Empirical Observations on Language

Abstract

Language is seen in different forms. Language and representations of knowledge using Language are identified. Structure and its inherent Duality are seen as fundamental to Language. Symmetry and Degrees of Freedom of a Language are introduced based on Duality of structures. Calculus is reinvented as Phase Transformations in Thermodynamics to govern changes between Fluid and Condensed Forms of a Language. For Mathematicians reading this: Language can be represented as composition of groups which may be further reduced to a representation in terms of simple groups.

Disclaimer

This is a write up meant for a lay person like myself. It used to be just a log for myself but I’ve had to turn it into this form as this time needs it to be (isolation + Phone and Laptop stopped working midway of typing my handwritten notes + felt sick and thought the end was near, but now I’m fine for a while and using a nearby laptop to type up my two notebooks).

I have never officially taken a course in linguistics. This write up may have nothing to do with linguistics. In fact, this may not have much to do with the usual sense in which languages are studied. I am interested in a broader category of Languages. I would like to use the word Language to be any structure used to represent knowledge. As in usual usage, when both language and knowledge are specified, a particular language will be used to represent the specific knowledge. The following note on terminology is for those with hard wired logic and maybe skimmed over for those who are comfortable with not so precise definitions, which will anyways become clear by the end.

• Terminology: If I may, at least for this write up, I would like to use the word Language to be any structure used to represent Knowledge. To clarify again: whenever I use the word ‘Language’ with a capital L, I mean anything that we use to represent knowledge. I loosely refer to anything specific (already represened in some Language) that we would like to represent (in another Language) as ‘knowledge’, with a small k.

• (Example). This means that relatively abstract knowledge like “the concept of a triangle” or “the concept of nothing” are not considered as ‘Knowledge’, but instead as representations within Language as they are defined by terms using the Language of Mathematics or some other Language. At the same time, these are considered as knowledge (with a small k) if they are to be represented by using relatively more specific Language such as the Language of Euclidean Geometry for “the concept of a triangle” or the Language of Set Theory for representing “the concept of nothing” as the empty set. This serves to demonstrate that

  • ‘knowledge’ and ‘representations in a Language’ are not exclusive. The same thing may be considered as knowledge or as a representation depending on the context. A commonplace example is: vapor may be condensed to liquid which may further be condensed to solid. In the earlier half, the liquid serves as the representation of vapor, whereas in the latter, it serves as the knowledge to be represented as the solid.

  • knowledge takes particular forms in a certain Language. This represented form exists as relatively abstract knowledge beyond the Language one sees it in, such as “the concept of triangle” as represented in Euclidean Geometry Language, exists in a relatively abstract form even in non-Euclidean geometries. The abstract form represented in a concrete Language corresponds to condensed form and the abstract form itself corresponds to the fluid form in the Language of Matter (in the previous example). More will be made clear through plenty of examples in the upcoming sections.

How it all started

Initially, I was more interested in the workings of my own mind, eliminating ignorance and knowing the nature of the self rather than the ways in which we represent what we know. However, since most of my interactions involved communication, I was interested in using these as a mirror to observe more about myself. Some years ago, I was reasonably satisfied with my initial quest (the observations can be verbalized by silence, but for some pointers, see Nirvana and Anattavada, they were through a philosophical approach), and what remained was to look at structures and their resulting dynamics. This is what led me to observe structures in language and communication. I used to say that given another chance, I would have studied communication in undergraduate and this quest has only strengthened my predisposition.

There are multiple forms in which the aspects of Language keep coming up. Here I highlight some which have appeared throughout the last few years.

In high school, I used to struggle with the notion of Physics as being the ultimate truth about nature. I felt that there were gaps in my teachers’ arguments. I never completely understood how Newtonian Physics explained everything about the world. For example, I never got a clear answer to what is mass and where does mass originate from? We kept doing mass balance in problems but there is no negative mass in the universe, the existence of the universe itself violated a much needed conservation principle. I thought, “What if these great scientists were also having gaps in their concepts? After all they are human too.”. Soon I was told about Einstein’s Theory of Relativity being better than what I had been taught in school. “Mass changes with velocity in Relativity”, they said. Well, what is to say this new theory is right either, I thought to myself? What if I reject the notion of mass altogether and come up with something new? What does it mean to understand nature? Can we ever truly know what nature is? All this time in high school, I was implicitly assuming that there is some true fixed nature out there which is waiting to be understood by humans. But how can we humans, who argue at the smallest of things, assert with certainty that the truth some person claims is the ultimate truth of nature? What if the smartest person in the world just convinces everyone else to accept what he says is true? That’s where science comes in as an empirical practice and protects objectivity. One has to verify the claims against experiments which are independent of who made the claims. How could Newtonian physics have been accepted when it was not the absolute truth? Well, it worked well for the experiments that they could do at the time. But soon there were experimental observations, Blackbody Radiation, Young’s Double Slit and Photoelectric effect to name a few, that couldn’t be captured by Newtonian Physics (Towards the end of this article, I refute this and attempt to show how Thermodynamics is enough).

I thought hard about why we should do theory. Zaheer Khan can swing the ball without knowing Bernoulli’s equations, we can predict how a pen will fall down from our hand onto the floor without knowing Einstein’s field equations. If all we can verify are the data points known by experiments, why try to extrapolate and formulate everything to obey some equations? What makes meaningful extrapolation of these data points possible in the first case? How can we be sure that there will never arise an anomalous data point to go against our general theory? There were much more deeper unanswered questions which I don’t go into here for brevity. These questions made me question Physics in ways that appeared as though I was a non-believer in science. I had the idealistic notion that even if I was a non-believer, the truth in Physics should be so compelling so as to undeniably convince me of it. Later I also questioned the act of verification by experiments. However, to my surprise, some anime came to the rescue and made me believe in the act of investigation by experiments. I started to visualize the act of understanding as a battle in which there is an enemy with unknown characteristics. If I did nothing and waited for the enemy to do something, I would either be defeated without knowing anything about the enemy or the enemy wouldn’t respond and I would still end up knowing nothing. The way to investigate the enemy was through experimental attacks from my side and analyzing how the enemy reacts. The battle was less about winning or losing but more about understanding the enemy. There’s constant observation and action. It is a dance. I borrowed this analogy of dance from cricket. I used to play professionally in high school and always thought of the game as a dance between the players through the ball and the bat. Now nature doesn’t seem different either, I started thinking of Physics as a dance between the mind of humans and the physical universe. If it wasn’t for this observation, I would have probably stopped thinking about Science in my second year of undergraduate.

On the more philosophical side, I had started to gather doubts about my own mind. When our mind is conditioned, we are only accepting certain systems of beliefs and certain forms of claims to be true. I saw plenty of misconceptions that I myself had, which convinced me of how my mind was restricted to understand things because of its conditioning. This led to the question, how can a conditioned mind understand the truths of nature? Isn’t it like a robot with faulty programming trying to do the job which should have been done by a perfect robot. Over the years this translated to the question of how can we, with our limited cognitive capacities, be able to completely grasp the objective truths of nature? What if the ways in which we experiment are also limited? Can we ever be clear of self deception?

For a while, I was trying my best to get rid of the faults in my mind but then I realized that I would never be able to do so. Imagine a faulty program trying to find methods to correct itself. In all likelihood, the correction itself would be faulty. There is no escape. I made my peace with it. I wanted to be the ideal dancer with no flaws who would completely match with any dance partner and adapt to any dance style, or the ultimate warrior who could understand any enemy that came in battle, after all nature seemed to be like this flawless absolute powerful enemy. But what if it wasn’t? What if nature itself was like any other dance partner that has flaws, that has its own limitations? This made me question if the so-called Laws of Physics were the absolute truth. What if nature itself didn’t have everything figured out? It could be possible that its laws are actually dynamic, that they develop and adapt as we do. What is the right compatibility for nature? What if we are supposed to observe and adapt to dance with nature? I immediately recalled how in the freshmen Quantum Physics class, solving the Schrodinger equation for Hydrogen atom was a mess in cartesian coordinates but suddenly turned into formulas when the coordinates were chosen to be spherical. What if Mathematical Physics was not the right Language in which to understand (dance with) nature? Maybe there was a Language which was better adapted to understand the physical universe.

I got very used to asking this question to any Physics enthusiast, student or faculty that I met. I got weird stares most of the time and unsatisfactory answers for the rest. I was surprised when one of my fellow labmates on an internship himself asked me a similar question: What if we will never be able to completely understand the universe? He continued to demonstrate how most of Physics had understood nature in terms of governing equations (laws) that couldn’t be solved exactly. Take Schrodinger Equation for example. We can only get explicit solutions in a handful of cases which we call toy models. In all the rest of the cases, the best we can do to understand nature is to solve them approximately by using some kind of asymptotic numerical method. This approximate solution is what bothered him. What if the theories were themselves approximate? What if we would always have some error term that keeps us away from the exact theory?

Once we establish that there does exist a way to represent a phenomenon mathematically, one may ask about whether it can be done in finite time or in a suitable finite dimensional parameter space or something else which is the complexity question. If the complexity question is answered by demonstration, it may be that we are capturing some approximate version of the phenomenon but not the phenomenon itself. But can we establish the representation question in itself? For example, recently Prof. Tao proved the Collatz conjecture for almost all numbers (in the logarithmic density sense), proving almost boundedness, by constructing an invariant measure on them but that didn’t get us close to settling the conjecture because the “almost all” doesn’t represent the complete dynamics of the Collatz map. Similarly mathematical models may be working for us to represent ‘almost all’ situations to make us feel they are an accurate model but in reality it may not be a ‘faithful’ representation.

I went down this track to study theories of knowledge and found nothing but ongoing debates in Epistemology (branch of Philosophy) and Philosophy of Science. This shifted my thinking beyond Physics. Following some ancient scriptures, I jumped back onto the idealistic stuff. Instead of the possibly imperfect universe, I had reasons to believe in the existence of an ideal truth which is independent of physical reality and an ideal mind which can gain this truth. Since this investigation was more focused on knowledge and our own mental structures itself rather than the particulars about the universe, I started studying Mathematics which seemed to have more absolute truths. For instance, look at Pythagoras Theorem. Once proved, there is no way that it changed with particular physical conditions. But soon I realized that the so-called theorems in Mathematics were just like the Laws encountered in Physics: they worked only in specific conditions. Pythagoras Theorem didn’t work in Non-Euclidean geometry. This made me see Pythagoras theorem as a tautology. It’s inbuilt in the way we define to measure distances. Soon all of Mathematics was a tautology. All that the equations did was change the form of the initial object to present it in some other way, all the while going along equalities which meant we never got anything new out of what we started with. Mathematics seemed no different than cooking. This was a naive view to hold and it has changed since then. I appreciate now that there can be extremely complicated behaviours of systems which can be simplified and stated elegantly using Mathematics. One of the reasons to write this is to illuminate this very fact. However, cooking itself turned out to be extremely complex and fun too as I got to know the first summer in which I had to cook myself. Making a round Roti stays till date as the most challenging tassk for me.

Differentiation and Integration

The investigation into thinking led to the following observations:

• Mental processes may be reduced to two fundamental processes- thought and abstraction.

• Thought is division. Thought is representing observations in particular form, ie. Language. Thought captures observations by naming. Creating Boundaries is needed to name things. Specifying something needs boundaries.

  • For example, to say something about “this” needs creation of a boundary which separates “this” from the rest. In the case of Real Numbers, we can talk about positive numbers only if we create a boundary stating that everything else is negative. Boundaries are needed to differentiate between things.

  • Mental structures or abstractions such as triangles can be distinguished from squares as one forms equivalence relations to define partitions among all Polygons which separate the triangles from the squares. Such partitions are boundaries differentiating all Polygons into various disjoint Subsets (which I will call Phases). In this sense, abstract thoughts (or structures) are creation of boundaries (Phase Boundaries).

• One may restate the above point as: thought is “Dualistic”, since creating a boundary implies creation of two sides. Positive numbers exist if only if negative numbers exist. In looking at one side, one misses out on the other. Division (Thought) destroys Symmetry. The same “Thing” gets Differentiated into two distingushable objects. Thought (Naming) Differentiates the two.

  • For example, in the process of Evolution, at one point, there’s only a single body but when it gets separated by a boundary of physical division, we name one as ‘This’ and the other as ‘That’. This symmetry breaking process gives rise to ‘Dual Phases’. Parent and Child both come into existence simultaneously. Positive and Negative, Higher and Lower, Left and Right, Rigid and Fluid, are all examples of Dual Phases.

• Abstraction is the “reverse” of thinking. Abstraction is dissolving the boundaries that were created by thought. Abstraction relases captured observations from particular forms (Language). Abstraction forgets naming, which requires dissolution of boundaries that create the names.

  • For example, even though the bodies may get separated by a boundary of physical division, but we share a common name for both the Parent and the child: the abstraction which dissolves the two different names (boundary) gives the notion of a Family. This symmetry restoring process Integrate the Parent and Child into one simultaneously.

• In other words, Abstraction and Thought are Duals to each other.

I would like to focus only on these two fundamental operations involving mental structures- thought and abstraction as explained above. To avoid much debate away from the focus of this write up, we introduce two new terms to replace thought and abstraction. Let us call the process of creating boundaries (thought) as ‘Differentiation’ and the process of dissolving them (abstraction) as ‘Integration’. Note that the capital D and I are used to avoid confusion with the terms from the corresponding operations used in the Language of Mathematics, which we will later see to be particular cases of the former. As said at the beginning, since any structure forms a Language, from now on mental structures shall be called a Language.

• Language Duality:

Consequences

• This thinking about thinking helped me to distinguish between psychological time and physical time. We can get caught up in mental structures for a long physical time. For example, while in a lecture, I may be Differentiating (creating mental boundaries) to represent Vector Spaces as a mental structure while the professor moves ahead with other geometric concepts in front of my eyes. Similarly, even though the examples that represent vector spaces may be clear to me, it may take time to Integrate (abstract by dissolving boundaries) how these examples all belong to the same ‘concept of a vector space’. Thereby, mental progress in time may be slower than physical progress in time. Hence I claim that psychological Time is emergent from structure. Differentiation and Integration are what create time. Only when one needs to Differentiate (represent something in a relatively rigid form) or Integrate (abstract out to relatively less rigid form), does one need time. This implies that observers with different Languages have different psychological time. What one records as Past depends on the Language in which one represents observations. For example: The memories of small sized organisms will be different than the memories of a human sized organism even if they are shown the same objects.

• In the absence of boundaries, there is no specific thing to think about. We need structures (formed by boundaries) to represent observations. Without these mental structures, there is no thinking, and hence there cannot be any psychological time. Time requires two distinguishable states for change to happen. This can take place only if there are divisions. In a space without divisions, there is no time since there is only one thing. One may say that Differentiation creates time whereas Integration collapses it.

• The investigation started with me trying to eliminate my ignorance but ended with me realizing the importance of ignorance. Unless we ignore something, we cannot think about something specific. Instead of trying to get rid of conditioning and dissolve boundaries, we must create the right structures by Differentiation and dissolve the unnecessary ones by Integration until we find the structure that is compatible with the specific thing that we want to represent. On the other hand, in knowing something specific, we also gain insight into what we have ignored. By knowing positive we know what is negative, by knowing light we know what is darkness. The differences come about only because of this division. If there were no two enemies, there would be no battle and hence no observations.

• The nature of the self cannot be known by thinking since there is division between the object and the observer. Whatever is observed cannot be the observer. The self is the observer which is beyond any observation.

• Complete Integration dissolves all boundaries, which causes end of thinking and hence end of time. In absence of any duality, there is only one. Nothing changes because there is nothing else to change to. In the absence of all boundaries there is ultimate symmetry. This cannot be captured by thinking because capturing is always in specific form. Hence there is no way to experience and/or record it. Complete Integration is not a finite psychological time process since dissolving all boundaries implies going beyond time.

• Boundaries in mental Structures create psychological Time and vice versa. In this sense, Structures and Time are dual notions just as positive and negative are. Integration of Structures and Time gives rise to a unified notion of Structuretime (this is similar to, the notion of spacetime referred to, in General Relativity).

Structure and Language

Now that we have reduced from mental process to boundaries and the resulting structures, let us investigate these structures and their resulting dynamics. I request that we replace the word ‘structure’ by ‘Language’ and accept to use them interchangeably for the rest of the write up.

• I shall use this opportunity to highlight my use of the word Language instead of structures for most of the rest of the write up. Usually structures make us think of very rigid objects. We often say that fluids don’t have much structure. This is what I want to avoid. Within the Language of Mathematics, there are structures like Sets which are less rigid than structures like Groups. But there are also Languages that are less rigid than the Language of Mathematics. The Language of English itself is one such Language. I want to address the structures that are within such less rigid Languages as well. In fact, I want to explore all kinds of structures, be it relatively rigid ones like Sets or those that are relatively less rigid like Fluids themselves. In a way, I am Integrating to unify the notions of structure and Language. This is motivated from my insight that Unconscious Thoughts or relatively unstructured thoughts can be thought of as a Fluid flow.

Following are some examples.

• (Example). In particular, structures that we identify through our cognitive capacity as belonging to the physical universe will also be called as a Language. Fluids and Condensed Matter are then both called as Languages, the former being less rigid than the latter. All the structures corresponding to the physical universe collectively form a structure which is considered as a Language, the Language of Physics.

• (Example). All boundaries that are built in a theory form a Language (structure). Number theory may be called as the Language of Numbers. Group Theory may be called as the Language of Groups. Type theory is a Language. Calculus is a Language.

• (Example). Music is a Language. Acoustic sounds also form structures that have multiple states that can be distinguished from each other.

• (Example). English, Hindi, Marathi, Latin, Greek are all usual languages and here they are also Languages.

• (Example). Mathematics is a Language. Symbols form structures and they are all part of Symbolic Language. Formal Languages are also Languages.

• (Example). Films are Languages. Visual structures whether in the physical universe or on a screen, form a Language. Similarly, Dreams are a Language.

• (Example). Any structure forms (or is part of a) Language.

• (Example). Chalk Sound forms a Language. The sound that is made when a Professor writes with chalk on board is a Language. I wonder if students pay attention to that, I find it interesting. It’s a form of Sound Language.

Theory of States of Matter- An Elementary Language

The simplest theory to pick and demonstrate the ideas involved in Language is probably the Language of States of Matter.

• Take a simple ( restrict to one dimension) system like Water which exists in different states of Matter. In this Language, Differentiation takes the form of Condensation and Integration takes the form of Evaporation. To keep consistency with the rest of the write up, we say that here Condensation plays the role of Differentiation and Evaporation plays the role of Integration. Starting from a Fluid form (water vapor or liquid water), Differentiation gives Solid form (ice) (hereafter I shall replace the word ‘Solid’ by ‘Condensed’). The Phase Boundary or the interface between the two is the Structure of this Language. Each form- Condensed or Fluid is a representation of Water in the Language of States of Matter.

• Differentiation in this Language gives the notion of Rigidity. The Condensed Language is called relatively more Rigid. We see that Differentiation (adding structure) leads to greater Rigidity whereas Integration leads to lesser Rigidity. Lesser Rigidity means less structure and hence the boundaries of fluids (gases and liquids) are not well formed. Instead of writing more rigid and less rigid, I shall write more Rigid and more Fluid.

• Differentiation destroys Symmetry, Integration creates Symmetry. This is because Differentiation is an additional boundary which differentiates between what should have been indistinuguishable.

• The Differentiated forms are now restricted by the additional boundary compared to its Integral form. This is decrease in the Degrees of Freedom. Integrated forms have less boundary which allows more freedom. This is increase in the Degrees of Freedom.

• Corresponding to every symmetry (or degrees of freedom) is an associated measure which gives its “amount (mass)” or “distance”. When the “amount” in one form matches that in another, they are in equilibrium. This is the behaviour that we want to see in all Languages. These are the common features (universal)- Structure as Duality or Language as a form of Duality, that we will look for.

• Thus we have seen Condensed-Fluid Duality:

The Language of Signs

Positive and negative.

The Usual Languages- English, Physics and Mathematics

• The above behaviour is analogous to the behaviour of the usual Languages that we know of. For example, English includes relatively loose definitions and vague statements and hence allows more fluid boundaries in its structure. Hence English may be considered analogous to Gaseous form Language. Physics is relatively more precise with defining quantities that it considers and hence has rigid boundaries which makes it analogous to Liquid form Language. Mathematics has the most rigid definitions and only allows statements that follow logic rules. Hence, Mathematics may be considered analogous to Solid form Language. I consider all these forms as states of the same Language and would like to show how Mathematics can be evaporated (Integrated) to give the Language of Physics up to some degree of freedom which in turn could be evaporated (Integrated) to give English Language. Conceptually it is clear that there are additional constraints in Physics than in English and similarly for Physics and Mathematics. There are also clear regions where these Languages ‘meet’. That is, most places where English and Physics meet are in the statements leading up to the mathematical formulation of various theories. There are also places where Physics and Mathematics meet: in the rigorously derived equations of Physics. In this sense, Physics is a two component phase Language formed by Languages of Mathematics and English. Hence it has a resulting rigidity that is in between those of its components. However, there are details that need to be shown one by one. That is the goal of this write up. In particular, what are the analogues of temperature and pressure that work as State determining Variables for these languages, what does the corresponding Phase Diagram looks like and how to formulate a Thermodynamics-type theory which determines the Phase Transformations in the context of these Languages remains to be seen. As we explore different Languages in the upcoming sections, these concepts will start taking much more concrete form. For example, In Physics, the Planck’s constant and the speed of light; the permittivity and the permeability, etc. are corresponding variation parameters to Temperature and Pressure (this will be seen shortly).

• Dynamics that appear in one Language may not appear in another Language that is different in Rigidity. In the Language of Matter, we see that Solids do not show all the dynamics that are seen in liquids and gases and this is true for any combination of two choices out of solid, liquid and gas. Difference structures give rise to different dynamics, hence the dynamics that can be seen in one Language such as English, namely those of emotions cannot be seen in another Language such as Physics or Mathematics that are more Rigid. Concepts like emotion and sadness are not allowed in the Rigid structure of the Language of Mathematics. A Language with more Rigid structure can be embedded in a Language with less Rigid structure. A Language with less Rigid Structure must be given Additional Structure (Differentiation) to be represented in a More Rigid Structure. Additional Structure means loss of Symmetry or Degrees of Freedom. This is similar to how Water needs to lose Degrees of Freedom (continuous translational symmetry) to be solidified to a Solid. The two Languages formed by Differentiation are hence Dual Languages to each other.

• Thus we have seen Math-English Duality

Language of Sets: The elementary Language in the Language of Mathematics

• In the Language of Sets, creating an Equivalence Relation (Reflexive, Symmetric and Transitive) plays the role of Differentiation and Quotienting Out plays the role of Integration. Creating an Equivalence Relation partitions a Set into many disjoint Phases with rigid boundaries, which corresponds to condensation (Differentiation) whereas the Quotient Set of a Set dissolves the boundaries and Identifies distinct elements of the Set, which corresponds to evaporation (Integration). The partitioned set is rigid like a solid whereas the quotient set is fluid like a liquid.

• Thus we have seen Partition-Quotient Duality:

The Fundamental Language of Thermodynamics- Phase Diagrams

• The phases involved in Thermodynamics of a single component system are:

• Here condesned form is called Solid and The Fluid Phase maybe further Differentiated to Liquid and Gas

• To create a picture of the different Phases (Solid, Liquid, Gas) and their states, we use what is known as a Phase Diagram. Phase diagrams are a ‘compilation’ of Phase Boundaries (Dualities) involved in a Language. The way to ‘compile’ various Phases is as w

• I’m assuming most people have seen the Phase Diagram of Water, which shows how the Different Phases: Ice, Water and Vapor exist with the Phase Boundaries representing the Partitioning with the Triple point being a point where all three- ice, water and vapour meet. Here is a similar Phase diagram:

• For a fixed Pressure, we see how Phase starts with Solid Phase at Lower Temperature (lower degree of freedom) then transitions into Liquid Phase by crossing the Black Phase Boundary Line and finally transitions into Gas Phase by crossing the Blue Phase Boundary Line. We may ignore the critical lines for now.

• Thus we have seen Phase-Phase Boundary Duality.

• The most common diagrams are the one, two and three component Phase Diagrams. Having more than one component which can mix with each other. The mixing percentage is itself a varibale and this means that there is increase in the number of Degrees of Freedom of the boundaries involved. For each Phase, the sum of fraction of components that are mixed together should add to 1, hence knowing the percentage of all but one component in the mixture determines its state uniquely. This implies that the Phase Diagram exists in dimension \(D\), where \(D= (#Components-1)\).

• Here is a two component Phase Diagram:

• In High School many of the function plots one sees in Mathematics are types of Phase Diagrams.

• The boundaries of each Phase represents the interface at which Phases on either side can come together to interact and stay at equilibrium with each other.

• For each Phase, we want to get a measure of how far it is from a Phase Boundary (Equilibrium) and this variable is called as The Gibbs Free Energy is used to Differentiate between Phases. When the Gibbs Free Energies of Phases are equal, they cannot be distinguished and they are in Equilibrium.

• Thermodynamics is an Empirical Language which measures the variable macroscopic physical quantities of materials. These include Temperature, Pressure, Volume, Entropy and so on. Thermodynamic Language captures the Duality that exists among these Variables so that all of these are not independet.

• We look at the Differentiation of Intensive and Extensive Variables. Intensive Variables do not change on scaling the system, Extensive Variables change on scaling the system. Intensive Variable is Rigid like a solid (Differential form) whereas Extensive Variable is Fluid (Integral form). Examples of Intensive Variables are Pressure, Temperature whereas those of Extensive Variables are Volume and Entropy.

• Thus we have seen Intensive-Extensive Duality:

• Energy Potentials in Thermodynamics are formed by Intesnive-Extensive Duality. The Intensive Variable is conjugate to an Extensive Variable. These conjugate Variables are composed together: side by side to form contribution to give the fundamental relation. This expresses the Potential in terms of independent Extensive Variables called natural variables. Here we take \(P, T\) to be the natural variables. Any set of natural variables can be obtained by Legendre Transforms which maintain the symmetry among all natural variables.

• When the total derivative of \(G\) is expressed using the multivariable chain rule, we obtain: \(dG = \frac{\partial G}{\partial P} dP + \frac{\partial G}{\partial T} dT\)

• Using the relations, \(\frac{\partial G}{\partial P}=V\) and \(\frac{\partial G}{\partial T}=-S\), we obtian: \(dG = V dP -S dT\)

• For Equilibrium, set \(dG =0\), to obtain the equation of Phase Boundary:

• This is of the form: Extensive Intensive = Extensive Intensive

• Maxwell’s Relation is obtained by taking Partial derivatives of the cross conjugate variables:

• A Component is a fundamental unit (irreducible) such as the Water Molecule which is common to all Phases in the Language. Water itself is reducible to atoms but that is part of another Language which is not the focus here.

  • (Example). Prime Numbers are the components in the Number Theory Language.

  • (Example). Elementary Particles are the components in the Particle Physics Language.

• A useful result is the Gibbs Phase Rule. This states a relation among the number of (#) Phases (like solid, liquid, gas)-\(P\), components (like two or more fundamental things which make up each and every Phase)- \(C\), degrees of freedom (symmetries)-\(F\) and variation parameters (Temperature, Pressure, etc.)- \(N\). The relation when each of them is finite is given by:

• When they are not finite, the much more fluid notions of Cardinality or Measure replace the rigid notion of number (#).

• We may check this relation in the Phase Diagrams depicted above. The first one was only for one Component- #Components (C) = 1. There are two variation parameters- Temperature and Pressure. Thus N=2. This determines the right hand side of the Gibbs Phase relation to give:

• This implies that in the region where there is only one Phase, P=1 means F=2, that is, every single Phase region has Two Degrees of Freedom in the diagram.

• In the region where there are two Phases together, P=2 means F=1, that is, a Two Phase region has one Degree of Freedom in the diagram. This is why Phase Boundaries have lesser Degree of freedom than the Phase in the Interior of these boundaries.

• In the region where there are three Phases together, P=3 means F=0, that is, a Three Phase region has no Degree of Freedom in the diagram and hence reduces to a point which cannot be observed. This is why the Triple Point has lesser Degree of freedom than the Phase Boundaries.

• Wherever there is interaction of Phases with each other, there is compromise in the Degrees of Freedom.

• Composition: Amount of each component that uniquely determines every point on the Phase diagram in the absence of any variational parameter.

• Lever Rule: Conservation of mass describing the Decomposition of a mixture of Phases to the Pure Phases at the boundaries that make it up.

• Terminology: For the rest of the write up I would like to use the words ‘more Fluid’, ‘less Rigid’, ‘more symmetric’, ‘higher Entropy’, ‘additional degree of freedom’ interchangeably as they are all various forms of the same knowledge.

• Thus we have seen Phase A-Phase B Duality

• Phase Diagrams are the Language : Since any Language is made up of Phases and Phase Boundaries, we see that their compilation into a Phase Diagram is Fundamental to all Languages. The Phase Diagram itself is all the Structure which we call the Language.

Calculus: The Language of Languages within the Language of Mathematics

• In the Language of Calculus, differentiation plays the role of Differentiation and integration plays the role of Integration. The derivative of a function is more rigid whereas the integral of a function is more fluid. This is seen by looking at the extra degree of freedom that the integral of a function has in terms of the integration constant. This additional degree of freedom may also be interpreted as an additional symmetry, that is the differential comes into existence by breaking the symmetry (differentiating between) the integral which unites the various functions.

• In general, Differentiation in any Language is adding structure, hence it increases Rigidity. In the geometric interoretation of Calculus, this is seen by looking at differentiation as condensing a Nonlinear form (Fluid) to its Linear form (Rigid), that is, a nonlinear curve gets condensed to tangent line of the curve. This notion of “tangent line” becomes more fluid (Integrates) and becomes the Gradient for surfaces and further generalizes (Integrates) to become the Jacobian Matrix in multivariable Calculus. Differentiating or Linearizing restricts the degrees of freedom of any Mapping to that of a Linear Map.

• In general, Integration in any Language is destroying structure, making it less rigid and hence it increases Degrees of Freedom. For any curve (1 Degree of Freedom), integrating the curve gives the Area (2 Degrees of Freedom) enclosed by the curve. Similarly, Integrating a Surface (2 Degrees of Freedom) gives the Volume (3 Degrees of Freedom) enclosed by the Surface and so on.

• The differential form corresponds to the condensed form whereas the integral form corresponds to the fluid form. Stokes theorem is analogous to the Phase Boundary between liquid and solid where they meet in the phase diagram. This equation depends on the Manifold with respect to which the integration is done. As another interpretation, Stokes theorem is the interface between the differential form (boundary) and integral form (interior). Thus, Stokes Theorem may be seen as a Phase Boundary condition appearing in Thermodynamics.

• For example, the differentiation of a non-zero degree polynomial reduces its degree by one. This means that Differentiation creates an additional rigid structure and loss of one degree of freedom each time it is done. Ultimately, one reaches the Zero Polynomial which is the Zero Entropy Language in the Polynomial Language.

• Thus we have seen Differential-Integral Duality:

• Mirror Symmetry: The Invariants of Calculus : Another example in the Language of Calculus is that of trignometric functions. These are special functions in the Language of Calculus since they are not affected by differentiation or integration. This is immediately seen by looking at Euler’s Relation (relating the Exponential function with the Trignometric functions) and the fact that the Differentiation of an Exponential function is itself with some scaling factor. Whenever something is not affected by Differentiation or Integration, it must be the case that it ‘exists on the boundary’, which is to say that they retain their symmetries (degrees of freedom) and that they cannot be Differentiated by the operation of differentiation. This is similar to Mirror reflection in which objects at the boundary stay as they are and the structure of an object (maps isomorphically) is retained in the reflection.

  • To see that they satisfy an invariance under Differentiation, look at the equation of the form \(d (f) = c (f)\), where \(c\) is some constant and d stands for differentiating. Integration of this equation gives that the solution is an exponential function (or it may be defined as the solutions). That is, Exponential Function is an Eigenfunction (name given to functions which satisfy such an invariance) under differentiation. To reinterpret, Eigenfunctions form the symmetries of the Operation (whether it is differentiation or something else). Using these Invariant functions to Represent other functions helps us to see what parts of the function will remain invariant under the operarion. If we can represent a function in terms of invariant functions, then these form the units of the given function which remain invariant under the operation. These fundamental units will be called the ‘Fundamental Invariants’ or the ‘Basis’ of the Langauge since they span the boundaries which create the entire structure of the Language. Trignometric Functions form the Basis of the Language of Calculus. If there exists a Basis, since all the Phase Boundaries of a Language (structure) are part of the Basis, any Phase in the ‘Interior’ of the Phase Boundary can be represented using this Basis. Such a representation of a Phase using Basis will be called ‘Composition’. The Phase when decomposed in terms of Basis boundaries, gives an ‘Amount’ corresponding to each boundary This is called Projection. Since the Basis Lie at the Boundary, they only have one degree of freedom. Any given Phase is formed by composing the degrees of freedom of the Phase boundaries surrounding it. (This is treatment of multi component systems under linear composition).

Thermodynamics and Calculus

We shall see how the Laws of Thermodynamics can be Reinvented in terms of the Language of Differentiation and Integration introduced.

• Zeroth Law as Differentiation: Zeroth Law states that if two thermodynamic systems are each in thermal equilibrium with a third one, then they are in thermal equilibrium with each other. Thus, Thermal equilibrium determines the Phase boundaries which Differentiate systems from one another. That is, thermal equilibrium is an equivalence relation on the set of thermodynamic systems which partitions them into classes labelled by a quantity called Temperature. This partitioning based on Temperature values is creation of boundaries and hence Temperature plays the role of Differentiation. Thermal flow takes heat from Higher Temperature system to Lower Temperature system. Lower temperature system corresponds to the solid form whereas higher temperature system corresponds to fluid form.

• Thus we have seen Low Temperature-High Temperature Duality or Cold-Hot Duality:

• First Law as Differentiation : The First Law states that the change in internal energy (ΔU), of a thermodynamic system is equal to the energy gained as heat (Q) and the thermodynamic work (W) done by the surroundings on the system. Thus, First Law Partitions the Internal Energy into Heat and Work. This Partition corresponds to an equivalence relation in the Set of Energy which we have already seen in the Language of Sets to be a Differentiation process. This makes Temperature a Differential form.

• Thus we have seen Work-Heat Duality:

• Second Law as Integration : The second law states that Heat cannot spontaneously flow from a colder location to a hotter location. If Heat could flow spontaneously from colder to hotter, it would further Differentiate the Cold from the Hot. This means that Heat flow plays the role of Integration by breaking the Differentiation between Hot and Cold systems that was set up by Temperature in the Zeroth Law.

• Thus we have seen Thermal Equilibrium-Heat Flow Duality:

• Third Law showing Entropy as an Integration : The third law states that as the Temperature of a system approaches absolute zero, all processes cease and the Entropy of the system approaches a minimum value. In all Languages, our intuition based on Physics of materials is that the most Differentiated (condensed) form is the closest to the Zero Language which is absolutely rigid: no degree of freedom whereas the most Integrated form is closest to Empty Language which has complete freedom. This intuition works because of the Third Law. This Law recognizes assigning Entropy values (or degrees of freedom) as the Dual concept to assigning Temperature (Differentiation) in Physical systems. Hence Entropy plays the role of the Integral form.

• Thus we have seen Temperature-Entropy Duality:

• I must add that heat flow being seen as a fluid (like a wave) in nature makes the Temperature (and hence Entropy) take continuum values. There is an additional duality in the relation \(E= k_bT\) where on fixing the Temperature, the Energy increases in discrete steps. This is a form of Wave-Particle Duality which is also seen in \(E= \hbar \omega\). When a particular Temperature or Frequency is fixed, one degree of freedom is lost and Energy loses its Wave-like character. This makes Energy act through the degrees of freedom provided by the corresponding Particles that are discrete. Particles and Waves are distinguished by their discrete and continuum degrees of freedom respectively. This Duality will be seen in detail soon.

Composition or Addition as a Language

• Composition is the process of compiling separate Structures into one. In Physical Language, it may be seen as Mixing. It may involve both Integration and Differentiation.

• One form of Mixing is Addition. Addition is a structure that can be imposed on Sets and hence it is a Language of Differentiation.

• In its most rigid form, Addition is seen as addition of integer numbers. This rigid form of addition is naturally inherited from the Language of rigid particles in Physics. Rigid Particles are discrete, like numbers. Adding numbers is counting together two groups of Particles. Addition of fractions is obtained without much change in the rigid form of addition: the only difference getting the denominators to match first so that the fractions can be added meaningfully. A fundamental change in this rigid form of addition is obtained by going from a discrete (rigid) degree of freedom to a continuous (fluid) degree of freedom. In other words, Addition from its Rigid discrete form gets Integrated to Addition in its Fluid form- which we call ‘Scaling’ or ‘scalar multiplication’.

• I realized this while teaching at an NGO to underprivileged kids. One of the bright kids asked me what Multiplication was and I quipped back with the usual “repeated addition” answer. She immediately asked how to add 4, \(\pi\) times and I was speechless for a whole ten minutes before realizing that the “repeated addition” does not have enough degrees of freedom to account for multiplication. The right answer should be Multiplication is just addition in a more fluid form, that is with a continuous degree of freedom obtained from the scalars (Real numbers) used for multiplying.

• Thus we have seen Addition-Multiplication Duality:

• In the case of Complex Numbers, Addition has both its Rigid and Fluid forms inherited componentwise (treating the real and Imaginary pats as independent) from the Real Numbers except that the Addition is even more fluid by virtue of Two Degrees of Freedom (plane) of the Complex Numbers compared to the One Degree of Freedom (line) in the Real Numbers. That is, Complex Addition and Multiplication have more degrees of freedom and are more fluid than Real Addition.

• Beyond Complex Numbers, Addition is also inherited componentwise to that in Vector Spaces (more popularly known as the Parallelogram Law of addition, that is fluid as the degree of freedom is that of the dimension of the Vector space. This fluid form of Addition is also the Language of Addition which is natural to Waves (Fluid-like) in Nature and hence results in ‘superposition’ as the Language form of Addition for Waves. So superposition of waves is the Integrated form of Addition of rigid objects. Which means, to ‘Add’ Particles, the Language to be used is Addition in its Rigid form- Addition of Real Numbers; and to ‘add’ Waves the Language to be used is the Language of Addition in its Fluid form- Vector Addition or Superposition.

  • For example, if we take two Real Numbers \(x\) and \(y\), the Rigid Addition gives as a result \(z\) which is a Real Number given by \(z= (x+y)\).

  • However, when we Complex Add a Complex Number to another Complex Number, say for example, \(x\) is to be added to \(y\), with angle between them being \(\theta\), the Fluid Addition gives as a result \(z'\) which is a Complex Number given by the Parallelogram Law of Addition, \(z'= x+y\) which has additional Degree of Freedom. This Additional degree of freedom is manifestly seen by comparing the Magnitudes of \(z\) and \(z'\). The Absolute value is \(\lvert z \rvert = x+y\) for Real Addition whereas it is \(\lvert z' \rvert = \left(x^2+y^2+ 2\lvert x \rvert \lvert y \rvert \cos{\theta}\right)^\frac{1}{2}\) for Complex Addition. The \(2\lvert x \rvert \lvert y \rvert \cos{\theta}\) Non-Linear term is what shows the Additional Degree of freedom in Complex Addition or Superposition, which is not present in Real Addition. This is exactly the term which makes Light Wave added with Light Wave to allow Bright and Dark zones in Young’s Double Slit Experiment. That’s just the Language of Waves. Quantum Mechanics is a more Fluid Theory obtained by Integration of Classical Mechanics to account for Wave Language of Matter, which is why Complex Addition (Fluid Addition) replaces Real Addition (Rigid Addition) and the complex number \(i\) enters the Schrodinger Equation. It is this additional degree of freedom in Complex Numbers which is what makes it useful to handle more Fluid Wave-like behaviours. The Language of Complex Numbers provides the Additional Degree of Freedom required to represent Wave Phenomenon.

  • I believe if we were to hypothetically give numbers such as \(1,2\) to birds and ask them to “add” them, they would say that the addition is \(1^2+2^2+2 (1\cdot 2) \cos{\theta}\).

  • We should see easily that a Language based on Quaternions (a + b i + c j + d k) which has more degrees of freedom than Complex Numbers can replace Quantum Mechanics as its Integral Theory and represent much more Wavelike behaviour. However such high Degree of Freedom also makes the Phase Diagram more complicated with many possible Compositions (mixing of Phases) that can take place.

  • In fact, Rigid Addition can be made more Fluid to higher degrees of freedom by Integrating it to Fluid Multiplication (Powers). That is, replace \(z= x+y\) by \(z=(x^n +y^n)^\frac{1}{n}\).

• Thus we have seen Addition-Superposition Duality:

• Addition can take various other forms:

  • In Sets, Language of Addition takes the form of disjoint union.

  • In Abelian Groups, Addition becomes word concatenation.

  • In Categories, Addition takes the form of coproduct.

  • Composition in Phase Diagrams.

  • Superposition in Waves.

  • Nonlinearity in Multiplication Powers

• Addition Structure can be reinterpreted as ways to Rigidly Partition a Countable Set. Corresponding to every Partitioning of a countable set, replace the Partitions by + signs. Unlike Fluid Addition in superposition which mixes waves and hence Entropy stays the same, Rigid addition decreases the Entropies by not allowing mixing to take place across the Rigid Partition.

  • Fermat’s Last Theorem for positive integers \(x,y,z\) and \(n>2\) involves the Equation: \(z^n = x^n + y^n\), which can be reinterpreted in this Language to say that a Highly Nonlinear Fluid (\(z^n\)) cannot be Rigidly Partitioned into Two Fluids of Equal High Nonlinearity (\(x^n + y^n\)). In other words, my intuition says that the Degrees of Freedom of Two Highly Nonlinear Fluids Added Rigidly are too low to for a Fluidly Added (completely mixed) Highly Nonlinear Fluid. In other words, the entropy of Left Hand Side scales as \(n\), whereas that of the Right Hand Side scales way less than n due to the Rigid Addition (which is a partition of the Nonlinear fluid and hence increases its rigidty, meaning loss of entropy). This relatively high entropy of a completely mixed fluid on the Left Hand Side makes it very difficult to Rigidly Partition the Integers (Countable) and yet maintain the entropy required on the Right Hand Side. As will be seen in the next point, Bringing the entropy of Right Hand Side to scale as \(n\) requires addition of \(n-1\) very nonlinear mixing terms. Without such entropy of mixing, the Partitioned Fluid on Right Hand Side will not be able to match the Entropy of the completely mixed Fluid on the Left Hand Side. This reasoning works as it we are limited to integers that are rigid.

  • Fluidized version of Fermat’s Last Equation: Suppose that instead of Rigid Addition, we were to do Fluid Addition of the Nonlinear Fluids involved in Fermat’s Last Theorem, then the Fermat’s Equation equation Integrates to a more Fluid Form: that is \(z^n = x^n +nx^{n-1}y+\ldots+nxy^{n-1}+y^n\) which gives \(z^n = (x+y)^n\) which is nothing but \(z= x+y\), independent of n for the case of positive integers. This is always satisfied since, any two positive integers \(x, y\) will always give another positive integer \(z\) on Rigid Addition. This shows why it is Natural for Fluids or Waves to superpose rather than add Rigidly. The non-linearity in their behaviour requires additional Degrees of Freedom to compose with each other.
  • The observation to keep in mind is that Fluid Form of a Language interacts with Fluid form of other Languages. As seen in the above example, Making the Fluid form (Waves) interact with Rigid form of another Language (Rigid Addition) doesn’t go well. This is exactly what is seen in the Language of Quantum Mechanics when both Particles and Waves need to be treated together.

Quantum Mechanics: Language of Particles and the Language of Waves

• The Language of Particles corresponds to the solid form whereas the Language of Waves corresponds to the liquid form. Stokes theorem or the Phase boundary appears in many contexts relating the Particle and Wavelike behaviours. One such form is the De-Broglie relationship between Particle and Wave forms. Here, in the Phase Diagram of Particles and Waves, the variation parameter is the Planck’s constant which corresponds to variation parameters such as temperature and pressure in the usual phase diagram of materials.

• Particles and Waves are distinguished by their discrete and continuum degrees of freedom respectively. This is seen in the relations of Energy with Mass, Temperature and Frequency as \(E= c^2 m, E=k_bT, E=\hbar \omega\) respectively. When a particular Mass, Temperature or Frequency is fixed, one degree of freedom is lost and Energy loses its Wave-like character. This makes Energy act through the degrees of freedom provided by the corresponding Particles that are discrete.

  • For example as Light Wave of one frequency interacts with Light of another frequency, they are both Wavelike (fluid) and hence add by superposition. On the other hand when Light of a frequency interacts with particles such as an electron, which has only discrete degrees of freedom (atomic levels that an electron can occupy are discrete), then it has to act in the Language of Particles. This makes the Wave assume Particle like character- a photon with discrete degree of freedom. Hence, relations of the sort \(E=\) constant \(\cdot\) Continuous , capture Wave-Particle Duality by giving Energy both - continuum degree of freedom (Wave-like) and a discrete degree of freedom (Particle-like).

Language of SpaceTime: The Language of Space and the Language of Time

• The Language of Time corresponds to the solid form whereas the Language of Space is analogous to the liquid form. Stokes theorem or the Phase boundary corresponds to the Spacetime Metric (structure that is imposed on the 4 dimensional manifold) or the interval equation that integrates Space and Time in the Phase Diagram. Here, in the Phase Diagram of Particles and Waves, the variation parameter are the metric values (including \(c\), the speed of light) which corresponds to variation parameters such as temperature and pressure in the usual phase diagram of materials.

The Language of Electro-Magnetism

• The Language of Electricity corresponds to condensed form whereas the Language of Magnetism corresponds to the fluid form when seen from one perspective- This is because Magnetic Language has more symmetry (which is additional degree of freedom) than the Electric field- namely the absence of Magnetic Charge corresponding to Electric Charge. Positive and Negative Electric Charge arises out of symmetry breaking. On the other hand, Magnetic Field does not have Time Reversal symmetry whereas Electric Field does. Stokes theorem or the Phase boundary between Electricity and Magnetism corresponds to one half of Maxwell’s Equations which relate Electricity and Magnetism by Differentiation and Integration. Or, the Elecro-Magnetic Faraday Tensor gives Integrates Electric and Magnetic Fields by having them satisfy the covariant formulation of Maxwell’s Equations. This also involves Wave-Particle Duality in the form of Fields and Charge which is given by the Gauss Law and its corresponding Magnetic version. They show that Charge is obtained by Differentiation (Gradient) of the corresponding Field. The variation parameters are Permittivity (\(\epsilon\)) and Permeability (\(\mu\)) corresponding to Temperature and Pressure in the Phase Diagrams. These get combined in the Maxwell’s Equations to give a single parameter- the speed of Electromagnetic Waves- \(c\).

Languages in the Language of Physics

• Classical Mechanics, Quantum Mechanics and General Relativity are all Languages (Phases) within the Language of Physics. There are Phase Boundaries (interfaces) where the Languages meet: the Correspondence Principle, where Quantum Physics and Classical Mechanics meet, and the Equivalence Principle, where General Relativity and Classical Mechanics meet. There must be Phase Boundaries in the Phase Diagram of Language of Physics. These Phase Boundaries are what I shall delineate and make explicit.

• I see General Relativity and Quantum Mechanics as relatively more fluid Languages obtained by the Integration of the relatively more rigid Language of Classical Mechanics. That is, General Relativity and Quantum Mechanics have more Degrees of Freedom or they have more Symmetry than Classical Mechanics. Schrodinger’s Equation Plays the role of Integration in the Language of Quantum Mechanics, whereas Metric unification of Space and Time plays the role of Integration in General Relativity.

• Quantum Mechanics is more fluid than Classical Mechanics as it does not differentiate between a particle (condensed form) and its corresponding wave (fluid form) to be two disconnected entities. To restate: Classical Mechanics is a Partial Derivative (with respect to Wave-Particle Duality) Language (theory) of Quantum Mechanics since Classical Mechanics treats particles and waves separately (symmetry breaking). Quantum Mechanics is an Integral form of Classical Mechanics since it allows Wave-Partile Duality or so to say restores the symmetry of particles and waves by not distinguishing between them as disconnected entities. This adds degrees of freedom to Quantum Mechanics that are not present in Classical Mechanics. In the Language of Calculus, this corresponds to the addition of degrees of freedom to the solutions of a differential equation by integration constants ‘\(c\)’. On Differentiation, of these solutions one looses the symmetry of ‘\(c\)’ and restricts to one particular value of ‘c’, which is specified by initial condition.

• General Relativity is more fluid than Classical Mechanics as it does not differentiate between Time (condesnded form, since it has less degrees of freedom than space) and Space (fluid form, since it has more degrees of freedom than time). To restate: Classical Mechanics is a Partial Derivative (with respect to Space-Time Duality) Language (theory) of General Relativity since Classical mechanics treats Space and Time separately (symmetry breaking). General Relativity is an Integral form of Classical Mechanics since it allows Space-Time Duality or, so to say, restores the symmetry of Space and Time by not distinguishing between them as disconnected entities. This adds degrees of freedom to General Relativity that are not present in Classical Mechanics. General Relativity also has an additional Mass-Energy symmetry (\(E=mc^2\)).

• The above two suggest that there exists an Integral Language (Theory) which integrates General Relativity and Quantum Mechanics. Either Integrating Space and Time that are treated separately in Quantum Mechanics or Integrating Waves and Particles that are treated separately in General Relativity will give this Integral Language. This theory must have both Space-Time symmetry and Wave-Particle symmetry.

• This Language is analogous to the Language of Electro-Magnetism.

• There the phase Languages were Electricity, Magnetism and Wave Optics.

• Here the phase Languages are Quantum Mechanics, General Relativity and Classical Mechanics.

• There the Intergral Language was Electro-Magnetsim (E-M), here let us call the Integral Theory (X) to be seen as Quantum-Gravity.

• Electro-Magnetism recognizes Electric and Magnetic Languages to be dual to each other, that is, they can be related to each other by Differentiation and Integration- changing (Differentiating) Electric fields produce Magnetic Fields and changing (Differentiating) Magnetic fields produce Electric Fields. This is slightly different from the previous cases in which a single Differentiation of a fluid form gave us the corresponding condensed form. This is because E-M involves two Variables- Space and Time (which themselves have a hidden Symmetry that we will see later).
  \(\begin{eqnarray} \label{eq:M1} \boldsymbol{\nabla} \cdot{(\boldsymbol{\varepsilon}E)} & = & 0 \quad \textrm{Charge-Field Symmetry: Charge is Differential form (condensed) of Field (Fluid)}\\ \label{eq:M2} \boldsymbol{\nabla} \cdot{(\boldsymbol{\mu}H)} & = & 0\\ \label{eq:M3} \boldsymbol{\nabla} \times{E} & = & - \boldsymbol{\mu}\,\frac{\partial H}{\partial t} \\ \label{eq:M4} \boldsymbol{\nabla} \times{B} & = & \boldsymbol{\varepsilon} \, \frac{\partial E}{\partial t} \end{eqnarray}\)

• Electro-Magentism recognizes the \(\begin{eqnarray} \label{eq:M1} \boldsymbol{\nabla} \cdot{(\boldsymbol{\varepsilon}E)} & = & 0 \quad \textrm{ Symmetry of Charge (condensed) and Field}\\ \label{eq:M2} \boldsymbol{\nabla} \cdot{(\boldsymbol{\mu}H)} & = & 0\\ \label{eq:M3} \boldsymbol{\nabla} \times{E} & = & - \boldsymbol{\mu}\,\frac{\partial H}{\partial t} \\ \label{eq:M4} \boldsymbol{\nabla} \times{B} & = & \boldsymbol{\varepsilon} \, \frac{\partial E}{\partial t} \end{eqnarray}\)

• Here, we have already seen that Differentiation of Quantum Mechanics (with respect to Wave-Particle) and General Relativity (with respect to Space-Time) gives the same Language- namely Classical Mechanics. Thus, corresponding to Electric (\(E\)) and Magnetic Fields (\(B\)), we have the Wavefunction (\(\Psi\)) and the Space-Time Metric structure (\(S\)).

• We must recognize the Wavefunction and Metric to be dual to each other, that is they can be related to each other by Differentiation and Integration, Changing Space-Time Metric structure (\(S\)) should produce changes in the Wavefunction (\(\Psi\)) and changing the Wavefunction (\(\Psi\)) should produce changes in the Space-Time Metric structure (\(S\)).

• In Electromagnetism (taking the case of charge free region), when we relate the Electric (\(E\)) and Magnetic Fields (\(B\)) by Differentiating the required symmmetry breaking and Integrating the broken symmetries, we get the Maxwell’s Equations which combine to give Electro-Magnetic Wave Equation. In Quantum-Gravity (taking the case of mass free region), we must relate the Wavefunction (\(\Psi\)) and the Metric (\(S\)) by Differentiating and Integrating (these operations will take a different form in this Language), we will get the corresponding Maxwell’s Equations which will combine to give Quantum-Gravity Wave-Type Equation.

• Here, \(i\) times Planck’s constant (\(\hbar\)) and Speed of Light (\(c\)) correspond to the variation parameters Permittivity (\(\epsilon\)) and Permeability (\(\mu\)) (corresponding to Temperature and Pressure) in the Phase Diagrams. These get combined in the Quantum-Gravity Equations to give a single parameter- the speed of Quantum-Gravity Wave-Types.

• The subtlety lies in handling the right symmetries in the right forms. Just like \(E\) and \(B\) in Electromagnetism, the Wavefunction (\(\Psi\)) and the Metric (\(S\)) are not directly related by a single Differentiation or a single Integration of symmetries, there are multiple symmetries involved. One may guess the corresponding equations, with the appropriate Space Differentiation forms instead of Divergence \(\boldsymbol{\nabla}\cdot\), curl \(\boldsymbol{\nabla} \times\) and appropriate Time Differentiation instead of \(\frac{\partial}{\partial t}\), to be:

• However Quantum Mechanics does not have Mass-Energy Symmetry. Moreover, unlike in the case of Maxwell’s Equations, the Schrodinger Wave Equation is not Lorentz invariant- no Space-Time symmetry. Quantum Mechanics is a Linear Theory whereas General Relativity is a more fluid Nonlinear theory. Hence we try to replace it with the corresponding relativistic Hamiltonian giving the Dirac Equation, which is where Quantum Field Theory (Fluid form of Quantum Mechanics) comes in. In any case, General Relativity doesn’t have a \(\hbar\) anywhere so it need additional equation to include Wave Particle Duality. Simply using the Dirac Equation to serve as this additional equation doesn’t work. This leads to incredibly messy amount of Phases boundaries which make it difficult to track which Phase Language is to be obtained by which symmetry. The degrees of freedom involved are: Space-Time (Linear and Nonlinear), Mass-Energy, Particle-Wave.

• There are real materials with extremely complicated Phas diagrams and I expect the one involving many symmetries such as those involved in Quantum Mechanics and General Relativity to be complicated. To go beyond these Phase Languages, and unify the Languages, one may have to abandon macroscopic observations and build a new microscopic Language by considering what are the irreducible features in each Language and increase the degrees of freedom. String Theory is one such Language. Instead of point particles that have no degree of freedom in Quantum Field Theory, String theory treats string-like objects which have one degree of freedom. I am yet to see a theory in which Mass and Charge are treated with symmetry, that is Charge-Mass symmetry.

• The variables Planck’s constant (\(\hbar\)) and Speed of Light (\(c\)) are the corresponding variables for temperature and pressure in the Language of Physics. These corresponds to the Gibbs Free energy type relations that have to hold at Phase Boundaries. Parameters like Temperature which appear in the equilibrium equation are the ones which can be varied to reach different regions in the Phase Diagram. This is why Quantum-Gravity effects are seen only below the Planck length or near Black Holes. One must also note that we have only seen the corresponding solid and liquid phases, a much more fluid Gaseous Language of Physics with more symmetry and much higher degrees of freedom may be yet to be seen or it may happen that it becomes the Ideal Gas type Language or even the Empty Language.

• General relativity seems to be more fluid than QM. I would estimate GR to be gas-like whereas QM to be more liquid-like. Technical details need to be worked out.

Language of Fourier Analysis

• Fourier Transform plays the role of Integration and Inverse Fourier Transform plays that of Differentiation. Fourier series provides fundamental forms of different Temporal Rigidity (frequency) so that a given function with varying amounts of Rigidity can be suitably represented by a mixture of these different fundamental forms. This gives rise to Representation Theory which Integrates (generalizes) Fourier Analysis to Harmonic Analysis.

Representation Theory as a Language of Languages within Mathematics

This brings into question representation theory: How much does the condensation map (Differentiation) capture?

• For instance, if a condensation map is a bijective linear map between two vector spaces, there is complete transfer of information and the two are identical except for changing the symbols used to write them. No change in rigidity, if we ignore the change in rigidity of the symbols involved. For Mathematicians, they are not concerned with the symbols used to represent these and only consider the relatively abstract things that they represent as important. Hence Mathematicians call this is an “isomorphism map” and not a true condensation map.

• For a condensation map to really be called a condensation map (a differentiation), it should increase the rigidity of the target to be more than that of the input. That is, decrease entropy of input. In this sense, the mathematical term “derivative” used to refer to the product obtained by differentiation is very well coined since it really captures the notion that the product of differentiation is just obtained by putting additional structure on the original, hence retaining the English notion of ‘derivative’ as something that is obtained by adding things to the original. The tangent map at Identity is a Differentiation (condensation map) which linearizes the Lie Group to its Lie Algebra. The exponential map which takes Lie Algebra to its Lie group is an Integration (evaporation map): it “delinearizes” the vector space. The linear vector space being determined by its basis is much more rigid than a nonlinear group.

• We see in representation theory that we can take nonlinear group operations which have complex dynamics and condense them to linear operators. This is pursued in a more general way through the Langlands Program in Mathematics.

Category Theory as a Language of Languages within Mathematics

Similar to Representation Theory above, category theory generalizes this idea and uses different rigidities in the target space: categories instead of absolutely rigid vector spaces. Since we can’t know the objects directly, we represent them in more rigid or less rigid forms to be able to control them better. Hence study of objects can be replaced by study of maps from the object to others.

Equivalence of Languages: Intrinsic and Extrinsic Geometry

Geometry and Topology are two phases of the same Language. Geometry is obtained by Differentiation of Topology. In this Language, Differentiation means adding Metric Structure. All theorems in Topology on Differentiation take particular forms in the Language of Geometry. However within Goemetry there are two approaches: Intrinsic and Extrinsic Geometry. Language of Intrinsic and Extrinsic Geometry were shown to be equivalent by the famous Embedding Theorems. Can we do the same for other Languages? Embedding a Language into another. Languages with relatively higher rigid structure can be embedded into Languages with relatively more fluid structure, that is to say that a rigid Language can be embedded into a fluid Language however the reverse is not true. In the most rigid Phase of a Language it has physical form as symbols existing in spacetime. This Language of Symbols is naturally embedded in Language of Physics. Hence all notions can be inherited from the Language of Physics onto the Language of Symbols. This means that the thermodynamic calculations involving entropy, volume and other quantities are all the same just by considering the Language symbols as part of the physical world.

Empty Language

Knowledge with a capital K that was referred to in the beginning may be seen as the empty Language- a Language with no form or structure. It cannot be visualized or represented in any Language. One may get arbitrarily close to it through Integration.

Empirical observations

• Differentiation and Integration are fundamental to creating structures. Hence, Calculus (Differentiation and Integration) is fundamental to Language.

• For each Language we can define a concept such as Rigidity. In other words, there is a Rigidity or the opposite concept of Entropy (the more thermodynamic term) that depends on the structure.
  • Languages have different sublanguages, that is, they have multiple different phases that make them up which may create a range of Rigidity or Entropy for a given Language. Just like in Nature, a fluid has a range of entropy but it is usually much higher than that of a solid. -For example, the Language of Mathematics is more rigid than the Language of English. That is, English Language has higher Entropy than Mathematics. On the other hand, Programming Languages are more rigid than the Language of Mathematics. Formal Languages are more Rigid than Natural Languages.

• In the limiting case of Languages, we obtain Empty Language which is no Language at all (is formless), and the Zero Language which is absolutely rigid or has No degree of freedom or zero Entropy. Any given Language lies between these two limiting cases. Hence, the Empty Language can only be obtained by Integration and the Zero Language can only be obtained by Differentiation. In a sequence or continuum of Phases of a Language with different rigidity, the Language closest to the Empty Language will be called the “Universal Integral Phase Language” or the “Limit Language” whereas the Language closest to the Zero Language will be called the “Irreducible Language” or the “Colimit Language”.

• Corresponding to each Language, there are dynamics which arise in the possible substructures within each particular Language. For example, in the Language of Numbers, there exists a Language of Prime numbers which represent their own particular dynamics.

• Conservation principles: Conservation Principles are seen as a result of symmetry. Stated in the Language of Mathematics, this knowledge is represented as Noether’s Theorem. The breaking of symmetry is what leads to formation of distinct states. This ensures that the ‘same amount’ (to be interpreted fluidly as the notion of ‘same amount’ changes depending on the Language) of both distinct forms are formed. Knowledge is conserved. Knowledge does not appear out of nowhere. Different Languages conserve knowledge that they represent in different forms. Specific knowledge can be represented in different forms using different Languages. In the Language of Particle Physics, conservation principle takes the form of particles and antiparticles.

• Each Language is observed as a phase. Study of transitions between Languages is nothing but the Theory of Phase Language transitions which is based on spontaneous symmetry breaking, which in turn is what the initial observations on thinking were based on.

• Language Duality: A Language cannot be isolated. That is, a Language cannot be only interfacing with the Empty Language. This is because if the Empty Language surrounds a Language from all sides, there is no way to interact with it through any other Language. Hence there is no way to Integrate or Differentiate such an isolated Language to a Language that is known. This implies that Languages appear in pairs. A Language may interface with the Empty Language on one side but it must interface with another Language that is not an Empty Language. That is, any Language has at least two interfaces, one with its Differentiated form and the other with its Integrated form (which may or may not be the Empty Language). This may be restated as: Any Language when Differentiated gives a Language that is not the Empty Language. This is equivalent to saying that Empty Language cannot be obtained from a Language by Differentiation, which is obvious since it can only be obtained by Integration of any other given Language.

  • (Example). The prime example of Language Duality is Electromagnetism Language. Electric Language interfaces with Empty Language (empty physical medium) but its other interface is Magnetic Language. Similarly, Magnetic Language interfaces with the Empty Language but it interacts with the Electric Language. This does not mean that these Languages don’t exist independently. Just like condensed and fluid phases of matter, both Electric and Magnetic Languages exist in their own right but they must interface with each other. The way to go from one Language to the other is by Differentiation and Integration. This interface between Electric and Magnetic Fields is given by one half of Maxwell’s Equations. The other half of these Equations correspond to another Language Duality: Wave-Particle Duality, which is similar to the Fluid-CondensedMatter Duality. The Electric Field is a fluid form of the relatively condensed Electric Charge and analogously for Magnetism. This other half of Maxwell’s Equation shows the Electric Charge as the differential form of Electric Field and analogously for Magnetism. The subtlety is that Magnetic Language includes an additional structure: that of Charge Duality in Local Space: for every Magnetic Positive Charge, there is a corresponding Negative Charge locally. Hence the Net charge over Space acts as an Empty Charge Language. So this Maxwell’s Equation says that the Differentiation of Magnetic Waves (Field) gives Empty Charge Language.

  • (Example). Calculus: Language of Differentiation is dual to Language of Integratiation. That is, Integration is a form of Differentiation. It’s just reversed. Loosely speaking, Integration is the mirror reflection of Differentiation. Or Integration is the Fluid form of Differentiation which is the condensed form. Another way to put it is: Calculus emerges from symmetry breaking, which is consistent with our observation that all Languages emerge from symmetry breaking. Differentiation is creation of condensed form by condensation and Integration is the creation of fluid form by dissolution.

  • (Example). Wave Particle Duality: Language of Waves (fluid form) is dual to Language of Particles (condensed form). That is, waves can be obtained by Integration of Particles, in other words Particles are the Differential form of Waves. Space Time Duality: Language of Space (fluid) is Dual to Language of Time (condensed).

  -(Example). Equations in Mathematics: All equations display how to unify the two different forms of the same thing- the left hand side and the right hand side. That is, equations restore the symmetry that is broken by imposing the boundary to distinguish between the left hand side and the right hand side. This is why symmetry pervades throughout Mathematics and Physics making Aesthetics inevitable in Science.

  Examples to keep in mind:

• Ice

  water

  vapour

  • Rigid

    Less Rigid, More Fluid

    Fluid

• Rigid (condensed)

  Fluid (Gas)

• Differentiation

  Integration

  • Calculus:

    differential form (condensed)

    integral form (additional degree of freedom in ‘c’, the integration constant).

  • Nature:

    Condensation

    De-solidification

  • Thermodynamics:

    Temperature

    Entropy

• Particle

  Wave

  • Charge

    Field

  • Mass

    Energy

  • Localized in Space

    Delocalized in Space

  • Momentum

    Wavelength (Space Frequency)

  • Addition of Real numbers

    Addition of Waves (Superposition)

  • Intensive

    Extensive

• Lower Degree of Freedom

  Higher Degree of Freedom

  • Time (1 degree of freedom)

    Space (3 degrees of freedom)

  • Reality

    Dream

  • Addition of Real numbers

    Addition of Waves (Superposition)

  • Boundary

    Interior

• Linear

  Non-Linear

  • Vector Space

    Group

  • \(x\)

    \(x^2\)

  • One degree of freedom

    More than one degree of freedom

  • One Symmetry

    More than one Symmetry

• representation

  knowledge

• Boundary

  Interior

  • Intensive

    Extensive

  • Pressure

    Volume

  • Temperature

    Entropy

  • Chemical Potential

    Particle Number

  • Force

    Displacement

  • Stress

    Strain

  • Differential

    Integral

• Programming Language (Code) or formal language

  Mathematics

  English or natural language

• Wave Optics, Electricity and Magnetism

  Electricity

  Magnetism

  • Waves, Electric field and Magnetic field

    \(E~q, \epsilon\)

    \(B~0, \mu\)

• Classical Mechanics

  Quantum Mechanics

  General Relativity

  • particles, waves, space, time

    \(\Psi\) (Wave-Particle)

    \(S\) (Metric Tensor) (Space-Time)

• Language of Time:

  Past

  Present

  Future

• Representation theory: Groups (nonlinear) condense to vector spaces (linear), Lie groups condense to Lie algebras

• Category theory: Functors between different categories. Differentiation Takes a Higher Category to a Lower Category. Forgetful Functor takes a Lower Category to a Higher Category.

• Cohomology theories: d^2 =0 is that boundary of a boundary is empty.

Fluid Flow to model unstructured thought and solid structures to model cognitive biases/representations/language.

Unconscious thought keeps flowing and seems to be unstoppable by rigid boundaries. By the intuition developed in terms of fluids, are there cracks in our mental structures? What about Godel’s incompleteness theorem? Are all axiomatic systems restricted to be incomplete? Rigid boundaries are formed by Condensation of this very fluid which it tries to contain. That is, conscious thoughts are condensation of flow of unstructured concepts. Language is condensation. Understanding language is evaporation. One must let the rigid structures of language or mathematical symbols to evaporate into abstract concepts which have fluidity. For example: addition is a very rigid operation when it is taught in school. Throughout the computations done to explain addition, there are rules to follow, algorithms and memorization like carry over of digits. Later there’s addition of fractions, then the concept gets more fluid to adapt it to addition of vectors. In category theory, fluidity is on display as addition takes several forms- in the category of sets: disjoint union, in the category of vector spaces: direct sum, in the category of abelian groups: word concatenations. Fluid takes the shape of its container, similarly the same abstract concept takes different forms based on the structures within which they appear. The more universal a language the less structure it has.

Representation and Modeling

• Representation through a Language is inherently limited. Only structures that are isomorphic to the Language can be represented by the Language completely. This means that all the boundaries that determine a structure must be replicated in the Language. However, such a Language can be modified to accommodate the given structure. Formation of condensed boundaries or evaporation of those not required requires infinitely slow equilibrium processes to be reproduced accurately. Anything else is a non-equilibrium process which leads to a structure which is not stable. Thus, representation of a thing is not the thing itself. Representation may capture essential aspects but never create an exact replica. Having different SpaceTime coordinates itself also serves to distinguish the two and hence not be the ‘same’ in all sense. In fact, creation of an exact replica of something must reduce the two (the thing and its replica) into one indistinguishable form. This process is analogous to creating a mirror which does not reproduce an exact replica but a just a reflected form of the object. Moreover, different observers (Languages) represent the external structures (Languages) in different internal Languages. Their particular forms are different but they all represent essential aspects of the same thing. The Duality of observer and object is maintained as long as they are separate. Duality is symmetry breaking which means that there cannot be equivalence of the observer and the object. Reinterpreted, this means that the observer and the object are different forms of the same thing. Take for example, Electricity and Magnetism or Positive and negative numbers. Classical Mechanics and General Relativity also represent different forms of Physics. Their interface (or mirror) is the Equivalence Principle where both Languages meet. General Relativity reduces to Classical Mechanics is like saying Liquid condenses to its rigid form. Representation helps us to see different forms of the same thing. Hence we may never be able to replicate the Physical Language of Nature, but represent it in a form (Language) which we call the Language of Physics.

Phase transformations

The study of Phase Transformations in the Language of Physics boils down to Spontaneous Symmetry Breaking. This is a universal Property seen in Phase Transformations of all Languages.

Fluids

Techniques from the study of Fluids in Physics may be imported to study the Fluid Phases of other Languages. This makes study of Fluid Dynamics very critical. Because of the enormous entropy (degrees of freedom or lack of structure) in Fluids, Navier Stokes Equations have been famously difficult to study.

Theory of structures as Languages

• Music is the same but it comes in different forms in different instruments. The operations are different but that doesn’t change the fact that the instruments all produce the same thing that is called music. We don’t argue about the existence or uniqueness of a thing called ‘music’ then why do we find it hard to think about “consciousness” or a universal fluid or whatever one may call the single abstract object which takes different forms? Why can’t we argue that sound and light are the same thing- waves just seen in different structures. They show different behavior, have different rules (satisfy different partial differential equations) but that’s just because they appear in different structures. Just as the concept of coproduct appears to follow different rules in different categories. What about forces of nature? Gravity, EM, QFT? All can be formulated as a field theory. Need to evaporate rigid structures of each theory. Within natural languages, we have English, Hindi, Marathi, etc., but all of them are languages and we can create a theory of language-linguistics. Similarly, all physical theories can be evaporated to a fluid abstract theory. Can we make a theory of condensation and evaporation itself? Theory of how theories condense or evaporate. Particular examples are : Quantum Mechanics (as a more fluid theory) with much greater Degrees of freedom which on condensation gives classical mechanics or General relativity which also on condensation gives classical mechanics. Lift up (that is evaporation) of a theory (structure) to another theory (structure) may not be unique: fluids can have different rigidities as in liquid or gas states. Similarly, general relativity and quantum theory may be the evaporation of classical mechanics to different degrees of rigidity. Phase diagram of theories: regions where theories meet. Thermodynamics is what governs the phase diagram of theories as well. We have all the concepts ready made in thermodynamics. Gibbs free energy, entropy, enthalpy. Where the Gibbs free energy of two theories is the same (that is, both the structures have the same free energy), we have stable equilibrium and the two theories meet each other. Thermodynamics is the metatheory of how theories (structures) behave. Calculus is an example of meta theory (form of thermodynamics) within mathematics. Differentiation and integration being the condensation and evaporation process respectively. The integral form has more degree of freedom (additional degree of freedom being encoded in the integration constant) than the differential form.

• Each phase of a Language has Sub-Languages which are sub-Phases. Differentiation and Integration takes a more rigid structure at the sublevel for SubLanguages. That is, Calculus as a Language itself can be differentiated to go from the Calculus of a higher level Language (more fluid) to a lower level Language (more rigid). For example: The differentiation operation on Distributions is more fluid than the differentiation operation on functions. In other words, The differentiation operation on functions is obtained by Differentiating the differentiation operation on Distributions.

• How to assign or measure temperature, enthalpy or entropy of theories? Every theory has its structure which is very much like a physical object. For assigning a scale, all we need is a way to compare the two. This can be done. In fact a theory may correspond to a phase rather than a fixed state. States of a theory may change temperature with time as the theory develops but only when there is a phase transformation, do we see a completely new theory (language). Set a standard reference and then start “measuring temperatures” with respect to this standard. For example theory of Lie algebras (differential) is at a lower temperature than the theory of Lie groups (integral). In physics, the triple point of classical mech, quantum mech and general relativity in the phase diagram where h (planck’s constant) and c (speed of light) are the two parameters corresponding to temperature and pressure, can be taken as a standard. This is just the theory of how phases transform. One must go to statistical mechanics to get a microscopic theory. Theory of quantum phase transitions may also be analogous to something.

• Atoms of a Language: Integrated forms of the concept of atoms exist in all Languages as the “Irreducible Language” or the “Colimit Language”.

• All forms of Language interact with each other at the interfaces. Just as Vapor, Liquid and Solid can be in dynamic equilibrium with each other and interact with each other through interfaces, soul interacts with our mental body which interacts with our physical body. The interface of soul and our mental body is difficult to capture because of its fluidity but the interface of mental and physical body is seen in the brain and to some extents in various parts of the body. Just as condensation of Liquid to solid needs the right thermodynamic conditions, Differentiation and Integration need the right analogous conditions at the interface. Our brain has the right conditions for Differentiation of mental body to the physical body and Integration of physical body to the mental body. Humans have an extraordinary capacity for Differentiation and Integration (Language of Calculus) which separates it as a species from the rest. The rest of the world seems to have only a small range of structures (range of Languages) that can be created by them.

• Mental world is Integration of Physical world. Spiritual world is the integration of Mental world. Physical world is solid, Mental world corresponds to Liquid and Soul corresponds to Vapour. One may be able to Integrate or Differentiate further. For example, Mental structures can be Differentiated to Physical bodies. Our physical bodies are not as rigid, and evolve over time. Further differentiate to give genes or DNA which are much more rigid. Language of Religion refers to these Phases as Hell (rigid like a solid and has least degree of freedom, Earth (liquid which is in between solid and gas) and Heaven (which is fluid like a gas and has the highest degree of freedom). Similarly, Fields can be differentiated to give Elementary Particles that are way more rigid. Wave-particle duality in Physics is the particular form of the statement that knowledge and representation are not exclusive, that is the same object acts as differential form or an integral form depending on the Language in which it is represented (method of observation).

• Time is a language. Past is solid, Present is liquid and Future is gas. Time is emergent from Structure. Observation involves representation of outside in a compatible Language (structure) inside. We start with the Structure that is compatible with our internal Structure (Language) and then evolve to Integrate and Differentiate our internal Structures accordingly to match the other dynamics seen in the external Structure. This is analogous to Heat flow in which a heat flows from regions of higher temperature to regions of lower Temperature until there is a match (equilibrium). The corresponding quantity to Heat flow here is Time flow. That is to say, in our own terms what is immediately representable becomes condensed into rigid memories as the Past. The one which is slightly more fluid relative to our mental structures, but about to be condensed is the Present (liquid). That which is relatively most fluid is the Future. This implies that observers with more rigid mental structures than one observes more rigid structures earlier than one. Similarly, observers with more fluid mental structures than one observes more fluid structures earlier than one.

• Universal Calculus: Every Language has forms: Solid, Liquid and Gas. They are related by Integration and Differentiation.

• Recording is Differentiation. Differentiation leads to compression. More structure means more boundaries are being made. These are formed by condensation of fluid. Fluid takes more volume because of its fluidity. On condensation, the volume decreases and we have compression. The condensed form has different dynamics than the fluid form. For example, the abstract thoughts we have are fluid but they are condensed into symbols which show different dynamics than the fluid thoughts. Hence knowledge (fluid) and its representation (condensed) each have their own existence relative to one another.

• Recording does one half of the mental process. That is, condensing the observations into a form is done by recording. However one needs evaporation to enable the representation to be flexible like a fluid and adapt to different forms. Calculus involving both Differentiation and Integration is what makes thinking complete. It is the Duality of Calculus which allows humans to make Languages with such a broad range of rigidities that correspond to diverse Structures.

• Mathematics is a natural language with more rigid structure than English or some other natural language, in the sense that the definitions and ways of arguing are more rigid. In English, approximate knowledge plays a very important role and that is highly subdued in math. There is some sort of Structuretime Duality going on here- the more we want to make a concept independent of time, the less independent it can be in structure, that is, it has highly well defined structure in space (localization)- one could say this is like uncertainty in fourier analysis (or its corollary Heisenberg uncertainty in QM). So math uses a more rigid approach to definitions than usual language. In this sense, the dynamics of english language beyond that rigidity cannot be captured in math, similar to how proofs in formal language logic don’t capture the intuition in mathematical natural language based proof.

Consequences

• Not all Dynamics that are available to occur in fluids can be seen in solids. By analogy we see that one particular language cannot capture all there is to see. For any given Language, one may relax the structure slightly to obtain a more fluid concept. The rigidity of Languages affect their scope of capturing dynamics.

  • More fluid Languages like English allow us to talk about things like emotions- love, sadness etc. which cannot be defined mathematically, math being rigid cannot capture these emotions and hence cannot answer questions pertaining to them either.

• The reverse is also true: all dynamics in solids are not seen in fluids.

  • So the kind of dynamics we see in mathematical proofs are not available in English or a more fluid language. This allows us to see that Mathematics doesn’t model the world as an exact but instead, it condenses everything in rigid terms: condensation is just decreasing entropy by restricting Degrees of Freedom to be able to observe better. We try to condense everything into rigid terms to be able to capture and control it: the amount of randomness in a fluid is too high to be captured or controlled.

• An interesting observation here is that of scale. On large enough Time scales, solids are actually no different than a fluid. Glass (amorphous) is considered to be a solid (compared to a usual liquid) just because of its relative rigidity and kinetics at human timescales. It actually flows just like a liquid but the rate of flow is very small (it takes millions of years to observe significant mass flow). So an observer who observes once every million years would call glass as a fluid. Similarly in mathematics, although the language is relatively rigid, concepts like continuity, addition/product, have evolved slowly over the years to reveal their fluidity. Time scale is itself an Observer (Language) which sees different things from different points. This is Duality: what is Differential at one scale is Integral at the other.

• The Unreasonable effectiveness of Mathematics is not unreasonable at all: Mathematics is just the appropriately condensed version of relatively fluid ideas which otherwise stay abstract knowledge. Mathematics is prevalent in sciences because it has the rigidity to allow capturing, analyzing and manipulation and then it allows fluidity of those concepts to generalize. It is this control over condensed matter that makes reproduction and transport of ideas easy and the fluidity of concepts that allows it to be applicable so widely. Actually, logic (or formal language) is more rigid than mathematics. Why don’t we use this for controlling information? It takes much more time for condensing/evaporation by the human mind. The rigid aspect may be better for computers but not for humans. It takes way more time to reproduce. That’s exactly why coding is not a natural language. Similarly, within Mathematics, Linear Algebra is prevalent because we want to formulate everything in the most condensed form. A linear map is known completely by the values it takes on the basis. This rigid vector space structure allows highly condensed forms of representing knowledge.

• I believe there’s more rigid languages too which haven’t been discovered yet. In physics, an analogue would be the Bose-Einstein condensate. I think if we can find an analogous way to form a ‘bose einstein condensate type of language’, it will be a breakthrough in compression technology (and hence artificial intelligence- http://prize.hutter1.net/, https://en.wikipedia.org/wiki/Hutter_Prize, https://analyticsindiamag.com/hutter-prize-data-compression/). It certainly has to do with information entropy. This also raises the question of the scope of quantum computers as they have additional dynamcis due to non-abelian statistics of anyons.

• How do we think about abstract concepts? Approximate boundaries: There is no strict delineation of the notion of a chair. One may vaguely say that chair is something we use to sit on. But then, can the floor be called a chair? How do we generalize the concept to a situation that we have never encountered before? It’s by allowing fluidity of concepts. Another example ‘addition’ in categories- direct sum, disjoint union, etc. Instead of having exactly defined boundaries, approximate notions that are like fluids can also form part of our knowledge.

• Observing the abstract form of something may not help in seeing everything about it’s particulars just like Observing the dynamics of a fluid form of a substance may not help in knowing the dynamics in its solid form.

• Mathematics is a rigid language in terms of the rigidity of allowed boundaries but it has sublanguages that may be quite fluid in terms of the low amount of structure: for example, category theory assumes very less structure compared to usual natural languages. As a result, it can be used to represent relatively abstract knowledge quite well. The behaviour of gaseous states (most fluid) of all matter is very similar since all have less structure and high flexibility- they all behave like an ideal gas. Hence the most fluid Language of Mathematics is very close to an Ideal Gas and can be used to represent most abstract knowledge. I am not sure if Category theory or any other such Language of Mathematics is as close to the Ideal Gas like Language form. For instance, I do not expect it to be able to represent dynamics of human emotions. The boundaries involved must be very fluid to allow such fluid knowledge. On the other hand, the Language of Poetry seems to represent human emotions quite well.

• One one hand, knowing the abstract conditions is enough to know all possibilities. An example is dynamical systems: One may understand all the rules of chess but becoming an expert chess player requires detailed study of structures that may arise within play. Details within a simple framework of rules may be much more complicated than one may think. Think about classification problems in a category. Groups, Topological Spaces, etc are hard to classify. Probability and ergodic decompositions. Complex flows arising out of Navier Stokes, quantum dynamics from Schrodinger equation, piano compositions from key notes, billiard flow dynamics. Starting with rigid boundaries doesn’t restrict their interior to be rigid. The rigid boundaries evolve to fluid structures inside these boundaries. In a Phase Diagram, the boundaries of a Phase have less degree of freedom than the interior region of a Phase. Or to put it the other way, the boundaries are formed by condensation of the fluid phases to allow a rigid interface between them. Similarly, mathematical equations that correspond to these boundaries are rigid and compact but they allow a vast amount of fluidity to arise within those boundaries.

• A single Phase of a Language is a SubLanguage. For example English is a phase. It interacts with more fluid Languages such as Mathematics. For English Language as a Phase, one can see the different states that it takes: the state of English in text messaging is more fluid than the state of English used in a Book or a Journal Publication. It is most fluid in Poetry. These different states may give a feel of Pressure and Temperature of English changing in different States.

• The Integrated forms of Entropy and Temperature may be easier to detect in rigid Languages like Information Theory (Look at Shannon Entropy for instance) or formal Languages. On the other hand, Pressure and Volume will be easier to detect for fluid Languages like English/Latin.

• This view of Mathematics as a Language answers the question of whether Mathematics is invented or discovered. It is both. To condense a fluid (universal) concept in a particular structure, one feels constructive hence closer to invention whereas when one evaporates (Integrates) a rigid concept to a more fluid (universal) one, it feels as if it was already there and it was just discovered. Calculus is the Language of Duality.

• Exterior Differentiation is the Integral form of operations found in Vector Calculus: Gradient, Divergence and Curl.

• Zeta and Mobius functions in different structures.

Exercises

• Thermodynamics as Calculus : The Language of Thermodynamics becomes the Calculus of all Languages in this interpretation. To know the regions in which different forms of a Language meet, one must do a Thermodynamical calculation and formulate the corresponding conditions in terms of Macroscopic Observables of a Language.

• Statistical Mechanics : The Language of Thermodynamics is an Integral Language. The theory to explain these starting from Microscopic laws is given by Statistical Mechanics. Statistical Mechanics type theory would be the general microscopic theory for Languages. Statistical Mechanics is the Differential form of Thermodynamics. Every Statistical Mechanics Law has a corresponding Dual Law in Thermodynamics.

• Mass : Protons and electrons don’t have the same mass, all the words or each letter in the alphabet don’t have the same mass, how does one define an inertial mass like concept for a Language? What are the indivisible units of a Language? Prime numbers in Language of Number Theory? Elementary Particles in the Language of Particle Physics, Quantum Fields in Quantum Field Theory Language? See others for yourself.

• Entropy : Does information theoretic equivalent of Entropy work for information languages? How to generalize (Integrate) this notion to the rest of Languages? This is done by seeing Entropy as an Integral form corresponding to Temperature.

• Temperature : How to define Temperature for a Language? Is the analogue of Heat flow, symbol flow in written Language? That is communication between two people stops when they both have the same rigidity in their Language of contact. In the case of Language of Nature, this reduces to the usual heat flow and temperature notions. In terms of talking about abstract knowledge, is it possible to define that a person can communicate with someone else the abstract knoweledge only if they are both able to represent it in the same Language property called “temperature”. ‘Successful Communication’ would correspond to equilibrium of the zeroth law of thermodynamics as no more communication would be required after that point.

• What Type of Phase diagrams are allowed in a Language? For Matter, there are rules like the Gibbs Phase Rule which is easy to formulate for some other Languages. What forms does Thermodynamics take in different Languages? Integrated or Differentiated Stokes Theorems in different Languages correspond to their Phase boundaries. Gibbs Free energy equivalence is the kind of Stokes Theorem in Language of Matter. Integration of Differential forms (the enthalpy and Entropy expressions that are differential forms) inside the domain (vapour) should equal the Integration of Differential forms on the boundary (that is the Liquid). The type of Phase Diagrams that are allowed are the dynamics that we see in Nature and any Language.

• How to define the State of a Language? Look for universal (Integral) Properties (corresponding to Volume, Pressure Temperature, etc.). All Phases of a Language will exhibit some universal properties which can be used to define the state of the Language. These Universal Properties must be “irreducible”, that is, they cannot be Differentiated further to get a Non Zero Language since they appear as the rigid Boundaries of that Language.

• What are the Generalized Laws of Universal Thermodynamics? They are Fundamental Laws.

• What about masless structures like Light (photons), EM fields? Energy to replace mass? How to do thermodynamics on Light (Bose Statistics)?

• Music as a Language: there should be an analogous concept of mass and entropy in the Language of Music. Is Vocal Units the corresponding concept for number of moles. Volume is the size of time interval (duration of sound). In recorded form, the Language is different and the corresponding size of a recorded music is its file size. In physics, the frequency determines energy (which is equivalent to mass) of a vibration, is it the same for Language of Music? How does this connect to the Language of String Theory? Why can’t they do the same in Math? Or maybe they do and I haven’t seen enough Math to feel emotions.

• Language can be Interpreted as more than Physical Symbols: There are physical symbols to represent everything in a compact form. These symbols are the most rigid aspect of any Language. However they can be used to represent extremely fluid abstract knowledge such as “nothing” (in Sets as null, in numbers as 0, in Euclidean Geometry as Origin, etc). Symbols may be the most condensed form of any abstract knowledge. The word emotion is as rigid as any other symbolic word in English but it represents quite a fluid concept, at least its boundaries in my mind are not at all rigidly defined. What about information rate and density, no. of words used, syllables, coverage of meaning and vocabulary?

• We see in representation theory that we can take nonlinear groups and condense them to linear operators. Similarly, Category theory generalizes this idea and uses different rigidities in the target space: categories instead of absolutely rigid vector spaces. Since we can’t know the objects directly, we represent them in more rigid or less rigid forms to be able to control them better. Hence study of objects can be replaced by study of maps from the object to others.
• Bose Einstein Condensate would be ultimate compression of entropy
• What about plasma and superfluids?
• Size compression may not be intelligence but instead entropy compression.
• What makes a solid a solid?

• How to assign rigidity to a language? Does relative entropy compression act as a good measure?

• How do we see Integrated forms of Symmetry in Language? Fluids have continuous Translational symmetry. Condensed forms have discrete Translational symmetries and may have continuous or discrete rotational symmetries. These are fundamental to structures and should be seen in any Language.

• The Universe Thinks and Communicates : If we consider the structures arising in Nature as Language, then it certainly thinks and communicates, the Language which we best adapt to understand its Language is that of Physics. General Relativity and Quantum Physics seem to have been found as the Languages that are the condensed form of the Language that Nature primarily uses but there are quite a few areas (such as the Black Holes and Early Universe) in which a different fluidity of structure is required to match Nature’s Language. At least this interpretation, matches my idea of Nature being a dynamic partner in dancing with us. Nature evolves its Language just as we do.

• Language and Dynamics: How uniquely is the dynamics defined by a Language? How to know the limits of what kind of dynamics can a Language represent? Music theory seems pretty good for representing emotions, which a rigid language like Mathematics fails to do. I wonder why. In some sense music notes are rigid too but the way they are played probably adds the flexibility in the Language. Why can’t we do the same in Mathematics? Or maybe we are not looking for such dynamics in Mathematics.

• Our Mental structures are evolutions of Physical Structures: Most concepts we have are rooted in some form of Physical origin. The symbols we use to represent Language are physical. This makes me conjecture that all the mental structures we create have evolved by Differentiation or Integration of physical structures. If this is true, Physics should be the easiest Language to formulate. Or maybe that is what makes it so hard, since we perturb the natural Language of Physics in the process of intellectualizing it. Or could it be that the Language of Physics that we formulate is just semantics of the Nature’s Language?

• Monkeys outperform humans in Cognitive flexibility: Monkeys don’t use Languages that are as rigid as human natural Langauge. The more rigid the Language we use in thinking, the less flexible we can be.

• Physics is a two component Language made up of English and Mathematics.

• Composition may be seen as a form of Differentiation in the Language of Binary Operations. Addition, Mixing, Composing, Bilinear maps, etc. all lead to more boundaries. What about superposition of two Fluid Languages? Superposition is an Integrated form of Addition.

• Given a theory, can we find its integral? Treating the given theory as a vector field use Calculus to get the Integral theory.

Empty Language

Knowledge with a capital K that was referred to in the beginning may be seen as the empty Language- a Language with no form or structure. It cannot be visualized or represented in any Language. This is the Identity.

Theory of States of Matter - An Elementary Language Revisited

The simplest theory to pick and demonstrate the ideas involved in Language is probably the Language of States of Matter.

• Take a simple ( restrict to one dimension) system like Water which exists in different states of Matter. In this Language, Differentiation takes the form of Condensation and Integration takes the form of Evaporation. To keep consistency with the rest of the write up, we say that here Condensation plays the role of Differentiation and Evaporation plays the role of Integration. Starting from a Fluid form (water vapor or liquid water), Differentiation gives Solid form (ice) (hereafter I shall replace the word ‘Solid’ by ‘Condensed’). The Phase Boundary or the interface between the two is the Structure of this Language. Each form- Condensed or Fluid is a representation of Water in the Language of States of Matter.

• Differentiation in this Language gives the notion of Rigidity. The Condensed Language is called relatively more Rigid. We see that Differentiation (adding structure) leads to greater Rigidity whereas Integration leads to lesser Rigidity. Lesser Rigidity means less structure and hence the boundaries of fluids (gases and liquids) are not well formed. Instead of writing more rigid and less rigid, I shall write more Rigid and more Fluid.

• Differentiation destroys Symmetry, Integration creates Symmetry. This is because Differentiation is an additional boundary which differentiates between what should have been indistinuguishable.

• The Differentiated forms are now restricted by the additional boundary compared to its Integral form. This is decrease in the Degrees of Freedom. Integrated forms have less boundary which allows more freedom. This is increase in the Degrees of Freedom.

• At the Phase Boundary (which is the interface), both the Condensed and Fluid forms meet. There is “Equilibrium” at the Phase Boundary if there is “No Differentiation”. That is, the Language of the Condensed Form matches the Language of the Fluid Form. To reiterate: The Structure of Condensed Form must match the Structure of Fluid Form for “Equilibrium” at the Boundary. Since there is no way to Differentiate between the two Structures at Equilibrium on the Phase Boundary, there are no Boundaries of this Language (Structure) within the Phase Boundary at equilibrium. Thus, when restricted to a Phase Boundary at Equilibrium, within itslef there is No Differentiation of Phases in this Language. Simply stated, this means that one cannot distinguish between ice (Condensed Phase) and Water (Fluid Phase) at the interface where there is Equilibrium between the two phases. There maybe sub-Boundaries and sub-Phases of the given Phase

• The act of differentiating once creates a boundary and further differentiating within the equilibrium boundary gives nothing (no further differentiation is possible). Since Differentiation in general, decreases the Degrees of Freedom, it must be the case that the final differentiation is not possible since the Degrees of Freedom have already all been reduced to nothing. That is, the Equilibrium Language at the Phase Boundary must be the most Rigid Structure before all its Degrees of Freedom vanish on Differentiating further and it becomes the Language with no Degree of Freedom (Absolutely Rigid Language).

• To capture The above Paragraph using symbols, let \(=\) represent the words ‘is the same in Structure as’ let \(d\) represent differentiation particular to the Language whose Phases are being distinguished by it, let the symbol \(0\) denote the Language with no Degrees of Freedom with respect to this Language, that is the absolutely Rigid Language, let the symbol \(E\) stand as a variable for all the various possible Phases within this Language, and let the symbol \(E*\) denote the Equilibrium Phase at the Phase Boundary of the Language. In symbols the above paragraph says:

  • \(- + = 0\) or \(-dd+ =0\)

  • \((d E)_{E*} = 0\).

  • The first equation above, holds as a property of the Language of Differentiation. That is, be it any Language (Structure), one cannot create a Boundary of that Language within a Boundary already made in that Language. This is clear since, any sub-Boundary within itself within two Phases on either side, any other boundary which also Differentiates the two or some other structure (Language), it always holds. The second

• (As a sidenote) The above paragraph is an example of a language differentiation process which takes English Language and reduces its degrees of freedom by compressing it to rigid symbols. That is, a statement of some length (one dimension) was reduced to a Rigid symbol of zero dimension, which is a loss of degree of freedom. This differential form of English Language is called Language of Mathematics (this will be seen later).

• To clarify: this doesn’t mean that there is no structure in equilibrium at the phase boundary, there is an Equilibrium Phase (\(E*\)) at the Boundary which is a structure (compilation of boundaries), just that with respect to this Language (Structure), there is no further differentiation (boundary) possible at equilibrium. Or, there is no way to create another boundary of this Language within the Equilibrium Phase at the Boundary. To state in other words, differentiation (the particular form of Differentiation within this Language) of the Structures in this Language within the Phase Boundary is not possible, which is equivalent to saying that the Difference between two Phases of this Language doesn’t exist at the Equilibrium Phase boundary.

• Let’s Start with some given Phase (SubLanguage) of a Language. We may obtain another Phase in this Language by Differentiation, that is increasing boundaries (decreasing degrees of freedom) or Integration (increasing degrees of freedom). The more the degree of Differentiation, meaning more boundaries created in a structure, the lesser there are degrees of freedom in a Language. This is going away from Equilibrium. The more the degree of Differentiation, meaning more boundaries created in a structure, the lesser there are degrees of freedom in a Language. This is going away from Equilibrium. This is why the degree of Differentiation in one Phase must Match the degree of Differentiation in its Dual Phase. This takes the form of Maxwell’s Equations or Stokes Theorem in Calculus.

• Corresponding to every Phase (or degrees of freedom) there are as many ways to partition the Energy. This is what is known as Equipartitioning of Energy. The Measure of Energy given to each Dual Phase across the Boundary must be equal however they take different forms. This is the behaviour that we want to see in all Languages. These are the common features (universal)- Structure as Duality or Language as a form of Duality, that we will look for.

• Thus we have seen Condensed-Fluid Duality:

Intensive-Extensive Duality

• I shall reinterpret The Language of Thermodynamics in terms of Language Duality introduced above. The Language of Thermodynamics Differentiates Physical Variables into Intensive and Extensive Variables. In this Language, Intensive Variable and Extensive Variable are Dual to each other.

• Intensive Variable plays the role of Differential Form (Condensed) whereas Extensive Variable plays the role of Integral Form (Fluid).

  • For example, Pressure is an Intensive Variable (Condensed Form). Volume is its corresponding Dual Extensive Variable (Fluid form). Volume when Restricted by a Boundary (decrease in Degrees of Freedom) gives Pressure. Or, Pressure is obtained by Differentiation of Volume. Volume is obtained by Integration of Pressure, to give it additional Degrees of Freedom.

  Pressure

  Volume

  • For example, Temperature is an Intensive Variable (Condensed Form). Entropy is its corresponding Dual Extensive Variable (Fluid form). Entropy when Restricted by a Boundary (decrease in Degrees of Freedom) gives Temperature. Or, Temperature is obtained by Differentiation of Entropy. Entropy is obtained by Integration of Temperature, to give it additional Degrees of Freedom.

  Temperature

  Entropy

  • Force-Extension

• The “Amount of Mass” in Thermodynamics is just the Energy. All Laws of Thermodynamics say that Energy is conserved no matter what forms it is divided into.

A small Push is all it needs for Gravity to take one towards Equilibrium

(This is incomplete and currently being updated regularly, please be patient for the final version)

Duality

This section can be read without trying to make much sense of it. At the end, one can come back and go through it again.

• Identity is.

• Identity is not.

• Neither of the statements above make any difference since Identity is and is not. Identity is Everywhere and Nowhere. One can proceed with either statement or no statement at all and it makes no difference, as we shall see later. Since existence or non-existence of Identity does not make any difference, please allow me to make the following statements about Identity which amount to nothing.

• Identity cannot be captured entirely and yet all we end up doing is capture Identity entirely.

• Identity is Dual to Duality.

• Identity is a Differential form of Duality and Duality is an Integral Form of Identity.

• Duality is Creating and Destroying Boundaries to Partition the Identity.

• Every Language (Structure) is based on Duality.

• Boundaries and their corresponding Dual Phases are the Language (and that’s all there is to a Language).

• Boundaries

• A Boundary and its corresponding Dual Phases together form a Composition.

Composing Energy-mass and Space-time gives Einstein’s Identity

• The origin in the above diagram is the Space-time Identity, that is \(c^2\). Einstein’s Identity connects Energy-mass as another symmetry with \(c^2\) as the Identity.

The Identity by Einstein \(E=mc^2\) is seen as a composite of Energy-Mass Duality and Space-Time Duality, where \(E\) is Energy, \(m\) is mass and \(c\) is the Speed of Light. We rearrange the equation as

Recognizing the multiplication operation (\(\cdot\)) as \(compose^i\) and the division operation (\(-\)) as \(compose_i\), that is, multiplication is identified as dual of division, we see

The two expressions on either side of the “equal to sign” are recognized as reflections of each other. Bring them on the same side of \(=\) by Identifying them with the orientations \((-) compose^d\) and \(compose_d (+)\) (which are individually identity but duals to each other), to the two duals on the right hand side. They are brought to the left hand side side as:

Recognizing Speed of light as expression of Space-time Duality at equilibrium, that is \(c_0= \left( \,\,(Space)\,\, compose_i \,\,(time)\,\, \right)_0\) and \(c^0= \left( \,\, (Space)\,\, compose_i \,\,(time) \,\, \right)^0\), (in this case \(c_0=c^0=c\)), we see

In symbols, Recognize Energy as \(E\), mass as \(m\), Space as \(X\), time as \(t\), and compose as \(\circ\), we see

• For keeping track, in usual language, the above equation reads as \(\frac{E}{m} \left(-\,\, (\frac{X}{t})_{eq} \,\, (\frac{X}{t})_{eq} + \,\,\right) = 0\), where \((\frac{X}{t})_{eq} = c\).

Back to where we were, since composition here is associative, we may drop the brackets (paranthesis) one by one, starting from inner to outer, that is from the signs \((-)\) and \((+)\), we see

Now removing the brackets from \((X)\) and \((t)\), we see

Further peeling gives

In Dirac’s Bra-Ket notation, this is

This is the most reduced form which is equivalent to \(\left(\frac{E}{m}\right)(-c \cdot c+)=0\) or \(\frac{E}{m}-c \cdot c+=0\).

• This is interpreted as: Energy-mass Duality and Space-time Duality are dual to each other. The Energy-mass duality axis is Orthogonal to the Space-time Duality and they meet only at the Light Cone which is where Space and Time are at equilibrium with each other. Hence \(E=mc^2\) is a two Phase Equilibrium between (Energy-mass) as one Phase and (Space-time) as another Phase. It may be reduced further to Energy, mass, Space (\(X = (x_1,x_2,x_3)\,\,\)) and time as its components.

• The Space-time Metric is what decides the amount (measure) of how far from Equilibrium the Space-time fabric is. The Light Cone in the resulting Phase Diagram gives the Equilibrium Phase Boundary between Space and Time as the Phases. Einstein’s Field Equations are the corresponding Maxwell’s relations for the Thermodynamics of Energy-mass and Space-time.

• We see that \(E = mc^2\) is a special case (Equilibrium) of a general Thermodynamic Potential Energy \(E\), which may be defined as \(E = mc^2 \,\,-x_1^2-x_2^2-x_3^2 + c^2t^2\)

• In the Language of Group Theory: \(c^2\) is the identity, \(-\) (Division) is the group operation. Energy (\(E\)) and mass (\(m\)) are inverses of each other with ; Space (\(X\)) and time (\(t\)) are inverses of each other.

  • That is, \(\frac{E}{m}= c^2\) and \(\frac{X}{t}= c^2\). If we Identify \(c^2\), the identity with the symbol \(e\), then this reads \(\frac{E}{m}= e\) and \(\frac{X}{t}= e\). That is, Energy and mass are reflections of each other and Space and time are reflections of each other.

  • The other operation \(+\) with 0 as Identity allows defining Energy to be the other identity. This makes the group into a Field.

The above representation as a group only happens at equilibrium. When there is no equilibrium, the measure of how far from equilibrium is given by the Thermodynamic expression \(E= mc^2-x_1^2-x_2^2-x_3^2 + c^2t^2\), where \(E\) acts as a functional on \(m\) and the metric (-,-,-,+) acts as the Integral to compose space and time as duals. This composition of a Group with addition and subtraction to give a thermodynamic type scalar quantity, and multiplication to give inverses with \(c^2\) as the Identity can be represented as a Field. A field allows addition and subtraction; multiplication and division. Thus, Energy-mass and Space-time can be seen as corresponding to Field of Complex Numbers (Provided all three dimensions of space are identified as one). In other words, Complex Numbers represent the Phase Diagram of Energy-mass and Space-time with the axis as their Phase boundaries.

• The above may seem to be dimensionally wrong but I request us to be patient and Identify all forms of energy as mass and all forms of distance as mass as well to see all equations as an Equilibrium Mass Balance.

If Energy and Mass are identified, and Space and Time are Identified, the above graph of a two dimensional Euclidean Plane is seen as two Circles orthogonal to each other about a common axis. This represents the Globe, with Space-Time as the Equator and Energy-Mass as the Prime Meridian. Their meeting point at the South Pole as the Identity which is called \(c^2\), whereas the North Pole to be the dual Identity.

Identifying Identity as Equilibrium

• Note that writing \(=0\) at the end of lines in the scalar equations of energy is for completeness of mentioning it as the identity or to represent that the left hand expression is in equilibrium. This shows that assuming existence of identity (0) is not needed, we can simply call the expression to be balanced, but we choose to do so as a pointer that all equilibriums are forms of expression of a common abstract identity.

Dualities are Fundamental Laws of a Language - Every Fundamental Law is an Equilibrium Phase Diagram

• Duality is Differentiation and Integration.

• The Fundamental Law of Degree \(0\) is the Identity of Composition.

• Every Fundamental Law is a Composition Law of Duality. This means that every Fundamental Law is obtained by Composing Dualities. This is just a restatement of the fact that every Fundamental Law is an Equilibrium Phase Boundary. In particular, every Mathematically correct Equation is an Equilibrium Phase Boundary which can be obtained by composing Dualities.

• This is why Fundamental Laws are Natural: They Appear and Disappear naturally at the same time as expressions of Duality.

• Every Equilibrium Phase Diagram is a Fundamental Law and all that is a Fundamental Law is a Phase Diagram. The Universe itslef is a Phase in a Phase Diagram. On the other side of the Equilibrium Phase boundary of our Universe is our Dual Universe. Differentiation and Integrations connects all Phases to one another and hence allows us to go to Higher Degrees of Freedom.

• The Fundamental Laws themselves have Degree of Freedom lesser than the Degree of freedom at which level they are the Fundamental Law. This is obvious as Fundamental Laws are the Phase Boundary where two Phases of a level higher in Degree meet.

• Differentiating a Fundamental Law of Degree 1 gives two Fundamental Laws of Degree \(1/2\) one corresponding to Even Parity and another corresponding to Odd Parity. Similarly, on Differentiating further, one may obtain Fundamental Laws of degrees of freedom that are positive powers of \(1/2\). Since they have lesser than one Degree of Freedom, they cannot be seen in the usual Positive Integer Valued Dimensional Spaces. However, they have more than Zero Degrees of Freedom. To go below Zero degrees of Freedom, one needs Differentiation of an entirely Different Degree- at a higher level than the Degree of Freedom of Differentiation.

• Dark Matter has Negative Degrees of Freedom and is a Dual Phase to our entire Universe. Their existence is obvious since every Phase must has its dual, so our Universe has its Dual too. We are separated from the Dual Universe by an Equilibrium Phase Boundary hence we do not have the Degree of Freedom to go there as we are. However, if we Differentiate ourselves, we can go there leading to cancelltion with our Dual Phase. This is similar to how positive numbers when composed with Negative Numbers gives 0. Or Look at any equation where numbers are to be moved from one side to the other, there is opposite sign and then cancellation. This can be Imagined by seeing the Reflection Symmetry in Mirrors or composition in Numbers. Any Equation be it Stokes Theorem or Pythagoras Theorem is a display of this.

  • For example, + and - are Fundamental Laws in the Language of Numbers at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

  • For example, Differentiation and Integration are Fundamental Laws in the Language of Duality Calculus at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Integration corresponds to Even Symmetry, Differentiation corresponds to Odd Symmetry.

  • For example, Fermions and Bosons are Fundamental Laws in the Language of Particles at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Bosons correspond to Even Symmetry, Fermions correspond to Odd Symmetry.

  • For example, Left and Right are Fundamental Laws in the English Language of Space at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Left corresponds to Even Symmetry, Right corresponds to Odd Symmetry.

• Each Fundamental Law has its Dual Law. They are Related by the Duality Relations, that is, one may be obtained by Differentiating the other may be obtained by Integrating. This can be restated as a statement of Mirror Symmetry: Fundamental Laws are Mirror Reflections of Each other.

  • For Example: All n-Spheres for \(n\) are Fundamental Laws in the Language of Euclidean Measure. For \(n\) greater than 0, \(S^n\) is a Fundamental Law at the the degree of freedom \(n+1\) in the Language of Euclidean Measure. \(S^0\) is a Fundamental Law at the the degree of freedom \(1\) in the Language of Euclidean Measure. Each of them individually have Degree of Freedom \(1/2\). Point of \(S^0\) on the Right side corresponds to Even Symmetry, Point of \(S^0\) on the Left side corresponds to Odd Symmetry.

• The condition for a Law (E) to be a Fundamental Law (\(E^{*}\)) at the Degree of freedom \(i\) is:

• Fundamental Law in a Language, at the degree of freedom \(i\) are possible because of the Fundamental Law in the Language of Calculus:

The Fundamental Laws in the Language of Number Theory

• Whole Numbers are the Fundamental Laws in the Language of Number Theory. The number \(n\) is a Fundamental Law at the degree \(n+1\).

The Fundamental Laws in the Language of Euclidean Geometry

• \(1\) is the Fundamental Law of Degree \(0\). Hence, \(1\) is the Identity of Composition.

• + and - are Fundamental Laws at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

• \(\sin\) and \(\cos\) are both Fundamental Laws in the Language of Euclidean Geometry. Each of them individually have Degree of Freedom \(\). \(\sin\) corresponds to Even Symmetry, \(\cos\) corresponds to Odd Symmetry.

The Fundamental Laws in the Language of Powers

• \(e\) is the Funndamental Law of Degree \(0\).

The Fundamental Laws in the Language of Complex Numbers

• \(1\) is the Fundamental Law of Degree \(0\).

• + and - are Fundamental Laws at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

• x, y

The Fundamental Laws in the Language of Groups

• Identity (\(e\)) is the Fundamental Law of Degree \(0\).

The Fundamental Laws in the Language of Physics

• Energy (\(E\)) is the Fundamental Law of Degree \(0\). It gets Partitioned into various Dualities: Mass, Temperature, Time, Space, all quantities represent Duals.

Abstract

Languages from Physics and Mathematics are represented in the Language of Duality. This reduces the known Languages to be a Form of “simple mass balance”. Every Fundamental Law in Physics, Mathematics or any other Language is to be seen as a form of Duality. This is further reinterpreted as: it is this unique conserved quantity (Identity) which appears in all kinds of possible forms of Duality.

“Knowledge is Conserved” (The Fundamental Law of Language Duality)

• I want to state: “Knowledge takes Form as Duality” or even better: “Knowledge is”, or its Differential form: “Duality is”. But it will take us some time to get there.

• For now, let us state the Fundamental Law as: Knowledge takes various forms but the “Amount of Knowledge” is the same or “Knowledge is conserved”.

• To some of us this might seem not to be a law at all, after all we have an entire education system which gives us more knowledge. I shall request that we be patient. I will first work out several detailed examples for us to follow.

• This might seem trivial and is very much so but it is for this very reason that its Dual can take non-trivial form.

• I shall represent everything I know as a form of the Law - “Knowledge is Conserved”. That is, I shall Differentiate the Law to its Differentiated Forms. Since the Knowedge I know “amounts to nothing”, I will appreciate if anyone would like to show me the Laws they know, so that I can see those Laws as a form of this Law too.

• At the end I shall make a statement for “why all this Duality?”. This statement “amounts to nothing”. It will then be clear why all “Fundamental Laws” are called “Fundamental Laws”. In fact, it will be clear that All “Fundamental Laws are statements that amount to nothing”.

• All known Fundamental Laws will be shown to be “Absolutely Rigid in the Language at their repsective Degree”, that is, “Fundamental Laws of a Language form the Irreducibles of a Language” or “Eigen-Laws of the Language”. This means “Fundamental Laws are Equilibrium Statements” or “Fundamental Laws are Equilibrium Phase Boundaries of a Language”.

• By virtue of all Fundamental Laws being the Equilibrium Phase Boundaries of a Language, all the above made statements will follow. It will also be clear that all “Fundamental Laws” have no Degree of Freedom with respect to their Language at their Degree, that is, they cannot be Differentiated further in that Language at that Level. This is why they are Fundamental Laws: they are Invariants of the Language. This is not to be confused with: Fundamental Laws cannot exist in various forms- they can take as many forms as possible, and in fact, they do- “There is a Fundamental Law at every Degree of Freedom in a Language”. The Key Point to note is that the Differentiation used needs to be in the Language of which they are the Equilibrium Phase Boundary and at the respective Degree of Freedom. Fundamental Laws of a Language are related to each other by Differentiation and Integration.

• The condition for a Law (E) to be a Fundamental Law (\(E^{*}\)) at the Degree of Freedom \(i\) was derived in Equilibrium Thermodynamics :

  \[d_{i}(E^{i+1})_{E^*} =0\]

• Fundamental Laws in a Language, at the degree of freedom \(i\) arise from The Fundamental Law in the Language of Calculus:

Mass Balance as a form of “Knowledge is Conserved”

• The typical example of “Knowledge is conserved” is the law of mass balance. This law states that mass stays conserved no matter “how we divide it into different forms”. In the Language of Duality, this corresponds to “Knowledge stays conserved no matter how we think (divide) it into different Languages”.

  • We see this appearing throughout in Nature. No matter what system we take, we find that dividing it into pieces doesn’t change its mass.

Projection Duality as form of “Knowledge is Conserved”

• Sterographic projection or any kind of projection is a form of Differentiation. The Angle is the dual (differential form) of the Radius, a Line is the Dual of a Circle, the Plane is the Dual of the Sphere and the \(n-Sphere\) is the dual of \(n\) Dimensional Plane, which is its differential form. In this form, the “Knowledge stays conserved no matter how we divide it into or Line”. This gives the Fundamental Laws :

• Length along line = \(\theta r\)

• Length along plane (Area) = \(\theta r r\theta\)

• Length along \(n-1\) dimensional plane = differential form of \(S^n\).

Electric Field-Electric Charge Duality as a form of “Knowledge is Conserved”

• The well known Law relating Electric Field to Electric Charge is just the statement “Knowledge stays conserved no matter how we represent (divide) it into different Languages”. Knowledge in this case is the “measure”

• The way to “measure” the “amount of Mass” at the degree in which Charge exists is simply its scalar value or its Fundamental Law (Scalar Values will be later seen to be the Fundamental Laws at the Degree 1). The way to “measure” the “amount of Mass” at the degree in which Electric Field exists to meet with its dual Charge is its Flux. This is the Fundamental Law which Gauss had also discovered.

• The Fundamental Law itself can be differentiated to give its Differential Form, which states that Charge is the Dual of Electric field, which is another way of stating a conservation law, here Flux (from its Integral) becomes divergence (Differential form):

Magnetic Field-Magnetic Charge Duality as a form of “Knowledge is Conserved”

• The well known Law relating Magnetic Field to Magnetic Charge is just the statement “Knowledge stays conserved no matter how we represent (divide) it into different Languages”. Knowledge in this case is the “measure”

• The way to “measure” the “amount of Mass” at the degree in which Charge exists is simply its scalar value or its Fundamental Law (Scalar Values will be later seen to be the Fundamental Laws at the Degree 1). The way to “measure” the “amount of Mass” at the degree in which Magnetic Field exists to meet with its dual Charge is its Flux. This is the Fundamental Law- Gauss’s Law for Magnetism had also discovered.

• The Fundamental Law itself can be differentiated to give its Differential Form, which states that Charge is the Dual of Electric field, which is another way of stating a conservation law, here Flux (from its Integral) becomes divergence (Differential form):

• Magnetic Monopoles do exist but not in our Universe. They exist in a Degree of Freedom that is Dual to our Universe. Hence, they cannot be seen. However they are Dual to Electric Charges. That is, Magnetic Monopoles create Electric Monopoles and vice Versa. The Orthogonality of meeting points of Magnetic and Electric Fields also shows that they are Dual to each other. Orthogonality is a form of Duality (Fundamental Law).

Space-Time Duality as a form of “Knowledge is Conserved”

• Space and Time are duals of each other.

• Measure of space is equal to measure of Time is seen at the Equilibrium Phase Boundary of Space and Time: The Light Cone. This is what Einstein discovered in the Metric. The Einstein’s Field Equations are the set of Maxwell’s Relations that can be formed by composing Dualities.

• Another way to see this is by seeing that Space and Time meet “orthogonal” to each other. This is a sign of Duality.

• Space-Time Duality composed with Electro-Magnetic Duality produces Light or one may look at it in the Dual direction: Light produces Space-Time and Electro-Magnetism. Light is the Identity for Space Time and Electromagnetism.

Quantum Mechanics as a form of “Knowledge is Conserved”

• Particle and Wave are the dual forms. Operators and Spectrums are the corresponding Duals in Mathematics.

• The Duality of Fundamentals \(i\) and \(\hbar\) gives rise to \(i\hbar\) being conserved.

• Amount of Energy in Particles plus the amount of Energy in waves must stay conserved or together (Dual). Amount of Mass in Space Duality between Space-Time is stated in Qunatum Mechanics along with composing the Particle-Wave Duality. Hence this Equation is to be read in two Dual Directions- one involving space time duality and the other involving Particle-Wave Duality which is represented by the Wave function. The “New Space” produced by composing the two Dual directions is the Energy.

• \[(-d_{Space}d_{Space}+) \Psi =(i\hbar) d_{time}(\Psi)\]
• The above equation is a Maxwell’s Relation (Thermodynamics) corresponding to the Equilibrium Phase Diagram of Particle-Wave Duality.

Quantum-Gravity

• If Quantum Mechanics and Gravity are to be treated as Independent Variables (Dual Varibales): A new Energy (\(E\)) must be defined having degrees of freedom in Space-Time Duality as well as in Particle-Wave Duality, That is the new Hamiltonian now acts on \((\Psi, S)\) as the Independent Variables. The metric (64 dimensional: 8 cross 8 matrix) then takes the form

• The condition for Equilibrium is then \(dE =0\) and it gives

• This results in a new wave with a new fundamental speed of propagation in the direction orthogonal to the Space-Time and the Quantum mechanical wave. The new fundamental speed \(\S\) is given by \(\S^2= \frac{1}{c \hbar}\). The corresponding Mawxell-Type Equations are obtained by composing the Duality of \(\Psi\) and \(S\). If reduced to derivatives in terms of \((x_1,x_2,x_3,t)\), there are \(2^8\) Maxwell-Type Equations (same as those seen in thermodynamics). The equations would involve curvature of curvature.

• A new force with these extra dimensions, a total of 8 (to combine both \(\Psi\) and \(S\)) which is the Integral version of Gravity (of space-time) must be named. This force is then Dual to a Potential Energy of 64 dimensions.

The Integral Theories to Quantum-Gravity

• When the Equation above was written it automatically sets up its dual equation which can be further composed with this one to go to higher and higher Degrees of Metrics. This is no different than doing Euclidean Geometry in Higher and Higher Dimensions. The fundamental Law of Each Dimension are just the Coordinate Axis and the corresponding phases are \(2^d\) Maxwell’s relations.

Duality is a Fundamental Law - Every Fundamental Law is a Duality (an Equilibrium Phase Diagram)

• Every Fundamental Law is a Duality.

• Every Fundamental Law is a Composition Law of Duality. This means that every Fundamental Law is obtained by Composing Dualities. This is just a restatement of the fact that every Fundamental Law is an Equilibrium Phase Boundary. In particular, every Mathematically correct Equation is an Equilibrium Phase Boundary which can be obtained by composing Dualities.

• This is why Fundamental Laws are Natural: They Appear and Disappear naturally at the same time as expressions of Duality.

• Every Equilibrium Phase Diagram is a Fundamental Law and all that is a Fundamental Law is a Phase Diagram. The Universe itslef is a Phase in a Phase Diagram. On the other side of the Equilibrium Phase boundary of our Universe is our Dual Universe. Differentiation and Integrations connects all Phases to one another and hence allows us to go to Higher Degrees of Freedom.

• The Fundamental Laws themselves have Degree of Freedom lesser than the Degree of freedom at which level they are the Fundamental Law. This is obvious as Fundamental Laws are the Phase Boundary where two Phases of a level higher in Degree meet.

• Differentiating a Fundamental Law of Degree 1 gives two Fundamental Laws of Degree \(1/2\) one corresponding to Even Parity and another corresponding to Odd Parity. Similarly, on Differentiating further, one may obtain Fundamental Laws of degrees of freedom that are positive powers of \(1/2\). Since they have lesser than one Degree of Freedom, they cannot be seen in the usual Positive Integer Valued Dimensional Spaces. However, they have more than Zero Degrees of Freedom. To go below Zero degrees of Freedom, one needs Differentiation of an entirely Different Degree- at a higher level than the Degree of Freedom of Differentiation.

• Dark Matter has Negative Degrees of Freedom and is a Dual Phase to our entire Universe. Their existence is obvious since every Phase must has its dual, so our Universe has its Dual too. We are separated from the Dual Universe by an Equilibrium Phase Boundary hence we do not have the Degree of Freedom to go there as we are. However, if we Differentiate ourselves, we can go there leading to cancelltion with our Dual Phase. This is similar to how positive numbers when composed with Negative Numbers gives 0. Or Look at any equation where numbers are to be moved from one side to the other, there is opposite sign and then cancellation. This can be Imagined by seeing the Reflection Symmetry in Mirrors or composition in Numbers. Any Equation be it Stokes Theorem or Pythagoras Theorem is a display of this.

  • For example, + and - are Fundamental Laws in the Language of Numbers at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

  • For example, Differentiation and Integration are Fundamental Laws in the Language of Duality Calculus at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Integration corresponds to Even Symmetry, Differentiation corresponds to Odd Symmetry.

  • For example, Fermions and Bosons are Fundamental Laws in the Language of Particles at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Bosons correspond to Even Symmetry, Fermions correspond to Odd Symmetry.

  • For example, Left and Right are Fundamental Laws in the English Language of Space at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Left corresponds to Even Symmetry, Right corresponds to Odd Symmetry.

  • For example, \(\sin{\theta}\) and \(\cos{\theta}\) are Fundamental Laws in the Euclidean Geometry of Space at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). \(\sin{\theta}\) corresponds to Even Symmetry, **\(\cos{\theta}\) corresponds to Odd Symmetry.

• Each Fundamental Law has its Dual Law. They are Related by the Duality Relations, that is, one may be obtained by Differentiating the other may be obtained by Integrating. This can be restated as a statement of Mirror Symmetry: Fundamental Laws are Mirror Reflections of Each other. Hence, at the dimension \(n\), there are \(2^n\) Fundamental Laws.

  • For Example: All n-Spheres for \(n\) are Fundamental Laws in the Language of Euclidean Measure. For \(n\) greater than 0, \(S^n\) is a Fundamental Law at the the degree of freedom \(n+1\) in the Language of Euclidean Measure. \(S^0\) is a Fundamental Law at the the degree of freedom \(1\) in the Language of Euclidean Measure. Each of them individually have Degree of Freedom \(1/2\). Point of \(S^0\) on the Right side corresponds to Even Symmetry, Point of \(S^0\) on the Left side corresponds to Odd Symmetry. Poincare Conjecture is a Fundamental Law because 2-Sphere is the equilibrium Phase Boundary of \(R^3\).

The Fundamental Laws in the Language of Number Theory

• \(0\) and \(\infty\) are Dual to each other.

• \(0\) and \(\infty\) are the Fundamental Laws at the Degree 1.

• The above Fundamental law is stated as : Duality of \(0 \quad \infty\).

• Whole Numbers are the Fundamental Laws in the Language of Number Theory. The number \(n\) is a Fundamental Law at the degree \(n+1\).

• This is interpreted as Differentiating the \(\infty\)$ Language at the Degree \(i\) requires going beyond the Degree

The Fundamental Laws in the Language of Complex Numbers

• Fundamental Laws are +,-. This composed with the Differentiation operator \(d\) produce the Fundamental Laws \(\sin\) and \(cos\).

• The Riemann Hypothesis is true and its a Fundamental Law in a Sub-Language of the Language of Complex Numbers. This can be proved by showing that the Composition of Fundamental Laws generated by Duality of negative evens with the Duality of \(\frac{1}{2}+it\) produces the Zeta Function. Atiyah’s proof is the other way round. It is correct if interpreted properly.

The Fundamental Laws in the Language of Euclidean Geometry

• 1 is the Fundamental Identity.

• + and - are Fundamental Laws in the Language of Euclidean Geometry. 1 and -1 are Fundamental Laws in the Language of Euclidean Geometry.

• \(\theta\) is a Fundamental Law of Measure \(2\pi\).

• \(\sin(\theta)\) and \(\cos(\theta)\) are both Fundamental Laws in the Language of Euclidean Geometry obtained by composing the Duality of . Each of them individually have Measure \(2\) since they are formed between the duality of \(-1\) and \(1+\), that is they Integrate to \(2\).

The Fundamental Laws in the Language of Powers

• \(1\) is the Fundamental Identity.

• \(-\infty\) and \(\infty\) are Dual to each other. This Duality creates (Integrates) to \(\log\). Simply place together \(-\infty\) and \(\infty\) -this horizontal composition line (or axis) is the degree of freedom within which Logarithm exists. Logarithm is the “Gravitational Force” between \(-\infty\) and \(\infty\).

• Just like we see movies on a screen, the Integral versions of the Universe can see their Differential forms. Each of us act as rigid objects-particles or numbers for the Integral Universe. That is, we are the Fundamental Laws of our Integral Universe.

• \(0\) and \(\infty\) are Dual to each other. This Duality creates (Integrates) to \(\exp\).

• All remaining Fundamental Laws are formed by composition of these dualities. Composition of Dualities is how one Integrates.

• This is how Ramanjuan and other geniuses “generate Laws out of nowhere”- anything we see are forms of numbers interacting with each other. All equations are pictures of Equilibrium Phase Diagrams and all of these are generated out of Identity simply by composing the Dualities involved. English and Art(drawing) are Integral versions of Mathematics but the same applies there too. The Classics are all Fundamental Laws which exhibit symmetry.

• All of us are Dual forms of each other. Recognizing each other composes us together to form an Integral.

• Duality exists because it does not exist. Duality exists because Non-Duality does.

Summary

• Introduced universal concept of Language to refer to all structures.
• Introduced a universal Calculus with notions of Differentiation and Integration to study transformations of different Phases of a Language. This concept of Phase Transformations of a Language generalizes Thermodynamics as a prototype Empirical Theory of Languages. A microscopic theory (Statistical Mechanics) and Kinetics of Language may be introduced separately.
• Each phase of a Language has Sub-Languages which are sub-Phases. Differentiation and Integration takes a more rigid structure at the sublevel for SubLanguages. That is, Calculus as a Language itself can be differentiated to go from the Calculus of a higher level Language (more fluid) to a lower level Language (more rigid).
• Thermodynamics governing Phase Transformations is seen as The Language of Calculus governing transformation from differential to integral forms and vice versa.
• Language can be represented as composition of groups which may be further reduced to a representation in terms of simple groups.
• Art is to be recognized as the Inegral Language of Science and Science as the Integral Language of Mathematics. They are all Dual to each other.

Postscript

No matter how much is tried, there is a particular Language being used to represent these ideas which limits the scope of what is to be said. This implies that no justice can be done to create a theory or a “Universal Language” covering all Languages (or what is more popularly called “the theory of everything”). Simply introducing an object of rigidity outside the bounds found in the particular Language will give us a counterexample to claims of a “Universal Language” that covers everything. A trivial example to refute any such claims is the Empty Language which can only be obtained in a limiting case. It is up to the observer to take their time to evaporate or condense the ideas represented here to their own liking.