Your question (``Does this mean that older forms of mathematics will disappear?’’) suggests math has an evolution or selection--of--the--fittest history in human discovery.

Science has precipitated out of philosophy to be that part of human knowledge that predicts observations. Does science extend as far as metaphysics? Likewise math as we know it today has become the usefulness of...

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Your question (``Does this mean that older forms of mathematics will disappear?’’) suggests math has an evolution or selection--of--the--fittest history in human discovery.

Science has precipitated out of philosophy to be that part of human knowledge that predicts observations. Does science extend as far as metaphysics? Likewise math as we know it today has become the usefulness of counting and geometry. That is, math is the result of a selection of the useful methods (evolution). You noted logic has not been as useful as counting math. Math and logic have not yet reconciled (yes, I know about Russell and Whitehead’s book but I like others think he had inadvertently assumed the counting process in his set development.)

One of the characteristics of today’s usage of math is the concept that a (counting) number that represents a physical object can be negative. The number system I used as a hunter was one, couple, few, many. We had more need to characterize snow conditions than numbers. Farmers such as in the Bible tend to use 40 to mean many but uncounted. The later development of merchants resulted in the concept of assets as positive number and liabilities a positive number. By forming an equation whose right hand side (rhs) is real/measured quantities with an operation (assets minus liabilities or clock and rulers converted to a geometry) and whose left hand side (lhs) is an abstract (transformation) quantity (net worth, space, time). The interpretation of the rhs is easy. The interpretation of the lhs has great difficulty that admits a negative physical object.

We are much too careless in the interpretation of the lhs, negative objects, division, zero, and infinity (unbounded). Perhaps the hunters were right - when we reach the limit of our concepts we should say ``many’’ and leave it at that.

However, set and, in particular, group theory seems to have some usefulness in identifying patterns in the stable points of energy/mass assembly (particles). Like with the periodic table, identifying holes in the group predicted particles and the particle characteristics. This math suggests a still finer structure.

Again, I like the idea that the math we use is a result of the selection from among nature’s characteristics.

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As I am much involved in the foundations of mathematics, I noted some inaccuracies in your essay on this topic:

"Bertrand Russell asked what would happen if you have a set of sets that does not include itself"

A consequence of the axiom of regularity of ZF is that no set is ever member of itself. And as in this theory all objects are sets, every set is a set of sets that does not...

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As I am much involved in the foundations of mathematics, I noted some inaccuracies in your essay on this topic:

"Bertrand Russell asked what would happen if you have a set of sets that does not include itself"

A consequence of the axiom of regularity of ZF is that no set is ever member of itself. And as in this theory all objects are sets, every set is a set of sets that does not contain itself, and none of the sets it contains contain themselves either. Instead, the reasoning of Russell's paradox starts by considering THE set of ALL sets that do not contain themselves. (I prefer the word "contain" rather than "include" as the latter may be confused with the inclusion relation, which is not involved in the Russell paradox).

Where you you take this from :"if [a list] does list itself it must list that it lists itself, which means it must list that it lists that it lists,...and so forth" I neither saw any reasoning like this, nor can find any logic according to which any list of lists should also have to "list that one of its lists, lists something". Wondering where you take that from, I see you refer to a writing by Russell in 1903. I'm not going to check if that reference actually contains that fuss or not, but anyway that's a very old reference, and thus not one that I guess any specialist in the foundation of maths would refer to nowadays when trying to explain what the paradoxes of set theory really are, unless of course they would otherwise know that a given idea turns out to be correct and worth quoting, instead of simply trusting what was written at that time. The only thing I see related to the structure of your sentence, is the axiom of regularity which forbids such loops of constructions, however it is only very remotely related and we would need to completely rewrite and reinterpret your sentence in very different ways in order to actually make such a link, and I don't feel like developing this now.

"Turing demonstrated that no Turing machine can emulate all other possible Turing machines to determine if it halts. To do this it must emulate itself emulating all possible machines, which gets one into the same conundrum that Russell found"

Sorry, this is definitely not the way the reasoning goes. There is no problem to make an algorithm emulating all other algorithms, including itself emulating all algorithms including itself and so forth. The problem is not with emulation, but with finitely proving a claim of impossibility for some algorithm (or equivalently an emulation of it) to ever stop. In other words, there is no problem to determine that an algorithm halts, in the case it will halt : we just need to run it "long enough". What we cannot do, is to find a general method that will surely happen, sooner or later, to establish that some other algorithm will NOT halt, in case it will not halt. Because no matter how long a simulation we make, the problem is that, in case it will actually never halt, we can continue emulating it indefinitely and not see it halt but we will remain unable to know if the reason we did not see it halting, is because it will actually never halt or because we did not emulate it long enough to see it halting. The claim "This algorithm will never halt" cannot be proven by running the said algorithm any amount of time, but would require a formal proof of that claim, which is something very different from the act of emulating the algorithm. Even if the said algorithm will never halt, the search for proofs that it never halts, is a very different algorithm (which depends on the precise chosen axiomatic system of set theory), which might never halt either.

"Kurt Goedel's proof that no mathematical system can ever prove all possible statements as theorems about itself"

This sentence is confusing, grammatically ill-formed and I cannot see any known result that looks like this. One thing he proved instead, for example, is that no algorithm can ever prove all true arithmetical statements, unless it is self-contradictory (also wrongly proving wrong statements). And also, that among the true statements that no algorithm can prove, is the statement that this same algorithm will never prove any contradictory statements (in case it is true).

I understand that the foundations of mathematics may not be your area of specialization, so that you may be confused about what the results exactly say and how they may be proven. I also understand that, when something is subtle and complicated, it may be uneasy to explain them in a short essay. However I consider that in case you choose to sketch the explanation of something, you should care to do it right ; and if you can't then you should not even try.

"The diagonal elements in the list when increased or decreased by one can be formed into a string of numbers that does not exist in the list"

Hmm, which list are you talking about, and to prove what ? Even while I remember there exists a result whose proof involves something like this, you should care to correctly specify which is this result obtained by which kind of argument formed by involving which list of stuff, instead of casting a randomly shuffled list of possible results mixed with possible reasoning involved in the proofs of some results.

"the existence of unprovable propositions, and further that these statements effectively declare their unprovability"

It would not be so interesting to find an unprovable proposition, unless this proposition also happened to be true. Also, not any unprovable statement effectively declares its own unprovability, but only a specifically constructed statement does.

"The second Goedel theorem is that these statements must be true, because their falsehood would contradict the statement declaring its own unprovability."

If you could prove that a statement is true because its falsehood contradicts something we know, then the statement would be provable, in contraction to what you just mentioned. So there is an argument why it is true anyway, but it cannot be a strict proof of truth. It is something more subtle. The problem is, when an argument is subtle, it needs a clear enough explanation so as to give a proper idea what it is about. Because the very point of paradoxes is that they are about subtle ideas, that require careful distinctions between concepts which feel similar but are in fact different, until we prove that they are indeed sometimes not equivalent in some cases.

"the cardinality of the continuum, thought to be larger than countable infinity, is not decidable, but where one can construct models independent of the axioms of set theory"

You meant : that its value (known to be larger than countable infinity, but coming after how many other infinite cardinals) is not decidable but independent of the axioms of set theory, as we can show by constructing diverse models where it takes different values.

"We have an intuitive sense of numbers and the inductive reasoning for why if there exists the integer N then the integer N + 1 must exist. Goedel tells us that something goes wrong with this; there is something in basic arithmetic that is not computable".

I never heard about Goedel claiming there would be anything wrong with the idea that if an integer N exists then the integer N+1 exists too. I even never heard of what it might formally mean for the integer N+1 to not exist.

"We might then ask the question: do all the numbers between these two large numbers exist?"

It depends what you mean by "exist". Mathematics has its own concept of existence, by which all these number indeed exist, but which is independent of any concept of physical existence.

"This means if they exist in some meaning according to computation there must be a machine that performs any calculation"

Here again, it depends what you mean by "exist". We can mathematically consider the mathematical existence of "machines" more powerful than this universe, with the only defect that we can only know the results of some specific cases of its calculations, those that can be deduced by a much shorter method.

"the Kolmogorov entropy" : it seems you mean the Kolmogorov complexity, which is indeed a concept of entropy, to not be confused with what is called the name of Kolmogorov entropy but that is quite different.

There is indeed a theorem by Chaitin about Kolmogorov complexity, which is about finding the smallest possible program making a given output, or optimal data compression algorithm, and is inspired from the Berry paradox. However even though I know this theorem and its proof, I could not relate to it the bits of sentences that you sketched and that seemed to me totally incoherent.

"We know that between 10^10^10^10 and 10^10^10^10^10 there are numbers that have enormous complexity, but we cannot know what is the smallest of these numbers that has no such description"

Any very big number can always be described as having an enormous complexity ; the question is whether it also admits a simpler description, with complexity smaller than some amount. If n is a reasonable number (such as n=500) and K is a large number that is "quite complex" in the sense that it is not known as having any relatively low complexity (say, < n+100), then we cannot know what is the smallest number larger than K that has a description with complexity < n.

However your example completely fails : among all numbers between 10^10^10^10 and 10^10^10^10^10, we do know what is the smallest one with a low complexity, and that is 10^10^10^10 itself (since we could define it in a simple manner !)

In the ordinary double slit experiment, the paths do not wind around the slits, as any contribution of paths that wind around are usually neglected in descriptions of this experiment (they are small corrections from quantum field theory, and do not affect the basic paradoxical aspect of the experiment).

I can admit a theoretical concept of super-Turing machine which "solves the halting problem" of ordinary Turing machines. However I think you gave a wrong example here : "the problem of whether a light switch that is turned on in the first second, then off in the next half second, then on again in the next quarter second, then . . . , and whether the light switch is on or off at the end." The obvious answer here is neither. Generally for any algorithm that may turn on or off a switch along time depending on some computation, a super-Turing machine may tell whether there is a time after which it will stay on, or a time after which it will stay off, or neither, i.e. that it will keep alternating endlessly (for any time there exists a later time with different result). But if you explicitly assume endless alternation at the start then you no more need any super-Turing machine to discover that you will get endless alternation.

Unfortunately, this abundance of inaccuracies I found in what I could decipher in your essay as I know the topics, does not leave me quite optimistic about what I cannot decipher, on topics I am not familiar with (especially HOTT)

A simple web search seems to indicate that there is no such thing as "Polish set theory".

We have multiple theories for the foundations of mathematics, with possible variants of set theory, but I would not take this as if it meant "we have no particular foundations to mathematics". There is a picture of hierarchy and interdependence between versions of set theory, and there are clear reasons for this. The main reason is that there is not one unique mathematical universe, but an endless hierarchy of bigger and bigger ones, that fit different descriptions.

Chaitin's work insisted on pessimistic aspects of the foundations of mathematics. This does not mean everything in mathematics is baseless and happens by chance, "by no logical reason". Actually, the incompleteness theorems themselves are examples of remarkable successes of mathematics to handle its own foundations - because they are mathematically proven results !

You incoherently conclude "It is very difficult to understand how this could be scientifically demonstrated, yet maybe regularities in physics described by mathematics exist for no reason at all. Mathematics and physics have this curious relationship to each other for purely stochastic or accidental reasons; there ultimately is no reason for this"

You mysteriously remove the "maybe" on the way, replaced by an "ultimately"... seemingly for no reason at all. What can allow you to you positively claim this "ultimate" absence of reason, as if you got a proof for this, while at the same time you say it is very difficult for you to "understand" the possibility to prove it, either that such a proof should exist but you visibly do not know it (otherwise you would understand it) or anyway you strongly believe that this absence of reason is a fact (why ?) ?

But the worst form of incompleteness, in my opinion, is that of the incomplete understanding of how the foundations of mathematics look like and what do the diverse incompleteness theorems actually say and for what reason, that may be due to a lack of clarity in the way these things are usually presented. I invite you to visit my site settheory.net where I cared to explain as clearly as I could the main concepts and paradoxes at the foundations of mathematics. Maybe you will find there that the real picture of the foundations of maths is more coherent than what you now think.

Finally, I invite you to read my own essay where I discussed how quantum physics avoids the incompleteness of the infinity which the continuity of physical space naively seems to contain.

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Dear Sir,

Ancient Indian texts describe in detail about number theory including what is a number, what is zero, what is infinity, what are negative numbers and irrational numbers, the difference between one and many, why one is the first number, why two follows one, why three follows two, why four follows three, why zero comes after nine, why the number system repeats thereafter, why these...

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Dear Sir,

Ancient Indian texts describe in detail about number theory including what is a number, what is zero, what is infinity, what are negative numbers and irrational numbers, the difference between one and many, why one is the first number, why two follows one, why three follows two, why four follows three, why zero comes after nine, why the number system repeats thereafter, why these numbers are called one, two, etc. Some of it you can see in our essay. They also knew calculus (called chityuttara). These are not primitive, but highly advanced theories related to the physical world. What you call distance between objects is actually space, which is the ordered interval between objects, just like time is the ordered interval between events.

Our instruments for perception/measurement have limited capacity both in content and time. Thus, what we measure depicts a temporal state of a limited aspect over limited period. The problem comes when we generalize our limited information. We impose our ignorance or inability to know on the Universe and call it fuzzy. Every quantum system (including superposition, entanglement, spin, etc) has a macro equivalent. Unfortunately, instead of looking for it, modern physics chases fantasy like extra-dimensions, dark energy, etc. We have discussed these in our essay including Russell’s paradox. Computers are GIGO – garbage in garbage out. It cannot overcome limitations of programming (apart from physical and energy constraints), which is done by a person with limited knowledge. Thus, they cannot answer all questions.

We have discussed Gödel’s incompleteness theorem and Wigner’s paper elaborately to show their inherent deficiencies. Regarding two-slit experiments, we have repeatedly wondered why no one has conducted the experiment with protons. That would show the fallacy. Though we know much about electrons, we still do not know “what is an electron”. This ignorance leads to fantasy and we call that a theory! The same problem bugs the concept of event horizon that is said to encode the causal structure of Spacetime. Each event in Spacetime has a double-cone attached to it, where the vertex corresponds to the event itself. Time runs vertically - the upward cone opens to future of this event. The downward cone shows past. But if the light pulse radiates in all directions, it should show concentric spheres and not a double-cone. The trick is done by first taking two dimensions and time as the third dimension. But even then it will be concentric circles and not a conic section. Event horizon is the limit of our vision. The recently debated black hole firewall paradox arises out of such misleading manipulations.

As Carl Popper remarked, modern science is more concerned with the cult of incomprehensibility than finding the truth. There is a need to review and rewrite physics.

Regards,

basudeba

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Hello Lawrence,

As a layperson I am not going to pretend to understand these new approaches and the maths being utilised but in your conclusions you state interestingly,

"Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent whole-

ness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos...

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Hello Lawrence,

As a layperson I am not going to pretend to understand these new approaches and the maths being utilised but in your conclusions you state interestingly,

"Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent whole-

ness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos [21].

This chaos is a set of statements that are purely self-referential and have truth or falsehood by no logical

reason.

Possibly the quantum vacuum is similar. It may be a tangle of self-referential quantum bits, where

some sets of these exist in logical coherent forms. These zones of logical coherence might form a type of

universe. These logical coherent forms are then accidents similar to Chaitan0 s philosophy of mathematics.

It is very dicult to understand how this could be scientically demonstrated, yet maybe regularities in

physics described by mathematics exist for no reason at all."

Basically, the mystery of the quantum vacuum is really expressed as the 'hierarchy problem'. I think if you stare at the so called hierarchal gap then that is where the state of physics is at today. I like the idea of entanglement defining geometry somehow and this may be inherent in the blank space (void) where the particle desert exists.The Big Gap (or whatever you may call it) maybe the potential pool from which the multiverses are generaed or at least some indication that the multiverse do indeed exists. Maybe GUT convergences are different for different Universes. Also, maybe it is a matter of physics becoming more aligned with mathematics toward the quantum gravity realm. In that we live in a 4D world our physics (From Greek physika) is generally found to work (empirically) from the platform of the sensory (sensation) because of the physical nature of things at our low energy. Sensory as apart from intelligence. At some point the physics may blur into the mathematics and the sense-data (empirical) will be left behind. This implies that intelligence will discover things by 'explanatory power' and that the empirical apparatus (machines) no longer yield anything useful in upward and onward explorations. Maybe 4D is unique in that it implies a physical world in which physics is a workable context. If so then there might be an " end of the machine(s)' such as the LHC where it just cannot yield the pure mathematical result as a physical (sensation-observation). Then we are at the "end of physics" in a different sense and a brave new world of mathematics (explaining thins) is awaiting. (Maybe the meaning of life is not 42 but 4D ;-) ) If you look at the hierarchal gap as a 'zen koan' then there is more there than meets the eye.

mark

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Dear Michel,

Of course I am aware of orbifolds with respect to superstring theory. The vertex operator algebra with partition function p(q) =tr q^N = Π_{N}1/(1 - q^n) is related to the Dedekind eta function. The trace results in the power [p(q)]^{24} In this there is a module or subalgebra of SL(2,Z), eg S(Z) ⊂SL(2,C), that forms a set of operators S(z)∂_z. This module or subgroup...

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Dear Michel,

Of course I am aware of orbifolds with respect to superstring theory. The vertex operator algebra with partition function p(q) =tr q^N = Π_{N}1/(1 - q^n) is related to the Dedekind eta function. The trace results in the power [p(q)]^{24} In this there is a module or subalgebra of SL(2,Z), eg S(Z) ⊂SL(2,C), that forms a set of operators S(z)∂_z. This module or subgroup is then over certain primes, such as either Heegner primes or maybe primes in the sequence for the monster group. This is of course related to the Kleinian groups and the compactification of the AdS_5.

The AdS_5 compactification issue is something I started to return to. I gave up on this after the FQXi contest over this because it did not seem to gather much traction. The AdS_5 = SO(4,2)/SO(4,1) is a moduli space. The Euclidean form of this S^5 =~ SO(6)/SO(5) is the moduli space for the complex SU(2) or quaterion valued bundle in four dimensions. The AdS_5 is then a moduli space, and the conformal completion of this spacetime is dual to the structure of conformal fields on the boundary Einstein spacetime. This moduli is an orbit space, and this is the geometry of quantum entanglements.

If this is the case then it seems we should be able to work out the geometry of 3 and 4 qubits according to cobordism or Morse theory. My idea is that the Kostant-Sekiguchi theorem has a Morse index interpretation. The nilpotent orbits N on an algebra g = h + k, according to Cartan’s decomposition with [h,h] ⊂ h, [h,k] ⊂ k, [k,k] ⊂ h

N∩G/g = N∩K/k.

For map μ:P(H) --- > k on P(H) the projective Hilbert space. The differential dμ = = ω(V, V’) is a symplectic form. The variation of ||μ||^2 is given by a Hessian that is topologically a Morse index. The maximal entanglement corresponds to the ind(μ).

In general orbit spaces are group or algebraic quotients. Given C = G valued connections and A = automorphism of G the moduli or orbit space is B = C/A. The moduli space is the collection of self or anti-self dual orbits M = {∇ \in B: self (anti-self) dual}. The moduli space for gauge theory or a quaternion bundle in 4-dimensions is SO(5)/SO(4), or for the hyperbolic case SO(4,2)/SO(4,1) = AdS_5. The Uhlenbeck-Donaldson result for the hyperbolic case is essentially a form of the Maldacena duality between gravity and gauge field.

With your presentation of the ψ-problem and the connection between the Bell theorem and Grothendieck’s construction, you push this into moonshine group Γ^+_0(2). This leads to the conclusion or conjecture, I am not entirely clear which, that the moonshine for the baby monster group is coincident with the the Bell theorem. The connection to the modular discriminant is interesting. This then gets extended to Γ^+_0(5). Your statement on page 7 that g(q) = φ(q)^24 is much the same with what I wrote above. There is a bit here that I do not entirely follow, but the ideas are intriguing. I would be interested in knowing if the hyperbolic tilings of Γ^+_0(5) have a bearing on the discrete group structure of AdS_5.

You may be familiar with Arkani-Hamed and Trka's amplitudhedron. The permutations arguments that you make give me some suspicion that this is related to that subject as well. This would be particularly the case is the Γ^+_0(5) is related to the tiling and permutation of links on AdS_5 given that the isometry group of AdS_5 is SO(4,2) ~ SU(2,2) which can be called the twistor group. This is connected with Witten's so called "Twistor-string revolution."

Thanks for the paper reference. That looks pretty challenging to read. I am not quite at the level of a serious mathematician, though I am fairly good at math and well versed in a number of areas.

Cheers LC

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There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled ‘earth-centric’ idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably...

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There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled ‘earth-centric’ idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably then it seems natural numbers ought to have an equal standing as any other numbers, irrational, complex, or even numbers yet to be invented.

Is there any physical underlying reason for natural numbers’ special status? Or are the natural numbers just a convenient way for people to count and were invented by macro intelligent beings like us?

Since all natural numbers are mere derivatives of the number ‘1’, so let’s look closely at what this number one really means. There are two broad meaning of the number one. First it registers a definitive state of some physical attribute, such as ‘presence’ or ‘non-presence’. We can find its application in information theory, statistical physics, counting and etc. The second interpretation of number one is that it denotes the ‘wholeness’ of an entity.

In physics, natural numbers virtually have no sacred places prior to the establishment of quantum mechanics. After all, we don’t need any natural numbers in our gravity functions or the Maxwell electro-magnetic wave functions. Some sharp observers would argue that the ‘R squared’ contains a natural number 2. However on close examination the number 2 is merely a mathematical notation for a number multiplying by itself, and it has no actual physical corresponding object or attribute. The fact that there is no natural number in the formulae represents the idea that time-space is fundamentally smooth. For instance there is no such law in physics that requires 7 bodies (non quantum mechanical) to form a system in equilibrium.

Had we obtained calculus capability before we can count our fingers, we probably would have been more familiar with the number e than 1-2-3. We might have used e/2.718 to represent the mundane singletons. There is no logical requirement that we couldn’t or shouldn’t do it. It is all due to the accident that people happened to need to count their fingers earlier than the invention of calculus. There is no physical evidence that the number ‘2’ is more significant than the any other numbers in the natural world.

However with the standard model of quantum mechanics, energy is quantized, that is, it can only take natural numbers. This idea profoundly altered the status of natural numbers in physics and is a direct contradictory of the notion of ‘no center in the universe’ principal. In this sense it is far more unorthodox than the two relativity theories combined because the latter in fact enhance the ‘no center in the universe’ law. Why does the quantum have to be integer times of a certain energy level, and not an irrational number like square root of 17, or the quantity e? Does it really mean there are aristocrats in the number world, where some are nobler than others? Were the ancient Greek mathematicians right after all, who worshiped the sacredness of natural numbers and even threw the irrational number discoverer into the sea?

From this standpoint we can almost say that quantum theory has some bad taste among all branches of natural science.

Before the quantum theory got its germination, actually people should have noticed the unusual role natural numbers play in rudimentary chemistry. For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule? If scientists are sharp enough back then they ought to be able to be alarmed by the oddity underlying the strange status of natural numbers. It could almost be an indirect way to deduce the quantized nature of electrons.

Fundamentally if natural numbers indeed play a very unusual role in nature, then nature resembles a codebook not just from a coarse analogy standpoint. It is the ultimate codebook filled with rules for a limited number of building block codes. The DNA code is an excellent example.

If it’s a codebook, inevitably it takes us to surmise if information itself is the ultimate being in the universe. It is probably not electrons, strings, quarks or whatever ‘entities’ people have claimed. It is the information that is the only tangible and verifiable entity out there. Everything else is a mirage or manifestation of some underlying information, the codebook.

In this sense physics has somewhat gone awry by focusing on the wrong things, the ‘attributes’ such as momentum, position and etc. Instead, information is what contemporary physicists talk about and experiment with. Otherwise, the physicists would have no right to laugh at the medieval scholars who based their intellectual work on the measurement of the distance between a subject and God’s throne.

The nature has revealed her latest hand of cards to us. It looks like it’s the final hand but no one can be sure of course.

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