Relationship of Human Brain, Mind, and Consciousness with Physical Science and Mathematics:

The intricate interplay between the human brain, mind, and consciousness bears a profound relationship with the domains of physical science and mathematics. This connection illuminates how these fundamental aspects of human existence find common ground with empirical investigation and quantitative analysis. Here, we explore the multifaceted relationship between these facets of human cognition and the exacting disciplines of physical science and mathematics:

1. Neurological Underpinnings and Physical Science:

Brain as the Physical Substrate: The human brain, as the epicentre of cognitive processes, is fundamentally rooted in physical science. Neuroscientists employ physics and chemistry to uncover the intricate neural networks and electrochemical interactions that underpin consciousness. Technologies like functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) reveal the neural dynamics responsible for cognitive phenomena.

Brain as a Biological System: Physical science provides the framework to comprehend the brain as a biological system. Principles of thermodynamics, kinetics, and electrostatics are applied to elucidate the energy demands, reaction rates, and electrical properties of neural processes, offering insights into the biochemistry of cognition.

2. Mathematics as the Language of Brain Function:

Quantitative Analysis of Brain Activity: Mathematics serves as the lingua franca for interpreting the brain's functional patterns. Through mathematical models and statistical analyses, researchers quantify the neural correlates of consciousness, allowing for rigorous comparisons and predictions. Concepts like Fourier transforms help analyse the frequency components of neural signals in techniques like spectral analysis.

Connectomics and Graph Theory: Mathematical graph theory is pivotal in modelling the intricate connectivity patterns within the brain. It enables the characterization of brain networks, shedding light on information flow, modular organization, and functional specialization. Graph theory, in conjunction with network theory, is crucial for understanding how different brain regions interact and contribute to consciousness.

3. Mind and Mathematical Logic:

Logical Reasoning and Abstract Thinking: The mind's capacity for logical reasoning is closely aligned with mathematical logic. The ability to deduce, infer, and discern patterns of thought represents a form of abstract, mathematical reasoning. This intellectual capacity enables the mind to engage in systematic analysis, problem-solving, and the formulation of logical arguments.

Mathematics as a Tool for Complex Thought: Mathematics provides a structured framework for organizing and expressing complex thoughts. The precision and rigor of mathematical language empower the mind to tackle intricate concepts, make precise predictions, and develop sophisticated theories. The synergy between mathematics and the mind extends to diverse fields, from philosophy to the natural sciences.

4. Emergence of Consciousness and Complexity Theory:

Complexity Theory and Cognitive Emergence: The emergence of consciousness from neural processes is a complex phenomenon. Complexity theory, a branch of mathematics, explores how intricate systems, such as the brain, exhibit emergent behaviour. It sheds light on how individual neurons collectively give rise to conscious experiences, transcending the sum of their individual activities.

Mathematical Approaches to Consciousness: Mathematical models, such as the Integrated Information Theory (IIT) and the Global Neuronal Workspace (GNW) model, offer formal frameworks for understanding consciousness. These models quantify the extent to which information is integrated across different brain regions, linking mathematical concepts to the fabric of consciousness.

In summary, the intricate relationship between the human brain, mind, and consciousness intertwines with the disciplines of physical science and mathematics. These domains provide the tools and methods for unravelling the neural underpinnings of consciousness, quantifying brain activity, facilitating logical reasoning, and modelling the emergence of consciousness as a complex phenomenon. This interdisciplinary synergy underscores the profound connections between the essence of human cognition and the rigor of empirical investigation and mathematical inquiry.

5. Mathematical Presentation:

5. Mathematical Presentation:

F = (Mᴏʀᴅ + Mᴅᴍ + (-Mᵃᵖᵖ))⋅aᵉᶠᶠ, or equivalently:

F = (Mᴍ + (−Mᵃᵖᵖ))⋅aᵉᶠᶠ 

This can be expressed as:

F = Effective mass (Mᵉᶠᶠ)⋅aᵉᶠᶠ

where: aᵉᶠᶠ ∝ 1/Mᵉᶠᶠ and Mᵉᶠᶠ = (Mᴏʀᴅ + Mᴅᴍ + (−Mᵃᵖᵖ)).

Thus, aᵉᶠᶠ generates −Mᵃᵖᵖ.

Total Mechanical Energy (Eᴛₒₜ):

Eᴛₒₜ = PE + KE

This can be expressed as:

Eᴛₒₜ = (Mᴏʀᴅ + Mᴅᴍ + (−Mᵃᵖᵖ)) + KE, or equivalently:

Eᴛₒₜ = (Mᴍ + (−Mᵃᵖᵖ)) + KE

Where: F  ∝ aᵉᶠᶠ and F generates KE. 

6.1. Effective Acceleration Generates Apparent Mass:

6.2. Forces Generate Kinetic Energy:

6.3. Mass-Energy Equivalence: Extended Classical mechanics.

6.4. Equivalence of Apparent Mass and Dark Energy's Negative Effective Mass:

6.5. Kinetic Energy with Negative Effective Mass:

6.6. Negative Mass in Mechanical Energy vs. Positive Mass in Nuclear Energy:

6.7. Implications of the Relationship between Effective Acceleration, Apparent Mass, and Mechanical Energy:

6.1. Effective Acceleration Generates Apparent Mass:

In extended classical mechanics, effective acceleration (aᵉᶠᶠ) is inversely proportional to effective mass (Mᵉᶠᶠ). This relationship can be mathematically expressed as:

aᵉᶠᶠ ∝ 1/Mᵉᶠᶠ

​This means that as effective acceleration increases, there is a corresponding decrease in effective mass. This dynamic interaction leads to the generation of negative apparent mass (-Mᵃᵖᵖ). As effective acceleration increases, the effect of apparent mass becomes more pronounced, creating a unique and significant relationship between acceleration and mass within this framework.

The notion of apparent mass, particularly when it takes on a negative value, introduces a novel perspective on the behaviour of objects under acceleration. In this context, as an object's effective acceleration increases—potentially due to external forces or influences—the effective mass must decrease in order to maintain the equality described in the proportionality. Consequently, this decrease in effective mass manifests as a more pronounced negative apparent mass.

This relationship underscores a crucial aspect of extended classical mechanics, suggesting that the dynamics of motion and mass are interlinked in ways that deviate from classical interpretations. The generation of negative apparent mass illustrates how accelerated systems can exhibit behaviours that challenge traditional notions of mass and inertia. It reflects a deeper understanding of how effective forces interact with mass in a non-linear fashion, leading to counterintuitive outcomes, such as reduced resistance to acceleration or even increased responsiveness to applied forces.

In summary, the interplay between effective acceleration and apparent mass in extended classical mechanics reveals a complex relationship that enriches our understanding of mechanical systems. As effective acceleration increases, the resultant behaviour of apparent mass not only emphasizes the significance of acceleration in determining mass properties but also challenges established principles, paving the way for further exploration into the mechanics of motion.

6.2. Forces Generate Kinetic Energy:

In mechanical systems, the acting forces (F) are directly proportional to the kinetic energy (KE) produced within the system, represented as F ∝ KE. This fundamental relationship illustrates that when a force is applied to an object, it results in a change in the object's state of motion, leading to the generation of kinetic energy.

When a net force acts on an object, it causes the object to accelerate, which is quantitatively described by Newton’s second law. As the object accelerates, it gains velocity, and thus, its kinetic energy increases. This interaction emphasizes that the magnitude of the applied force influences the extent of kinetic energy generated: greater forces result in higher accelerations, which in turn produce more kinetic energy.

Additionally, this relationship underscores the principle of energy conservation within mechanical systems. The work done by the acting forces is converted into kinetic energy, illustrating that energy can transform from one form to another while remaining conserved. In this context, it is essential to consider not only the magnitude of the forces but also their direction and application, as they ultimately dictate the efficiency of kinetic energy generation and the overall dynamics of the system.

In summary, the proportionality between forces and kinetic energy is a cornerstone of classical mechanics, illustrating how the interplay of forces and motion underlies the generation and transformation of energy within mechanical systems.

6.3. Mass-Energy Equivalence: Extended Classical mechanics.

Concept of Mass-Energy Equivalence:

In the realm of extended classical mechanics, the principle of mass-energy equivalence takes on a nuanced perspective. Here, apparent mass (−Mᵃᵖᵖ) is not merely a theoretical construct but is fundamentally equivalent to kinetic energy (KE). This relationship underlines a crucial assertion: mass can be transformed into energy and, conversely, energy can manifest as mass. This interplay is reflective of a deeper connection between mass and energy that transcends traditional boundaries.

Understanding Apparent Mass:

The apparent mass, represented as −Mᵃᵖᵖ, is characterized by its negative value, suggesting distinct behaviour within the framework of extended classical mechanics. Rather than being a mere derivative of standard mass concepts, apparent mass embodies unique dynamics that arise from effective acceleration (aᵉᶠᶠ) and the contributions of various mass components, including normal matter and dark matter, while directly representing the effective mass derived from dark energy (Mᴅᴇ).

Kinetic Energy as a Transformative Agent:

Kinetic energy (KE) represents the energy of motion, mathematically defined as KE = 1/2·M·v², where M is mass and v is velocity. In extended classical mechanics, the kinetic energy generated by the interaction of forces can be directly related to the apparent mass. The equation −Mᵃᵖᵖ = KE encapsulates this relationship, reinforcing that the apparent mass can be understood as a manifestation of kinetic energy under specific conditions of acceleration and motion.

Interconnected Dynamics:

This equivalence suggests that as a system experiences changes in effective acceleration, there is a corresponding transformation of energy into apparent mass. For example, increased effective acceleration might lead to greater kinetic energy, thus enhancing the apparent mass in the system. This reciprocal relationship emphasizes that energy is not merely a by-product of motion but is fundamentally intertwined with mass, influencing how we understand the dynamics of systems in motion.

Consistency with Classical Mechanics:

While the notion of mass-energy equivalence is often associated with relativistic physics, its implications within the framework of classical mechanics cannot be overlooked. The assertion that -Mᵃᵖᵖ = KE provides a clear, classical interpretation of how mass and energy interact. This framework suggests that mass can be manipulated through the application of forces and accelerations, and as a result, energy dynamics emerge as an essential component of mechanical systems.

Conclusion:

In summary, the relationship -Mᵃᵖᵖ = KE within extended classical mechanics not only reaffirms the principle of mass-energy equivalence but also illustrates the intricate interplay between mass and kinetic energy. This relationship reveals deeper insights into the nature of forces and motion, highlighting how mass can be conceptualized as a dynamic quantity influenced by energy states and effective accelerations. Through this lens, we gain a more profound understanding of the mechanics governing our universe, bridging classical interpretations with modern concepts of energy and mass.