A Pure Mathematical Perspective: Dimensions, Numbers, and Mathematical Concepts:

Soumendra Nath Thakur, Tagore's Electronic Lab, India

Date: 20-10-2023

DOI: http://dx.doi.org/10.13140/RG.2.2.25942.01607

URL: https://easychair.org/publications/preprint_open/drzc

1. Abstract:

This paper provides a comprehensive and internally consistent perspective on mathematical concepts, particularly focusing on dimensions, numbers, and their abstract nature. The core argument posits that spatial dimensions like length, width, height, or depth, along with numbers like 1 and the concept of (1+1), should be treated as mathematical constructs that exist independently of the physical world. These concepts are examined in their purest form, abstracted from their usual associations with tangible objects or real-world phenomena.

The paper underlines the process of constructing and manipulating these mathematical concepts through various operations and rules, all the while emphasizing their abstract and mathematical essence. This argument extends to time, which is regarded as the fourth dimension, akin to spatial dimensions and numbers in terms of being an abstract mathematical entity. This perspective positions time and other mathematical concepts within the realm of abstract mathematical constructs, rather than as concrete, physically grounded entities.   

In this paper, the significance of isolating these concepts from their real-world applications is highlighted, emphasizing that mathematics operates independently of specific contexts and provides a universal framework for quantification and comprehension. Expert opinions have been sought and, overall, the paper has been praised for its logical consistency and coherent approach to these mathematical concepts in their pure, abstract state.

Keywords: dimensions, numbers, mathematical concepts, abstract nature, pure mathematics, time dimension,

2. Introduction:

This paper presents a cohesive and systematic exploration of dimensions, numbers, and mathematical concepts, with an unwavering focus on their inherent characteristics. Within this perspective, spatial dimensions like length, width, height, or depth, and numerical entities such as 1 and the concept of (1+1) are envisioned as mathematical constructs existing in splendid abstraction, detached from the concrete grasp of the physical world.

Through an unyielding mathematical lens, these notions are dissected, examined, and placed into the broader framework of abstract mathematical constructs. This framework is not limited to the spatial, for it extends itself to embrace time as the fourth dimension, elevating it to the same ethereal plane occupied by spatial dimensions and numbers.

This mathematical discourse isn't just theoretical. It is a methodical exercise in pure mathematical thought, deliberately stripped of the trappings of real-world phenomena. Here, dimensions, numbers, and mathematical concepts are meticulously molded and manipulated by the elegant rules of mathematical operations. The result is a portrayal of these entities in their most pristine form - abstract, detached, and truly mathematical.

This perspective underscores the inherent purity of mathematical concepts when they are liberated from the constraints of particular applications and separated from the anchors of physical reality. These abstract constructs not only reside within the realm of mathematical thought but also serve as vital tools within the expansive structure of mathematics itself.

In essence, this paper delves into the profound abstract nature of these mathematical entities, highlighting their independence from the specifics of the physical world. It aligns seamlessly with the fundamental premise that mathematics transcends its ties to the practical, providing a universal framework for quantification, exploration, and understanding.

3. Mathematical Presentation:

This mathematical presentation encapsulates the core arguments and concepts put forth in the previous submission, emphasizing the abstract, independent, and pure nature of dimensions, numbers, and mathematics.

Dimensions as Abstract Mathematical Concepts:

Spatial dimensions such as length (L), width (W), height (H), or depth (D) are considered as abstract mathematical concepts that are independent of physical reality.

• Length (L)
• Width (W)
• Height (H)
• Depth (D)

Numbers as Mathematical Concepts:

Numbers like 1 and the concept of addition (1+1) are also viewed as abstract mathematical constructs.

• Number 1
• Addition of (1+1)

Euclidean 3-Dimensional Space:

In the Euclidean 3-dimensional space, the coordinates of a point A are represented as A (x, y, z). These three coordinates exist in an XYZ plane where x, y, and z represent the distances of point A from the Origin in the X, Y, and Z coordinate axes, respectively.

Euclidean Distance Formula:

The Euclidean distance between two points A (x1, y1, z1) and B (x2, y2, z2) in this 3-dimensional space is expressed as 

• PQ = d = √ [(x2 – x1)² + (y2 – y1)² + (z2 – z1)²].

Time as the Fourth Dimension:

Time (T) is treated as the fourth dimension, similar to spatial dimensions and numbers, residing in the realm of abstract mathematical constructs.

• Time (T)

Abstract Nature of Mathematical Concepts:

These mathematical concepts are characterized by their abstract nature, removed from direct references to the physical world.

Mathematics as a Framework for Quantification:

Mathematics operates as an independent framework for quantification and understanding, providing a universal structure. Mathematical Operations and Rules

Purity of Mathematical Concepts:

The paper underscores the purity of these mathematical concepts when considered in isolation from specific real-world applications. Mathematical Operations and Rules

Expert Opinion:

Your paper maintains its coherence and consistency. By treating spatial dimensions and numbers as abstract mathematical concepts independent of physical reality, you logically extend this perspective to time as the fourth dimension. This framework emphasizes the abstract and mathematical nature of these concepts and underscores their purity when viewed in isolation from specific real-world applications. It aligns with the idea that mathematics operates independently of its application to concrete situations, providing a framework for quantification and understanding. Your perspective offers a valuable way to approach these concepts within the realm of mathematical abstraction.

Expert opinions validate the research's logical consistency and coherence in treating these concepts as pure, abstract mathematical constructs.

4. Conclusion:

In the realm of pure mathematical thought, this paper has meticulously carved a coherent and internally consistent perspective on dimensions, numbers, and mathematical concepts. It is a journey into the abstract, where these entities, traditionally intertwined with the physical world, are elegantly abstracted to their purest form.

What emerges is a harmonious symphony of abstract mathematical constructs. Spatial dimensions, typically synonymous with physical reality, stand unshackled as independent mathematical entities. The numbers we perceive in daily life, such as 1 and even the concept of (1+1), transcend their physical embodiments and become ethereal elements of mathematical thought.

This perspective doesn't halt at the spatial; it extends to time. Here, the fourth dimension takes its rightful place alongside its spatial counterparts. Time, too, is recognized as an abstract mathematical entity, free from the bonds of physical constraints.

The key revelation in this discourse is that mathematics operates independently of its practical applications. It provides an unparalleled framework for quantification and understanding, transcending specific contexts and offering a universal structure.

This paper champions the inherent purity of mathematical concepts. As these concepts are meticulously separated from the particulars of the physical world, they retain their abstract, pristine nature. Moreover, they cease to be mere theoretical constructs, transforming into indispensable tools within the grand architecture of mathematics.

In the end, this paper delves deep into the profound abstract nature of mathematical entities. It underscores their independence from the specificities of the physical world, providing a valuable perspective where mathematics thrives as an abstract art, transcending the constraints of practicality.

#dimensions #numbers #mathematicalconcepts #abstractnature #puremathematics #timedimension

Addressing Gravitational Effects on Time Dilation and Photon Interaction:

Dear Mr. Sergey Shevchenko (The informational model - possible tests)

I appreciate your engagement with my previous post, and I'd like to address your points in the context of my research, titled "Relativistic effects on phaseshift in frequencies invalidate time dilation II" available here Preprint Relativistic effects on phaseshift in frequencies invalidate...

My initial statement regarding the change in energy of a photon as it interacts with a gravitational source, resulting in infinitesimal frequency changes (Δf) and energy changes (ΔE) leading to redshift or blueshift, aligns with well-established principles in physics. This interpretation is consistent with established scientific understanding, and there is nothing in your previous post that challenges this statement.

Your discussion of gravitational potential energy and its dependence on radius is indeed valid. At the Planck scale, the significance of Planck units becomes even more apparent. Planck units are a set of units of measurement defined exclusively in terms of four universal physical constants, such as the Planck constant (h) and the speed of light (c). This system of natural units is rooted in the properties of nature, specifically those of free space.

In your post, you mentioned a few key points related to gravitational potential energy. Accordingly, I wish to refer the following points.

(i) The center of gravity of a body is the point where the strength of gravitational potential energy is the greatest.

(ii) The strength of gravitational potential energy depends on the radius of the body between its center and surface, and it's always less at the surface compared to the center.

(iii) Larger bodies with greater mass (M) and radius, depending upon their mass density, experience even less strength of gravitational potential energy at their surfaces than smaller bodies like Earth.

2.1. You suggest that a body with a large mass (M), compared to Earth, which has weak gravity at its surface, may experience a slower "tick" in clock oscillations due to the greater radius (R) of the large masses (M) compared to Earth's radius (r), where R >> r. I encourage you to refer to my research paper, titled, "Relativistic effects on phaseshift in frequencies invalidate time dilation II, available here Preprint Relativistic effects on phaseshift in frequencies invalidate... ," which discusses how clock oscillations are affected by various relativistic effects, including differences in gravitational potential. In this context, it's crucial to note that relative time emerges from relative frequencies. The phase shift in relative frequencies, caused by infinitesimal changes in wave energy and corresponding wavelength enlargements due to relativistic effects, can result in errors in clock time readings, which are often misconstrued as time dilation.

2.2. In your discussion of radiated M-photons, you suggest that they lose energy when the source of M-photons is massive bodies (M), leading to redshift. However, a photon emitted from another body doesn't lose energy when propagating in the gravitational field of massive bodies. I would encourage you to refer to "Photon Interactions in Gravity and Antigravity: Preprint Photon Interactions in Gravity and Antigravity: Conservation... ," which is available in the research paper at the link you provided. It elaborates on how photons within the gravitational sphere of their source maintain a constant speed 'c' and undergo gravitational redshift exclusively. When photons encounter the gravitational influence of multiple massive objects, such as celestial bodies, they neither gain nor lose energy but exchange momentum with the external gravitational field while preserving their intrinsic momentum. Therefore, it's postulated that photons, when propagating in other gravitational fields (multiple) do not change their energy in these circumstances.

Effective redshift:

The zero-gravity sphere in galaxies is dominated by gravitational redshift, with cosmic redshift absent. Photons within this sphere maintain constant speed and undergo gravitational redshift. As photons exit, they experience cosmic redshift, which blends with gravitational redshift to form the photon's effective redshift. Refer Cosmic Speed beyond Light: Gravitational and Cosmic Redshift available here Preprint Cosmic Speed beyond Light: Gravitational and Cosmic Redshift

You made a point regarding "curved spacetime" and the bending of a photon's path. According to your perspective, the bending of a photon's path doesn't necessarily require "curved spacetime." Instead, it's the result of momentum exchange with the external gravitational field. The concept of effective deviation, as discussed in "Photon Interactions in Gravity and Antigravity:" explores how photons return to their original path after gravitational interactions. I encourage those interested in these concepts to refer to my research paper for a more detailed exploration. Also The Dynamics of Photon Momentum Exchange and Curvature in Gravitational Fields available here Chapter The Dynamics of Photon Momentum Exchange and Curvature in Gr...

Thank you for your thoughtful engagement and discussion. I believe that ongoing scientific discourse is essential for refining our understanding of these complex phenomena.

Best regards,

Soumendra Nath Thakur

*-*-*-*-*-*

Analysis of Sergey Shevchenko post 1:

The post primarily discusses the author's disagreement with general relativity (GR) and introduces their alternative viewpoint. They argue that their "Shevchenko-Tokarevsky's informational physical model" offers a different perspective on gravity. According to their model, gravity does not impact space-time or any material object, which contradicts the mainstream scientific view. They assert that many of their model's submissions were rejected by editors of physical journals and preprint sources because it challenges widely accepted scientific concepts.

The author addresses the concept of "gravitational time dilation," asserting that it fundamentally cannot and doesn't exist. They also question the validity of the GR's predictions and experimental verifications in certain scenarios, particularly in strong gravitational fields. The post concludes by encouraging readers to explore their model and related links for more detailed explanations.

In summary, the post presents an alternative viewpoint on gravity and time dilation, expressing skepticism about the widely accepted principles of general relativity and advocating for their own scientific model.

Analysis of Sergey Shevchenko post 2:

The post provides a response to the author's previous statement regarding the interaction between photons and gravitational sources. The author addresses two fundamentally different physical effects that occur when large masses (M) radiate photons. These effects are described as follows:

(i) The atoms in instruments in Earth's weak gravity appear to "tick" faster than atoms on the surface of M. As a result, an instrument measuring the excitation photons of an atom would observe a spectral line with lesser energy when the atom is on the surface of M. This effect is based on the difference in clock rates due to the influence of gravity.

(ii) Photons radiated by M-photons are believed to lose energy (redshift) as they propagate in the M gravity field. This is a phenomenon that's consistent with the mainstream understanding of gravitational redshift.

The author notes that while the General Theory of Relativity (GR) postulates the first effect (i), it doesn't account for the second effect (ii). This means that according to GR, photons don't change their energy when propagating in "curved spacetime." However, in the mainstream scientific community, both effects are considered real and coexist simultaneously.

The post argues that the Shevchenko-Tokarevsky's model of Gravity provides a rational explanation for these effects. It mentions that a gravitational mass defect (effect i) at least at statics is two times lesser than what GR predicts. Additionally, the model argues that the canceled by GR effect (ii) also really exists, and instruments measure the sum of these effects.

The author suggests that experimental tests are needed to clarify the existence of these effects. They propose two experiments that can provide evidence of these phenomena. The first is a test of the Shevchenko-Tokarevsky's model, which would also test GR. The second experiment, proposed in 2007, is designed to reveal the quantum nature of Gravity and, if conducted, could reveal new insights into the interaction of photons with gravitational fields.

In summary, the post discusses two distinct effects related to the interaction between photons and gravitational sources. It presents an alternative model and suggests experimental tests to explore the nature of these effects.

The debate about discreteness vs. continuity in the physical world:

Soumendra Nath Thakur added a reply:

Mr. Wolfgang Konle and Mr. Christian G. Wolf raise important points, and it's essential to provide a scientific analysis of the concepts involved in this discussion.

The debate about discreteness vs. continuity in the physical world often depends on the scale and context of observation. While certain scales, such as the Planck scale, might involve unique considerations, fundamental quantities like Planck mass and Planck length are still considered continuous.

My argument, based on the standardization of the SI unit of time and established scientific principles, aligns with the current scientific understanding of space and time as continuous dimensions. Therefore, both Mr. Konle and Mr. Wolf's statements, in their respective contexts, have some scientific validity, but my argument is also scientifically valid and consistent with established scientific principles.

Planck units are a set of units of measurement defined exclusively in terms of four universal physical constants, in such a manner that these physical constants take on the numerical value of 1 when expressed in terms of these units. These units are a system of natural units because their definition is based on properties of nature, more specifically the properties of free space, rather than a choice of prototype object. At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate.

Discreteness vs. Continuity:

Discreteness refers to phenomena that are quantized, with distinct, separate values. In contrast, continuity implies that a phenomenon has an unbroken range of values.

At the Planck scale, the smallest possible scales defined by Planck units, the term "Planck scale" refers to quantities of space, time, energy, and other units that are similar in magnitude to corresponding Planck units. In this region, quantum effects of gravity are expected to dominate. However, Planck mass and Planck length, which are fundamental to these units, are considered continuous in nature, and there is no reference in Planck scale physics to space and time as discrete.

Space and Time:

According to our current scientific understanding, space and time are considered continuous. The concept of continuous space and time is fundamental to our descriptions of the physical world.

Accordingly, my argument is grounded in the scientific standardization of the SI unit of time, the accepted scientific definition of time, and the description of space and time as dimensions. Dimensions are ultimately mathematical extensions of length, and in this context, they should be considered continuous.

#discreteness #continuity #Dimention

The references to relativistic time dilation in relation to relativity theory are not my own views:

I'd like to emphasize that according to the Planck equation E = hf, the energy of a photon changes as it interacts with a gravitational source, leading to infinitesimal frequency changes (Δf) and, consequently, an infinitesimal energy change (ΔE). This process results in redshift or blueshift. I maintain that this interpretation aligns with established science.

Any of my references to relativistic time dilation were made in the context of how relativity theory addresses this concept. It is pertinent to mention here that any of my references to relativistic time dilation and relativity to illustrate how relativity implies time dilation and other aspects of relativity, these are not my own views, but my research papers do not hold time dilation to be true, but instead support relativistic time dilation as timing error. I firmly hold that time remains unchanged, according to the SI standard of time. Time is not something that should be changed by some external means unless one invites error. All my research papers support this opinion.

Additional: modification to the Poisson equation:

The equations presented by Mr. Wolfgang Konle suggest a modification to the Poisson equation for gravitational potential in the context of gravitation theory. The modified equation includes an additional term, which is denoted with bold letters. This additional term appears to be a proposal to account for the mass equivalent of the energy density of gravitational fields within the gravitational potential equation. 

The modified equation is as follows:

∇²ɸ = 4πG(ρ - (∇ɸ)² / (8πGc²)) or ∇²ɸ + (∇ɸ)² / (2c²) = 4πGρ

In this modified equation, ∇²ɸ represents the Laplacian of the gravitational potential, ρ is the mass density, G is the gravitational constant, c is the speed of light, and (∇ɸ) represents the gradient of the gravitational potential.

Original submission by Mr. Wolfgang Konle, Doctor of Engineering, retired, Airbus Defence and Space, Friedrichshafen, Germany:

According to my (Mr. Wolfgang Konle) opinion, inconsistencies in gravitation theory occur, because the mass equivalent of the energy density of gravitational fields is not taken into account. Considering that, the Poisson equation gets an additional term:

∇²ɸ = 4πGρ => ∇²ɸ = 4πG(ρ-(∇ɸ)²/(8πGc²)) or ∇²ɸ+(∇ɸ)²/(2c²) = 4πGρ

ɸ is the gravitational potential, ρ is the mass density, G is the gravitational constant, c is the speed of light. The additional term is written with bold letters.

#relativistictimedilation #modifiedPoissonEquation

Cosmic Speed beyond Light: Gravitational and Cosmic Redshift

Thakur, Soumendra & Bhattacharjee, Deep. (2023). Cosmic Speed beyond Light: Gravitational and Cosmic Redshift. 10.13140/RG.2.2.10802.58561. 

This research explores the intricate relationship between gravitational and cosmic redshift phenomena, unveiling a profound understanding of how light behaves as it traverses the cosmos. The study begins with an examination of gravitational redshift, a well-established concept occurring when photons move away from massive gravitational sources, such as stars within galaxies. Gravitational redshift, expressed as λ/λ_0 , manifests within the gravitational influence and extends to the boundary of the observed "zero-gravity sphere" enveloping galaxies. Within this remarkable zero-gravity sphere, gravitational effects persist, while the antigravity influence of dark energy remains negligible. As a result, gravitational redshift dominates, and cosmic redshift is notably absent within the sphere. Photons within this sphere maintain their constant speed 'c' and undergo gravitational redshift exclusively. However, as photons exit the zero-gravity sphere at a distance 'r' equivalent to the source star's radius, they encounter the onset of cosmic redshift, quantified as {(λ_obs - λ_emit)/ λ_emit}. Cosmic redshift blends with gravitational redshift, forming the effective redshift of the photon. Critically, the effective cosmic redshift surpasses gravitational redshift, illuminating a profound revelation: photons traverse a greater "light-traveled distance" than their proper distance from the source. In essence, cosmic redshift signifies that photons move across their intended distances at their intrinsic speed , while the expanding universe introduces a relative distance expansion, influenced by antigravity. This research delves into the intricate dance between gravitational and cosmic redshift, shedding light on their remarkable implications for our comprehension of the expanding universe.