Stoic Physics, Gödel, and Quantum Mechanics

In this article I briefly present the case that Stoic natural philosophy provides the missing meta-language to finally make sense of quantum mechanics. Modern Stoic writers such as Lawrence C Becker dismiss Stoic physics as an embarrassment. Becker tries to cobble together a hybrid of Stoic ethics with what is essentially a modern version of Epicurean physics. The result is an Epicurean Stoicism, an oxymoron if ever there was one. To Chrysippus, the very core of ethics arises from the fundamental physical principles of matter. If Stoic physics is as Becker describes it merely a “flippant speculation about physical processes,” it is hard to see that Stoic ethics could possibly escape the same epitaph.

In Hellenistic times, the two dominant philosophical schools of thought were the Epicureans and the Stoics. Of course, there were also the Sceptics who sat on the fence advocating suspension of judgment as Sceptics do. Leaving aside the fence sitters, my interest is in the two opposing camps with the Epicureans on one side of the fence and the Stoics on the other. Of the two camps, the easiest to understand is the Epicurean philosophy. The reason is that, in so many ways, the Epicurean worldview corresponds pretty much to that of present day, modern science. Epicurean doctrine differs from modern science in that, like practically all natural philosophy in antiquity, it was non-empirical. Add empirical methodology, the associated quantification, and one ends up grosso modo with scientific methodology resembling modern physics. Charles Sanders Peirce picked up on this when he wrote that the philosophy of John Stuart Mills corresponded almost exactly with that of the Epicureans.

Epicureanism is materialist, determinist, and above all, fundamentally atomist. Epicureanism studies the reality “out there,” a reality that is assumed mind independent and behaving in a completely deterministic way – well almost in a completely deterministic way. Unbridled determinism leaves no place for free will and that poses a problem. To leave some slack for free will, Epicure added a fresh ingredient, He added an escape clause to his atomist, deterministic equation. Certainly, reality could be, in the limit, totally explained by the deterministic motion of the atoms making up the material universe. However, this motion was not totally deterministic. Apparently, every now and then an atom exhibits an imperceptible random “swerve.” If this were not the case, the universe would never have evolved beyond its point of departure. Instead, as a gross accumulation of random Epicurean Swerves, the universe nano-swerved into the state that we see it in today. And there you have it. With a bit of creative elaboration, this worldview could also even embrace Darwin’s theory of evolution. After matter nano-swerves to a certain state of affairs, matter starts micro mutating in such a way as to produce organic compounds, elementary life forms, amoeba, monkeys, and eventually us. We are all the end result of trillions of Epicurean swerves.

Classical nineteenth century physics has no need for the Epicurean Swerve as it saw a completely deterministic form of atomism, much like that of Leucippus and Democritus who preceded Epicure. But modern physics is not classical it is quantum. Unlike classical physics, quantum mechanics has been developed in order to explain its own version of the non-deterministic Epicurean Swerve. According to quantum mechanics, reality “out here” is not deterministic but permeated with its own versions of the Epicurean Swerve, The observed non-deterministic behaviours of nature at the quantum level are sometimes referred to as the “quantum mysteries” or even as examples of quantum “weirdness.” Specifically, this includes the questions of entanglement, Heisenberg’s uncertainty principle, the collapse of the wave function, the mysteries of the two-slot experiment, and Einstein’s comment regarding “spooky action at a distance.”

From Dirac Razor to Stoic Razor

With the advent of quantum mechanics, the chief casualty of classical physics is the concept of the mind independent reality. The isolation of the objective world of objects from the subjective world of the subject is unachievable in practice. Somehow, the lot of object and subject are intimately entwined and interdependent on each other. The most frank admission of the new reality comes from the Copenhagen interpretation of Quantum Mechanics. Expressed by Dirac, often referred to as the Dirac Razor. The Razor sates that the new physics was basically a formal scheme limited to the prediction of experimental results. Anything to say about ontological or other philosophical questions was strictly outside the realm of physics.

In other words, one should avoid sounding silly by claiming that something exists in “the world out.” Instead, the only objective knowledge about what exists is that which exists concurrent with the instant of measurement. This can be thought of as moment when object and subject are both present. Both share the same “now”, so to speak. It is in this idealised instant that we start to glimpse the need for a change of paradigm. Dirac’s Razor declares that the new physics requires a dramatic paradigm shift from classical physics. However, the new physics does not explain the new paradigm. The new physics simply cries out “Shut up and calculate.” Instead of the new physics leading to a clearer and more insightful insight into the nature of reality, it provides the opposite – a mindless, numbing mania of number, measurement and obscure abstraction. The curious public demand more, they demand an explanation.

The missing paradigm can be found by breaking away from the Epicurean style realism and changing philosophical camp to that of the Stoics. The Stoics can be said to have their own version of Dirac’s Razor and the primacy of the moment. The Stoic version states:

Entities in the past or the future do not objectively exist. The only entities that exist are those immediately present with the subject.

The Stoics exploited this principle in their ethics, teaching not to fear anything in the past or the future as such things do not objectively exist and so cannot exercise any powers on the present. The Stoics thus become heroes of the present, mastering the integrity of now.

The same principle underpinned their physics. To the Stoics, entities had to be material and corporeal capable of acting and being acted upon by other material corporeal entities. Of course, all such acting and acting upon only occurs in the present. Implicitly or explicitly present in the present must be the subject. Thus, the nowness involved in the Stoic Razor is that of the subject, the subject in question. This means that the principle must apply to the universe we live in, the universe as subject bathing in its nowness. Moreover, the same principle must also apply to any other subject such as living organisms and of course to human beings, be they slaves or freemen, man, woman, or child. Whether animate or inanimate, all creatures become heroes of their own present, an individual presence that is distinct but harmonious with that of Nature.

I interpret and express the underlying, universal principle of Stoicism in this form, as the Stoic Razor. The Razor is synonymous with the principle of life. The physics and logic of life must be based on this universal principle. It is important to note that present day computer controlled robotic systems violate this principle. The “present” or “nowness” owned by a robot is dominated by pre-programmed instructions. Instead of being a hero of the present and making its own way in the world, the robot is a slave to its past.

The Stoic Razor principle dictates all entities that objectively exist. The principle applies to all organisms, be they animates such as biological life forms, or inanimates like the universe we live in. All such organisms are dictated by this draconian condition. But here is the catch. Here is the rub. The condition is so draconian that it must apply to itself. The principle demands that an organism obey the principle at all costs then, in the same breath, it demands that the same organism must refuse to be dictated by any principle whatsoever outside its immediate presence. What this means is that the principle is non-programmable.

Gödel versus the Stoics

We see here the flip side to what could be called the Gödel Razor. The formalisation of the pre-programmed robot or Turing Machine is in the form of an axiomatic mathematical system. To be non-trivial, the axioms must include those of elementary arithmetic. For any such non-trivial axiomatic system A, Gödel’s first Incompleteness Theorem applies. The incompleteness theorem comes up with its own version of a principle G applied negatively to itself. Expressed as the proposition G:

G: The proposition G cannot be proved.

Now if G can be proven from the axioms A, the mathematical system must be inconsistent as G says that G cannot be proven. On the other hand, if we assume that the system is consistent then there must exist propositions in A which are valid but cannot be proven. Thus, Gödel’s incompleteness theorem effectively states that any consistent axiomatic mathematical system A can be cut down the middle into two sides with what I will call the Gödel Razor. On one side of the razor, the left side say, will be all of theorems provable from A. On the right side of razor is the murky side of the system made up of all the rest of the well-formed formulas of A some of which will be logically valid but unprovable from A. Gödel’s incompleteness theorem says that these valid but unprovable propositions must always exist no matter what. Many consider that Gödel’s incompleteness theorem to be the most important of the twentieth century.

Mathematics is interested in all of the propositions on the left side of the Gödel Razor These are all the provable theorems of the axiomatic system A. In principle, it is possible to program a Turing Machine to mechanically enumerate all of the theorems on the left side of the Gödel Razor. However, on the right side there are also some propositions that are valid. These are valid theorems of the system but are unprovable. Mathematicians may be interested in these unprovable theorems but Gödel has proven them to be out of bounds to the traditional paradigm of mathematics.

This is the point where the ancient Stoics can step in. We, as Stoics, can take a fresh look at proposition G above. We say that a proposition is true if it can be proven from the axioms A. The Stoics refer to this as the contingently true, in this case contingent on the axioms A. The Stoics made a clear distinction between the contingently true and the truth. The contingently true was considered incorporeal. That is fair enough. There is nothing more abstract and incorporeal than being contingent on a set of abstract axioms A. For the Stoics, just as the true was incorporeal, the truth was corporeal. This might seem quite odd. How can a truth be corporeal? What is a good example of a corporeal truth?

The best example I can come up with is none other than Gödel’s central proposition G where we add the proviso that G must be a corporeal truth! Following the Stoics, to be a corporeal truth G must involve material corporeal bodies acting on and being acted upon and all of this taking place in the present. Here we have left the world of abstract mathematics and have entered the world of an organism pushing and shoving, toing and froing, in such a way as to maintain the veracity and hence truth of a fundamental proposition, notably the proposition G. Somehow it seem that for this organism, assuring the truth of G is so important that its life depended upon it. The bodies immediately associated with or owned by the organism must act and be acted upon in such a way that the proposition G is valid. Imagine that this organism is fighting for life. The organism’s prime purpose in life is to assure that the proposition G corresponds to the truth. The organism will do anything within its physical powers for this to be the case. These are desperate times. Moreover, there seems to be no let up. It seems that this preoccupation will endure throughout its life right up to the day it dies. From cradle to the grave, for this organism G is and must be maintained as a fundamental truth. This is a self-justifying truth and hopefully for the organism, it’s going to work.

Now it is time to read the fine print. What does this organism-backed proposition G actually say? G states that G cannot be proven. Now G might conceivably contain some more fine print. This is of no concern to us but might be of some concern to the organism fighting for its particular mode of life. Extra specificity in the proposition G is permissible as long as it is free from any entanglement with nefarious activity outside the organism’s precious nowness.

In Concluding

I hope that I have written graphically enough to convey the central message. The epistemological foundation of present day science and its realist mind independent view of the world is Epicurean in Nature. Quantum Mechanics with its associated quantum mysteries and weirdness throws a spanner in the works. Stoic natural philosophy based on its logic, physics and even its ethics provides a way out of the conundrum. I sketch out how the Stoic paradigm is diametrically opposed to that of axiomatic mathematics. Everything provable in an axiomatic mathematical system can be enumerated by a Turing machine type computer. But Gödel showed that certain truths are out of bounds of formal mathematics. However, I claim that they are not out of bounds to another kind of formalism — that implicit in Stoic natural philosophy. Truths on the out of bounds side of Gödel’s Razor become accessible from within my interpretation of the Stoic paradigm.

The simple message I am trying to convey in this article is that the principle of Stoicism involves a universal life principle that underlies the organisation of all animate life as well as inanimate life like our universe as an organism in its own right. The organisational principle is the opposite to fromal mathematics. Deterministic systems like mathematics aspire to establish a chain of relations from the a priori to the a posteriori, from the axiom to the theorem, from cause to effect. This is the reason of the robot. The reason of life involves an organism hell bent on proving its own self-reliance by NOT being dependant on the a priori. The robot is driven by the a priori; The life form is driven from what it now is not by what it ever was.

However badly I may have explained it, just go back to the Gödel proposition G and see how Gödel handled it for mathematics. Then, take the opposite to that, and you have the Stoic paradigm in a nutshell…

Final Points

Objection:

Robots are pre-programmed and life forms are not. But every biological life form is programmed by its genome is it not? Are not life forms just robots pre-programmed in their DNA?

Response:

I explain elsewhere that the genetic code is not a programming language. Mathematical language and all computer-programming languages encode diachronic structures. The most elementary diachronic structure is the Peano successor function that both Russel et al and Gödel used to generate the natural numbers. I claim that the genetic code is a calculus of physics and logic that is without number and so free of any successor function. No diachronic structure is allowed. It only codes the present. You have to read my book and other work to get a fuller grasp of that though. The genetic code is a non-diachronic coding technology, not a language in the usual sense.

Objection:

Biological life forms are coded in the genetic code. How can the universe be seen as a life form when there is no sign of a genetic code?

Response:

I rename the genetic code the generic
code. Biological life forms I classify as animates. In animates the generic code is expressed as genetic material (DNA or RNA) separate from the functioning material. Organisms like our universe I call inanimates. In inanimates, there is no distinction between genetic material and functional material. It is all the one stuff. The four letter of the generic code, combined in triads correspond to the elementary particles in physics. On my web site, I have constructed an interactive database that shows this correspondence. Paper 4 is a draft of how that all works. (more is in the pipeline) In other words, the Standard Model of particle physics and much more, can be worked out from first principles by a reinvigorated Stoic natural philosophy as a kind of metaphysics.

Objection:

Stoic physics is based on the ancient four-element theory of Empedocles. This theory has long been debunked and replaced by modern physics.

Response:

Quantum physics has come to the same kind of conclusion as Empedocles and, in particular, Heraclitus. There are four kinds of tension, four kinds of fundamental force in particle physics and quantum mechanics as is well accepted. Foursomes occur regularly throughout physics and even in mathematics. In Category Theory, there are four kinds of morphism, epi, mono, bi and iso. Aristotle’s syllogistic logic was the first to provide a logical basis for a four-element aspect to logic. One can explore that in the four terms of syllogistic logic in my Aristotle Engine on my website. However, the Stoic five indemonstrables provide a direct statement of the logic behind the four-element theory of matter. The third syllogism actually corresponds to the fifth Stoic element pneuma and can be used to construct the other four classic elements.

The Scholastics used the letter AEIO to label the four terms of the syllogism. Nature uses the letters ATGC in the genetic cum generic code to code the basic building blocks of life. In my writings I show how this relates to Stoic logic and the ancient four element theory of matter as well as the modern four force theory of physics.

Weirdness In Physics

There are two kinds of physics one classical one weird. Classical physics based on the pleasing common sense notion that there is a realist mind-independent reality “out there,” just waiting to be measured and described by precise, deterministic, mathematical models. For terminological convenience, I will refer to this kind of paradigm as left side. In this paper, I address the complementary opposite, the right side paradigm. The right side leads to physics that is far removed from common sense. This is the natural domain of quantum theory, a domain full of the apparent weirdness such as the quantum mysteries of uncertainty, the wave-particle duality revealed by the double slit experiment, entanglement, and so on. The rational foundations of the classical left side paradigm have reached a relative maturity and demonstrates a high degree of rigour concerning its foundations. The same cannot be said for the right side paradigm. Quantum theory with its apparent counter-intuitive weirdness and many mysteries is in dire need of formalisation. The objective of this paper is to provide such a foundation and even a language based calculus. The calculus provides an alternative to the Standard Model and provides a much more detailed account of the elementary logical structure of matter independent of the need for any empirical observations.

Like Odin, the ancient Norse god of thought and logic, present day physics achieves rational clarity by a simple astuce, that of being one eyed. According to present day orthodoxy, there is only one scientific paradigm. Science is mono-lateral, not bilateral. The paradigm must be fundamentally left side and thus realist. The apparent weirdness thrown up by quantum theory demands a quest for further refinement of existing orthodoxy, not another radically different paradigm switch. There is no right side science. This is not the position taken in this paper. One can still remain true to the one eyed Western tradition illustrated by the monocular Odin, god of reason. All one has to do is to use the other eye but not at the same time of course.

I argue that science must be bilateral. There must be two separate, fundamentally opposed but complementary paradigms.

The idea is far from novel. Bohm argued his own form of bilateralism with his notions of the Explicate and Implicate orders. Dirac saw the two paradigms of physics as one fundamentally temporal and the other as fundamentally spatial. The philosophers see it as an opposition between physics and metaphysics.

The general characteristics of the left side paradigm correspond to what is called the Scientific Method. As well as being fundamentally realist, the methodology is reductionist, atomist, and dualist. If there is going to be a symmetry between the two paradigms, the right side methodology one would expect the right side paradigm to be non-realist, non-atomist, and non-dualist, whatever these terms might eventually mean once formalised.

Underlying these dichotomies between realism and non-realism, dualism and monism and so on, there must be a fundamental dichotomy from which all others arise.

An intuitive, informal idea of the fundamental dichotomy is that between object and subject. Both the left side and right side paradigms embrace this same dichotomy right at the very core of their respective formalisms. However, they treat the subject object dichotomy in quite opposite ways. These two ways to treat the subject object dichotomy establishes a further dichotomy between the left and right paradigms. The difference between the left and right paradigm treatments of the object subject dichotomy is as follows. The left side paradigm articulates the epistemological configuration of the traditional classical sciences.

The left side paradigm starts with the subject-object dualism taken at the macro level that demands a pure realism where the object of study is objectified by eliminating all reference to the subject, which remains forever invisibly off stage. In empirical science, the object of the science is embraced in an environment called the controlled experiment, a subject free laboratory. The subject, in this context, becomes impersonal and has sometimes been referred to operate as the God’s eye view and even the view from nowhere, empirical scientists would probably prefer the interpretation of the view of the objective, dispassionate observer. Traditional mathematics objectifies its each of its problem domains by a controlled rational environment defined by a set of axioms. The subject is nowhere to be seen in axiomatic mathematics, as all mathematical entities are objects. Like in the empirical sciences, the implicit macro level impersonal is invisible in the formalism. Given an axiomatic system A, the only macro dualism in mathematics is the notion of the mathematically dual system A`. However, the allusive subject is nowhere to be seen in A` as, just like system A, the mathematical entities of A’ are all objects. Axiomatic mathematics is a two-headed coin.

The strength of Western culture is to be like Odin, one eyed. But which eye? Right eyed is to be left-brained. That is the way of present day science. The secret towards an integrated scientific view of reality may be to be one eyed but to change from one eye to the other, depending on circumstance or lack thereof. Only to be one eyed at the one time. Present day science is fixed right eyed, hence left brained. The alternative perspective is ignored. What is needed is a bilateral approach to science. Present day science is mono-lateral. Odin was much wiser than that, surely.

Syllogistic Logic

Traditional sciences and mathematics is very “left brained” – abstract, dualist, empirical, atomist, and rely on a rhetorical form of reasoning. In antiquity, the Epicureans priveledged that form of thought. The Stoics favoured a non-dualist, non-atomist, dialectical form of reasoning. When it comes to Aristotle, such a dichotomy is not at all clear cut. 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 had 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.


Figure 1 The four kind of terms. The Scholastics later labelled them with four letters.

The Four Terms and the Left Side

Aristotle’s syllogistic term 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 is involved with propositions expressible in natural language. 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 as real entities. 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.

Both of these aspects, the abstract and static nature of the logic, are characteristics of left side thinking. By default, left side thinking has become synonymous with the modern.

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 in other sections of the blog. Firstly, obtain a pair of oppositions. Employ one opposition to define a left-right dichotomy and the other opposition for the front back structure.


Figure 2 Venn diagrams for the four terms of Aristotle

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 3 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.

Term Logic

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 Aristotle’s 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.

Aristotle’s syllogistic 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.

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 4 (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. One could say that the semantics go out the window and are left trivialised. 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. This allows modern logic to talk about things that are known not to exist, a characterising feature of abstraction.

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 4 (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 of Nature’s alphabet.

 

 

 

Where is the centre of the universe?

This is a post from the Stoic mailing list at Yahoo Groups. It touches on a central tenet of Stoicism.

Jan wrote:

It’s certainly traditional Stoic doctrine that somehow connected with the all-pervading Logos (=Zeus=Nature=Providence=designing fire) is the obligatory law of nature, aka jus nature; the mind of the (human) sage is, according to classical Stoicism, aligned with this law of nature. That’s a bit too mystical for me (although it was convenient enough for the ancient Stoics.)

. Continue reading “Where is the centre of the universe?”

The Shape of Mind

brain two hemispheress

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”

The Shape of Space

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”

The Dawkins Ignorance Hypothesis

socrates

Richard Dawkins’ polemic, The God Delusion, is an excellent example of left side reasoning and so it is not surprising that he presents a worldview that is totally at odds with the main thrust of our project. His polemic has been widely contested on many fronts but one of his core assumptions seems to have escaped criticism. His assumption can be paraphrased as the declaration:

You can’t get knowledge out of ignorance.  Everybody knows that!

From there he goes on to claim that God is nothing more than a placeholder for ignorance. He claims that we get nowhere by labelling our ignorance God. Once again, we could add the implied comment, “…and everybody knows that.”

Dawkins has descended down to the philosophical level of the ranting radio Shock Jock where what is wrong with the world is all so bleeding obvious. Dawkins claims that he is presenting the bleeding obvious and gets quite irate when people can’t seem to understand something that is as plain as the nose on your face. You simply can’t get knowledge out of ignorance, can’t you see that?

In his polemic, he is committed to the street language of the rant that has become so prevalent over recent times. This populist discursive style does not lend itself to considering the more measured and considered aspects of the problematic. However, someone of Dawkins’ education and statue would have come across Kant. In the Critique Kant addressed precisely Dawkins’ question:

How can you get knowledge starting without any a priori knowledge or experience whatsoever?

In other words, how can you get knowledge out of ignorance? For Kant, metaphysics was the science that was supposed to find answers to this question. Many philosophers argue that such a science is impossible. You simple cannot get knowledge out of ignorance. Towards the end, Kant might even have ended up with this view. If these philosophers are right then any notion of metaphysics or God must indeed be quite vacuous.

The key part of the riddle is that everyone knows that you can’t get knowledge out of ignorance. The ignorance at first glance becomes, on closer inspection, a kind of knowledge, particularly if everyone knows it. We find ourselves right in the middle of the Socratic Confession of Ignorance. It is this very paradigm that we use to construct the most awe-inspiring knowledge of all. However, to Dawkins this kind of awe is “silly,” declaring

I’d take the awe of understanding over the awe of ignorance any day.

Hawkins claims that you can’t get anywhere by labelling ignorance God. He particularly riles to the God label, but presumably his claim goes for any label. You won’t get anywhere by labelling ignorance, full stop.

Dawkins is providing a good service for us. He is providing a formal declaration  about ignorance and knowledge. We will call it the Dawkins Ignorance  Hypothesis. It simply states:

It is impossible to obtain knowledge by labelling ignorance.

This statement is meant to be obvious to the audience that Dawkins is addressing, namely the mythical god fearing, banjo plucking, pig farmers of the Appalachian mountains.  However, on closer inspection we see that not only is Dawkins a vehement hater of the Gods, he is also a hater of mathematics. For instance, the core notion of elementary algebra is to obtain knowledge by precsisely “labelling ignorance.” Let x be the unknown in the equation. The value of x is unknown. Solve for x. Presto, knowledge from labelling ignorance.

Now it might seem that we are just being disingenuous here. In fact, we are deadly serious because, in our work, we use this vary technique of labelling ignorance of the most profound kind, in order to reverse engineer the Code. This is exactly the same Code that Dawkins has spent so much of his professional energy on analysing from the bottom up. Our approach is top down. From pure ignorance we can obtain the algebra of knowledge, just as Kant called out for in the Critique.

As developed in our forthcoming book, there are two ways of labelling ignorance, one is with the feminine F and one with the masculine M. F labels the ignorance of the wildcard and M labels this ignorance as being the only singular bit of knowledge capable of dragging us out of this morass. Armed with these two ignorance labels, the four letters of the generic cum genetic code can be constructed.

This gender calculus takes us back to the time of Empedocles and his theory of the four roots or letters. The Stoics interpreted them as the four elements in their physics. For us, this leads to a generic algebra that can be interpreted as the reverse engineering of the genetic code. This genetic code, we call the generic code as it codes more than just the biological. It can code virtually anything. We find this literally awesome. Knowledge can be obtained out of ignorance. Also, whenever and wherever there is an M in the equation, there is the possibility of interpreting it as the finger of God.

The enigmatic key to it all is the Socratic confession of ignorance.

All I know with absolute certainty is that I know nothing with absolute certainty.

Who would think that this provides the ontological building block for the science of the generic? Nature is a little wilier than indicated by the Dawkins Ignorance Hypothesis.

Strife

To Karl Marx, change presents as the history of class struggle. According to Empedocles, change was the outcome of the incessant struggle between the forces of Love and Strife. Love unites the elements together to become all things. On the other hand, Strife brings about the dissolution of the one back into the many. The elements become unmingled, naturally attracted to their like. Division and morselisation intensifies.

Whatever it may be, the class struggle has been settled and it now appears Strife is everywhere and gaining the upper hand every day. Voracious corporate CEO’s, bankers, and money merchants are no longer constrained by social consciousness.  Free to gouge, they turn on their cringing critics, accusing them of a politics of envy. Apparently, the change now is dictated by the forces of greed and envy. Marx’s class struggle has become a fight between Greed and Envy.  All of this takes place in the fractured milieu of rampant, unrestrained, deregulated, Strife.

However, our principle concern is not the Strife that is rampant in our society, but the Strife that is ravaging unchallenged in our educational institutions and particularly in the sciences and mathematics. The historic critics of science and things at large, the Humanities, have disintegrated into enclaves of irrelevance. As like attracts like, scholars run to shelter to form their own hermetic communities on their chosen island of increasingly fractured speciality. Some still nostalgically ruminate over long lost causes. Others attempt to find inspiration in nihilistic Post-Modernist world views. For the rest, there may be meagre pickings at the bottom of the barrel. There must be at least one unsaid word yet to say about someone who once said something somewhat interesting about something or other. Surely.

Meanwhile, blinded by the success of technological revolutions, the sciences march on victorious as each island of specialisation breaks up into even more islands of specialisation creating the greatest oceanic world of knowledge ever known. A consensus develops that all is well. This is all there is. This half-world is the world. There is no other view but this one. This is it.

We pause for a moment to offer our thoughts of condolence to our brilliant and best young minds that are at this very moment being ushered into the half world of present day science, there to be trained to think with half a brain. Cloistered from the distractions of the real world, these young minds are being moulded to provide an army of intellectual technicians trained in the karate of abstract thought. This is the era of the abstraction technician. Abstraction brings together like with like. In the process, the abstraction becomes increasingly stripped of specificity. The supreme abstraction is the vacuous, the vacuosity of Everything. As like attracts like, this is surely leading to Strife.

Strife plays a major role in the traditional left side sciences. For example, axiomatic mathematics has for its very vocation, the creation of Strife. This is because such mathematics is fundamentally based on abstract generalisations, the lumping together of like with like. The advancement of mathematics thus follows a similar trajectory to the all the other left side sciences. It explodes into a myriad of every increasingly specialised islets of specialisation.

 

Work notes for forthcoming book

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.
Postscript
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.