We show in cited work that one can connect geometry with energy, through quantum entanglement, so that the curved geometry of spacetime can emerge naturally from the wave function.

the point of view of bohr concerning the laws of nature is in essence that of I.Kant. Kant said, that we, mankind, make the laws of nature. . what he meant, was, that the empirical data we gain, in the widest sense, by our senses, are are ordered and interlinked only in the axiomatic system of our awareness, which, according to Kant, is built up by the a priori categories. one problem with this concept is, that it seems to ''fall

from heaven'''. this was severely and correctly criticized by Nietzsche and Schopenhauer, who initiated Psychology.

HA: The Curry-Howard (CH) relationship provides a linking mechanism between structural logic and man-made code that seems "falling from heaven" and fits here -- to explain the apparently etherous and out-of-band link between thinking about Pi and finding GR with QM.

It is revealed, in studying the CH relationship in CS, that the link exits but not directly. It is created by categories which are common to both, and provide an interconnect mechanism that creates what seems like a "wormhole" between different universes, linking them. This is more basic than set theory, and can be a firm basis for all mathematics, as explored by Vladimir Vovodsky, in his work of developing a homotopy theory for algebraic varieties and formulating motivic cohomology. He used man-made computer code to verify and explain structural logic in the world of experiences.

As commonly stated, Vladimir Voevodsky was an algebraist with a deep understanding of topology who found novel ways to apply topology to algebraic geometry and to the foundations, and formalization of mathematics. His work on foundations was interrupted by his sudden death in September, 2017

Regarding the work of Vovodsky in this area, where we see the Curry-Howard relationship, a good reference and usable PDF is at JSTOR, in https://www.jstor.org/stable/26473954?socuuid=7cb46cfb-d6fb-4911-b9c3-366d5a9e8df5&socplat=email

In view of the notice below, I am at your disposal by PM. No comments will be made publicly. Thank you.

ADMINISTRATIVE NOTICE: It seems that the company RG has a 90-day window to address any issues before they are made public. This, may lead to excess work for RG, paradoxically, as a congestion to observe the RG ToS.

The question is even if we join GR and QM, how much would this help us.?

Say space time is curved, but it would take a very massive object to curve it noticeably in a mile say. In the mean time quantum phenomena may be taking place over at most 1 cm, exaggerating.

Then what kind of relevance, save in the interior of out most massive bodies in the universe, would this have.? So how close could we possibly get to validate such a theory?

By the way, the probability density implication of the wave function indicates it can have no effect on geometry. I have not heard sensible discussion yet.

QM assumes a locally eucledian space for good reason. The only unification so

far is with SR, at least partially.(the so called relativistic QM)

Mercurys orbit is thanks to GR alone, mixed with nothing. Alternatively a third object may perturb the orbit in the same way as the GR prediction.

So mathematicians should come down to earth.

JW. Thank you. No comments can be made publicly, due to RG lack of moderation. But PM is available.

Preprint A Heuristic View on the Composition of Space

Preprint A New View on the Composition of Matter

GF: you spam this and other questions without concern. Please delete. It is off-topic

New analysis is needed in mathematics. See

https://www.researchgate.net/post/Computer_science_has_no_epsilon-deltas

There are of course alternatives to the epsilon -delta limit proceedure.

One is historical and old, take it from the second term of a Taylor

expansion or equivalent algebraic expansion

Say

(x+y)^N= x^N + y N x^(N-1)+.......+y^N

known from basic algebra , as the binomial expansion, coincident with Taylor, gives the derivative of x^N as

N x^(N-1)

Similarly expansions for say sin(x+y) were known far before Taylor.

JW: Yes, but the Taylor expansion does not do away with the error limit, just makes the imagination go "as small as desired".

On the other hand, using Eq.(1) of

https://www.researchgate.net/post/Computer_science_has_no_epsilon-deltas

and integers make the error zero. And, allows one to use discrete numbers, with no fictions such as continuity.

Sometimes Taylor has a natural cut off instead of error cutoff term. It sometimes coincides with simple algebra.(case shown)

I dont understand your integers making error zero.Going to all integers impovrishes algebra a lot. What is eq (1)? You dont accept variables in the real domain either?

JW: I added the reference explicitly. It is Eq.(1) of

https://www.researchgate.net/post/Computer_science_has_no_epsilon-deltas

Regarding Taylor -- it is not zero if not linear, which would be general case. With integers, it is always zero, linear or not. All 4 arithmetic operations are valid exactly for integers, and all expresssions become mod p, using modular arithmetic. The AES (Advanced Encryption Standard) works this way, and must be absolutely exact, or it won't decrypt!

BTW, old astronomers such as the Mayan, Greek, Arab, etc., had no idea of planets and moons, and made no hypothesis for the movements they measured on the sky, with planetary (loop sided) trajectories. Yet, the Greeks used the Antikythera Mechanism and could foresee them, without Newton's theories, which was pioneering the use of modular arithmetic using gears. The Mayans used modular arithmetic directly, not gears, and could also foresee them with exact precision. The use of infinitesimals and Cauchy epsilon-deltas, or irrationals, was not needed. There is no fiction of continuity either.

Feynman explained exactly this orbit case, see the end of this short video: https://youtu.be/NM-zWTU7X-k, cited before.

Suppose you would try to work with real numbers in a digital computer, or using code. Then, you would have to treat real numbers in a computer. After a number of rigmaroles, originally, a number of long rolls of parchment full of “nonsense,” eventually as a long, unnecessarily time-consuming hassle, which are just a simulation of a simulation, the computer ends up doing the only thing it can possibly do, the inevitable -- integer arithmetic and number coding.

All other operations in arithmetic are defined in terms of addition and number coding. Multiplication is repeated addition. Subtraction is addition with encoding. Division is the inverse of multiplication.

Computers can only use integers. Then, using philosophy, one cuts through the rigmaroles, and uses Integers directly. With finite integer fields, using Galois numbers. And computers are being optimized to work with large Galois numbers. That is all they can understand.

No wonder that physics, also, can skip the rigmaroles, and only use finite integer fields with Galois numbers. No more the "7 fictions".

Ed

The reason for Taylor having a natural cut-off has nothing to do with linearity,

In the case I gave it is because successive derivatives of x^N end up at zero.

Sure, it has some error if it is an infinite expansion you cut off. Dont see linearity as relevant here.

Taylor was never even meant for integers, but for real variable functions...so

I dont even know what it is you are talking about.

Of course you are right that computers, have roundoff error dealing with real variables, but it is not all about computers, as you seem to assume.

That computers only deal with integers is false. For that matter they cannot deal

with integers of very high magnitude.

Using factorial representation you could store very large or small numbers.

The limitation of computers explains why much work is done analytically rather than numerically.

With your philosophy you would need to toss out practically all scientific

computer simulations made till this very day. You are just on your typical

sell job of Galois integers.

You learned in school that multiplication means repeated addition, but you wouldent I presume use that method to actually do a multiplication?

JW: Computers work exactly as I said, why would there be a reasonable reason to doubt it? Or, you can believe in the dark arts...

There is no round-off error with integers, and integers can be used to calculate anything. Physics, also, can only measure finite integers Anything else, is folly and waste of time. No knowledge is gained, anywhere else.

I dont think you explained how computers work (with bits), but I have no reason

to doubt your knowledge of that.

On the level of the user, you see thing differently. Symmbolic or algebraic calclation is possible.

Think of it this way, using only integers your minimum error is 1 unit.

Tiffany

Glad you find me entertaining.

Of course our thoughts should be ductile, not set in stone.

You are right, epsilon and delta belong to Calculus, not computers.

One last remark, you cannot calculate orbits using QM. The notion of orbit or path

is not contained in QM. It is also inapropriate to use QM for macroscopic stuff like the motion of planets.

JW: for functions that are y = C + ax, where C and a are constants, the Taylor expansion is exact linearly. Otherwise for x^N with N>2, it is an approximation. Nothing, however, is gained there. The expansion must follow integers or rationals, the only numbers that exist in nature. Anything else is fictional.

JW: you wrote, "One last remark, you cannot calculate orbits using QM.". Well, that is how QM started, calculating the electron orbits in Hydrogen.

JW: Some people think that working with bits, the minimum error is 1 unit.

Nothing could be more amiss. In integers, 0+0=0, no error. This goes on, and there is no error at all. 0+1=1, etc. There is a carry over, such that 1+1=10, but that is exact too. All operations are exact. For all operations. No error term.

And one can use an arbitrary number of digits, even though finite, like 2^4096. No calculation in cryptography, for example, can be done except with integers. Physics will just have to follow the same steps. And it will end up with Galois numbers as a finite mathematical field, exact, rather than an approximate number.

Here, the universe is a non-euclidean space, where the ratio of a circumference to its diameter has to be measured. It does not help much to use the value for Euclidean space, known as the irrational pi, as it cannot be measured.

We suppose that the desired value should stand as the ratio of two finite integers, a rational number, as it can approach pi as closely as desired. As Archimedes proved, the value is the ratio of the circle area divided by the area of the square with same radius.

What value do we get, taking account that the value is similar, but may change according to the local curvature?

The curvature is never zero and we can use finite digits for the ratio. Rather than a particular value, the quantum becomes a rule. For a non-euclidean space, QM meets GR.

JW and all: Here, finding the right balance between encouraging discussions and respect for other people’s opinions can be challenging.

We need to make difficult decisions and weigh concerns about free expression, personal, and community respect.

Reasonable minds can differ on whether particular content is objectionable or otherwise violates physics, math, or RG policies.

RG offers PM and immediate preprints that are very valuable for users. Considering that, and that all are aware of it, I have chosen to use disengagement as a resource here, when facing what I feel as systematic opposition.

Disengagement can take various forms. I may not answer except by PM, not answer at all, or answer later.

I am still here, read all posts, and the history of responses is available. Any reasonable question should already have been answered, and does not have to be reaffirmed. The cache principle applies. Everything helps, and the noise helps highlight the signal. Thank you for your participation!

Ed

There are two very different things: How the actual computer works, or how say a programmer may write in some high level language.

My comment is directed more to the second case.

In the first case you do not have much of an option, I would say.

Programmers make plenty of mistakes, before getting it right.

You cannot talk about abscence of error

Poor programmer if they could only declare INT variables.

About QM

No, it partially started (in the case of dilute hydrogen gas) looking at the

discrete spectra of lines produced under electric tension. There are many other experiments to consider also.

Up through the old QM you could still deal with the notion of path, but this definitely disapears in the modern version.

My feeling is that persons could see the opertnity to learn more from such exchanges.

Usually you dont learn if the response is simple agreement.

You can trust me on programming and on QM, have ample experience in both.

Tiffany

I dont like to get into unproductive argumentation, but pick up a book on Calculus and compare the space dedicated to this proper subject, and that that may be dedicated in the name of Galois.

Do you study Calculus in the BA program?

After aliens actually land I may be able to reconsider my position.

JW: As I said before, finding the right balance between encouraging discussions and respect for other people’s opinions can be challenging.

We need to make difficult decisions and weigh concerns about free expression, personal, and community respect.

Reasonable minds can differ on whether particular content is objectionable or otherwise violates physics, math, or RG policies.

I have chosen to use disengagement as a resource here, when facing what I feel as systematic opposition.

Disengagement can take various forms. I may not answer except by PM, not answer at all, or answer later.

Thank you for your participation!

Ed

OK, take f(x)=xxx

(x+h)^3 = xxx+3xxh+3hhx+hhh

by algebra

Then use Taylor

f(x+h) = f(x) +hDf +1/2 hhDDf +...

Dxxx= 3xx

DDxxx =6x

DDDxxx =6

DDDDxxx=0

You will get the same answer!

JW: As I wrote, "for functions that are y = C + ax, where C and a are constants, the Taylor expansion is exact linearly.." Your expression shows it.

And I continued, "Otherwise for x^N with N>2, it is an approximation." Your expression shows it.

Here, one can apply, "Kommt Zeit, kommt Rat!"

A PM asks about consequences of QM in GR. In calculating the orbit of Mercury according to QM, it becomes open [1,2]. See also this questions text.

While not changing local values very much locally, the long distance predictions can become very different.

REFERENCES

[1] https://www.researchgate.net/publication/339988557_On_the_Sun's_Gravity_and_the_Topology_of_Mercury's_Orbit

[2] Petersen, A. The Philosophy of Niels Bohr. Bulletin of the Atomic Scientists, Vol. 19, No.7, 1963.

Ed

You dident even read my answer

For x^3 I showed that the Taylor expansion is exact (That it coincides with the Binomial expansion). Similar for any x^N

If you are only talking about linear functions it is OK, but you switched topics.

I reiterate, that you dont calculate the orbit of Mercury using QM

JW: You wrote, "I reiterate, that you dont calculate the orbit of Mercury using QM." Is this a dogma? We showed otherwise, in theory, and the spacecraft BepiColombo may show it expermentally next years.

If you had any real understanding of Quantum Mechanics you would realize that it is an improper tool to use in this case. Classical Mechanics, including GR, is the correct path.( This problem has already been solved in relativity.)

Why? First of all the problem is macroscopic, and secondly implies the finding of a orbit or path, which QM wont give you.

Quantum Mechanics is mostly for things the size of a atom (microscopic) or less.

Wave functions or State vectors dont give paths.

JW: You may read the actual paper we wrote. But, as Leon Tolstoy said, “The most difficult subjects can be explained to the most slow-witted man if he has not formed any idea of them already; but the simplest thing cannot be made clear to the most intelligent man if he is firmly persuaded that he knows already, without a shadow of doubt, what is laid before him.” Thank you for contribution.

JW: The truth is the whole. To be positive or negative is subjective.

As VV says, it simply shows that our way of cognition is as contradictional as the reality, as we say in TR unavoidably.

Everyone, then, has the right to his/her point of view. And nobody has the right to denigrate the point (worse: person) with whom he/she disagrees, that is an ad hominem attack.

Here, we respect all apparently divergent viewpoints, even denigration of the self. Free speech is free. The noise just highlights the signal.

Dream away with your own thoughts, and learn nothing.

You also cross me plenty of times, thanks. It helps me focus better on some problems.

JW: Thank you. I suggest one can start answering questions, such as what has been called the "Tohu wa-bohu", by clearing the field. There is a lot of clutter to clean. I suggest 7 fictions:

Microscopic continuity, infinitesimals, irrationals, Cauchy epsilon-deltas, real numbers, random numbers, and limits.

As a consequence, the concept of "error" disappears, and everything helps. The noise, for example, highlights the signal. The real numbers become the union set of integers and their ratios, the rationals a/b. There is no half-hole, nothing is missing. Cantor result becomes elementary school stuff.

Rather than disprove the 7 fictions, I prove by showing that all of arithmetic, algebra, and calculus follow without them, which shows that they are not needed.

It is very hard for me to become that radical!....It means tossing out any normal text of Calculus. Perhaps your position is closer to Intuitionism (Brower)?

PS

I think most real numbers are irrational, and excluse their a/b expression.

If you want to fight something, you still have to retain the debated concepts

and teminology.

Intuitionism causes a great deficit in the number of provable results in mathematics.(versus formalists like Hilbert)

I think your position excludes any infinite expression as being unreal. However you seem even more radical than intuitionism. I very much doubt that you could rescue it all, as you say.

The epsilon-delta treatment of derivatives is a real pain to students, but it does have the merit of the geometrical notion of slope. Other than that, it could be just an algebraic notion, but you neeed both points of view.

JW: "Intuitionism" is a derogatory term ... in a deductive science, like math. Intuition is a clear force against logic, being meta-logic, outside of logic.

No, I prefer to use my own term, as Digital Constructivism (DC) -- linking to Constructivism in mathematics, but providing the idea of the quantum as an ontology.

In this idea, Tohu wa-bohu is what exists between quanta -- as nothing, where even the idea of nothing is too much. Formless and void is a better description.

In no way is Intuitionism a derogatory term as used in mathematics.

It is a well recognized movement of thought in math, together with formalism or logicism, but it does have connection with constructionism.

Logicism or deductive science for math is also a questionable description for math, many would describe math as more of an art form. (Or to some "what mathematician do")

Gödel started the tred to show incompletness in arithmetic.

By using the term digital you leave out ayone who does analytic calculation using the brain or paper and pencil, this is quite a limited view.

So I guess the ratio of the perimeter to the diameter of a circle has no meanig to you?

Sorry never heard of Tohu.....

from what you say sounds like nihalism.

Dont get the connection of quanta (as in QM) with this discssion.

You can use this term in liu of a finite jump, you can find this later concept say in Hegel.

JW: You and many others describe math as more of an art form. This would include the "ego" as the decider, and personal opinions would matter as much as politics in the January 6 events in the US would want.

But, in the same way that there is intersubjective truth, expressed in the "rule of Law", there is objective truth, expressed in the law of gravity, eletroweak theory, and in the Atomic Bomb.

Everyone, friend or foe, must obey an objective truth. Only enemies need to obey the rule of law, as the Romans concluded; while for the friends, everything.

But corruption should not have power over mathematics. Otherwise, one could restrict logic to binary logic, impose LEM (Law of the Excluded Middle), call Gödel, and rule, albeit with uncertainty.

Instead, one can calculate without error, and use tri-state or more logic. This is already done in commercial digital circuitry, and is an IEEE Standard, look up SystemVerilog.

JW: The ratio of the perimeter to the diameter of a circumference has meaning as "x" and was exactly calculated by Archimedes.

It is a number that depends on the curvature of space, and for non-euclidean spaces, it is rational, the ratio of two finite integers, two quantum numbers -- with nothing in-between.

Rationals can be calculated, counted, and measured by anyone, friend or foe, in finite time, and thus belong to reality. A reality that is quantum at the core, but does not exclude paper-and-pencil, analogue. We can see that in universality, where continuity is a fabrication, an illusion, of being too far removed to be known.

The method of the "new" calculus, DC, to obtain the derivative exactly, without fictional Cauchy epsilon-deltas, or requiring continuity?

It is similar to other ideas of solving difficult problems. First, one begins by transforming the problem to another space, the integral form, where the solution is easy, and continuity is not required. Then, by transforming it back.

Ed

Your story is inconsistent. Let us stick to plane or eucledian geometry.

So then tell me what exact value Archimedes found.(Presumably, to be consistent, you should tell me it does not exist, or, as you say, out of reality) How would you calculate pi without error?

Well, about the derivative, I already gave you the Lagrange method.

Cannot understand a good part of your spiel.

Who uses quantum numbers in liu of integers? (The meaning is fairly different)

JW: I have to respect your opinion, and be vague. I gave all the references. It is now a matter of public record. You can follow what you want.Thank you. This may be my last comment to you, as I have to leave space for other viewpoints. Michael Polaniy explains it.

In looking at irrational numbers, that cannot be measured nor counted, one has to be careful. Just like with the word "qubit", which many think "screams quantum and must be used", or the complex number square root of -1, which many think "is necessary for the Schrödinger equation", or bit in terms of communication theory.

But words have no intrinsic meaning, as Frege explained. There is no "theory of meaning" of a particular word. Words are a reference, and reference and meaning are just two independent variables.

In particular, instead of using complex numbers, and pretend they occur in nature, one can use ordered pairs of numbers.

Rationals and integers can then be used correctly, to represent all that the imaginary root does, in a not mysterious faction. It is just a 180 degree rotation. Less distracting, less fancy, more exact. The Cauchy residue theorem, then, becomes easier to explain.

How about pi? In the measurement of an angle, instead of using what one does not know as pi, one can simply measure the angle with a protractor and scale, and take the angle in a straightline as 180 degrees. Less distracting, less fancy, more exact.

Here one has to more careful with the "sharp shooter's" paradox.

It is a fallacy to use an irrational in an expression (necessarily as an approximation), and then find cases of that expression as "examples" in nature.

This happens with the golden ratio, Fibonnaci numbers, three phase currents, e, pi, etc. What cannot be calculated, should not be used in the first place.

As I referenced, all arithmetic, algebra, and calculus can be proven without approximation, or error, and using only integers (and rationals).

No need for any fiction, such as infinitesimals, Cauchy epsilon-deltas, or irrationals. Then QM can meet GR.

Almost none of the above has to do with "QM meeeting GR". You could only (hypothetically) solve this by doing Physics.

Right now the two seem just too different and address different problems.

JW: Current mathematics impose upon physics several fictions, 12 or more. With such large number of unnecessary approximations, no wonder that physics cannot represent the universe we see -- we estimate that only around 4% is represented in current physics.

In particular, I, Jason Gallas, and Robert O'connell, had to use a finite difference approximation to the Schrödinger equation, exact within a suitable set of functions, to escape from a continuity artificially imposed by the second derivative, and published in Physical Review Letters with important scaling laws for Rydberg atoms in magnetic fields. See:

Article Scaling Laws for Rydberg Atoms in Magnetic Fields

We want physics experiments to be replaced in a not so far future by computer simulations, especially in quantum computing and quantum mechanics. We need computer experiments to be realistic. Nuclear fusion and fission, with A and B bombs, are already developed by simulation in computers. We need that extended to civilian purposes, open to everyone.

Changing mathematics first is imperative, then QM naturally implies GR. We show in cited work that one can connect geometry with energy, through quantum entanglement, so that the curved geometry of spacetime in GR can emerge naturally from the wavefunction in QM.

Undergrads and grads frequently use reals and even complex numbers in computer code that can run for a long time, in their own PC or in the computer center -- while the computer will run on integers anyway. This is a waste of time, and resourced as "called for ".

Students should be motivated to use "x mod p" in integer arithmetic, instead. That is what DC does, and works for all 4 arithmetic operations, algebra, and calculus. There is no apparently "real number" or "complex number" being called for or generated.

Well, not all of us are tempted to take "the mathematical trip" , that I believe just gets in the way of Physics.

It partly has to do that good physics is getting very expensive, cetainly much more that just running a computer.

This kind of excuse wont tempt me. I try to keep my brain alive with analytical calculation, by the sound of it a dying art among new generations, digitized, but not really progressing.

Quantum computing is just still a myth...However the prerequisite is able to do QM

Sorry, but in my view QM will never imply GR. It is more likely that there will be somewhat of a shift in GR twards something else.

JW: This attitude sounds like dogma. Instead, let's consider facts. In particular, I, Jason Gallas, and Robert O'connell, had to use a finite difference approximation to the Schrödinger equation, exact within a suitable set of functions, to escape from a continuity artificially imposed by the second derivative in mathematics, and published it in Physical Review Letters with important scaling laws for Rydberg atoms in magnetic fields.

GR + QM is influenced by the same thought, that conventional mathematics is imposing on GR a continuity that does not exist microscopically. Instead, consider a different topology: not continuous but quantum. Everything works then, from the ground up.

As Feynman used in physics, the way to think about phenomena is not to think about mathematics. It is to observe experimentally first, observe nature, and then find mathematics to the problem. Two different mathematical descriptions that predict the same experiment are, on that light, indistinguishable.

Here, we are observing first, and realizing that the conventional mathematics tools at our disposal do not fit the observation. The phenomena is not continuous, so the mathematics must not include Cauchy epsilon-deltas. Computer science, moreover, does not have epsilon-deltas.

Then, we have hit the center of the problem, and space is mostly empty. Atoms cannot get "as closely as desired". There are no Cauchy epsilon-deltas to talk about. Mathematics is essentially, as a ground-rule, quantum.

There are particles, but there is nothing in-between particles, where even "nothing" is, perhaps, too much.

So, we must create a new mathematics, a new calculus, with no infinitesimals, no irrationals, no microscopic continuity, and so on ... no errors. This; is however the old mathematics of integers, calculated by x mod p, and modeled in gears. It was used by the Greeks in the Antikythera Mechanism. It was used by the Mayans in their calendars. It could prognosticate celestial orbits and eclipses, thousands of years away, with certitude. That is what we want. It was obscured by clutter. Integers obey all 4 arithmetic operations, exactly. Continuity is derived in the macroscopic, with universality. Things appear continuous but, actually, are made of far-appart, discrete, individual, atoms; and we are too far removed to see it directly.

This is how we manage QM to meet mathematics, physics, and GR. By using integer arithmetic, exactly. Galois numbers revealed themselves to be a natural language to express the mathematical field relations we needed. The algebraic approach imposed itself as a ready-made relationship. Change QM, then change GR. But it all begins by changing mathematics first.

Ed

What you call continuity is used also in QM, and very different mathematical approaches are used to find basically the same results. The PDE of Schrodinger and Matrix approach of Heisenberg.

Where do you think integers should enter? I can tell you this, that Galois integers play no part in QM.

Your ideas are simply too far away from how things really go in Physics...and that is one area I know about. Im ending discussion in this thread.

The areas in math you would need in QM are

Functional Analysis

Hilbert Space

Operators and operator algebra

Matrices

Partial Differential Equations.

You dont seem to handle any of this, and stuck with integers.

JW: Thank you for your contributions, but the fault may be with the self. Good bye.

TR (topology reduction) by the author can be understood as providing the topological basis -- TR means "that a continuous path in a higher dimension must be discontinuous when projected in a lower dimension".

TR does not change Wirklichkeit (nature as we oserve it), nor Natur (nature as unkown, unseen) as divided by Kant, in German. TR just changes how we observe nature. In that sense, TR changes physics -- Niels Bohr explained -- as physics concerns what we can say about nature, and there our description changes, although the facts remain the same.

A similar situation happens with the number Pi, currently. We realize Pi as (3.141592...) is not measurable. Does it exist in nature, nonetheless?

To give a short answer, NO. The universe is a non-euclidean space, where the ratio of a circumference to its diameter must stand as the ratio of two finite integers. The value is similar, may change according to the local curvature, but the curvature is never zero and we can use use finite digits for the ratio. Rather than a particular value, the quantum becomes a rule.

This … changes GR, to a quantum basis -- what we were long waiting for -- harmonizing GR and QM.

A disagreement here blocks for the self a new vision -- whereas the collective follows it already.

There is a complement to mass, lIke distance is a complement to time, in spacetime.

There is proper time, as there is proper mass, called mass. For comoving observers, mass exists and is measurable. Comoving observers also exist. They share the same wavefront, in a classical wave picture.

But what is the "classical wave" picture? It MUST not be a wave microscopically, but macroscopically, we conclude from this question.

The wave is still composed of molecules and atoms, of particles, but not of hypothetical "wavelets" as imagined using the Maxwell Equations.

There is no wave, microscopically speaking, even infinitesimal (which does not exist!), and we are at that lower level when we talk about particles.

There is no duality, but universality -- when at a higher level, we are too far removed and do not see the particles in their unit, just in their aggregate.

This provides a new basis for math, QM, and thus GR.

In physics, negative absolute temperature exists, in a LASER medium, and is hotter than infinity. Is Boltzmann wrong, thermodynamics defective?

No. Ramanujan explained it, and analytic continuation gives the mathematical theory.

In particular, 1 + 2 + 3 + ... does not go to infinity, he explained, but to -1/12. Exactly. Time for conventional mathematics to accept it.

If even that result can be accepted, why not that irrationals -- that no one can count or even measure in nature -- still bother us? We are afraid of illusions. That is one of many, including microscopic continuity in Cauchy epsilon-deltas. Time for conventional mathematics to end this folly, and introduce discrete solutions to the wavefunction in Schrödinger equation, in the "new" QM, with lots of new opportunities to mine the gold of truth.

To "exist" must be strictly interpreted as Brouwer proposed, to "be able to be constructed". We can accept missing materials, but not missing algebra.

As we eventually learn when teaching, many enriched educational approaches include pedagogy designed to cultivate imagination, creativity, and lateral thinking.

That is important at an early age, even in elementary school. But can adults choose to also broaden horizons or opt to keep a narrow view, or is this choice itself in the first place a function of what we generically can call Openness?

External motivations to increase Openness, for instance talking with someone more open-minded, or wanting to advance professionally, might lead individuals to try new things more than they would if left to their own devices. There is opportunity for everyone, and sometimes one just has to learn to wait.

One can use continued fractions, and stop at any point according to a well-defined error, with code and integers.

In particular, any square root can be defined by a continued fraction. This signals the end of mathematical error in mathematics, computer science (code) and physics. GR can meet QM.

We can calculate anything we want, and it will be exact, even when quantum mechanical -- the Heisenberg uncertainty was the observer's, not the target's.The computed reality is objective, precise, valid for friend or foe.

Time is endless. One can take the "blue" pill and live happily, even for many lives, in an imaginary reality.

But, one day, one decides to take the metaphorical "red" pill, and awake.

Computability becomes reality. It will be harder for computing to be limited by bias. Humans can proceed, and still be limited by ego, as online discussions show. Seraphic beings less, beings that can also exist.

Comoving objects, subject to no force, can experimentally separate. This happens under cosmic expansion (Hubble), which can also create speeds greater than c. But, what is the physical theory for cosmic expansion? So far, nothing serious under SR or GR.

But cosmic expansion can be seen under QM. To explain not only why the expansion, but why is the expansion accelerating? Can we do something about it? Or, about its effects? Or, benefit from its effects, as we learned to benefit from gravity, and model its effects -- by developing GR? Having GR and QM together?

GR, in the field equations of Einstein, includes a second derivative. This mandates the solution to be continuous, which is a mathematical burden, it is not physics.

The mathematical microscopic continuity of Cauchy is present there, as epsilon-deltas. Time to evolve, the topology is NOT microscopically continuous, but quantum. A different differentiation rule is needed, and provided, by digital constructivism (DC).

In the Russian language, MIR (Russian: Мир) is sometimes translated into English as "world," or as "peace," or as "village," but a single-word translation misses its full significance. Let's use PEACE.

With those wishes in mind, we focus on MIR as what everyone agrees to.

Not somehow "respecting" the differences, but identifying them and working to reach that common goal -- which leads more easily to union.

In the potential interest of dialogue, one can say:

"Sharp contradictions between quantum theory and general relativity may be due to mathematics rather than to physics.".

This what I have been addressing on/off since 1976, even before I completed my undergraduate course at ITA.

That's the reason my recent manuscript exists. Several publications after my undergraduate degree, in top journals, I agree that this phrase, which is not mine, represents what is exact.

Solving the GR/QM conundrum is not a matter of physics only.

Mathematics seems stuck in the 17/18/19 centuries, and needs to accept QM as part of nature.

Central to that is the definition of derivative, and mathematical real-numbers.

So, our "fight" is for mathematics, before physics, and I congratulate anyone for the realization that mathematics must change -- because of physics.

Mathematics, however, has changed. It is a "paradigm shift" as defined by Thomas Kuhn. It cannot be avoided or delayed, or stopped.