Dear Kevin,

Your essay is really fantastic and inspiring. You ask very important questions that I have not asked myself. Below I try to comment main aspects of your essay.

The aspect related to quantification.

I start from your example: “…2+1 defines 3 […] this is an axiom of measure theory, which means that it is assumed.” Obviously we can easily find examples where...

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Dear Kevin,

Your essay is really fantastic and inspiring. You ask very important questions that I have not asked myself. Below I try to comment main aspects of your essay.

The aspect related to quantification.

I start from your example: “…2+1 defines 3 […] this is an axiom of measure theory, which means that it is assumed.” Obviously we can easily find examples where 1+1+…=1 or defines 1. The examples are delivered by every superposition of waves creating a wavepacket. We perceive an electron as a countable object (1 piece), eternal and indestructible unit of matter. In reality it is a wavepacket. We perceive (with measuring instruments) its envelope and not single waves creating the packet. We perceive e.g. an apple and not elementary particles and interactions that create the wavepacket called an apple. So adding apples or electrons we add wavepackets. Adding many fermions and bosons we get 1 apple. After consumption this wavepacket is decomposed into smaller wavepackets up to vacuum. Laughlin said that “The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic aether . But we do not call it this because it is taboo.” So I call this elastic medium spacetime (as Einstein did).

The aspect related to ordering.

“…this should not be surprising since it has been generally believed that the laws of physics reflect an underlying order in the universe. It is explicitly demonstrated that some laws of physics not only reflect such order, but in fact can be derived directly from it.” I propose to find that underlying order in the Thurston geometrization conjecture, proved by Perelman. It states that every closed 3-manifold can be decomposed into pieces that each have one of eight types of geometric structure, resulting in an emergence of some attributes that we can observe. We can assign every fundamental interaction and matter to the proper Thurston geometry with metric.

“…here it is explicitly demonstrated that some laws of physics not only reflect such order, but in fact can be derived directly from it. This has enormous implications for the direction and progress of foundational physics in the sense that it enables one to see that common mathematical assumptions, such as additivity, linearity, Hilbert spaces, etc., while familiar, are most likely not fundamental…” That is right! In my opinion additivity, linearity, Hilbert spaces, etc. show the effectiveness of geometry rather than mathematics as a whole set of abstract structures. The additivity is what we perceive in macro or micro world (as I noticed earlier) and it presents our language rather than reality. Without that humans’ perception baggage there are only wavepackets and their superpositions that create constantly evolving and dynamic geometric picture. The geometry devoid of human language is universal language itself, comprehensive probably even for aliens or future supercomputers.

The aspect related to symmetries.

You claim…I believe that the answer lies in the deeper symmetries that various problems exhibit... and you quote Jaynes: “the essential content ... does not lie in the equations; it lies in the ideas that lead to those equations.” Please note that only geometric structures (objects) can show symmetry in transformation (technically an isometry or affine map) that maps the object onto itself. Thurston introduced his version of symmetries in geometry. That is too extensive to describe here. It would deserve a separate essay.

If you are interested you can find details in my essay.

I would appreciate your comments. Thank you.

Jacek

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Dear Kevin,

You have focused on how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics.

You have also taken your thought in line with Hamming.

You are trying to explain which one is more...

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Dear Kevin,

You have focused on how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics.

You have also taken your thought in line with Hamming.

You are trying to explain which one is more fundamental mathematics or physics?

Its indeed the clue what governs the mathematical equations,operators e.g. addition,multiplication etc. Its certain laws of invariance,symmetry order.Its not mathematics describing physics but the laws of invariance,symmetry,order behind those mathematical structures explaining that of physical reality. If we look at the algorithm of formation of numbers and operators e.g. +,-,*,/, we will find that the particular invariance, symmetry,order is drawn from the physical world reality itself e.g quantum structure of energy levels of atoms.

It is possible that the invariance,symmetry,order of classical mathematical structures e.g.hypotheses of geometry don't resemble the those characteristics of quantum world.This raises the deeper question in context of Riemann's geometry,that hypotheses of geometry don't at conform at quantum level.

So, this coherency and compatibility is the key why mathematics has been so effective in Physics. because my Mathematical Structure Hypothesis states that mathematical abstraction and physical reality both originate from Vibration. This is why sometimes mathematics explains physics and other times physical theories solve the mathematical problems.its a two-way interdependence but the different manifestations of the same Vibration.The mechanism of perception of Integers,addition all are its products deeply to be discovered ,how? Thats why e.g.entropy which was considered to be physical phenomenon also governs the mathematical structures as in Poincare,Geometrization conjectures.

Thats why the clue is to match the laws of invariance,symmetry,order of the two,otherwise it leads to mutual friction. We need to match the physical characteristics of mathematical abstractness with that of physical reality. Riemann Hypothesis is all about this laws of invariance behind functioning of these operators, complex analytical continuation.

In context of Skolem paradox,a particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong.

This is because that when mutual laws of invariance matches each other,it gives right result and otherwise wrong because of mutual friction.

Its equally important that as Wigner said that physical laws are conditional statements. What mathematics tries to do is superficially tries to find out the quantity relations between observables rather than why something happens.

What should be field of further research is as Richard Feyman said- The next great era of human awakening would come ,today we dont see the content of equations.

So,the nest era of research is to discover those laws of invariance which governs the mathematical equations itself and then match it with physics.

Anyway, you have written a great essay .

Regards,

Pankaj Mani

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Dear Kevin.

It was a real pleasure to read your essay, and I fully agree with your main idea we can summarize in the following terms: (i) There are mathematical edifices being interpretations of set theory specified by group theory, knowing that group theory by definition covers all aspects of symmetry, and (ii) physical laws stricto sensu are experience oriented interpretations of...

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Dear Kevin.

It was a real pleasure to read your essay, and I fully agree with your main idea we can summarize in the following terms: (i) There are mathematical edifices being interpretations of set theory specified by group theory, knowing that group theory by definition covers all aspects of symmetry, and (ii) physical laws stricto sensu are experience oriented interpretations of mathematical edifices ultimately formalizable in terms of group theory. Through all your essay, you advance relevant examples usefully condensed in table 1. Perhaps it could be interesting to add a counter-example: The so-called Clausius “law” is not a physical law stricto sensu. It is a pseudo-mathematical expression, as evidenced by the pseudo-differential without real mathematical signification figuring in it, and so not a physical law but just a kind of stenographical transcription of an increasing tendency. Despite of statistical/probabilistic/information theoretical ways allowing to circumvent this problem – to circumvent and nothing more (see below) – thermodynamics touching to irreversible processes continues to generate some malaise within physics. Much has been written about “law like reversibility v/s de facto irreversibility”; this discussion beginning with Boltzmann, Loschmidt, Zermello …is far from reaching its end. Anyway, for a physical law to be a law stricto sensu, it must be reversible, and so, symmetrical in prediction and retro-diction, and this would not be possible outside phenomena being formalizable by mathematical groups carrying all fundamental symmetries.

Just perhaps a little remark. Additivity is fundamental, I entirely agree with you on this point. But as you state very rightly that several Newtonian “principles” later were recognized as CONSERVATION LAWS, there is a group being more fundamental than additivity. I mean the Klein 4-group, a commutative transformation group F = {a,b,c,i; *} where a,b,c are transformations to be determined, i being the identity transformation and * an internal composition law so that any composition of transformations belonging to F is a transformation belonging to F. One shows easily – further details are in the semi-technical end note of my own essay – that within the Klein 4-group, a*b*c*i = i. In other terms, the Klein 4-group formalizes ultimately all systems remaining identical to themselves through all their possible transformations.

In my opinion, the Klein 4-group because of its absolutely fundamental aspect can be used to distinguish law like reversible physics and fact like irreversible phenomena added to physics, even if the latter can be approached in terms of statistical mechanics and/or information theory, both being formalizable additively as you say it in your table 1.

In a semi-technical end note of my own essay I touch briefly this point. Contrary to what common sense, intuition, and even simple grammar might suggest, irreversibility is not a direct negation of reversibility. In terms of group theory, these phenomena have nothing in common.

First an intuitive example. Consider an ideal watch without internal frictions etc. whose needles turn by their own inertia at a constant speed. This system, as long as nothing disturbs it, is reversible in terms of the spatial configuration of its needles; it will return to any configuration it occupies at a given moment. Under these conditions, the system (i) is characterized by an entropy variation equal to 0 and (ii) “remains the same” because it conserves its functioning mode. Now let us create an irreversible situation by projecting the system violently to the ground. This time the entropy variation is superior to 0, while the system – reduced to fragments – does not conserve its functioning mode. Nobody would seriously say that the fragments scattered on the ground are the “same” system as the ideal watch in operating condition. So reversibility PRESUPPOSES the conservation of the functioning mode characterizing the considered system, whereas irreversibility CONSISTS ON the transition [conservation of the functioning mode → non-conservation of the functioning mode].

The intuitive expression “functioning mode of a system” is certainly vague, but it can be formalized by the Klein 4-group where the combination of all the 4 possible transformations gives always the “identity transformation”. More details can be found in the end-note of my essay. But briefly speaking, the Klein 4-group formalizes ultimately all systems remaining the same through their transformations. Any physical law is in fine an interpretation I(F) of the intrinsically reversible Klein 4-group F. Irreversibility is the transition I(F) → non-I(F). So “real” physical phenomena are superpositions of IDEAL reversibility and DE FACTO irreversibility. Hence a gas initially in disequilibrium, composed of molecules with their movements dictated by reversible Newtonian mechanics remains ideally reversible but describes de facto an irreversible transition. Now information theory, because of its additive structure, can measure in a probabilistic-exact way the lack of information we encounter regarding the gas system, HOWEVER this exact approach relates not to the system as such, formalizable in terms of I(F), BUT to the system as it is contaminated by superposed irreversibility generating factors so that I(F) → non-I(F). In other terms, far from establishing irreversibility as a law like phenomenon, information theory, just measures the damage done by GIVEN de facto irreversibility to ideally reversible physical systems.

Well it is not sure that you will agree with all this. The reversibility v/s irreversibility issue is and remains controversial. But in this domain, group theory is good common basis for interesting discussions about controversial domains.

Best regards and happy Easter

Peter

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