On the Correspondence between Mathematics and Physics

John Karpinsky May 24, 2020

There is an ongoing question in Physics as to why Mathematics describes the physical universe so effectively. Mathematicians are mystified as well. This paper is an attempt to show an approach that might illuminate this relationship and enable progress toward some answers to this question.

Definitions

In order talk about this subject, We need some new words to describe Mathematics. The problem with most presently defined words is that they assume a tangible substance to intangible entities. Each of these words defines intangible corollaries to tangible entities.

The words and their corollaries are:

Inistence — -> Existence

Inist — — - → Exist

Inistent — → Existent

I think the definitions of these words are almost self evident from their corollaries, but I will try to define them anyway.

Inistence: (noun) the fact of not having objective or physical reality or energy

Inist: (verb) have a non-objective reality; unable to be touched or grasped; not having physical presence or energy; not constituting a physical object

Inistent: (adjective) not having physical reality or energy

I am sure that these definitions can be improved on, but they are a starting point. As we can see, we have a noun, a verb, and an adjective. The substitution of (In) for (Ex) makes the analogy that I think is appropriate. Internal vs External makes sense to me.

Inistence

Numbers do not exist. They inist. Inistence is parallel to existence and linked in an intimate way. The world of inistence is in the world of information. Information is constituted of the structure of physical things, numbers, numerical relationships, thoughts, algorithms, ideas, and abstractions. The structure of physical things is separate from the physical things themselves. I don’t know what links existence and inistence but the hypothesis of this paper is that inistence is formative of the world of existence. Anything that exists follows the rules of inistence. There is no space or time in inistence. It could be a zero dimensional entity, or it could be an n-dimensional entity. The fundamental characteristic of inistence is that it is logically consistent, just as we try to make mathematics logically consistent. I believe that when developments are made in the study of mathematics, they are discoveries of the inistent reality rather than inventions that create a new inistent reality. This makes our existent world operate in a logically consistent manner. That is why mathematics often times describes existent reality or the Universe. There is much mathematics that has no known correspondence to anything physical. The main characteristic of that type of mathematics is that the mathematician tries to make it logically consistent. It could correspond to something physical that has not been discovered yet. It has happened many times that mathematical structures discovered by a mathematician are found to describe something physical much later.

There will be difficulty in finding words to describe this relationship because our languages are developed for describing existence rather than inistence. There will be times when a word that is used for the discussion of inistence that belongs to the realm of existence. No doubt that in time, we will coin new words for this newly defined realm.

Forms of Inistent Entities

Numbers are the atoms of Mathematics. Mathematics is the relationship between numbers. It is also the relationship between other abstract entities. Symbols are the atoms of language, and language is also inistent. Other forms of inistence are ideas, thoughts, algorithms and other abstractions. This is not a complete list.

There are different characteristics of these inistent entities. This will develop as we discover how inistence is organized. For instance, Mathematics, as I have defined it, may be causative in the existent world. That is the subject of this inquiry. But thoughts that are created by a thinker, and are not necessarily causative, but we must be open to this possibility.

Mathematics

I use upper case for Mathematics because I want to distinguish the inistent entity from the study of mathematics using symbols. Mathematics controls how the physical world operates. What I mean by Mathematics are not the symbols by which we describe mathematics, but the underlying principles and operations that are being described. We may need a new word for Mathematics, but I will use that word until I figure out a better word. Just as a word is not the entity that it names, a mathematical symbol is not the underlying principle. Mathematics just inists and has no time or place. As far as we can tell, it is eternal and unchanging. That is why Pythagorus and others thought Mathematics was spiritual.

Abstractions

Abstractions, on the other hand often are created to describe something in the physical world. We don’t know if they are causative, but they help us make sense of existence and inistence. As far as I can tell, all the inistent categories are either Mathematical or Abstractions. Ideas, thoughts and algorithms are all part of the abstraction category. Even when we do mathematics, we are doing it in abstractions. Mathematics is eternal and unchanging, and Abstractions are affected by the existent Universes. But why are Abstractions and Mathematics both included in inistence? It is because neither of them have energy. That is what puts them out of existence and into inistence. The laws of Physics are abstractions of the Mathematics that controls the existent realm. I did not include mass in the definitions because that would be redundant. Einstein discovered that and developed his famous equation: E = MC².

So what?

Why does this even matter? If we are trying to explore how Mathematics controls existence, we need to have a set of concepts that clarifies the different categories. So far, I have included in inistence all that has no energy, but has a definite effect on our universe, or is affected by our Universe. So that seems to be the defining difference.

Information Theory

So how can we approach the problem of finding out why Mathematics has such profound consequences in the Universe? One way to start is to identify the areas where Physics meets inistence. One such area is in Information Theory. One of the successes of Information Theory has been discovering the limits of the transmission of information from one place to another depending on signal strength and noise in a transmission system. I don’t know if this avenue of inquiry will be useful, but it might. The information that is transmitted has no energy, but the signal that carries the information does.

Black Holes

Some Physicists have the idea that information cannot be destroyed. This is a central question in the study of black holes. Stephen Hawking and Kip Thorn had a famous bet with John Preskill on that very subject. Stephen and Kip said that information could be destroyed, and John said that information could not be destroyed. I am not an expert on that, and I don’t know what is true. Hawking, famously, conceded that John was right and he was wrong, and paid off the bet. But, if information cannot be destroyed, that would be consistent with the idea that information and inistence is eternal and unchanging. This could be true, but that implies that all information inists always. This is consistent with the concept of 4-dimensional spacetime and General Relativity. I think that the confusion caused by not having a clear distinction between inistent entities and existent entities could be a reason why this problem has been so difficult to solve.

For instance, when something physical falls into a black hole, it carries information into the hole with it. Is that information destroyed? Also, the form of that information could be important. If it is one atom, the information is in the form of the structure of that atom, such as, the mass, the spin of the nucleus, the spin of the electrons, etc. If a book falls into a black hole, the information is not only the structure of the atoms and molecules of the book, but the information encoded in the words and sentences of the book. Is that information destroyed? Are these two types of information the same, or do we need another distinction between these forms of information? They are both without energy, but one was created with the big bang and the other was created by a thinker.

We seem to have two forms of information, but both are without energy. When it comes to the information that falls into a black hole, it makes sense that the structure of the atoms, electrons, and photons is conserved. They are wave functions, and wave functions are never destroyed. However, the information on the structure of the object that falls into the black hole is more problematic. It could be that, because the object is probably destroyed by the tidal forces at the event horizon of the black hole, information on such structures is destroyed, even though the structure of the particles that make up the macroscopic object is conserved. Also, if a book falls into a black hole, the structure of the book will also be destroyed, and any information written in the book will also be destroyed. This is comparable to the information that we transmit from one place to another here on earth with electromagnetic waves that can get destroyed when the signal to noise ratio becomes less than one. Another way to say it is that the information theory dealing with electromagnetic wave transmission is different than the information contained in a quantum particle/wave. The failure to separate these types of information from one another may be causing some confusion.

So, if I am right, There are two types of information. The information in a wave function is never destroyed. But the information created by the assembly of wave functions into macroscopic things can be destroyed, and is destroyed by that object falling into a black hole. But one does not need to go to a black hole. A book falling into a fire will have both the information contained in the book and the structure of the book destroyed.

Two Types of information

So, what is the fundamental difference between these two types of information? Things made by the activity of a wave function cannot be destroyed because of the conservation of energy. They can only be transformed by an interaction with another wave function. But the energy is conserved, and thus the wave function is conserved, even if it is manifesting differently because of the interaction.

But for macroscopic structures, there is no separate wave function. The structures are formed by the interactions of the wave functions that make up the Universe. The information into how these structures are configured has no separate wave function. These structures are very fragile, and are often changing. In electromagnetic signals, the signal is easily overcome by noise, and great care must be taken to get a signal from one place to another. Because the information in the signal, and the information in the structure of a macroscopic object does not have a separate wave function and has no energy. So, my hypothesis is that any information with no energy is destroyed when the object carrying that information falls into a blackhole. Only the wave functions are conserved. And since the tidal forces at the event horizon are extreme, the objects themselves will be spaghettified. So, it is not surprising that information in the structure of the object is not conserved. Even stars falling into a black hole are spaghettified. No information on how that star was structured is conserved. Only the wave functions of the constituents of the star are conserved.

Ideas

Ideas also are in the inistent realm and have no energy, but when put or discovered in the right hands, changes something in the existent realm. This is relevant whether the idea is to go to the grocery store, or an idea a physicist has for a new theory. All of our thoughts or feelings are inistent. The idea that inistence is eternal has some disturbing consequences for the concept of predestination, because it implies that predestination is true. Every idea or feeling or thought must happen because it is present in spacetime. So our ideas always inist if this is true. This has always been an implication in 4-D spacetime, and General Relativity. But, as pointed out before, inistence has no time or place, but is dimensionless. Maybe it is really 4 dimensional. This is another area to explore in trying to understand inistence. It may be that discovering inistence dimensional properties could be a key to finding out how the existent Universe is connected to inistence.

Quantum Mechanics

The development of quantum theory was a major advance in our understanding of the physical world. Quantum theory was developed with contributions from many scientists all during the 20th century. Irwin Schrodinger made major advances in the development of quantum theory. He developed a Mathematical structure called the Schrodinger equation. It uses a mathematical entity called a “wave function”. The all physical systems have a unique wave function which describes how particles will behave and interact through space and time. These physical systems can be very large such as the whole universe or very small such as an electron or atom. A scientist defines the physical system and uses wave functions to describe it. One could think that the wave function is the inistent entity behind the existent physical structure. I like that concept very much because there is one to one correspondence between the inistent reality and the existent reality. Remember that the symbols used to do Mathematics are not the inistent thing itself but represent the inistent thing. I think that we have the smoking gun here that shows a link between the inistent and the existent. How the inistent entity representing the wave function creates the physical structure is still a total mystery, but at least we now have a clue. But it would explain much about the link between inistence and existence. It starts with the Mathematical entity creating the physical entity. From that, we can see that it is necessary that the physical entities must follow or obey the Mathematical structures.

To illustrate that the symbols representing the Mathematical structures are not the thing itself, Werner Heisenberg expressed the same thing as the Schrodinger equation using matrices. It is possible the describe the same Mathematical structure using different methods and symbols. It is just that using the Schrodinger equation’s wave function correspondence to the physical system is easier to comprehend.

Conclusion

This is just the beginning of the exploration of inistence. We need to find out how the Mathematical wave functions create the physical universe. It looks like this just pushes the mystery of how Mathematics controls the laws of physics up one level to how how Mathematical wave functions create our universe. But it does explain why Mathematics dictates the behavior of the existent universe. If the inistent-existent correspondence hypothesis is correct, the ability to use mathematics to understand and control physical systems follows directly. Defining inistence should clarify the difference between Mathematical entities and the symbols describing them. It should allow us to discuss these things with more clarity. This should help us solve the mathematical mystery by giving us a language that does not muddy the scientific waters.

References

[1] Hamming, R. W. (1980). “The Unreasonable Effectiveness of Mathematics”. The American Mathematical Monthly. 87 (2): 81–90.

[2] Wigner, E. P. (1960). “The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959”. Communications on Pure and Applied Mathematics. 13: 114.

[3] Marolf (2017), “The Black Hole information problem: past, present, and future,” https://arxiv.org/abs/1703.02143

[4] Mathur (2012), “Black Holes and Beyond,” https://arxiv.org/abs/1205.0776

I am a retired industrial physicist interested in the fundamental physics of the universe.

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