Foundations

Stoic Unification and Landing Leibniz’s Vision of a Qualitative Mathematics Without Number

My photo of the reconstructed Stoa in Athens. Time to reconstruct Stoic philosophy as well.
Abstract

In the advanced stages of composing a groundbreaking book, the author explores a revolutionary intersection of physics and metaphysics, guided by the ancient Stoics’ paradigms and Leibniz’s methodological innovations. This work ambitiously aims to transform metaphysics into a scientific discipline, with a unique focus on unveiling the deep semantics of the genetic code, likening it to a universal “generic code” paralleling physics. By formalizing this code’s semantics, the author envisions constructing a subatomic “periodic table,” revealing profound structures and potentially undiscovered entities, reminiscent of chemistry’s Periodic Table.

<a first section of the rough draft follows>

Contents

Prelude to the Prelude.

Prelude.

Introduction.

Aristotle: two kinds of science.

Kant’s Lament.

Bilateralism and the double logos.

A Note on Hellenistic Philosophy.

Epicurus and Evolution Theory.

The ancient Epicurean-Stoic divide as a modern template.

The Cynic origin of Stoic Thought.

I am in the latter stages of writing an important book. Some preliminary sections will be posted here. The dominant influences on this work are the underlying paradigm of the ancient Stoics, so misunderstood, and the methodology and unfinished project initiated by Leibniz.

The book proposes a revolutionary approach to physics by making metaphysics scientific, with profound implications. One facet involves deciphering the semantics of the genetic code, which parallels physics as a universal “generic code”. Understanding and formalising the deep semantics of this universal code, allows the construction of a subatomic “periodic table” to uncover deep structures and potential undiscovered forms, akin to the Periodic Table in chemistry.

The rationality required diverges from the monologic style of traditional science and analytic philosophy.

This project aims to foster a dialectical, synthetic way of thinking, embracing the concept of “complementary oppositions” often seen in quantum mechanics. Unlike Kant’s antinomies, these oppositions aren’t contradictory but mutually informative. This perspective extends to mathematics, where even conventional math represents just one aspect, the left adjoint of a broader, yet-to-be-revealed mathematical framework.

The tentative title of the book is yet to be decided but may be either “Foundations” or “Substance Theory.” This isn’t about a linear journey from A to B but a nuanced, circular exploration of ideas. Influenced by the universal science of the early Stoics and the methodological approaches of Leibniz, this work seeks to unify logic, ethics, and physics in a novel manner.

Prelude to the Prelude

Our next step is to present the Prelude to this work, but first, we will address a question regarding the role of conventional mathematics in this work. This section effectively becomes a “prelude to the prelude.” Initially, our perspective may seem perplexing, as it aligns with the Stoics on the role of mathematics in the fundamental theory of reality.

The founders of Stoicism lived during a time of intellectual flourishing characterized by vibrant mathematical activity, enriched by the works of Euclid and the great Archimedes. Euclid was a contemporary of Zeno, while Archimedes was a contemporary of Chrysippus. Other notable mathematicians from that era included the polymath Eratosthenes of Cyrene, known for his contributions to geography, mathematics, and astronomy, as well as Apollonius of Perga, a leading figure in Hellenistic geometry who laid the foundation for future developments in astronomy, optics, and mechanics. The Stoics would have been well-acquainted with this rich mathematical tapestry of their times.

However, despite, or perhaps because of, this mathematical richness, the Stoics rejected mathematics as playing a fundamental role in their foundational science. Their reasoning was simple: mathematics lacked wisdom. While they acknowledged its usefulness and technicality, they preferred logic as a means to make qualitative judgments and discern what was good, embracing wisdom. These ethical judgments were not viewed as mathematical problems but rather as resolvable challenges of logic.

The Stoics grappled with the same question that Bertrand Russell would later confront in modern times: which is more foundational, mathematics or logic? Russell’s aim in the Principia Mathematica was to demonstrate that all of mathematics could be reduced to logic. However, a sticking point emerged. Gödel, in his proof of the Incompleteness Theorem, devised a clever, albeit totally impractical, method to convert logical language into straightforward analytical mathematics, expressed in the language of numbers. The construct was essential for proving his incompleteness theorem. From Gödel’s work, the mantra of incompleteness arose: “there are true statements that can never be proven.” The conclusion from the incompleteness theorem was that Russell’s hope of completely reducing all mathematics to logic could never be realized.

This result has long been a source of wonder and speculation. It suggests an insurmountable limit to distinguishing the true from the false. Essentially, Russell and Gödel’s modern inquiries into the mathematics-logic relationship end up in a bit of a quandary, as they are rooted in an extremely limited conception of logic. Modern mathematical logic is situated within the realm of rhetoric, representing a technical form of linear, monovocal discourse. At its core, rhetorical rationality relies on syllogistic reasoning, where, as Aristotle stated, a conclusion follows from two premises. While rhetoric speaks with one voice, dialectic involves two and moves in the opposite orientation to rhetoric.

In modern times, it is often naively believed that dialectic operates on a similar paradigm to syllogistic two-to-one reasoning, except that the two propositions at the outset are contradictory rather than both true. Dialectic, following the thesis-antithesis-synthesis model accredited to Hegel, is also oversimplified and, in the limit, merely a rhetorical sham of the dialectic that the ancient Stoics would have engaged with. We will revisit this topic later, but for now, think of dialectic as moving in the opposite direction of syllogistic reasoning, from the One to the self-confirming Two. Critics may label this type of reasoning as akin to the realm of self-fulfilling prophecies, often found in religious doctrine. Nonetheless, in this work, we will delve into the significance of dialectic logic and its application in physics.

Prelude

It’s an incredible story, one that continues to amaze me. A close family friend, who has been a source of encouragement for decades, insists that the time has come to share it without delay. No more weaving intricate narratives that lead to this and that; it’s time to cut to the chase and lay it all out.

This work delves into the fundamental science of lifeforms, defining them as self-organizing substances capable of sustaining their own existence. We explore the principle that self-organization relies on a code that encapsulates a substance’s essence. While the organized substance may undergo changes, its organizing code remains a constant statement of its existential core. Remarkably, the code itself must be a substance to interact effectively with the substance it organizes. In biological organisms, the organizing code shapes and manages the substance, and we coin the term ‘bio-orgs’ for such lifeforms. There exists another kind, where the code substance and the organized substance are one and the same—these are ‘mono-orgs.’ In our work, we boldly propose that the universe we inhabit is a mono-org.

Our approach to advancing the life sciences centres on none other than the genetic code itself. Despite the continuous evolution of all life forms, the universal language that codes them remains absolutely unchanged over billions of years. If one seeks a fundamental means to explore and elucidate life, the answer lies in this universal language of Nature. The Central Dogma of biochemistry suggests that the genetic code is a mere transcription language. However, our project challenges this dogma. Our central goal is to reverse engineer the code from first principles, making it the operational calculus for explaining the organizational principles of life. This broad concept is not new; it was envisioned by Leibniz over three centuries ago. In a famous passage, he outlined his dream of developing a geometric algebra, free from numbers and based on just a few letters, to explain the natural form of things.

One could say that Leibniz anticipated the genetic code. However, his vision extended further; he believed this geometric algebra would possess logico-geometric semantics, making it truly revolutionary. Furthermore, he claimed that the same geometric algebra would explain not only animate but also inanimate entities. Today, we understand that the organizing “genetic” material in any biological organism is distinct from the functional material. In the case of inanimate ‘organisms’ like our universe, there seems to be no observable distinction between organizing substance and the organized. Thus, if Leibniz’s vision holds true for the inanimate, then the elementary particles of Particle Physics should be elegantly explained in terms of the four-letter algebra of the genetic code—now serving as a truly universal generic code. In the sphere of non- biological being, the organizing code material and the organized material are one and the same.

It may be that indeed biological and non-biological substance organisation are both code based, but this doesn’t forcibly mean that the code is the same in both cases. Our project provides argumentation for that the code actually is the same in both cases. Like the organic bio-orgs, the substance making up the universe itself, the mono-org, employs the same four letter code for their self-sustainment. That mono-org and bio-org organisation are both code based is probably easier to countenance than the necessity that the codes are instances of the same calculus, the same code, the same inner existential semantics. In this one-code scenario, biological life is not seen as a mere parasite that somehow randomly happened out of dead matter by mere chance. Rather, bio-orgs emerge from the host mono-org as a natural consequence. The heavy lifting entailed in mere being is accomplished by emergence of the mono-org. Being then becomes a fait accompli. The mono-org in isolation encompasses a contradiction. On the one hand, the mono-org is an expression of the code necessary for self-sustained existence, on the other hand stands the thing existing. The thing and its code are the same thing: Is this thing the coder or the coded? There is a contradiction here, a tension that calls out to be resolved.

One could launch into an evolution narrative of how somehow biological substance emerged from the mono-org to give rise to an explosion of bio-orgs. Alternatively, there is the possibility of an explanation based on emergence. This would involve some kind of dialectical argument where the internal contradiction between the thing as code, and the thing as body gives rise to a more elaborate form of organisation where the Code-Body mono-org entity separates into individual substances to form a higher symmetry other than just pure identity. Bio-orgs emerge as expressions of this new opposition between code and the coded. The mono-org, based on its Code-Body identity provides the possibility, the inevitability even, of the bio-org breaking free from the Cody-Body bondage. Code takes on other vocations than that of the pure substantial existence of the mono-org. However, the emergence of bio-orgs from the mono-org makes up a different topic to the present and will not be directly addressed in this in work. We will concentrate on the mono-org, its code and its body. Broadly speaking, our subject matter encompasses physics but from an entirely different perspective to traditional physics as will be seen.

Introduction

Aristotle: two kinds of science

Aristotle wrote that there were two kinds of science. There were the special sciences (to which we should include modern axiomatic mathematics) each characterised by having an object of study with determined genus. The other kind of science, which became known as metaphysics, can be characterised by an object of study of undetermined genus. But Aristotle avoided this obvious symmetry.

Aristotle actually pulled a sleight of hand here and instead of employing the notion of studying an object of undetermined genus as object of study he played on the noun/verb ambiguity of “being.” He replaced noun with verb. Instead of sticking to the essence of the matter he chose process. The object of study for the First Science was replaced by an amorphous reference to being qua being, In so doing he constructed a wayward path into the wilderness of an ontology free philosophical wasteland  that has waylaid many a thinker across the ages and particularly in the modern era.

 

Nevertheless, Aristotle was the first to clearly draw a demarcation line between the special sciences and metaphysics. His demarcation would have been clearer if only he had left Anaximander’s notion of apeiron, the totally unbounded and unqualified as the object of study for the First Science. But faced with the choice between a process philosophy versus an essence philosophy approach he, like so many of the moderns today, he chose the former. We remain true to the latter. We see metaphysics studying the thing of undetermined genus. We will be referred to it as the generic entity. The generic entity, so characterised, will thus be devoid of determined qualification including determined cardinality. Kant referred to the generic entity as the “thing in Itself”. The science, traditionally called metaphysics, could also be referred to as Generic Science, the generic science of the sciences, the science of the generic.

Since Aristotle, the special sciences have flourished but the same cannot be said of metaphysics. In early modern times, Kant made impassioned calls for that this science should see the light of day. In the Critique, he analyses the problem and the possibility of resolution. Later in Prolegomena to any Future Metaphysics his frustration becomes palpable. We cite the whole passage as the concerns of my project are the same as that so starkly painted by Kant in an important passage that we highlight as Kant’s Lament:zAfter several centuries,

Kant’s Lament

“If it be a science, how comes it that it cannot, like other sciences, obtain universal and permanent recognition? If not, how can it maintain its pretensions, and keep the human mind in suspense with hopes, never ceasing, yet never fulfilled? Whether then we demonstrate our knowledge or our ignorance in this field, we must come once for all to a definite conclusion respecting the nature of this so-called science, which cannot possibly remain on its present footing. It seems almost ridiculous, while every other science is continually advancing, that in this, which pretends to be Wisdom incarnate, for whose oracle every one inquires, we should constantly move round the same spot, without gaining a single step. And so its followers having melted away, we do not find men confident of their ability to shine in other sciences venturing their reputation here, where everybody, however ignorant in other matters, may deliver a final verdict, as in this domain there is as yet no standard weight and measure to distinguish sound knowledge from shallow talk.” (Kant, 1783)

Kant’s lament remains still unanswered. In fact, after more than two millennia the future for developing metaphysics as a science seem to be just as bleak as ever. However, there is a caveat. We raise the possibility that a great advance on the problem had already been achieved by the ancient Stoics working in the direct wake of Plato and Aristotle. To definitively prove such an achievement is difficult as there are no extant works from the founders of Stoic philosophy. Thus, a reconstruction of the Stoic doctrine is necessary.

Bilateralism and the double logos

The bilateral take on rationality is a persistent notion in this project. The traditional sciences employ a monolateral methodology combined with multilateral perspective of competing ideas and even “multiple worlds”. The bilateral perspective emphasizes complementary oppositions. Complementarity and bilateralism will be a constant architectural feature of our project and plays out at all levels of investigation, no matter what the scale. Even the relationship between left and right logos can be seen as a complementary opposition, albeit only comprehensible from the right side perspective.

A Note on Hellenistic Philosophy

For the special sciences, knowledge is presented in the form of theory and models. Theory development is reasonably straight forward as, for each science, there is a determined scientific object to conceptualize. One has a fair idea of what one is talking about. In metaphysics, the distinction between theory and object becomes problematic. Heraclitus introduced the term logos which, for our purposes, can be thought of as a generic expression of reality ranging over the gamut of rational substance to rational principle, ranging across any scale from the cosmic to the nano. The Stoics spoke of the double logos consisting of the logos prophorikos and logos endiathetos. At the individual, non-cosmic scale, the logos prophorikos was expressed in ‘uttered’ language and was considered deficient. The logos endiathetos was considered perfect but linguistically mute restricted to Plato’s notion of an internal language and even as the language of thought. (Chiesa, 1992);(Kamesar, 2004) Invoking the bilateral brain architecture metaphor, the prophorikos logos would merit interpretation as the left side logos, while the endiathetos logos would correspond to the right side logos. Similar to the two hemispheres of the brain, language is a speciality of the left side, while the right side is essentially mute, at least as far as any external communication is concerned. Plato attributed to it an internal language whereby the soul could communicate with itself.
In Hellenistic times, Epicureanism and Stoicism emerged as the two dominant philosophies. Both shared a foundational belief in materialism but diverged significantly in other aspects. Epicureanism, more aligned with modern scientific thought, was based on deterministic atomism. In this view, the universe consisted solely of matter, composed of countless atoms whose movements determined all change. However, Epicurus introduced a slight deviation from strict determinism with his concept of the ‘Swerve’ – a random, imperceptible deviation in an atom’s path, allowing for the unpredictability that led to the universe’s evolution.

Epicureanism, like much ancient science, lacked an empirical methodology. Yet, integrating such an approach would make Epicureanism quite compatible with modern scientific views, including its resemblance to quantum mechanics’ probabilistic determinism.

In contrast, Stoicism, viewed through a modern lens, must confront both ancient rivals and contemporary scientific paradigms. Many modern Stoic philosophers, recognizing the challenge, shy away from defending ancient Stoic physics. Some, like Becker, view Stoic physics with disdain, seeing it as an embarrassment to the philosophy.

My interpretation of Stoicism might not align with the orthodox view and could contradict the prevailing understanding. Nevertheless, the core of my argument should resonate, at least to some degree, with most Stoic philosophers.

Diverging from Epicureanism in nearly every paradigmatic aspect, Stoicism rejects dualism, atomism, abstraction, categorization, and realism. To grasp Stoicism’s unifying paradigm, understanding its anti-realist stance is crucial. Unlike Epicureanism and its scientific counterparts, which are grounded in realism and a dualistic worldview of independent objects existing ‘out there,’ observed by an objective, impersonal observer, Stoicism adopts a non-realist perspective based on different criteria of existence as will be developed in this work.

Epicurus and Evolution Theory

In passing, one could note that the Epicurus Swerve theory of change could be interpreted as an early version of the modern theory of evolution where atomic Swerve translates into genetic mutation. What is interesting here is that the Epicureans, like the Stoics, apply the same paradigm right across, ranging from the cosmic to the small.

This observation about the potential parallel between Epicurus’ Swerve and the modern concept of genetic mutation in evolutionary theory is an intriguing one. It highlights the often-overlooked depth and foresight in ancient philosophical thought, particularly in its application to natural phenomena.

Epicurus’ notion of the Swerve, as a random deviation in an atom’s trajectory, introduces an element of unpredictability into an otherwise deterministic universe. This unpredictability could be seen as a conceptual precursor to the random genetic mutations that are a key mechanism in the theory of evolution. In evolutionary biology, these random mutations in genetic material can lead to variations within populations, some of which confer advantages in survival and reproduction. Over time, these advantageous traits become more prevalent, leading to the evolution of species.

The parallel one can draw between the Swerve and genetic mutation is compelling in that both concepts introduce an element of chance or randomness that can lead to significant changes over time. In the Epicurean framework, the Swerve could be seen as the fundamental driver of change in the universe, from the macroscopic level of cosmic events down to the microscopic world of atoms. Similarly, in evolutionary theory, random genetic mutations drive the diversification and adaptation of life forms.

Furthermore, the application of a consistent paradigm across different scales, from the cosmic to the small, as noted in both Epicurean and Stoic thought, reflects a holistic approach to understanding the universe. This approach resonates with contemporary scientific efforts to find unifying principles that can explain phenomena at all levels, from the behaviour of galaxies to the workings of subatomic particles.

Our reference to this parallel not only underscores the historical relevance of ancient philosophies like Epicureanism but also invites a deeper consideration of how these ancient ideas might inform or be reflected in modern scientific concepts. The potential link between the Swerve and genetic mutation is a thought-provoking example of how the ideas of ancient thinkers, when viewed through a modern lens, can provide fresh insights and perspectives on the nature of change and the mechanisms driving the evolution of the universe and life itself.

The ancient Epicurean-Stoic divide as a modern template

In this work, we explore the ancient dichotomy between Epicureanism and Stoicism as a foundational precursor to a contemporary intellectual landscape. On one side of this modern interpretation lies empirical science, which draws parallels to Epicureanism with its emphasis on observation, experimentation, and a deterministic understanding of the universe. This approach aligns with the Epicurean focus on atomism and the material aspects of existence, underpinned by observation and logical deduction although Epicurus was oblivious to any empirical methodology.  On the other side, we propose a yet-to-be-fully-developed unifying universal science, inspired by Stoic philosophy. This novel approach seeks to transcend the empirical and deterministic constraints of Epicurean thought, aiming to integrate a more holistic and interconnected view of the universe. Inherent in this Stoic-inspired framework is a focus on the interconnectedness of all things, an emphasis on the underlying principles that govern the cosmos, and a perspective that considers both the physical and metaphysical aspects of reality. This Stoic-inspired universal science endeavours to bridge the gap between the material and the abstract, between the observed phenomena and the underlying principles. It challenges the conventional separation of observer and observed, seeking a more integrated and comprehensive understanding of the universe. This approach resonates with the Stoic belief in a rational, ordered cosmos, where every component is part of a greater whole. By juxtaposing these two philosophical schools as precursors to a modern intellectual debate, we aim to shed light on the enduring relevance of ancient thought in contemporary science and philosophy. This exploration is not just a historical inquiry but a forward-looking endeavour, seeking to inspire new ways of understanding and exploring the world around us. Through this lens, we aim to contribute to the development of a more holistic and integrated approach to science, one compatible with the empirical but offering a deeper appreciation of the interconnectedness and complexity of the universe.

The Cynic origin of Stoic Thought

The classical obstacle to developing natural philosophy as an exact science is the nature of its object of study. One inevitably finds oneself hopelessly confronted with the thing without any form of determined qualification whatever, and one is tasked with wringing some sort of science out of it, and from it alone.  Hopeless folly? Pre-Socratic philosophy thought differently and even sketched out pre-scientific hunches of what was involved.  Doubt resurfaced with Aristotle, but the Stoics who followed turned back to pre-Socratic origins to revive the quest for formalising natural philosophy as a unified and unifying science.   

No progress on this age-old problem is possible without a radical change in mindset. To this purpose, our attention turns to Zeno of Citium, the first of the three founders of Stoicism. Zeno started his philosophical education in Athens under Crates of Thebes, a famous Cynic philosopher of the time and a student of the legendary Diogenes, the unattested master and practitioner of pure unadulterated shamelessness. The Cynics emphasised the ethics of rejecting the sanctity of social rules and norms.  Instead, they argued that one should live according to nature, an adage that Stoicism borrowed. 

Taking a schematic stance on the matter, we state the Cynic dictum:

Cynic dictum P1. There are no absolute principles.

The Stoic philosophy that Zeno was destined to be one of the three founders stays true to its Cynic origins but adds a fundamental second proposition:

Stoic dictum P2. The Cynic proposition P1 is an absolute truth.

The difference with Zeno’s Stoic philosophy was that the Cynic dictum became nothing other than an absolute principle in its own right, a negation of the Cynic negation – the classic “negation of the negation”. To achieve such a radical step, Zeno needed a bit more nuanced intellectual chattels than the counter=culture preoccupations of the Cynics. Perhaps Zeno’s time spent studying with the Megarian logicians and dialecticians would have provided him with the necessary insight to recast Cynic rejection of a rule-based society to a rejection of a rule-based natural philosophy. In its place would be a universal natural philosophical understanding of the whole cosmos, unrestricted in scope or scale,

[Some other rough drafts might be added from time to time, before publication of the whole work.]

©Copyright 2024 Douglas J Huntington Moore

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