### ChatGPT - Benchmark - In order to understand the physical reality is mathematics required?

This document describes an actual conversation with the program "ChatGPT" as of 11 March 2023.
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### Reflection

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### Introduction

• A question is normally a follow up based on the previous question. The purpose is to get an explanation or more detailed about the previous Answer.
Both parties in the discussion are treated as equals.

### 4. In order to understand the physical reality is mathematics required

1. In order to understand the physical reality is mathematics required ?
Mathematics is a powerful tool for describing and predicting physical phenomena, but it is not necessarily required to understand physical reality. The natural world can be observed and studied without relying solely on mathematical models.
However, mathematics has proven to be an incredibly effective language for describing the laws of nature and making predictions about physical systems. For example, the laws of mechanics can be expressed using mathematical equations, which allow us to calculate the behavior of objects in motion. Similarly, the principles of thermodynamics can be expressed mathematically, which enable us to understand how energy is transferred and transformed in physical systems.
Mathematics can only be used for stable systems over a small period of time. In the range of 1000 years.

In many cases, the use of mathematics has led to profound insights and breakthroughs in our understanding of the natural world. For example, the development of calculus enabled us to describe the motion of objects in a precise and systematic way, which led to the formulation of Newton's laws of motion.
So while mathematics is not strictly necessary to understand physical reality, it has proven to be an extremely valuable tool for doing so, and has played a crucial role in advancing our understanding of the natural world.
2. But suppose I don't want to use mathematics how is it possible to understand that the earth revolves around the sun.
It is possible to understand that the Earth revolves around the Sun without using mathematics, although it may be more difficult to do so without relying on some basic concepts of geometry and measurement.
If you want to understand anything observations and measurement are required. With a telescope it was discovered that the Sun is a planetary system. The difition is a larger object which is a surrounded by a set of smaller objects, all moving in the same plane. Using that definition Jupiter can also be called a planetary system
One way to understand the Earth's motion around the Sun is through observation. Over the course of a year, we can observe that the position of the Sun in the sky changes relative to the background stars. This can be explained by the fact that the Earth is moving around the Sun in an elliptical orbit.
Another way to understand the Earth's motion around the Sun is through analogy. We can compare the Earth's motion to that of a ball on a string being swung around in a circle. The ball represents the Earth, the string represents the force of gravity, and the hand holding the string represents the Sun. Just as the ball is pulled towards the hand and changes direction, the Earth is pulled towards the Sun and changes direction as it orbits.
While these approaches may not involve advanced mathematics, they still rely on some basic concepts of geometry, measurement, and observation. However, it is important to note that mathematics can provide a more precise and detailed understanding of the Earth's motion, and has been crucial in developing our current understanding of the solar system.
Mathematics can be used to predict the future trajectories of the planets, but not our understanding. These predictions assume that the trajectories are stable and that there are no collisions and merges. This also requires that each object is physical stable. After each of these events the physical state of each object is different; every case is different and no mathematics can be used to describe the future.

3. The movement of earth around the sun is ''stable'' and can be described by mathematics, but that mathematics does not improve our physical understanding.
The mathematics used to describe the Earth's motion around the Sun is an essential tool for understanding the physical behavior of the solar system. The fact that the Earth's orbit is stable and can be described by mathematical equations is actually a key piece of evidence for our understanding of gravity and the laws of motion.
That the orbit of the Earth's is stable is in line with the observation that the orbits of all planets are stable. Also with the observation that they more or less, all move in the same plane and what is the most important that their trajectories don't cross each other. In due time all objects that are not in agreement with this rule are eliminated. As a result of each physical collision, the solar system becomes more stable.
The use of mathematics allows us to make precise predictions about the behavior of the solar system, such as the timing of eclipses, the position of planets, and the effects of gravitational interactions between celestial bodies. Without mathematics, we would not be able to make such accurate predictions, and our understanding of the physical behavior of the solar system would be limited.
Furthermore, the use of mathematics has led to many profound insights and discoveries about the nature of the universe, such as the discovery of new planets, the understanding of the properties of stars and galaxies, and the formulation of the laws of thermodynamics and quantum mechanics.
While it is true that a physical understanding of the Earth's orbit can be gained through observation and analogy, mathematics is an essential tool for refining and expanding that understanding, and has played a crucial role in advancing our understanding of the natural world.

4. But that does not answer the question how stable the Earth and the Sun are. Are two small black holes stable?
The stability of the Earth's orbit around the Sun and the stability of two small black holes depend on different factors and cannot be directly compared.
In the case of the Earth's orbit around the Sun, it is stable because the gravitational force of the Sun, which attracts the Earth towards it, is balanced by the Earth's tangential velocity, which keeps it moving in a nearly circular orbit. This balance between gravitational force and tangential velocity allows the Earth to maintain a stable orbit around the Sun.
As explained before the stability is not caused by gravitational forces, but by the elimination of all instable trajectories i.e., the trajectories that involved a merge between a small with a larger object.
The stability of two small black holes, on the other hand, depends on a number of factors such as their masses, spins, and separation distance. In some cases, small black holes can form binary systems in which they orbit around each other. These binary systems can be stable for long periods of time, but they can also be unstable and eventually merge, releasing enormous amounts of energy in the form of gravitational waves.
This raises the question, is it possible that there is a special event which trickers this change in behaviour
In general, the stability of celestial bodies, whether they are planets, stars, or black holes, depends on a complex interplay of gravitational forces, orbital dynamics, and other factors. The behavior of these systems can often be described and predicted using mathematical models, but the specific details depend on the specific properties of the bodies involved.
It is one or the other. IMO the most important one, are the physical detailed properties and not mathematics.
5 Assume that two revolving stars and two BH's have the same mass and are at the same distance. Itheir
If two stars and two black holes of equal mass are at the same distance from each other, their stability would depend on their initial conditions and the dynamics of their orbits.
For two stars of equal mass, their orbits around each other would be determined by their mutual gravitational attraction. If their initial velocities and positions are such that they are in a stable orbit, they will continue to revolve around each other indefinitely. If their initial conditions are unstable, however, their orbits may decay and the stars could eventually collide or merge.

For two black holes of equal mass, their orbits around each other would also be determined by their mutual gravitational attraction. In this case, their orbits could take different forms depending on the initial conditions, including stable circular or elliptical orbits, or more chaotic and irregular orbits. If their orbits are stable, they will continue to revolve around each other indefinitely. However, if their orbits are unstable, they may merge together, releasing enormous amounts of energy in the form of gravitational waves.
This sounds like the definition: a stable orbit is a stable orbit and an unstable orbit is an unstable orbit. Such a definition does not make sense. The real question is: in case that two objects merged, after that event, is the newly formed object a stable or an unstable object? That question cannot be answered using mathematics.
It is important to note that the dynamics of these systems are complex and depend on many factors, including the masses of the objects, their distances from each other, their initial conditions, and the effects of other objects in the surrounding environment. The behavior of these systems can be described using mathematical models, but the specific details depend on the specific properties of the objects involved.
6 Assume that two revolving stars and two BH's have the same mass, are circular and are at the same distance. Are the speeds the same?
In the case where two revolving stars and two black holes of the same mass are in circular orbits at the same distance from each other, their speeds would be the same.
This is because the speed of an object in a circular orbit depends only on the mass of the central object (in this case, the stars or black holes) and the distance between the two objects. The speed of the orbiting object needs to be such that the gravitational force of the central object is balanced by the centrifugal force of the orbiting object. This balance is achieved when the speed is given by the equation:
 v = sqrt(GM/r) (1)
Where v is the speed of the orbiting object, G is the gravitational constant, M is the mass of the central object, and r is the distance between the two objects.
Since the mass and distance are the same for the two pairs of objects, the speed of the orbiting stars and black holes would be the same.
This raises the possibility that all blackholes of identical masses are as stable as two stars of the same mass
7. But if small black holes are stable are larger black holes than also stable?
The stability of a black hole depends on several factors, including its mass, its rate of rotation (spin), and its electric charge. In general, larger black holes are more stable than smaller ones, but there is a limit to how large a black hole can be and still be stable.
The stability of a black hole is determined by its event horizon, which is the point of no return beyond which not even light can escape the gravitational pull of the black hole. The size of the event horizon is directly proportional to the mass of the black hole. As the mass of the black hole increases, its event horizon also becomes larger, making it more stable.
I have doubts if an event horizon has anything to do with the stability of a black hole. The same is true for the fact that stars are visible and blackholes are not.
However, black holes can also spin, and the faster they spin, the more distorted their shape becomes, which can cause instability. Similarly, if a black hole has a significant electric charge, it can also be unstable. The most important is what is the cause of this increase in spin. When this increasing is caused by a nearby star, it is this nearby star which causes the instability and not so much the internal structure of the black hole.
There is also a theoretical upper limit to the size of a black hole, known as the "Chandrasekhar limit", which is approximately 1.4 times the mass of the Sun. Beyond this limit, a star can collapse into a black hole, but the resulting black hole would be unstable and would continue to accrete mass until it collapses into a more stable configuration.
In summary, larger black holes are generally more stable than smaller ones, but the stability of a black hole also depends on its other properties, such as its spin and electric charge.
8. The question is if a binary system of two larger BH's is as stable as two small BH's (each of the same mass)?
The stability of a binary system of black holes (BHs) depends on a variety of factors, including the masses of the BHs, their relative separation, and their orbital parameters.
It is important to consider the difference how stable individual BHs are versus the stability of BHs based on mutual influences. This are two important differences
In general, the stability of a binary system of BHs is determined by the balance between the gravitational attraction between the BHs and the energy loss due to the emission of gravitational waves. The more massive the BHs are, the stronger their gravitational attraction is, which tends to make the system more stable. However, more massive BHs also emit gravitational waves at a higher rate, which tends to make the system less stable.
My understanding that the gravitational field of two objects of the same mass is the same at identical distances from the centre of the 2 objects. They also both emit the same gravitational energy.
Therefore, whether a binary system of two larger BHs is as stable as two small BHs of the same mass depends on the specifics of the BHs in question. If the larger BHs are sufficiently far apart and have relatively low orbital velocities, they may be as stable as two smaller BHs of the same mass. However, if the larger BHs are closer together and have higher orbital velocities, they may be less stable than two smaller BHs of the same mass.
In general, the stability of a binary BH system depends on a variety of factors, and it is difficult to make generalizations without more information about the specific BHs in question.
9. The question is if two stars like R136a1 (300 M☉) have the same stability as two BH of the same mass
The stability of a binary system of two stars like R136a1 (300 M☉) would be different from that of a binary system of two black holes (BHs) of the same mass.
R136a1 is a very massive star that is believed to be close to the theoretical upper limit for stellar mass. Such massive stars have relatively short lifetimes and tend to evolve rapidly, undergoing a series of evolutionary stages that can lead to strong mass loss and significant changes in their physical properties.
In contrast, BHs are formed from the gravitational collapse of very massive stars and are characterized by their extremely high densities and gravitational fields. BHs do not undergo significant changes in their physical properties over time, and they do not lose mass through stellar winds or other mechanisms. Initially the size of these new born blackholes are small
As a result of these differences, a binary system of two stars like R136a1 would be expected to be less stable than a binary system of two BHs of the same mass. The massive stars would be more prone to mass loss, tidal interactions, and other effects that could destabilize the binary system over time. In contrast, a binary system of two BHs would be expected to be more stable, as BHs are not subject to the same types of evolutionary changes that can affect massive stars.
It is worth noting, however, that the stability of any binary system depends on a variety of factors, including the masses of the objects involved, their separation, their orbital parameters, and other environmental factors. Therefore, the stability of a binary system of two stars like R136a1 compared to two BHs of the same mass would depend on the specifics of the objects in question and their orbital parameters.
10. What this description means is that is more complex to predict stability of 2 objects using
Yes, that is correct. The stability of a binary system depends on a variety of factors, including the masses of the objects involved, their separation, their orbital parameters, and other environmental factors. Therefore, it is more complex to predict the stability of a binary system of two objects, such as two stars or two black holes, compared to a single object.
For example, in the case of two stars, their evolution and the effects of their mass loss and tidal interactions can affect the stability of the binary system. In the case of two black holes, their formation and the effects of gravitational waves can also play a role in the stability of the binary system.
To accurately predict the stability of a binary system, detailed knowledge of the objects' properties and the environmental factors that may affect the system is necessary. Mathematical models and computer simulations can help in understanding the stability of binary systems, but they are limited by the accuracy of the input parameters and the complexity of the interactions involved. One important question to answer if two rotating black holes of the same mass are more prone to merging as two identical rotating stars with the same mass. The equations 1 and 2 don't emphasize this behaviour. A much more important parameter to consider is if a third object is involved
11. But the equations which describe the stability of identical BH's are difficult to verify, because the BH's cannot be observed.
That is correct. The equations that describe the stability of binary black hole systems are based on general relativity and are highly theoretical. They predict the behavior of black holes based on their masses, spins, and separation, but because black holes themselves cannot be directly observed, it is difficult to test these equations experimentally.
However, the recent detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) has provided strong evidence for the existence of binary black hole systems and has validated some aspects of the theoretical predictions. The detection of gravitational waves from the merger of binary black hole systems has provided information on the masses, spins, and orbital parameters of the black holes involved and has helped to refine the equations used to describe their behavior.
Although the equations themselves cannot be directly verified through observation, the predictions they make can be tested indirectly through the detection and analysis of gravitational waves. This has allowed scientists to gain a better understanding of the behavior of black hole binary systems and has opened up new avenues for exploring the properties of black holes and the nature of gravity.

### Reflection 1 - ChatGPT - Introduction discussion

The purpose of this discussion is to understand the importance of physics versus mathematics, in order to understand the evolution of the physical reality.
IMO the most important part are the chemical processes that take place. This is specific true for all biomedical processes involving life. The same is true for the evolution of stars including blackholes.
Mathematics is important for star configurations, which are stable in the sense that no collisions take place. The same is true for binary star systems. The stars circulating around the Black Hole in the centre of the Milky Way, also fall into this category.
That is not the case during the total evolution of a star when it evolves in a red giant and finally in a black hole. This is also the case for a binary star system, when there is material transport between both objects.

IMO the most important way to understand the physical reality is by performing experiments, observations and measurements.
Regarding the evolution of these processes time is an important issue. Time in the sense that it takes time for different individual processes, which are taking place at different locations to influence each other. This influences can be in the form of material transport or as radiation. Gravity in that sense can also be considered radiation. Light as such is also radiation.
In order to mathematical describe the evolution of a process it is important to describe positions of the subprocesses as accurate as possible. In order to do that you need one coordinate system and set of fixed ideal clocks, linked to a grid. The problem is that each clock is also a process, and to take care that they all tick at the same rate, they should not move. This is very important if you want to calculate the speed of an object as used in equation 1 and 2. There is more to say about this subject, but I will leave that for later

### Reflection 2 - mass

Equation 1 can be used to calculate the speed of two objects in circular rotation around each other.
That same equation can also be used, based on observations during a certain period of time, to calculate the masses of these identical objects.
Like:
 M = r*v^2/G (2)
Using these equations, based on previous observations, the future of the objects can be calculated, assuming the masses are constant. In case the masses of the objects more complex equations have to be used. A different issue to consider is the number the number of objects involved.

The conclusion is that certain parameters can be calculated based on mathematics, assumptions and observations, but that is no quarantee that these same parameters and can be used to predict the future.

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Created: 12 March 2023