Why are there Two Hemispheres?

Here, we are talking about the two epistemological hemispheres of left side and right side science, left side knowledge and right side knowledge  There are two kinds of take on reality. There are two kinds of knowledge. We leave implicit that this may also shed a lot of light into the biological arena concerning the two hemispheres, bi-lateralisation of brain function.

Our task is to find a fundamental, succinct answer to the following question:

What are the formal roles of left side and right side rationality?

In other words, exactly why must there be two takes on reality? At the start of writing this book two and a half years ago, the author could not provide a succinct answer to this question. Now he can and is profoundly pleased with the outcome. The answer comes in characterising the two modes of thought. The general consensus is that the left hemisphere is associated with analytical, verbal, linear, and intellectual thought processes and that the right is associated with holistic, spatial, non-linear, and intuitive processes. We now look at our more formal epistemological approach to this question.

Each paradigm is composed of two axes, one logical and the other semantic.

The Left Side Paradigm is based on Abstraction

Left side paradigm logic is based on second order logic. This is the upside. The downside is that left side reasoning is based only on first order semantics.

The Right Side Paradigm is Abstraction Free

Right side paradigm logic is only first order logic and so abstraction is impossible. This is the apparent downside. The upside is that right side reasoning is based on second order semantics.
There it is folks! In a nutshell, the abstraction paradigm underpinning all left side science is based on higher order logic but very flat semantics. The alternative 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 semantics, higher order semantics.
It appears that if you want a science of 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.
The two paradigms cannot be combined. They must forever stand apart. However, they can both be harnessed like the two horses of a chariot. It seems that they take it in turns to provide the working paradigm. How they cooperate and interact with each other falls outside the realm of this work and so is not considered here.
We start with first order logic. The prime example of first order logic comes from the Stoics. 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. Thus, they rejected Abstract generalisations do not exist. Aristotle’s species and genus 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 not is. 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. What is first order semantics? 
Firstly, who uses first order semantics? We blurt out 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.  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 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 that we know 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 have been based on first order semantic and only on first order semantic.
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 versions of space mathematics in mainstream 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 space-time 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 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; the simpler the better, according to Ockham’s razor. 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.
In this post we claimed that there are two kinds of knowledhge. Our more nuanced and more recent post points out that there are in fact two other “subtle” forms of knowledge, an FF type left side knowledge and an MM type on the right side.
The FF type of knowledge has no logic. It involves the interplay of first order and second order semantics, and is situated at the left hand rear part of the epistemological brain. The MM type knowledge has no semantics. It involves the interplay between first and second order logic  and is situated in the right frontal lobe of the epistemological brain. See  The Shape of Knowledge post for details.
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