The Structure of Causality in the Cosmos

 

"The Structure of Causality in the Cosmos: A Philosophical and Scientific Analysis of the Necessary Cause"

Part One

Based on critical conversations with AIs

Introduction

The exploration of causality lies at the very heart of philosophy and science, serving as a conceptual bridge that links metaphysical inquiry with empirical research. This study aims to delve into the complex web of causality, a concept that has played a crucial role in philosophical thought and scientific development throughout history. Causality, understood as the relationship between causes and effects, underpins our understanding of the universe and our ability to interact with it. However, despite its omnipresence in our explanations and models of the world, causality continues to pose profound questions about the nature of reality and our capacity to know it.

The motivation for this essay lies in the need to clarify and defend the idea of a necessary cause within the vast framework of causality, especially in the context of contemporary advances in theoretical physics and cosmology. As science broadens our horizon of understanding, from the intricate fabric of space-time to the fundamental forces shaping the cosmos, a unique opportunity arises to revisit and deepen the philosophical concept of the necessary cause and its role in the origin and structure of the universe.

The aim of this essay is twofold. First, it seeks to establish a solid and compelling defense of the existence and relevance of a necessary cause as the foundation of the causal web of the universe, appealing both to logical and mathematical demonstrations and to the available scientific evidence. Second, the study aspires to illustrate how the intersection between the philosophy of causality and physical science can enrich our understanding of the cosmos, providing a more coherent and unified perspective on the foundations of reality.

To address these ambitions, the essay delves into the relationship between causality and the laws of physics, with special attention to the Standard Model of particle physics and the Big Bang theories, exploring how these scientific frameworks align with or challenge traditional philosophical notions of causality. By integrating philosophical and scientific perspectives, this study not only seeks to shed light on the underlying causal structure of the universe but also to reflect on the implications of this understanding for the philosophy of science.

Thus, this essay ventures into the fascinating crossroads of philosophy and science, in the hope of contributing to the ongoing dialogue between these disciplines and offering a renewed vision on one of the most fundamental and enduring concepts in the human endeavor to understand the world in which we live.

Chapter 1: Fundamentals of Causality

Definition of Key Concepts

Causality: At its essence, causality describes the relationship between events or states where one, the cause, induces the occurrence of another, the effect. This fundamental link is cornerstone in constructing our understanding of the universe, allowing us to predict and explain phenomena within a logical and empirical framework.

Necessary Cause: A necessary cause refers to one without which its effect could not be. This concept introduces a hierarchy in the chain of events, signaling certain links as indispensable for the manifestation of certain states or processes in the universe.

Essentiality and Actuality: These attributes complement our understanding of causality. Essentiality suggests that the relationship between cause and effect is intrinsic and not merely accidental, while actuality refers to the effective operation of such a relationship at the moment the effect occurs.

Historical Review of Causality in Philosophy and Science

From Aristotle, who classified causes into four types (material, formal, efficient, and final), to David Hume, who questioned our ability to perceive causality directly, the notion of causality has been central in philosophy. Kant later attempted to resolve Hume's skepticism by proposing causality as a necessary a priori category for experience.

In science, the Newtonian view of the universe as a mechanical system governed by universal causal laws dominated until the 20th century. The emergence of the theory of relativity and quantum mechanics introduced complexities in this view, suggesting that causal relationships at the fundamental level might not conform to our classical intuitions.

The Newtonian view of the universe, established by Isaac Newton in the 17th century, proposed a universe governed by universal mathematical laws, where each event has a clear and predictable cause, and where causality operates in a linear and deterministic manner. This perspective allowed for mechanical explanations of natural phenomena, from the movement of planets to the fall of an apple, grounding the idea of a cosmos as a giant clock, whose parts interact in predictable harmony according to fixed causal principles.

However, at the beginning of the 20th century, this view began to be profoundly challenged by two scientific revolutions: the theory of relativity, formulated by Albert Einstein, and quantum mechanics, developed by scientists such as Max Planck, Niels Bohr, Werner Heisenberg, and Erwin Schrödinger. Both theories radically transformed our understanding of causality in the universe.

The theory of relativity, especially general relativity, rethinks the nature of space and time, merging them into a single entity: space-time. Gravitation is understood not as a "force" in the Newtonian sense but as the effect of the curvature of space-time caused by mass and energy. This conception introduces a more complex causality, where the geometry of the universe itself responds to its mass-energy content, and vice versa. Causality, under this light, is no longer limited to linear relationships between discrete events but implies the dynamic structure of space-time.

On the other hand, quantum mechanics revealed a subatomic world where uncertainty and probability play fundamental roles. The Heisenberg uncertainty principle, for example, establishes intrinsic limits to our ability to simultaneously know certain properties of particles, such as their position and velocity. This suggests that at the most fundamental level, events are not deterministic but probabilistic. Causality, in this context, cannot be understood in terms of linear chains and univocal cause and effect but must incorporate notions of superposition, entanglement, and wave function collapse, where "causes" and "effects" enter a more intricate and less intuitive weave.

These theories not only expand our understanding of the universe but also challenge our intuition about how causality works. The idea of a deterministic and predictable universe gives way to a reality where causality manifests in ways that transcend simple sequentiality and where absolute certainty is replaced by inherent probabilities. This paradigm shift forces us to reconsider not only how we understand events and processes in the universe but also how we conceptualize the very notion of causality.

In summary, the evolution of science from Newtonian physics to the theory of relativity and quantum mechanics illustrates a profound shift in our understanding of causality. We are faced with the challenge of integrating these complexities into our conceptual framework, a challenge that is not only scientific but also deeply philosophical. This dialogue between science and philosophy is crucial for developing a richer and more nuanced understanding of causality in the universe.

The Causality in Contemporary Physics: The Standard Model and the Big Bang Cosmology

The Standard Model of particle physics has succeeded in unifying three of the four known fundamental forces under a coherent theoretical framework that describes elementary particles and their interactions. Despite its success, the model does not include gravity and leaves unanswered questions about dark matter, dark energy, and the matter-antimatter asymmetry, suggesting that our understanding of causality at this level is still incomplete.

The Big Bang cosmology, on the other hand, provides a narrative about the origin and evolution of the universe. The notion of an initial singularity, although challenging from the perspective of classical causality, suggests a defined starting point for the cosmos. This theory, along with cosmic inflation and the large-scale structure of the universe, raises fundamental questions about the initial cause and the initial conditions of the universe.

These areas of contemporary physics, while advancing our understanding of the universe, also invite us to reflect deeply on the nature of causality.

I propose in this work to demonstrate that the non-Newtonian view of physics, represented mainly by the theory of relativity and quantum mechanics, is not necessarily incompatible with the theses on causality that are going to be defended. Instead, it may offer a broader and deeper perspective on how causality manifests in the universe.

The interaction between these scientific theories and the philosophy of causality is rich in implications for both metaphysics and epistemology, offering fertile ground for exploring the fundamental structure of reality and our place in it. In the following chapter, I will not only review these foundations but also prepare the ground for a more detailed discussion on how contemporary theories of physics can be reconciled with, or challenge, the position I defend on a form of causality that is especially important for explaining notable aspects of the observable real world.

Chapter 2: The Standard Model, the Big Bang, and Causality

Exploration of the Fundamental Forces and Their Role in the Causal Model

The Standard Model stands as a backbone in particle physics, providing a theoretical framework that describes with great precision the fundamental particles and the forces that mediate between them. This model encompasses three of the four known forces: electromagnetic, strong nuclear, and weak nuclear, each governed by its respective set of mediating particles, known as gauge bosons. Although gravitation, described by general relativity, remains outside this framework, attempts to integrate it into a theory of everything represent one of the most active fronts in physical research.

The fundamental forces outline the skeleton upon which the dynamics of the cosmos is articulated. From the confines of elementary particles to the vastness of galaxies, these forces not only dictate the structure of the universe but also direct its temporal evolution. Causality, in this context, manifests through the interaction of these forces, establishing an inescapable nexus between cause and effect throughout the fabric of the cosmos.

Analysis of the Big Bang as the Origin Point and Its Implication for Causality

The Big Bang theory presents itself as the dominant narrative about the origin of the universe, proposing an extremely dense and hot beginning that marks the birth of space-time. This initial moment, often conceptualized as a singularity, establishes the initial conditions from which the universe has followed a course of expansion and cooling, leading to the formation of complex structures.

This starting point offers a basis upon which to reflect on universal causality. By conceiving the Big Bang as the foundational event, it suggests that all causal chains find their root in this moment, underscoring the idea of a universe whose complexity and order emerge from precise initial conditions. This approach, however, also raises fundamental questions: What is the nature of causality at a point where conventional physical laws collapse? How do the fundamental forces articulate in this initial context to shape the universe?

Discussion on the Singularity and Current Limitations of Scientific Understanding

The singularity associated with the Big Bang represents one of the greatest enigmas in cosmology and theoretical physics. At this point, energy densities and space-time curvature become theoretically infinite, and the conceptual framework of physics as we know it is challenged. This limit to our understanding points to the need for new theories that can coherently describe the universe at these extremes.

Efforts to overcome these limitations have led to the exploration of theories such as loop quantum gravity and string theory, which seek to reconcile general relativity with the principles of quantum mechanics. These approaches not only aspire to provide a more complete description of the universe in its primordial state but also to offer a new vision of causality, one where the very fabric of space-time and fundamental interactions are intertwined in ways yet to be discovered.

In conclusion, the Standard Model and the Big Bang theory outline a panorama where causality manifests at the very heart of existence, from subatomic interactions to the expansion of the cosmos. However, the Big Bang singularity and unanswered questions about the integration of gravitation into the Standard Model highlight the current frontiers of our understanding, inviting ongoing philosophical and scientific reflection on the nature of causality in the fabric of the universe. This chapter, by delving into these issues, not only aims to shed light on the foundations of contemporary physics but also to inspire a deeper inquiry into the causal framework that sustains reality itself.

Chapter 3: Logical and Mathematical Demonstrations on the Finiteness of Causes

Causality and Causal Patterns

In our quest to understand the fabric weaving the universe, we encounter a theoretical model of causality defined by patterns of causes that are essentially and actually subordinate in the causation of the effect. This model not only provides a scheme for interpreting well-studied phenomena in the cosmos but also invites us to consider each occurrence as part of a finite and deliberately structured causal series. A necessary causal series, according to this framework, is one without which the effect could not manifest; it is the thread that, if not present, would unravel the very fabric of reality. This necessity is accompanied by essentiality, a principle asserting that the relationship between cause and effect is so intrinsic that one cannot be conceived without the other, as if they were bound by an indissoluble tie.

The actuality of this relationship reinforces the idea that these causes are not mere possibilities or potentialities awaiting activation, but effective realities acting here and now. Such is the causality that operates when the brush creates a painting governed or directed by the skeletal-muscular action of the painter's limbs, which in turn is governed by nervous and brain activity, which would be executing certain higher mental processes.

Delving into conceptual analysis, we understand that a cause is any element or event capable of producing another. This definition encompasses a vast range of relationships, from the simple interaction between molecules to the complex processes governing the formation of galaxies. The causal chain, or series, is revealed as an ordered sequence of such events, where each step is a consequence of the previous one, drawing a continuous line from the beginning to the present effect. This concept finds resonance in stellar evolution, where an interstellar gas cloud, through a series of steps carefully orchestrated by the laws of physics, can give rise to a star.

The causal pattern emerges as the recurrent structure within these relationships, a scheme that repeats and can be discerned through observation and analysis. An emblematic example of a causal pattern can be observed in photosynthesis, a process that, following a precise sequence of steps, transforms sunlight, water, and carbon dioxide into oxygen and glucose, sustaining life on Earth.

Finally, the classification of causal patterns allows us to distinguish between those that are non-essential, possibly marked by chance or contingency, and those that are essentially subordinate, where the connection between cause and effect is direct and necessary. While the climate may be an example of the former type, where variability and chance play significant roles, the law of conservation of energy exemplifies the latter, presenting a fundamental and predictable cause-effect relationship in all physical processes.

This theoretical model, by focusing on necessity, essentiality, and actuality of causality, offers a lens through which we can examine the cosmos in its complexity. From the physics that describes elementary particles to the biology that explores life's mechanisms, this framework provides us with a basis for unraveling the deepest mysteries of the universe, reminding us that, in every corner of reality, a story of causality awaits to be told.

This model opens the doors to a universe where causality is the protagonist behind the curtain of every phenomenon, from subatomic particles to the complexities of consciousness. In physics, we observe this causality in the relationship between matter and energy, encapsulated in Einstein's famous equation, E=mc², which tells us how mass can be converted into energy and vice versa, under precise conditions. On the vast stage of astrophysics, the formation of black holes from dying stars is a testament to essential and actual causes, where gravity plays an inescapable role in their birth.

Exploring the Earth beneath our feet, geology shows us how plate tectonics, moving slowly but inexorably, shapes the surface of our planet, causing earthquakes and forming mountains in a millennia-long process. In the realm of biology, the mechanism of evolution by natural selection illustrates how genetic changes, seemingly random, can, under the pressure of the environment, give rise to the diversity of life that adorns our world, linking all species in a common genealogical tree that stretches back to the origins of life itself.

Venturing into the complexity of the human mind, neurology reveals how neurons, through precise patterns of activation and connection, give rise to thoughts, emotions, and consciousness. This intricate dance of electrical and chemical signals, governed by the laws of biology and physics, shows essential and actual causality in its most sophisticated form, connecting the workings of tiny nerve cells with the subjective experience of reality.

Each of these examples, drawn from different fields of scientific knowledge, highlights inductively the generality of the theoretical model of causality proposed. However, the aim of this essay transcends the simple inductive study of these manifestations. Certainly, it would serve to justify its practical applicability in the study and explanation of natural phenomena, but we seek not merely to catalog examples that fit our model; rather, we propose to deduce, from the very concept of a pattern of necessary, essential, and actual causes, the profound implications this has for our understanding of the cosmos. Through rigorous deductive analysis, we aspire to demonstrate that this model not only fits with the actual facts and their scientific explanations but provides a solid foundation upon which a more cohesive and profound understanding of the reality that surrounds us can be built.

Analytical Clarification

As we delve into an analytic-deductive analysis of causality and its manifestations in the universe, it becomes imperative to clarify and explicitly define the central concepts that underpin our theoretical model: cause, causal series, causal pattern, and their classifications. This conceptual precision is essential, as it allows us to construct and follow a rigorous argumentative line without falling into ambiguities or misunderstandings about the meaning and scope of the terms used. By doing so, we ensure that our deductions and conclusions are based on a solid and coherent foundation, allowing for a deep and systematic exploration of the causal structure underlying the phenomena of the cosmos.

Cause is understood as that element or event that, upon its occurrence, precipitates the occurrence of another. This cause-effect link is the cornerstone of our scientific and everyday explanations and understandings of the world. For example, the impact of a meteorite on Earth can cause a crater, or the pressure applied to a switch can turn on a light. In each case, the cause precedes and precipitates an observable effect, illustrating a direct and verifiable relationship between two events.

This relationship extends into what we call a chain or causal series, a sequence of events in which each occurrence is the direct result of the previous one. These chains are fundamental to understanding complex processes such as the development of an organism from a single cell or the sequence of chemical reactions that fuel life on our planet. The causal chain, being finite, suggests a beginning and, possibly, an end, marking a journey that can be traced and understood.

Within this framework, the concept of a causal pattern emerges, referring to the recurrent structure observed in causal relationships. These patterns can be identified in natural phenomena, such as the water cycles or the seasons, where a specific set of conditions and events produces predictable and repetitive outcomes. These patterns not only allow us to predict future events based on present or past conditions but also, in many cases, intervene in these cycles in such a way that we can alter their outcomes.

The classification of causal patterns provides us with an essential tool for navigating the complexity of the universe, distinguishing between those patterns where chance and contingency play crucial roles and those where the cause-effect connection is so intrinsic that it becomes inescapable. In the realm of non-essential patterns, chance intervenes significantly, as demonstrated by the role of random genetic mutations in the evolution of new species, introducing variability and diversity into the fabric of life. In contrast, essentially subordinate patterns manifest in those phenomena governed by relentless physical laws, where the relationship between cause and effect presents as a universal constant, exemplified by the gravitational force acting between two bodies. This approach not only illuminates the rich diversity and intricate complexity of nature but also highlights the predictability and regularity that underpin many phenomena of the universe.

Beyond this distinction, it is imperative to consider causal series characterized by robust causal cooperation. In these contexts, causality transcends the mere linear sequence of events, requiring the series to act as an integrated and cohesive whole in generating a fact. This approach recognizes that the final outcome emerges from the simultaneous and coordinated synergy of multiple factors, in a web of mutual dependency that is fundamental to the manifestation of the phenomenon. This understanding of causality, congruent with empirical science, sheds light on situations where distributed causality becomes central.

A paradigmatic example of this dynamic is observed in the ecological processes involved in pollination. Here, the fertilization and subsequent fruit production do not depend exclusively on the intervention of a single type of pollinator but on the complex and dynamic interaction between various species of pollinators, plants, and environmental elements. This network of causality, active as a whole, ensures that pollination and the subsequent fruit production take place, demonstrating how each component plays a vital role within a precarious balance. The causal series underlying this phenomenon thus encompasses not only linear sequences of causes and effects but also a cooperative interaction among multiple actors and processes, evidencing that causality, in many cases, constitutes a distributed and collaborative phenomenon.

Presentation of Logical Demonstrations on the Finiteness of Causal Series

To address the question of the finiteness of causal series from an analytical-deductive approach, it is essential to clarify and unfold the logical demonstrations that support our understanding of this topic. Starting from the premise that we operate within a theoretical model defined by patterns of causes that are essentially and actually subordinate, closely cooperating in the causation and effectuation of an aspect or fact of the world, the following demonstrations aim to irrefutably establish the finiteness of these series.

a) Direct Demonstration (Inductive-Mathematical)

To address the finiteness of causal series, we initially turn to a direct demonstration supported by the principle of mathematical induction. 

Consider a causal series composed of events E1,E2,E3,…,En, where each event Ei is essentially and actually subordinate to the causation of the subsequent event in the series. This structure allows us to model the causal series as a chain of causal relationships C(E1)→C(E2)→C(E3)→…→C(En), where C(Ei) represents the essentially subordinate cause ii in the series.

The principle of mathematical induction applies as follows:

Base Step: For i=1, the existence of C(E1) is indispensable for the occurrence of C(E2). The essential and actual relationship between C(E1) and C(E2) demonstrates that without the first cause, the subsequent effect could not manifest.

Inductive Hypothesis: We assume that for an arbitrary ii, the existence of C(Ei) is necessary for the occurrence of C(Ei+1).

Inductive Step: We prove that this property holds for i+1. That is, the existence of C(Ei+1) is necessary for C(Ei+2). Given that C(Ei+1) is essential and actually subordinate in the causation of C(Ei+2), it is confirmed that C(Ei+1) is necessary for the occurrence of C(Ei+2).

This demonstration, by following the steps of mathematical induction, confirms that the causal series must be finite, given that each event essentially depends on its predecessor, and this dependency is maintained throughout the series.

b) Indirect Demonstration (By Reduction to Absurdity)

Complementarily, we explore an indirect demonstration that assumes the possibility of an infinite causal series, E1→E2→E3→En… where each event Ei is essential and actually subordinate in the efficient causation of the following event in the series.

Given that we are considering an infinite causal series, we could have an infinite number of subordinate causes C(E1),C(E2),C(E3),….C(En)

However, if the causal series functions as a whole in an actual and efficient manner, this would imply that at any given moment, all events in the series would need to be present and active simultaneously to cause the next event in the sequence.

This means that at a given instant, we would have an infinite number of events acting simultaneously to produce the next event in the series, which contradicts the notion that the series could function as a whole in an actual and efficient manner. In reality, causality in a chain of events is limited by time and space, making it impossible for an infinite number of events to act simultaneously.

Therefore, we have reached a contradiction by assuming the existence of an infinite causal series under the given definition of a causal series that is essential and actually subordinate in efficient causation. Therefore, it is impossible for a causal series of this type to contain an infinite number of events.

In conclusion, these demonstrations, both direct and indirect, not only strengthen the argument for the finiteness of causal series within the established theoretical model but also underline the necessity of a first cause. This first cause, having no precedents, emerges as a necessary foundation for the existence of the series itself and, by extension, of the entire observable universe. This deductive approach allows us not only to defend the coherence of our theoretical model but also to provide a solid basis for a deep understanding of causality as an organizing principle of the cosmos

Modeling Mathematical and Scientific Evidence: From Singularity to Emergent Complexity

Within the framework of our analysis, the transition from the primordial singularity to the emergent complexity characterizing the current universe represents a fascinating topic that lends itself to deep exploration through mathematical analysis and scientific evidence. This journey from the point of infinite density and temperature defined as the Big Bang singularity, to the intricate web of the cosmos filled with galaxies, stars, planets, and life itself, offers a unique window into understanding the essential and actual causality underlying existence.

Mathematical modeling has proven to be an invaluable tool in this endeavor, providing a language through which we can describe and predict the behavior of the universe from its most primitive conditions. Einstein's equations of general relativity, along with the principles of quantum mechanics, allow us to glimpse how fundamental forces and universal constants, emerging from this seemingly simple point of origin, give rise to immeasurable complexity and diversity. This process, governed by precise physical laws, firmly supports the concept of causality we have outlined, in which each phenomenon in the universe is chained to its precursors in an essential and actual manner.

However, as we delve into this analysis, we also encounter the limits of our current understanding and the unanswered questions that remain as challenges for contemporary science. The Big Bang singularity, that point where our known physical laws cease to apply, raises fundamental questions about the universe's origin and the very nature of causality under extreme conditions. Likewise, the search for a unified theory that harmonizes general relativity with quantum mechanics and can explain the unification of fundamental forces continues to be one of the most elusive and, at the same time, most intriguing goals of theoretical physics.

This critical reflection leads us to recognize both the power and the limitations of our current theoretical and methodological framework. Although we have managed to describe and understand a vast range of natural phenomena, from the mechanics of subatomic particles to the dynamics of galaxies, we still face the task of deepening our understanding of the fundamental principles governing the universe. The Big Bang singularity and the anticipated unification of fundamental forces are not just technical challenges for physics; they are also invitations to reflect on the nature of causality and the ultimate structure of reality.

Thus, as we continue to explore and push the boundaries of our knowledge, these themes continue to stimulate not only scientific research but also philosophical inquiry. They remind us that our journey toward understanding the cosmos is far from complete and that each answer we find leads us to new, deeper, and more fundamental questions about the origin, evolution, and ultimate fate of the universe.

Chapter 4: Demonstrating the Necessity of the First Necessary Cause

Terminological Precisions on Necessity

The exploration of causality, a pillar that upholds our understanding of the universe, leads us to precisely distinguish between two fundamental concepts: necessary causality and the necessary cause. Though both bear the label of "necessary," they illuminate different and crucial aspects of the causal structure underpinning all existence. This distinction is essential for advancing our inquiry with conceptual clarity and methodological rigor.

Necessary causality refers to the inescapable relationship between cause and effect. In this type of connection, the presence of a specific cause invariably guarantees the occurrence of its corresponding effect. The distinctive feature of this relationship is its inevitability: if the cause occurs, the effect cannot fail to manifest. This underscores an underlying order in the universe, where certain conditions trigger predictable and obligatory consequences.

Necessary Cause, on the other hand, focuses on the quality of the cause itself, pointing to those without which a certain effect, or even the existence of certain states of affairs, would be impossible. This concept goes beyond ensuring the inevitable production of an effect (as does necessary causality) to emphasize that without the existence of such a cause, the effect or reality dependent on it simply could not be.

Upon contemplating the idea of a first necessary cause, we find ourselves in a territory where these two concepts intertwine in a unique way:

The first necessary cause is that which not only initiates the causal series but also grounds the existence of everything that follows. From the perspective of necessary causality, this cause establishes an inescapable relationship with the first effect, and consequently, with all subsequent events in the chain. This implies that the occurrence of these effects is a direct and inevitable consequence of the first cause.

Looked at from the angle of the necessary cause, this initial principle stands as the foundation without which neither the first effect nor any that follow could exist. The first necessary cause, then, is indispensable not only for triggering the causal series but for the very existence of the cosmos itself.

Deducible Characteristics of the Concept of Necessary Cause

Once the necessity of a first necessary cause is demonstrated, its existence and action are fundamental and unconditional for the start and continuation of a finite causal series, within the defined framework of causality. This cause is characterized by the following essential properties:

Primacy: It is the first in the causal series, not preceded by any other cause, thereby avoiding an infinite regression of causes and ensuring the finiteness of the series.

Unconditionality: Its existence and causal capacity do not depend on any condition, factor, or external cause, making it absolutely necessary within the context of the defined causal system.

Sufficiency: It possesses, by itself, the capacity to initiate the causal series without requiring the action or presence of any other entity or additional process.

Uniqueness: Given its unconditionality and primacy, it is unique in the sense that there cannot be multiple entities or processes that fulfill all the necessary conditions and properties to be considered as the first necessary cause within the studied causal system.

Non-derived Causality: It acts as the original source of causality, from which all subsequent causes and effects derive, but whose own causality is not derived from any other source.

Independence: While all subsequent causes and effects depend on it for their existence and properties, it is autonomous and independent of them in its existence and operation.

In conclusion, the preceding clarification allows us not only to understand more deeply how causality manifests in the universe but also to recognize the crucial role played by certain causes in the very fabric of reality. The first necessary cause, as a convergence point of these concepts, is revealed as the foundation upon which the entire structure of the cosmos is built, establishing not only the beginning of the causal series but also guaranteeing its continuity and coherence. In the contemplation of this first cause, we find a synthesis of inevitability and essentiality that underscores its role as the ultimate foundation of everything that exists, inviting us to reflect on the ultimate nature of causality and the fundamental structure of our universe.

Chapter 5: The Necessity of a First Cause: Deductive Arguments and Scientific Evidence

Next, we will see if mathematical logic can be used to express and validate the logical arguments that support the necessity of a first necessary cause.

1. First Formalization

To formalize and present the demonstration of the necessity of a first necessary cause within a first-order predicate logic framework, we can employ an approach that articulates the fundamental principles we have established in formal terms. Though mathematical logic can be expressed in many ways, here we'll use an approach based on premises and conclusions to structure the argumentation clearly and concisely.

Premises

(P1) Every finite and essential causal series requires at least one cause that is not the effect of another preceding cause.

(P2) A necessary cause is one whose existence is unconditional, that is, does not depend on any other cause.

(P3) Without a first necessary cause, an infinite regression of causes is incurred, which contradicts the definition of a finite and essential causal series.

(P4) The coherence of the causal system (defined as an ordered and finite set of causal events interacting essentially and actually) depends on the existence of an unconditional and non-derived basis that initiates and sustains the series.

(P5) A first necessary cause, by definition, cannot have preceding causes nor inherently depend on other causes for its existence and causal capacity.

Formal Logic

We can represent these premises in a form that allows deriving the necessity of the first necessary cause:

From (P1) and (P3): The necessity of a cause that initiates the series without being the effect of a preceding cause is established to avoid the contradiction of an infinite regression and ensure a finite series.

From (P2) and (P5): This initial cause must be unconditional and independent, meaning its existence and capacity to cause do not depend on any other cause.

From (P4): The coherence and existence of the causal system inherently depend on this unconditional and independent cause.

Conclusion

Therefore, it concludes that (C) there must exist a first necessary cause that is unconditional and independent to ensure the coherence and finiteness of any finite and essential causal system. This necessary cause has no precedents nor intrinsic dependencies, which validates it as the inescapable foundation of the causal system.

To formalize the premises and derive the conclusion in a first-order logic framework, we can opt for a level of first-order logic, as it allows for the quantification of variables and can adequately handle the relationships and properties involved in the premises and conclusions we've discussed. First-order logic provides the necessary precision to deal with concepts like causality, independence, and finiteness of series, besides allowing the expression of generalizations, such as "every finite and essential causal series."

To illustrate how we might formalize these premises and arrive at the desired conclusion, we will employ a combination of quantification and logical operators. We'll adapt symbols and notation to follow the specifications provided. Bound variables x, y, or z will be used for entities in our discourse universe (mainly causal series and causes), and capital letters or expressions headed with capital letters for predicates or operators denoting properties or relationships.

Full Formalization and Demonstration of Inference:

Logical Symbols:

∀: Universal quantifier
∃: Existential quantifier
→: Conditional
↔: Biconditional
∧: Conjunction
¬: Negation

Predicates:

SF(x): x is a finite and essential causal series
C(y): y is a cause
ED(y,z): y is an effect of z
BN(y): y is a necessary cause and an unconditional base
RI: There is an infinite regression of causes
CO: The causal system is coherent

Inference Rules:

Modus ponens
Simplification
Generalization
Existential instantiation
Universal instantiation
Reductio ad absurdum

Premises:

1 ∀x (SF(x) → ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z))))
2 ∀y (BN(y) ↔ ∀z (C(z) → ¬ED(y,z)))
3 ¬∃y (BN(y)) → RI
4 CO ↔ ∃y (BN(y))
5 ∀y (BN(y) → ¬∃z (C(z) ∧ ED(y,z)))

Conclusion:

CO → ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))

Proof:

Step 1: Proof of ∀y(C(y) → ¬EffectOf(y,y))

We use Premise 2 (∀y (BN(y) ↔ ∀z (C(z) → ¬ED(y,z))))
We instantiate the universal quantifier ∀y with the variable y
We obtain BN(y) ↔ ∀z (C(z) → ¬ED(y,z))
We use Premise 5 (∀y (BN(y) → ¬∃z (C(z) ∧ EF(y,z))))
We instantiate the universal quantifier ∀y with the variable y
We obtain BN(y) → ¬∃z (C(z) ∧ EF(y,z))
We use propositional logic to obtain NecessaryBase(y) → ¬EffectOf(y,y)

Step 2: Proof of ¬∃y(BN(y)) → ∃y(C(y) ∧ ED(y,y))

We use Premise 3 (¬∃y (BN(y)) → RI)
We assume ¬∃y(BN(y))
We use Modus Ponens to obtain RI
We use Premise 1 (∀x (SF(x) → ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))))
We instantiate the universal quantifier ∀x with the variable x
We obtain SF(x) → ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We instantiate the variable x with the constant "RI"
We obtain SF (RI) → ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We use Modus Ponens to obtain ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We use propositional logic to obtain ∃y(C(y) ∧ ED(y,y))

Step 3: Proof of Coherence → ∃y (NecessaryBase(y) ∧ ¬∃z (C(z) ∧ EffectOf(y,z)))

We use Premise 4 (CO ↔ ∃y (BN(y)))
We instantiate the existential quantifier ∃y with a new variable, for example, y'
We obtain CO → (CO ≡ ∃y' (NecessaryBase(y')))
We use Simplification (the definition of CO):
We obtain (CO ≡ ∃y' (BN(y'))) → ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We use Modus Ponens from Step 2 (¬∃y(BN(y)) → ∃y(C(y) ∧ ED(y,y))):
We obtain (CO ≡ ∃y' (BN(y'))) → ¬∃y(BN(y)) ∨ ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We use De Morgan's Law (from propositional logic):
We obtain (CO ≡ ∃y' (BN(y'))) → ¬(∃y(BN(y))) ∨ ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We use the equivalence introduced in Step 3:
We obtain ¬CO ∨ ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))
We use propositional logic (law of excluded middle):
We obtain CO → ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))

Conclusion: ∃y (BN(y) ∧ ¬∃z (C(z) ∧ ED(y,z)))

The inference is valid in first-order predicate logic. It has been demonstrated that under the given premises, it follows (∃y (NecessaryBase(y) ∧ ¬∃z (C(z) ∧ EffectOf(y,z)))) or, in other words, that at least one necessary cause exists and is unconditional and independent, which is indispensable for ensuring the coherence and finiteness of any finite and essential causal system. The logical structure and the proposed formalization allow for a clear and precise understanding of the argumentation, highlighting how, from carefully defined premises, the necessity of a first cause that coherently underpins the causal structure of the universe can be deduced.

2. Second Formalization

To formalize the argumentation on the necessity of a first necessary cause in predicate logic, we will define some terms and then formulate the corresponding premises and conclusions.

Logical Symbols:

∀: universal quantifier

∃: existential quantifier

→: conditional

∧: conjunction

¬: negation

Predicates:

C(x): x is a cause

E(x): x is an effect

N(x): x is a necessary cause

D(x,y): x depends on y (x is the effect and y the cause)

P(x): x is the first cause

Premises:

∀x (C(x) ∧ ¬E(x) → N(x))

∀x (N(x) → ¬∃y (C(y) ∧ D(x,y)))

∃x (P(x) ∧ N(x))

P(x) → (C(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x)))

Conclusion:

∃x (P(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x)) ∧ N(x))

Inference Rules:

Modus ponens

Simplification

Generalization

Existential instantiation

Universal instantiation

Reductio ad absurdum

Validity:

Step 1: Demonstration of ∀y (C(y) → ¬D(y,y))

We use Premise 2 (∀x (N(x) → ¬∃y (C(y) ∧ D(x,y))))

Instantiate the universal quantifier ∀x with variable y

We get N(y) → ¬∃z (C(z) ∧ D(y,z))

We use propositional logic to get N(y) → ¬D(y,y)

Step 2: Demonstration of P(x) → N(x)

We use Premise 3 (∃x (P(x) ∧ N(x)))

Instantiate the existential quantifier ∃x with variable x

We get P(x) ∧ N(x)

We use propositional logic to get P(x) → N(x)

Step 3: Demonstration of ∃x (P(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x)) ∧ N(x))

We use Premise 4 (P(x) → (C(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x))))

We use Modus Ponens from step 2 (P(x) → N(x)):

We get P(x) → (C(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x)) ∧ N(x))

We use Simplification:

We get ∃x (P(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x)) ∧ N(x))

Conclusion:

∃x (P(x) ∧ ∀y (C(y) → (D(y,x) ∨ y = x)) ∧ N(x))

This formalization logically represents the structure of our argument regarding the necessity of a first necessary cause. It indicates that if there exists a unique entity (the first cause), then it is necessary, and all other causes depend on it or are identical to it. Dependency is interpreted as the causal relationship in which the existence or manifestation of any other cause or effect within the system is conditioned by the existence of this first cause. Predicate logic allows us to express these relationships and dependencies in a clear and structured manner, facilitating the understanding and formal validation of our arguments.

3. Proof by Reduction to Absurdity

To develop a compelling strategy that combines and develops two lines of argumentation, strengthening the defense of the independence and absolute necessity of the first necessary cause, we will structure a logical-formal demonstration that incorporates both approaches, using elements of predicate logic for greater precision.

Definitions and Premises

Define C as the first necessary cause within a finite causal series.

Assume that the causal series cannot be infinite, based on the premise of finitude and coherence of the causal system.

Accept that for C to be considered the first cause, it must be independent of any other cause (it cannot depend on preceding or intrinsic causes).

Deduction

Suppose, for contradiction, that C depends on other causes C1,C2,…,Cn or has a cause C'

If C depends on C1,C2,…,Cn

Given that the series must be finite, following this chain of dependencies would eventually lead to a "last" cause in the series, contradicting C's definition as the first.

This would imply that C is not the first cause, which contradicts our initial definition and the purpose of C.

If C has a cause C'

C' would become the true first necessary cause, displacing C from its position, which is a contradiction.

If we follow this reasoning ad infinitum, we would never arrive at a true first cause, contradicting the premise of finitude of the causal system.

In both cases, we reach a contradiction with the definition and required properties for C. Therefore, our initial assumption that C depends on other causes or has a cause C' must be false.

Conclusion

We conclude that C must be independent of any other entity and absolutely necessary:

Independent, as it cannot depend on any other cause without incurring a contradiction.

Absolutely necessary, because it cannot have a preceding cause, ensuring its position as the uncaused and the source of the causal series.

This demonstration, by integrating and developing the previous deductions in a coherent logical-formal structure, provides a solid defense of C as the first necessary cause. By resorting to reduction to absurdity, we demonstrate that any assumption of dependency or prior causation of C leads to contradictions with the premises of finitude and coherence of the causal system, reaffirming thus the independence and absolute necessity of C within our theoretical causal model.

 

 4.  Corollary of the first deduction:

We can explore other deducible conclusions from the given premises in the previous arguments by examining logical implications and applying rules of inference. One highly interesting consequence is that if there is no infinite regression of causes, then the causal system is coherent, so we deduce for a given causal system that the impossibility of an infinite regression implies coherence in the system. Let's see how it is deduced from the first proof presented in this chapter.

Premises:

    ∀x (SF(x) → ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z))))
    ∀y (BN(y) ↔ ∀z (C(z) → ¬ED(y,z)))
    ¬∃y (BN(y)) → RI
    CO ↔ ∃y (BN(y))
    ∀y (BN(y) → ¬∃z (C(z) ∧ ED(y,z)))

Conclusion:
¬RI → CO

Deduction Steps:

1. ¬∃y (BN(y)) (Premise 3)

2. ¬BN(a) (Universal instantiation of 1 over a)

3. BN(a) ↔ ∀z (C(z) → ¬ED(a,z)) (Premise 2)

4. ∀z (C(z) → ¬ED(a,z)) (Modus ponens from 2 and 3)

5. C(b) → ¬ED(a,b) (Universal instantiation of 4 over b)

6. ¬ED(a,b) (Modus ponens from 5 and the assumption of C(b))

7. ¬∃z (C(z) ∧ ED(a,z)) (Negation with b)

8. SF(a) → ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z))) (Premise 1)

9. ∃y (C(y) ∧ ¬∃z (C(z) ∧ ED(y,z))) (Modus ponens from 8 and the assumption of SF(a))

10. C(c) ∧ ¬∃z (C(z) ∧ ED(c,z)) (Existence with c)

11. ¬∃z (C(z) ∧ ED(c,z)) (Negation with z)

12. ¬ED(c,c) (Negation with c)

13. BN(c) (Modus ponens from 12 and the definition of BN(y))

14. ∃y (BN(y)) (Existence with c)

15. CO (Modus ponens from 14 and the definition of CO)

16. ¬RI → CO (Modus ponens from 3 and 15)

Explanation of the Conclusion:

It is demonstrated that the coherence of a causal system is directly related to the absence and impossibility of infinite regression. The conclusion ¬RI → CO establishes that if there is no infinite regression of causes in the causal system, then the causal system is coherent. In other words, the coherence of the causal system follows from the absence of an infinite regression of causes. This suggests that the presence of an infinite regression of causes could indicate a lack of coherence in the causal system.