problem of mind. One could be tempted to say that it is a brain problem but, other than sounding a bit crass, the epistemological brain we are constructing is more based on a metaphor than sticky grey matter.
Thus, to resolve Chalmers’ Hard Problem, we are faced with the challenge of developing the right side science. Using the biological brain as a metaphor, this requires understanding right side reasoning, a totally different kind of reasoning from left side reductionism.
We have a fair idea of how linear, reductionist left side reasoning works. The student can start off with elementary logic, truth tables, Venn diagrams and so on as an introduction to symbolic logic. The abstract exercise can be combined with practical applications, so that at least some semblance of contact with the real world is inferred. This is all part of the Easy Problem.
What is the corresponding right side way of reasoning? We have already provided a preliminary response to this question in previous sections. Right side reasoning works with oppositions. The only way to understand something is in opposition to something else. In left side reasoning, it suffices to give a label to something in order to get a conceptual handle on it. What’s more, as general linguist Ferdinand de Saussure pointed out, the label can be completely arbitrary. This is first order semantics in action; labelling technology. This does not work for right side reasoning. Arbitrary labelling is not allowed. Ferdinand de Saussure stayed clear of the Hard Problem and stayed at home on the left side, the easy side.
Unlike left side rationality, labels form an integral part of right side reasoning and do so in an incredible way. However, that most exciting and positively overwhelming part of the story must wait until the later part of this work.
For the moment, we must work in a label free world. Rather than say “Let A be such-and-such, consider A”, our first examples were based on oppositions of cardinality, the opposition between One and Many. This is not the most fundamental opposition. It is too simplistic. However, the One-Many opposition is useful for an introduction. We then introduced a second opposition, another version of the same One-Many opposition. The second opposition was opposed to the first. The first was assumed to apply spatially from left to right, the second from front to back, as shown in Figure 3.
Now here is the rub. Something has been cut into four with these left-right and front back cuts. However, what has been cut into four? Nothing is really being cut in this first application of the semiotic square. What is being established is simply a unique frame of reference from which to comprehend reality. We build this tennis court-like structure in the middle of the Cosmos and demand that the whole Cosmos gyrates around it. From this unique pedestal for viewing the world, we have a ready-made reference frame of what is left and right, as well as what is front and back. This is all set in the polarity convention shown in Figure 3. We have discovered the location and shape of the centre of the universe! In fact, it has the same shape as the centre of your universe.
Right side science must be simple and simplifying, whilst continuing to climb out of the trap of appearing simplistic. Granted, our square-shaped mind situated in the centre of the Cosmos might appear a little simplistic. However, the situation can be saved by this egotistical mind-sprite admitting that there might be other entities in the Cosmos that enjoy the same viewing rights as itself. In this less determined context, the centre of the Cosmos becomes not that entity but any entity whatsoever, the true centre of the universe. One might argue that maybe only one such entity has the necessary four-part brain to join in the fun. This would not be an impediment, provided the consciousness in question could imagine itself in the place of any one of the other mindless entities and would thus see that same thing as the mindless (that is, if it had a mind). However, even that requirement could be weakened because the single mind might lack the capability of imagining changing places with another. In that case, it would not matter, as long as the same result would have occurred if it had such a capability.
At this point, we pull the ripcord even though we have not finished the story. These little naïve adventures into right side reason can be like a voyage into insanity. The author thinks that such exercises may be beneficial for students as long as they do not rote learn anything. The benefit for the student is probably to wean them off a dependence on left side linear thinking and on to binomial thinking. It should be kept in mind that similar tortuous adventures can be entered into by, for example, simply explaining in words something like the clock paradox in the special theory of relativity. In applying the theory mathematically, the formal methodology works quite smoothly and effortlessly. Right side relativity, once endowed with its own formalism, a relativistic relativity rather than the classical, should also be smoother and effortless.
It is time to look at some more practical examples of semiotic wholes.
The intention here is to provide a gentle introduction to right side science via practical example of the semiotic square. The approach is informal and intuitive at this stage.The semiotic square is an informal way of understanding wholes. A whole is Totality looked at from a particular perspective. Any thinker contemplating reality in a fundamental, non-abstract way is lead to semiotic squares of some kind. We have already seen this with the case of Kiyosaki, the uneducated but “rich dad” who thought holistically about the rationale of generating cashflow. Kiyosaki thus sneaked into the ranks of the great philosopher’s like Hegel. In fact, these ranks are full of autonomous autodidacts like Kiyosaki. Unlike Kiyosaki, Hegel was highly educated, but both these figures shared one thing: an aversion for abstract thought. Abstract thinking is left side thinking. Both Hegel and the entrepreneurial Kiyosaki empha-sised right side thinking. They reasoned in terms of wholes. Wholes are not abstractions, as they include the subject. This is right side thinking. The abstract thinker gives way to the generic thinker, a much more powerful breed.
The whole examined in this section is more qualified than the theological variety. Instead of subject as the impersonal self, with all of its theological overtones, we are going to consider subject as personal self. We are going to consider the human mind from the perspective of psychology. What is the basic generic architecture of the psyche? Our response will be in the form of Freud’s semiotic square interpreted from a viewpoint somewhat like that of Freud’s student, Jung, Once again we will start from scratch.We start with the left right divide of reality as conceived by modern present day science. Modern science splits the Cosmos into two sides. On the left side can be found objects which are completely untainted by subjectivity of any kind. The Cosmos itself is sometimes referred to in hushed and hallowed tones as the Laboratory. In between the objectified objects on the left side and the other side of the laboratory is a glass wall. On the right side of the glass wall is the observing subject. This subject is not like any ordinary subject as he is the Supreme Scientist, beyond and above all other. The Scientist, sometimes represented iconically as being dressed in an impeccably white dustcoat, a sure sign of divine objectivity, is completely fair, dispassionate and unbiased in any way. This means that he is devoid of any determined viewpoint or favoured perspective. The Scientist is endowed with the unique ability of being able to see everything from literally nowhere. He has the God’s eye view. These characteristics form the essential ingredients for being the Supreme Scientist, Lord of the objective universe.
Ordinary, everyday, scientists that have to work for a living aspire to emulate the Supreme Scientist and obtain his God’s eye view. Frustratingly, they never quite achieve their objective. Some scientists are so impressed that they take on the Supreme Scientist as their personal god. Like George Berkeley, they believe that you cannot have a Spectacle without an omnipotent Spectator, and that even applies to the lonely tree on a hill spectacle. Other scientists are completely unaware or refuse to embrace the existence of any scientist clever than themselves. These are the atheists who spend all their time on the Left Side and parasitically enjoy the fruits they find there.Once again, we have made a literary excursion into the realm of the great left right dichotomy. It paves the way to looking at the great left right cleavage of the biological brain used as a metaphor for understanding the personal Self.
|Figure 5 Freud’s semiotic square of the personal Self in the form of the human Psychic Self.|
The material in this section is probably better suited for discussion in a tutorial situation with a small group of students. It involves an exercise in lateral thinking across several semiotic squares. The importance here is to have some fun as well as perhaps getting a deeper understanding, without actually learning anything in particular. Our fascination is in the generic shape of knowledge and less in specific content.
Figure 6 shows two semiotic squares, the Freudian square and one for parliamentary democracy. To avoid any diplomatic incidents, the democratic square has been grounded in Australian democracy, hence the flag. Freud’s square has been grounded in the psyche of a person of undetermined nationality. We will now spend a few moments explaining the democratic square as a subterfuge for explaining Freud: the author knows only a little about Freudian psychology. Like most people, he knows more about democracy and particularly how it works in his home country.
As the author started to fill in the details of the left side of the square, as reported below, he inexorably slid into a mode of thinking that he can only describe as Zinovievian (but without the talent!). The world starts to take on a Yawning Heights (Zinoviev, 1979) character. Despite having read most of Alexandr Zinoviev, he is not really an influence, but represents rather a syndrome. It is a kind of disease, except you don’t know who has got it.. Describing left side reality from a right side perspective seems to be the catalyst.
One way of explaining the democratic square is to exploit a few Buddhist insights. This turns our subterfuge into a double subterfuge, but it can shorten a long story. Besides, everyone likes Buddhism.
Take Parliament for instance. From a Buddhist perspective, Parliament can be thought of as the house of suffering. All suffering ends up here. The house is full of suffering because of the craving. Craving stems from the Cravers down below in the bottom left side slot. The role of Parliament is to try to appease the Cravers, which presents a perennially difficult problem; hence, the suffering and angst.
Parliamentarians publically refer to the Cravers as Voters, which gives the impression that somehow the Cravers control Parliament by voting for it. Nothing could be further from the truth. Voting is compulsory in Australia. The main reason people vote is to avoid a fine and so have more money to spend on their cravings. However, sometimes they will vote for a Parliamentarian who seems to identify with their particular craving. In private, Parliamentarians refer to the Cravers as the “It.” The word “It” might refer to the Electorate, but more commonly, the word is used generically. Those clever enough to translate the word into German and creatively back into English might end up referring to the Cravers as the Id.
The Id is a teeming mass of opposing desires. Down-river irrigationalists confront up-river cotton farmers. Talk back radio Shock Jocks inflame the airways, railing against the boat people arriving on shore. Indigenous people writhe in the consequences stemming from when the forbears of the Shock Jocks arrived in boats on what used to be their shores. Greenies battle against loggers. Every complex, every syndrome imaginable will be found here amongst the craving Id.
That completes this little section on the left side of the Freudian psyche, written in Zinovievian mode. Coming over to the right side of the Freud square, the desire to write in Zinovievian mode vanishes. Actually, it feels a little bleak on this side as all we have is the Self in the frontal lobe and a thing called the Super Ego equipped with some powerful jurisdictional and moralising capability. There also seems to be some law enforcement capability as well. Super Ego seems to be full of lawyers and law enforcement officers.
Although we could pursue this topic at length, that is not on the agenda. So far, we have developed some experience in semiotic analysis and hopefully had some fun. The author has used these informal semiotic forms of analysis over many years in his profession developing software systems and computer languages.
The Dialectic of the One and the Many
Figure 3 The Theological Brain. A semiotic square for the four takes on reality based on cardinality oppositions.
The One is Multiple (Judeo-Christianity)
The Multiple is One (Islam)
The Multiple is Multiple (Buddhism)
The Theological Square of Squares
Key Phrases: Semiotic square, genetic code, generic code, DNA, start codon, left right hemispheres, the divided brain, epistemology, anti-mathematics, masculine, feminine, gender differentiation, Generic Science, Semiotic structure
To bring this subject onto centre stage, as a full-blown science requires a certain kind of mathematics, this will be covered elsewhere. In the meantime, we can make do with some elementary apparatus accessible to any mind curious enough to go along with the flow. The kind of reasoning is not traditional reductionist, analytic reasoning of the ordinary sciences. Such reasoning is an “open loop” form of thinking that necessarily involves labels and meaningless symbols. Allocating arbitrary names and symbols to things is a shaky start in the search for the deeper truths.
We have been referring to this kind of thinking as left side. What interests us is right side thinking. Right side reasoning is closed loop thinking where concepts are expressed in the form of oppositions and oppositions between oppositions. Some call this dialectical reasoning but no one has yet succeeded in formalising such reasoning. This is one of our objectives. The basic ideal of such reasoning is that everything is determined and understood in reference to something else, something that it clearly and absolutely is not. The opposition, the dichotomy, express such references. The semantics of object is lost without a present subject and so this leads to the fundamental opposition between subject and object, each giving meaning to each other.
We paint the picture with broad brushstrokes. However, even before the brushstrokes there comes the canvas. The canvas has four corners and is sufficient for an artist to paint a whole picture. So far, we have looked at a number of wholes and found that, as a whole, they can be painted on a four-cornered canvas. In previous sections, we saw that the “rich dad” Kiyosaki painted his cashflow quadrant on his four-cornered canvas. We saw that Freud accomplishes the same thing for the architecture of the psyche. We discussed the functioning of Freud’s mechanism by talking about another semiotic canvas, parliamentary democracy; to demonstrate that by talking about one thing you can be really talking about another, a favourite pastime of artistic expression. As an attempt at some dangerous semiological acrobatics, we talked obliquely about the personal psyche in terms of the political psyche with a dose of Buddhist philosophical semiotics thrown in. It appears that we have stumbled on the universal language of the artist, a language that can talk across the board. Rather than just describe the scene, we can describe the canvas, the common ground for any painting. It also provides the elements for a common universal language that can operate across the board.
The canvas can be understood in the form of a semiotic square that encapsulates the two kinds of subject with the two corresponding worlds of objects. This semiotic structure is based on the opposition of two oppositions. The first opposition, termed the left right opposition, was seen as that between the impersonal subject on the right and the impersonalised objects on the left. Empirical scientists dreams of this dichotomy where a pure, dispassionate, non-entangled subject surveys a non-disturbed world of object-ve objects. Such a subject has the highly sought after “view from nowhere”, the God’s eye view, the holy grail of empirical science, the unachievable dream. This was the first understanding of what constitutes a whole, an amorphous mass of objects together with the necessary but totally undetermined impersonal subject.
In order not to be stranded in the domain of the unachievable dream, a second kind of subject must enter the scene, the real world, determined subject, the personal subject. This leads to a second opposition that we referred to as the second dichotomous cut across the canvas, the front back opposition. In the frontal lobes resides the epistemological domain of the personal subject. The rear is the epistemological domain of the other side of the whole, all that is not personal subject. This is the personalised object domain. The result is a canvas cut up into four regions. These regions are not spatial divisions. One could say that they represent epistemological regions describing the four aspects of a whole, any whole. This is ground zero. We have considered a number of examples already that share ground zero. The content has changed but the ground has been constant throughout.
A natural question is to ascertain where ground zero is located. It all depends on where the personal subject is located, and that can be literally anywhere. Everyone possesses his own ground zero. It is usually located somewhere in the region between the ears and behind the eyes. This is your own personal canvas for picturing the universe. Functioning correctly, it will be aligned with the impersonal version. It is split into left and right sides that in turn are split into front and back. This, in itself can be an immense source of wonder. However, we have not yet finished with the technicalities.
The generic ground for any entity taken as a whole can be understood in terms of the semiotic square. The square is generated from an opposition applied to itself. We have already interpreted this opposition in a number of ways. There was the opposition between subject and object. Another version was the opposition between the One and the Multiple. The most fundamental version of the opposition is that conveyed by ontological gender, the opposition between the masculine and the feminine. Gender will be revisited in more detail and precision later. Here we simply consider the masculine feminine opposition as involving a more generic opposition than the cardinality opposition between the determined One and undetermined Many. Gender is not limited to cardinality and goes right across the board from the quantitative to the qualitative. In all cases, the masculine appears as the determined singularity, that which is determined as singularity. The masculine is the only certainty in the equation. The feminine, on the other hand, is a totally unknown quantity. The best way to understand the feminine, albeit from the masculine viewpoint, is that it is a total wildcard. And this is the key. There is nothing wrong about knowing nothing about something as long as that is certitude. Here we find the Socratic confession of ignorance as the lynchpin of a whole algebra of the Cosmos! The ignorance is encapsulated in the feminine wildcard. The absolute certitude of knowledge that this wild card is a wild card is encapsulated in the masculine. The singularity of absolute certitude meets absolute uncertainty. This is the ultimate Principle of Uncertainty. What ‘s more, it provides the two letters capable of coding the whole Cosmos, any Cosmos.
We still have not come to the author’s object of wonder, but we are slowly moving in that direction.
The Four Letters of Antiquity
The above material will be revisited at a more leisurely pace in later sections. What we wish to retain here is the notion of a two-lettered generic alphabet. Intuitively we can say that these letters are M for masculine and F for feminine. These letters have semantic implications. The two letters have meaning as has been explained above. For example, the feminine F is the wildcard and is totally devoid of determined meaning, which, when you think about it, is really loaded in meaning. In a recent seminar given by the author, apparently a woman in the audience was taking notes and wrote down the letter F and then the word “wildcard” followed by a string of exclamation marks. So F seems to have meaning of some kind!!!
The Generic Square
The physics of the ancient world was not based on empirical left side thinking but rather an intuitive version of an embryonic right side science. In later sections, we will endeavour to reconstruct the ancient science and move it to a more rigorous and potentially formal footing. Of fundamental importance is the concept of gender, the most fundamental of any ontological principle. At present, we are content with an intuitive understanding of the concept. Figure 12 shows how gender coding can be used to provide the elementary algebraic expression of the ancient four elements. The table includes an additional column that describes how the same gender coding codes the genetic code. It is a relatively simple exercise to actually determine the exact match between the genetic code and the gender coding. Suffice to say that there are so many constraints to the puzzle that only one combination stands out. We do not go into these details here.
The Genetic Code viewed Left and Right Side
The Full and the Half Paradigm
There are two takes on reality. One is a full take and the other a half take. Left side science is based on the half take. It appears that the biological brain is similarly inclined. Thus, before investigating the scientific ramifications and avoiding any abstract musings, we look at the personality traits and competencies of the human brain when operating on a single hemisphere. What is the difference between the take of the left brain operating alone from the take of the right brain acting alone?
When only the left hemisphere is effectively operational, the subject suffers from “hemi-neglect”, as McGilchrist explains.
Because the concern of the left hemisphere is with the right half of the world only, the left half of the body, and everything lying in the left part of the visual field, fails to materialise … So extreme can this phenomenon be that the sufferer may fail to acknowledge the existence of anyone standing to his left, the left half of the face of a clock, or the left page of a newspaper or book, and will even neglect to wash, shave or dress the left half of the body, sometimes going so far as to deny that it exists at all. This is despite the fact that there is nothing at all wrong with the primary visual system: the problem is not due to blindness as ordinarily understood. If one temporarily disables the left hemisphere of such an individual through transcranial magnetic stimulation, the neglect improves, suggesting that the problem following right-hemisphere stroke is due to release of the unopposed action of the left hemisphere. But you do not get the mirror-image of the neglect phenomenon after a left-hemisphere stroke, because in that case the still-functioning right hemisphere supplies a whole body, and a whole world, to the sufferer. (McGilchrist, 2009)
Hemi-neglect is a characteristic of left side thinking, whether it be the biological brain or the scientific mind. The left side is aware of only one half of reality, whereas the right side must be aware of its domain of specialisation and the other side as well. After all, its specialisation is in terms of whole wholes, not half wholes.
Hemi-neglect runs right across the left side sciences. It always manifests itself in a binary way of thinking. We have already seen this in logic where left side reasoning is based on the Law of the Excluded Middle. Something is either true or false. Analytic Philosophy is full of it. The Mind Body, the imaginary and the real dualities are pet pre-occupations. When it comes down to linguistics and semiotics there will usually be two versions. One will be the left side version and is always dualistic. For example, the left side perspective on the nature of the sign is expressed by Ferdinand de Saussure as a dyadic opposition between signifier and signified as illustrated in Figure 13. Right side thinking employs a second opposition leading to a semiotic square with one “real” component determined by a triad.
The Tower of Babel
The central theme of this book is that this genetic code, this generic code is the language of wholes. As such, it is the natural language of Mind, the mind conscious of itself as a whole.
However, it appears only one half of mind, the right side, is based on the thinking in wholes and the corresponding 4-letter generic code. Here we come to the second theme of the book. We assert that the left side is not based on this four-letter alphabet. Rather than four letters, it only uses two, the two letters on the left as illustrated in Figure 14.
The right side thinks in terms of wholes and needs the full four letters of the generic code. However, the letters C and G relate to the Singular and the Universal. They express the requirement that the One must be One and the Multiple must be One. These are regulatory requirements. Such a mechanism can be restrictive. Like the free market economist who abhors legislation and regulation of the marketplace, left side reasoning dispenses with such travesties against individual freedoms. It becomes open loop and tries to go it alone. It doesn’t need any Cosmic Reason to figure out what should be done. It just needs a notepad of rules and a belief in Providence. Totally unaware of the guiding hand of the right side, an incomprehensible entity at best, the left side thinks that it is master of the world.
Figure 14 The generic mind: The right side is conscious of the whole. The left side has dispensed with the regulating machinery of the right side and has become open loop, relying on learnt rules. It has dispensed with the generic code and speaks the local patois. Faster, agile, and focused, the left side is unaware (Genesis) that the right side even exists, sometimes to its own peril.
All Rights Reserved. @copyright Douglas J. H. Moore 2011 Phrases: Semiotic square, genetic code, generic code, DNA, start codon, left right hemispheres, the divided brain, epistemology, anti-mathematics, masculine, feminine, gender differentiation, Generic Science, Semiotic structure