What is Gender?

Aphrodite

There is no construct in science more fundamental than gender. The ancients knew this but the moderns have long since forgotten it.

This post will explore the epistemological and ontological potential of gender in providing a unifying foundation for science and mathematics. In this respect, the structure of the French language provides a first glimpse of the relationship between knowledge and gender. French tends to explain concepts in terms of oppositions, often expressed across opposing genders. For example, French for knowledge is the feminine term la connaissance. The natural corresponding opposition in French is the masculine le savoir. Someone with a lot of specialised connaissance or knowledge is a connasseur. The most extreme kind of connaisseur.is the legendary idiot savant, the one who can digest the contents of the Yellow Pages in one sitting. On the opposite side of the fence is the savant of the non-idiot kind. The most gifted savant of all time was the equally legendary Socrates who had no reliable knowledge whatsoever as expressed in his Confession of Ignorance. However, he knew that fact with absolute certainty, a mark of the true savant. It is quite ironic that the Socratic Confession of Ignorance provides the key principle in developing algebra capable of integrating pure ignorance with pure certitude in a tractable manner, as we shall see.

Including axiomatic mathematics, all of the traditional modern day sciences are of the ordinary, common sense, analytic, fact-based, “connaissance” style of scholarship. These sciences are all well known as deductionist, abstract, atomist, and dualist. Employing the metaphor of the biological brain, we will refer to these sciences as instances of the left side scientific paradigm. The position we take in this paper is that left side paradigm is totally unsuited to provide a foundational science. Any unifying foundational science must be based on savoir, not connaissance. The savoir kind of scholarship we refer to as right side science. Our first task will be to explain the central role of gender in right side science.

Different natural languages implement gender in various grammatical ways. For example, Tagalog of the Philippines is remarkable for its complete absence of grammatical gender Even personal pronouns are neuter and so do not explicitly expose the sexual gender of the respondent. At the other end of spectrum is Jingulu, an Aboriginal language of Australia that has four genders. It is also interesting to note that Jingulu, like other Aboriginal languages, does not categorically distinguish nouns from adjectives, they all collapse into a broader category of nominals. In this paper we introduce the study of a code like language where even the categorical distinction between nominal and verb. and any other grammatical category, all such distinctions evaporate. The syntax becomes so generic that it virtually disappears. We call this language the generic code. We propose this language as the calculus for right side science. All natural languages are left side languages. There is only one right side language, the generic code. We will show how the semantics of this generic code can be reverse engineered from generic principles. With great trepidation, we also claim that this reverse engineered language provides the semantic foundations of the biological genetic code. In other words, the genetic code is an instance of a totally universal, generic code. This generic code is not subject to evolution. It must be in place right from the very beginnings of whatever might start to begin. We will show that the most salient feature of this generic cum genetic code is that, like Jingulu, the language is based on four genders.
Before attempting to tackle the problem of developing a generic language, we must look at the generic problem domain in which it is to operate. Generic language is to provide the calculus for a generic science. What is the nature of such a science?
Continue reading “What is Gender?”

The Shape of Knowledge

string pupputs

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 illustrating the four kinds of knowledge
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.

Side Note

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.

 

Is there an alternative to Abstraction?

A few years back I was listening to an interview on the radio with a female intellectual from the Middle East. She was asked the question “What do you find is the most seductive thing about Western culture?” Her response was direct and succinct. “Abstraction,” she replied, without any hesitation. Her reply stuck with me and added to my torment on this question. Is abstraction a fundamentally Western construct? Is not abstraction the highest form of thought? If not, what is the alternative? Does abstraction have a sibling?

I’ve been concerned about abstraction for many years. My approach has been to attempt to find an alternative paradigm. Back in the early eighties, I was attending a philosophy course given by François Châtelet at Vincennes in Paris, who sadly passed away a few years later. I volunteered to give a seminar on what I conceived at the time, to be an alternative mode of thinking to abstract thought. Instead of abstract theory, I rather naively proposed concrete theory, theory you could construct. I was greatly influenced by my work at the time in Computer Science and had started thinking that alternative to abstract theories were theories you could build with computer programs. Rather than the theory being abstractly described in static pages of a book in the library, surely we can construct theories with computer programs and actually execute them, instead of just reading them. I gave the seminar. I though it went off rather well but to my horror, and to the amazement of the class, Professor Châtelet became extremely agitated and attempted to destroy my argument in the most emotional terms. After the seminar, a group of students of Middle East origin, mainly Algerian, came to my defence saying that they understood what I was saying and were in complete agreement. Like me, they couldn’t understand why it sent Professor Châtelet off the rails.

Since then I came across Hegel’s public lecture Who Thinks Abstractly? (Hegel, 1966 (1808)), His speech was a real gem. Hegel remarks that it is often thought that abstraction is the affair of the educated and cultured man and that it is “presupposed in good society.” Hegel observes that the community:

at least deep down, it has a certain respect for abstract thinking as something exalted, and it looks the other way not because it seems too lowly but because it appears too exalted, not because it seems too mean but rather too noble,

Having played his audience one way, the rueful Hegel cuts to the chase:

Who thinks abstractly? The uneducated, not the educated. Good society does not think abstractly because it is too easy, because it is too lowly …

Skirting the outrageous, Hegel must come up with proof; it is not far away:

The prejudice and respect for abstract thinking are so great that sensitive nostrils will begin to smell some satire or irony at this point; but since they read the morning paper they know that there is a prize to be had for satires…

… and yes, just as in Hegel’s time, a mere glance at the front page of today’s tabloid provides ample testimony to Hegel’s claim. Every day headlines trumpet out the most abstract of abstract catch phrases for consumption of the masses. Hegel provides an example of such high abstraction, the Murderer.

A murderer is led to the place of execution. For the common populace he is nothing but a murderer. Ladies perhaps remark that he is a strong, handsome, interesting man. The populace finds this remark terrible: What? A murderer handsome? How can one think so wickedly and call a murderer handsome; no doubt, you yourselves are something not much better! This is the corruption of morals that is prevalent in the upper classes, a priest may add,…

Having taken in Hegel’s little gem of wisdom we are now able to answer the question, “What does a radio Shock Jock and a theoretical physicist have in common?” The answer, of course is – abstraction.

However, this doesn’t answer the question as to the alternative to abstraction. Our Western universities have become abstraction factories. Is there an alternative product? Is the alternative complementary? The purpose of this blog, and my book soon to be published, is to present the natural sibling to abstraction. I call it the generic. Instead of thinking abstractly, think generically. However, what is the generic?

Follow up post: The Alternative to Abstraction

The Alternative to Abstract Thinking : the Generic

Continuing from the previous post Is There an Alternative to Abstraction?
Having taken in Hegel’s little gem of wisdom we are now able to answer the question, “What does a radio shock jock and a theoretical physicist have in common?” The answer, of course is – abstraction.
However, this doesn’t answer the question as to the alternative to abstraction. Our Western universities have become abstraction factories. Is there an alternative product? The purpose of this book is to present the natural sibling to abstraction. I call it the generic. Instead of thinking abstractly, think generically. However, what is the generic?  Equally, what is abstraction for that matter?
Two Fundamental Questions
Our aim is to move towards a formal knowledge of knowledge. There are two kinds of knowledge. On one side, there is what we call left side knowledge, which is dependent on a priori information. On the other hand, right side knowledge expounds on what can be known, without any a priori information. Each kind of knowledge answers a different question. Thus, two very precise questions characterise each of the two sciences. We can simplify much of philosophical and scientific tussling over different answers if we recognise that there are two different questions behind the scene. The questions are in a natural opposition and antonymic symmetry with each other.
The domain of discourse for each question is totally disjoint. The questions are so distinct that they can be imagined as being “orthogonal” to each other. The first question, suitably schematically simplified, was posed by Kant in the Critique of Pure Reason:

Q1.

What knowledge can be achieved without reliance on any experimental  evidence whatsoever?

The answer would fall under the rubric of metaphysics. This question is familiar to all modern philosophers but is still waiting to be answered.
To some, like Karl Popper, the question is summarily dismissed. The problem posed by Kant is “not only insoluble but also misconceived.” (Popper, 1963) The problem was insoluble as we all know from Hume that there is no such thing as certain knowledge of universal truths. The only possibility was knowledge gleaned from observation of singular or particular instances. The inescapable truth is clearly “that all theoretical knowledge was uncertain.”
According to Popper, the problem was misconceived because Kant, even though he mentioned it, was not talking about metaphysics, but was really talking about what he didn’t mention; notably the pure natural science that had burst on the scene in his day, the science embodied in Newton’s gravitational theory. Newton’s theory has since been shown not to be the infallible exercise in pure reason that so impressed eighteenth century thinkers like Kant, but rather “no more than a marvellous conjecture, an astonishingly good approximation.” With the passage of time, Newton comes crashing down to earth and brings Kant’s question down with him. This demonstrates Kant’s misconception.
Popper concludes his demolition by replacing Kant’s bold question with his own languid alternative. “His question, we now know, or believe we know, should have been: ‘How are successful conjectures possible?’”
In this book, in order to arrive at a refutation, we actually go much further than Popper by bringing in some modern arguments to prove more convincingly that Q1 is insoluble. This is accomplished by showing that it is out of bounds of all formal mathematical reasoning.
To answer Q1 no axioms are allowed. Not only are operators of all sorts dispensed with – the commutative, the non-commutative, the associative, the non associative, even operator composition is declared a no go area. Traditional mathematics simply becomes non-operational in this zone. This is the domain where nothing can be said to proceed or succeed anything else.
In the business world, there is nothing more enticing to the entrepreneur than the accepted wisdom that something simply cannot be done. The proposition becomes even more enticing when learned abstract thinkers like Popper claim to have proven that it cannot be done.
Kant’s question Q1 viciously casts us into this apparently hopeless ultimate state of undetermined chaotic ignornace, However, by Popper arguing the futility of the enterprisse, the question becomes so well defined that surely there must be an answer. After all, it is only when the prisoner is actually placed in the confines of the four concrete walls of his cell can he really start plotting his way out. You cannot escape until you are locked up. Kant built the prison, Popper slams shut the door and rams home the bolt. It’s time to get out of this hell hole.
Once Q1 is clearly shown to be absolutely mathematically insoluble beyond any shadow of a doubt, we are then in possession of our first truth arrived at from pure reason alone. This is achieved without recourse to any experimental evidence whatsoever. We thus arrive at our first negative fact. We could call it a neg fact. The exercise then becomes one of building metaphysics out of neg facts in some way. This is obviously not an exercise in formal mathematics but an exercise in another genus of formalisation. We call it formal anti-mathematics. This and the remarkable results flowing from anti-mathematics eventually leads to code; a kind of “DNA of the Cosmos” so to speak. This is the principle theoretical contribution of this work and clearly the most enigmatic.
We now come to the second question, diametrically opposed to the first. It reads:

Q2.

What knowledge can be achieved with only reliance on experimental evidence?

The question is very brief and needs to be expanded somewhat in order to convey the intent. What kind of knowledge can be achieved under the assumption that only what is measured is real and only what is real is that which is being measured? What knowledge can be obtained by totally excluding counter factual reasoning?  Stated this way the answer to the question is probably already apparent as will be seen below. The implication is that the moon only exists if you are looking at it. What kind of knowledge can imply that?
In the first question the only discernable real was that discerned by all embracing pure reason – the Parmenidean real, the big picture. Q1 addresses the uppermost confines of the top down reality bucket barrel. On the other hand. this second question, Q2 imposes the opposite sense of what is real. It demands the ferociously materialist atomism and absolutist one to one nominalism that only the Epicureans ever had the audacity to contemplate to the fullest degree. To each sensation there is something, to each something a sensation. There is nothing else. For the Epicureans, this was the way the world ticks. For modern science, it becomes a particular scientific methodological paradigm. It’s the way the world ticks from a particular viewpoint. It’s the view from the bottom up. What kind of knowledge can be achieved within the confines of such a dogmatic straight jacket?
In this case the answer historically came before the question was ever seriously posed. The ancient answer was the physics of the Epicureans complete with their deterministic atoms moving along Bertrand Russel like causal lines but armed with an occasional, unpredictable, and at that time, indiscernible “swerve.” The modern answer is in the form of quantum mechanics, Heisenburg’s uncertainty principle, and in particular the classical Copenhagen interpretation of quantum mechanics.
The Epicurean ontological straightjacket implicit in Q1 limits the knowledge quest downwards to the minute, indivisible “Epicurean atoms” of reality: the elementary subatomic particles of modern physics. The only difference is that the atoms of Epicurus we assumed ot have extent. Modern physics is more radical in this regard. The elementary particles have no extent whatsoever. They are assumed to be point like. Such particles have nothing in the their interior. They simply don’t even have an interior. If there is something in the interior, your particle is not elementary. You haven’t reached rock bottom of the reality bucket.
The brutal minimalism of QM is succinctly expressed in Dirac’s razor principle.

Dirac’s razor

Quantum mechanics can only answer questions regarding the outcome of possible experiments. Any other questions, philosophical or otherwise, lie beyond the realms of physics.
This is the declaration that QM is a philosophical desert. QM declares that it is fundamentally a philosophical, metaphysical, epistemological, ontological, theological, spiritual vacuum. This is not a weakness, it is strength. It is this that gives it its rigour and even its vigour.

The Entanglement Problem

A situation arises in QM that there can exist minute particle systems which are non-localised. Consider the case of a pair of entangled photons produced by a photon splitting in two. Pairs of such photons can be produced in experiments. The polarisation of one entangled photon will be the opposite to that of the other. According to QM the actual polarisation for each photon would be indeterminate until the polarisation of one of the photons was actually measured. The measurement performed on one particle would flip its polarisation to say horizontal or vertical. According to QM, the polarisation of the other photon will instantly become the opposite polarity irrespective of how far away it is.
Einstein didn’t like the indeterminacy aspects of QM – “God doesn’t throw dice” but it was this “spooky action at a distance” that really bothered him. In the famous EPR paper, written with Podolsky and Rosen, he argued his case. QM conflicted totally with the classical view of physical reality that Einstein adhered to. According to his view a theory must allow for the simultaneous existence of “elements of reality” which are independent of measurement. The EPR paper gave a very concise and lucid definition of elements of reality:
If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.
The EPR paper then put forward a thought experiment that revealed a paradox in the QM theory of entangled particles. The EPR paper argued that each of the “entangled” photons would possess their own element of reality and have their polarisations determined at the time of the pair’s  creation, not at the time when one of them was measured. The measurement of  the polarity of one wouldn’t affect that of the other as its polarisation had already been determined and couldn’t be altered by any “spooky action at a distance,” as predicted by QM.
Basically the EPR paper argued for what is sometimes called “local realism”. The two fundamental principles are that there exist elements of physical reality or “hidden variables” and that this realism be local. The locality principle demands that theory must adhere to the principles of relativity (causes cannot propagate faster than the speed of light). Thus the measurement on one member of an entangled pair of particles should not effect any measurement carried on the other member.
The simplified argument is that either the locality principle and with it the special theory of relativity was violated or the elementary particles harboured internal “hidden variables.” In the first case relativity theory is proved wrong. Alternatively in the second case there are aspects of reality not accounted for by QM. QM is not proved wrong but is proved “incomplete.”
With the passage of time thanks to the ingenious theorem of J. S. Bell and the experiments devised by A. Aspect et al and others, it has been demonstrated that the EPR paper’s proposed construct of local hidden variables could not possibly explain particle entanglement. This left the possibility that QM entanglement explanation would violate relativity theory. However, that is not a problem either as there is no determinate causal relationship between the particle pairs. The process cannot possibly be exploited for signalling and thus does not violate relativity theory.

Popper on Quantum mechanics

We have used Karl Popper as a point of reference for the first of our reality barrel questions, the one stemming from Kant. He dismissed the question outright with scant regard to any possible answer. For symmetry we should consider the other side of the reality barrel where we found an already existing answer in want of a suitable question. We provide the question but what would Popper think of the answer? The answer was in the form of a twentieth century science called quantum mechanics. Would Popper in fact agree that quantum mechanics was a proper science? As is well known Popper had great difficulty accepting many of the tenants of QM. For a start, QM would have to abide by his falsifiable criterion in order to be acceptable as a science. This allows provisionally valid propositions to be deemed scientific provided that there existed the potentiality for the propositions to be proven false. To Popper, all that was admissibly scientific was uniquely constructed from such potentially falsifiable propositions.
If one takes the long view at what Popper is saying here, one can easily get the impression that Popper is more concerned in fighting political dogmatism on the campus, than engaging in real science. He was more intent on arguing that what was inherently anti dogmatic was inherently scientific. But was hard core science itself inherently hard core non dogmatic?
This question takes on great importance when we consider quantum mechanics, the most fundamental and deepest of the empirical physical sciences. The difference between quantum mechanics and all other empirical sciences is not expressed in the details of the subject matter addressed, but in the fact that it is the only pure empirical science. Being purely empirical, methodologically pushes its subject down to the very bottom reaches of reality. It means that quantum mechanics is the only empirical science which tolerates absolutely no “elements of reality” which exist independently of the actual act of measurement.  In order to achieve this goal it must dig down to the bare, nude essentials of reality.
It is the only such science. To put it another way, quantum mechanics is dogmatically empirical. To put it even more bluntly, quantum mechanics is empericism as an absolute dogma. This dogmatism is most clearly expressed with its Epicurean like dogma of the one to one relationship between the sensation and the real. Quantum mechanics theory of the real is that only what is measured is real. This science, located at the very bottom of the bucket of reality, where is nothing is deemed below, expresses itself in empirical tautologies. The measured and the real are two sides of the one thing. As such, this most reliable, accurate, and most dogmatic of the empirical sciences is inherently unfalsifiable at its core.
All the same, Popper stuck to his guns and had no alternative but to reject some of the essential tenants of quantum mechanics as being, in his terms, “unscientific”. In so doing, he ignored one of the two most fundamental questions one can ask concerning knowledge of reality. In the case of Q2, the knowledge is not only true, but measurably so. After all the Copenhagen dogma declares only that which is measured is real. What is real is only that which is measured.
This has lead to a tautology, an implicit “analytic judgment”. Kant would have found that fascinating. Moreover, this fundamental tautology appears not on the transcendental side of the equation but on the empirical. Even more fascinating is that this fundamental construct defines the pure empirical itself. The pure empirical is, well…, purely empirical. Such is the fundamental nature of quantum mechanics as declared in the Copenhagen interpretation.

Is There a Fundamental Level?

There are two takes on reality. There are tow fundamental questions Q1 and Q2 that express the fundamental opposition between the two fundamental perspectives on reality. The fundmanetl opposition reveals itself in many ways. An important consideration concerns whether ther is a fundamantal level of reality.
Is there a fundamental level? Jonathan Schaffer asks the question and summarises the fundamentalist response. “The fundamentalist starts with (a) a hierarchical picture of nature as stratified into levels, adds (b) an assumption that there is a bottom level which is fundamental, and winds up, often enough, with (c) an ontological attitude according to which the entities of the fundamental level are primarily real, while any remaining contingent entities are at best derivative, if real at all.” He lists the physicalist, epiphenomenalist and atomist variants on the theme. Finding plausible the hierachial view of nature in (a) as being compatible with the discoveries of science, Schaffer homes in on (b) as the problem area, which he remarks has been almost entirely neglected. Concerning the primacy of what is real, the fate of (c) is linked to (b) as a reasonable but not inevitable conclusion.
And so is there a fundamental, bottom of the bucket, level in Reality?
In our preceding discussion of quantum mechanics we argued, with scarcely camouflaged glee, for a dogmatic interpretation of the science findings which would seem to place us firmly in Schaffer‘s camp of fundamentalists. We were advocating the bottom of the reality bucket theory. On the face of it we supported without reservation all three tenants of the fundamentalist argument. At the risk of seeming, or even blatantly being, excessively schematic we identified the ontological approach of quantum mechanics as smacking of pure Epicureanism, a natural logical set of conclusions resulting from a pure unadulterated atomistic,
uncompromisingly blunt materialism and one to one nominalism evolving down from Democritus, a thinker not particularly notable for his subtlety and dexterity of thought, At least Aristotle didn’t seem to be very impressed., advocating at one time that Democritus’s books should all be burnt. These Epicurians, and by implication the author, certainly seem to resemble Schaffer’s bottom feeding fundamentalists.
But the Epicurians should not be treated too harshly. They were, after all, primarily engaged in a peaceful quest for happiness in this life. They had identified perhaps the greatest obstacle to leading a happy life, notably fear of the gods and the accompanying troublesome predisposition towards deep, contemplative ways of thinking. An anti-metaphysical, anti-philosophizing, theologically bland, and some would say, anti-thinking creed called Epicureanism was the result. Few would have predicted that this creed would one day serve as the ontological stalwart of the successful and accurate modern sciences of today. Modern physics can even mathematically describe, at least probabalistically, the dynamics of Epicure’s mysterious micro-physicalist “swerve”. Strong on empirical scientific prediction and mathematical accuracy on one side, a self declared philosophical, ontological desert on the other. It aims to describe it all but can explain nothing. It’s as they say in the classics, you can’t have everything. At least not at the same time.
We return to the question. Is there a bottom fundamental level? We have answered in the affirmative. In so doing we have sided with a kind of metaphysic which, as Schaffer points out, is not particularly palatable for the more reasonable and civilised of people. Painfully it appears that we have excluded ourselves from such a community. Self declared metaphysical pariahs, we must face the dire consequences of our apparently foolhardy prise de position.
However, as we have argued throughout this work, there are two takes on reality, not just one. Hence we have assented to the proposition that there is a fundamental layer. This corresponds to the left side science take on the world, the simple, rather simplistic, abstract, naïvely realistic view of the world.
The right side science take on reality has a different vocation to its uncivilised and rather uncouth partner in crime. Right side science must not merely be content with describing the qualities that a thing has, it must explain what a thing is.
From a historical perspective, we argue that the ancient exponents of the left side take on reality were the Epicureans. In our sometimes desperate attempt to gain some traction for a right side science, we have singled out the Epicurean’s nemesis, the Stoics.
Of immediate concerns to our current discussion is the Stoic view on whether or not there is a fundamental level. The general view amongst the Stoics was that there was no bottom fundamental level. In some way reality was infinitely divisible, at no matter what level. This was also a position held by Leibnitz who made pains to add the nuance of being infinitely divided rather than infinitely divisible.
As for the Stoics, Chrysippus is credited with saying:
A key contribution of this work is to indicate how this genetic code is in fact a generic code applicable to anything. In the appendix the approach is applied to show how the generic code applies to particle physics.
As to answering the two fundamental questions Q1 and Q2, we can claim to have dealt with Q2, but the enigmatic Kantian question Q1 still remains to be answered. Nevertheless, we are starting to see what needs to be done. Rather flippantly we can say that all we have to do is to revamp ancient Stoic physics and logic and make it scientific. Let’s hope that we don’t die trying. So many have.

A later post is more to the point in answering this question,
see The Shape of Knowledge
See also What is Gender?