Thank you for this question Andre. I believe this kind of question is key to answering fundamental philosophical questions of Quantum Theory. My view is that irrationals DO exist in Nature, BUT, whenever we make a measurement we can only get a rational answer. Think of the best tapemeasure in the universe; its graduations can only be rational, Further, I would say existence of the irrationals can never be proved nor disproved. I would say also, that complex quantities exist in Nature also, but their existence can never be proved or disproved either. If you'd like to look further into this, take a look at the Soundness and Completeness Theorems of Model Theory, a branch of Mathematical Logic. Together, these two Theorems show that: under the Field Axioms (the rules of the game for scalars) existence of rationals is provable, but existence of irrationals and complex numbers is neither provable nor disprovable. That is to say, they are logically independent of the Field Axioms --- or, mathematically undecidable in the language of Kurt Gõdel.

One way that has been used to think about this sort of question is to take the position that numbers, of whatever type, whether rational or irrational, are used as a way to describe observations of 'nature'. Whether that means that numbers, of whatever type, "exist in nature" might depend on what is meant by "exist in nature".

Thanks for your message. That's one point that was missing from my explanation. We can tell for sure that natural numbers and rational numbers exist in nature because we can see them with our eyes. For instance, I can define as the number one, a rod with length of 1 unit (could be meters, centimeters, ...). So the number two corresponds to two rods, one half corresponds to half a rod, and so on. Now, the same is not true for irrational numbers. It is impossible to build a square of length 1. That is, it is impossible to take four rods and make a perfect square, since the diagonal would be sqrt(2), which is irrational.

But the number pi=22/7 exists and its irrational.

Didn't you mean 22/7>pi? By the way, 22/7 is a rational number

Oh yes I get it. You are right pi is not equal to 22/7

Let's, as you say, define as the number one, a rod with length of 1 unit (could be meters, centimeters, ...), then the number two corresponds to two rods.... That is equivalent to saying that two rods is twice one rod, or they are in the ratio 2:1.

In the same way, the ratio of the diameter of a circle to its circumference is 1: pi

As such, if rational numbers are deemed to "exist in nature", then so do irrational numbers.

It all depend on what is meant by "exist in nature".

Ok. The meaning of exists in nature is: Can you design an experiment from which you could measure an irrational number? This question seems rhetorical since the obvious answer is no. My point here is, we start with natural and rational numbers because those are things we can see. Moreover, we also start with geometrical ideas, since geometrical figures resembling squares, circles, etc also exists in nature, i.e., we can see them. Then we start studying the properties of those numbers and those figures. We are creating math using those fundamental concepts all familiar to us in our daily experiences. When doing this, in our idealized version of those figures, we stumble upon irrational numbers, as Pitagoras did with sqrt(2). He realized that such number could never be written as a ratio of two integers. As such, the figures that Pitagoras was studying could never actually by built by us, since any irrational valued length should inevitable be approximated by a rational number for anything we can actually build or draw on a piece of paper. Therefore, irrational numbers were invented by us, since they come up from mathematical objects that were we also invented. The same is true for the notion of continuity, differentiation, and all the rest for that matter.

By the way, it seems that all fundamental constants of nature are irrational numbers. That might by indicating that our theories, too, were invented and not "discovered".

As I said earlier, one way that has been used to think about this sort of question is to take the position that numbers, of whatever type, whether rational or irrational, are used as a way to describe observations of 'nature'. The same perspective can be taken about all other aspects of mathematics.

That is the point. However, is it the case that the irrational numbers are there but we just don't have the precision required to measure them? We can compare it with the uncertainty principle. It is not that we lack enough resolution to simultaneously make accurate measurements of two noncommuting observables. It is in fact a fundamental principle of quantum mechanics prohibiting such simultaneous measurements. Could the same be said about irrational numbers? If irrational numbers really don't exist in nature, what does it say about the most fundamental properties of our universe?

We don't have the precision required to measure anything. When you said earlier about a rod with length of 1 unit (could be meters, centimeters, ...), any such rod in 'nature' is only approximately 1 unit. Only in mathematics is 1 unit precisely 1 unit.

The most fundamental properties of our universe are used as a way to describe observations of 'nature'.

Assuming that we know what those fundamental properties are. For instance, every time someone looks for a violation of the uncertainty principle due to the existence of a minimum length in the universe, the Planck length, then they are already considering that the uncertainty principle is not a principle of "nature" at all, since if it was a true fundamental principle, it could never be violated assuming that the universe is governed by a single universal theory.

Right now, the two answers: "yes, we can have irrational numbers in nature, but we can never measure them exactly", and "no such things can exist in nature" are equally likely to be true. Would there be a theory whose fundamental principle is one of the previous answers? The point of my question boils down to this question.

I would guess that the point of your question is related to Heisenberg's uncertainty principle.

Yes, it is indeed very similar in its essence.

And today I just stumbled upon this video :

https://www.youtube.com/watch?v=XnEqfTjp66A

It very much encapsulates what I was trying to convey.

Perfect spheres, circles do exist in nature. rain drops acquire the spherical shapes as they tend to accommodate maximum volume in the minimum surface areas, thereby justifying the existence of pi. If two unit line segments are inclined at right angles at one extremities, the other two are at a distance of sqrt(2). Being student of science and mathematics, we have plenty of examples to conceptualize the existence of irrational numbers in nature.

I do agree with Mr. Kattel !

I disagree! Unless you can design an experiment which could observe those perfect geometrical shapes, you can not claim with certainty that they do exist in nature.

Do straight lines exist in nature? Do circles exist in nature? No. According to Plato they exist somewhere as archetypes,. Is "somewhere" part of nature?

At this point, this is a purely philosophical question

Only in the sense that the mind is part of nature. Irrational numbers are mental constructs. They cannot be measured in nature.

Is it that we can't measure them because we just lack the precision required to do so or do they really don't exist in nature? That's the conundrum!

Among others, the so called Feigenbaum constant is irrational number. Of course, one can measure a given quantity, with certain accuracy. For example, the paper: M.P. Hanias, Z. Avgerinos and G.S. Tombras, "Period doubling, Feigenbaum constant and time series prediction in an experimental chaotic RLD circuit", Chaos, Solitons and Fractals, vol. 40, 1050 - 1059 (2009), is devoted for an experimental determination of the value of this constant.

This is potentially very interesting, specially considering that all the fundamental constants of physics so far look like irrational numbers. I have never heard of this. Thanks for the reference.

Dear Sydney,

Thanks for your answer. It is true that our models of nature require the existence of irrational numbers. I pointed out earlier on this topic that without irrational numbers, there can be no continuous curved trajectories. This implies that nature really would be "solving" problems using discrete methods (i.e., finite difference schemes for differential equations). The real question is, does our postulate that spacetime is a continuous manifold really holds true in "nature"?

Dear Sydney,

Thanks for your message. I intentionally used the word spacetime because my question refers to things we can either perceive (3 dimensions of space + passage of time) or observe (particles, trajectories, etc...). On the other hand, neither quantum fields nor classical fields can be either perceived or observed. Since the ontological status of quantum fields is still unknown, I prefer to leave them out of the discussion for the sack of keeping things simple and intuitive.

As I said before, the only viable approach to answering my question is to postulate our way out of the conundrum it poses, as you did by invoking Wheeler's conjecture.

Best Regards

This goes back to an age-old question: Are the number systems, eg. real numbers, complex numbers, etc. inventions of humans or discoveries of what exists in nature? Since the human mind is part of nature does this mean its activities and concepts are also?

This video is interesting too, and nicely relates with the discussion

https://www.youtube.com/watch?v=S4zfmcTC5bM&t=134s

I just discovered it recently.

Resolve this paradox. In order to pass through a ten second period of time, you must first pass through pi seconds. Time is a measure of change. If pi does not exist in nature you cannot pass through pi seconds. If you cannot pass through pi seconds, then change cannot be continuous; it must be discrete.

We don't have clocks accurate enough to measure pi seconds. So close to pi would we have to get in order t see any deviation from a continuous process? What we need is an indirect way to find out if irrational numbers exist or not.

The paradox has nothing to do with measuring pi seconds. The point is, it must pass through it. Similarly with the other irrational n umbers. There is a resolution of this paradox.

If the gap between all the rational numbers due to the non existence of irrational numbers is too small for us to detect, then any deviation from a continuous behavior wont be seen by any experiment we can conceive.

By the way, I agree with you that it would solve the issue if we could observe such small effect

This video might help to clear out some mystery here

https://www.youtube.com/watch?v=REeaT2mWj6Y

Thanks!

To wrap it up, only experiments can give a definitive answer to my question.

Let us not get too wrapped up in pi. The paradox exists for any irritional number between 0 and 10. In fact, if a random number could be picked between 0 and 10, the probability is zero that it would be rational. We thus say that almost all the seconds between 0 and 10 are irrational. The irrational numbers do not exist in nature because they are constructed in buiding the real numbers by the axiom of completeness. This is a mental construction; it occurs nowhere in nature except in the mind.

This is a valid point of view. In fact, the same can be said about every model of the world we have ever created! All our theories and models can be said to be merely mental constructions which only exists in the human mind. However, this solipsistic point of view is not what I had in mind when I started this discussion. I can even rephrase my question. Do irrational numbers (and also rational numbers, natural numbers, etc...) have a counterpart in nature?

Right. Even the natural numbers are not so natural. The positive integers are based on an equivalence class of sets and the postulate of induction. The postulate of induction is a human construct and a set is an undefined concept. The postulate of induction leads to an infinite set of integers. I doubt if infinity is part of nature. Your question was a very interesting question and opens up a lot of thought,. We know a lot about numbers, but what do we know about nature?

Thanks for your comment. We know that whatever nature is, it seems to comply reasonably well with the expectations of our models. For instance, we can send rockets to the moon, to Mars, and to other corners of the solar system with a remarkable degree of success. Moreover, we can build skyscrapers, cars, computers, as well as treat and cure diseases. We design experiments based on our models, and almost all such experiments behaves exactly as we expect. In fact, based on this enormous success, quite often we tend to see nature as being exactly like our models describe it. Thus, which parts of our models have counterparts in nature and which parts exist only in our minds?

Dear Andre,

I was thinking of a similar comment. Isn't it amazing that applied mathematics is so useful?

The working tool of the mathematician is the theorem. The working tool of science (the study of nature) is the experiment. Otherwise they are mostly alike in their use of logic and definitions.

Experiments , whether to establish theory or to confirm theory, lead to quantitative results. The bijection between the real numbers and the straight line leads to coordinate systems, geometry, and consequently space and time. Isn't this beautiful!

Thanks again for opening up this area.

Tom

Numbers do not exist in nature. They are human constructions. I agree that five items can exist in nature, but five cannot. Regarding nature, a number can be used as an adjctivec but not as a noun.

I am on the same page as Carter here. I am almost convinced that what we call mathematics do not exist in nature. However, it does have deep connections with the workings of reality ( this from the perspective of realism) . We, and all other animals and even insects for that matter, are born with some notion of numbers that seems to be deep routed in our brains, and seems to capture some fundamental aspects of reality. What we call math, and numbers and all that, is our generalization of that notion we were born with.

I will give an example of what I mean. A long time ago I was reading a book about prime numbers. The author mentioned the story of autistic twins that, to any given list of numbers, they would only react to prime numbers, however large those numbers happened to be. At the time it seemed to me that they had some extra knowledge about numbers and how to effectively factorise them that we hadn't figure out yet. This naive perspective was based on the belief that math exists in nature, and we are just discovering it as we go along. As appealing as it may sound, later I found out there is another way of looking at this. What if those twins had instead a wider notion of that basic numerical skills I mentioned we were all born with, and instead of knowing some marvellous simple formula for detecting prime numbers, they were just factorising them mechanically like we do, but way faster. This point of view make sense since there are accounts of people with amazing calculating skills which, in order to calculating complex integrals or whatever, they did it activating areas of the brain responsible for eye movements (something ordinary people can't do), which are known to respond super fast to stimulus.

Wulf, I also have the ability to conceive of a unicorn, but that doesn't mean it exists in nature. The human concept of a unicorn may exist in nature inasmuch as it exists in the human mind and the mind is part of nature, but that in no way means that there is a real referent for my concept in the world. That would be a category mistake.

I'm not sure how much of this conversation is the result of sloppy semantics, and I don't really care to tease it out. But you might be interested in what are called "Orchard Problems:" https://www.youtube.com/watch?v=p-xa-3V5KO8

Were we to describe ratios found in nature USING human concepts, we would be hard-pressed to find any that are rational.

Dear Wulf,

I bet you haven't read all that was posted here, so let me paste bellow one of my previous posts that I hope will make it clear for you what is my opinion on this.

Fellow Carter wrote:

"Let us not get too wrapped up in pi. The paradox exists for any irrational number between 0 and 10. In fact, if a random number could be picked between 0 and 10, the probability is zero that it would be rational. We thus say that almost all the seconds between 0 and 10 are irrational. The irrational numbers do not exist in nature because they are constructed in buiding the real numbers by the axiom of completeness. This is a mental construction; it occurs nowhere in nature except in the mind."

My response was:

"This is a valid point of view. In fact, the same can be said about every model of the world we have ever created! All our theories and models can be said to be merely mental constructions which only exists in the human mind. However, this solipsistic point of view is not what I had in mind when I started this discussion. I can even rephrase my question. Do irrational numbers (and also rational numbers, natural numbers, etc...) have a counterpart in nature?"

Dear Wulf,

I bet you haven't read all that was posted here, so let me paste bellow one of my previous posts that I hope will make it clear for you what is my opinion on this.

Fellow Carter wrote:

Let us not get too wrapped up in pi. The paradox exists for any irrational number between 0 and 10. In fact, if a random number could be picked between 0 and 10, the probability is zero that it would be rational. We thus say that almost all the seconds between 0 and 10 are irrational. The irrational numbers do not exist in nature because they are constructed in buiding the real numbers by the axiom of completeness. This is a mental construction; it occurs nowhere in nature except in the mind.

My response was:

This is a valid point of view. In fact, the same can be said about every model of the world we have ever created! All our theories and models can be said to be merely mental constructions which only exists in the human mind. However, this solipsistic point of view is not what I had in mind when I started this discussion. I can even rephrase my question. Do irrational numbers (and also rational numbers, natural numbers, etc...) have a counterpart in nature?

Dear Wulf,

I bet you haven't read all that was posted here, so let me paste bellow one of my previous posts that I hope will make it clear for you what is my opinion on this.

Fellow Carter wrote

Let us not get too wrapped up in pi. The paradox exists for any irrational number between 0 and 10. In fact, if a random number could be picked between 0 and 10, the probability is zero that it would be rational. We thus say that almost all the seconds between 0 and 10 are irrational. The irrational numbers do not exist in nature because they are constructed in buiding the real numbers by the axiom of completeness. This is a mental construction; it occurs nowhere in nature except in the mind.

My response was

This is a valid point of view. In fact, the same can be said about every model of the world we have ever created! All our theories and models can be said to be merely mental constructions which only exists in the human mind. However, this solipsistic point of view is not what I had in mind when I started this discussion. I can even rephrase my question. Do irrational numbers (and also rational numbers, natural numbers, etc...) have a counterpart in nature?

Dear Wulf,

I bet you haven't read all that was posted here, so let me paste bellow one of my previous posts that I hope will make it clear for you what is my opinion on this.

Fellow Carter wrote

-Let us not get too wrapped up in pi. The paradox exists for any irrational number between 0 and 10. In fact, if a random number could be picked between 0 and 10, the probability is zero that it would be rational. We thus say that almost all the seconds between 0 and 10 are irrational. The irrational numbers do not exist in nature because they are constructed in buiding the real numbers by the axiom of completeness. This is a mental construction; it occurs nowhere in nature except in the mind.

My response was

-This is a valid point of view. In fact, the same can be said about every model of the world we have ever created! All our theories and models can be said to be merely mental constructions which only exists in the human mind. However, this solipsistic point of view is not what I had in mind when I started this discussion. I can even rephrase my question. Do irrational numbers (and also rational numbers, natural numbers, etc...) have a counterpart in nature?

I had the same problem as carter did

I disa

I disagree that the point of view I have been advocating here is detrimental to the development of science. I instead believe that it is important to draw a line between our models of nature and nature itself. Telling that I have 5 apples or V apples does not qualify as the definition of what 5 ( or V) is. Many animals can tell the difference between 5 apples and 3 apples, although they can't do the same with the symbols '5' and '3'.

Thank you for this question Andre. I believe this kind of question is key to answering fundamental philosophical questions of Quantum Theory. My view is that irrationals DO exist in Nature, BUT, whenever we make a measurement we can only get a rational answer. Think of the best tapemeasure in the universe; its graduations can only be rational, Further, I would say existence of the irrationals can never be proved nor disproved. I would say also, that complex quantities exist in Nature also, but their existence can never be proved or disproved either. If you'd like to look further into this, take a look at the Soundness and Completeness Theorems of Model Theory, a branch of Mathematical Logic. Together, these two Theorems show that: under the Field Axioms (the rules of the game for scalars) existence of rationals is provable, but existence of irrationals and complex numbers is neither provable nor disprovable. That is to say, they are logically independent of the Field Axioms --- or, mathematically undecidable in the language of Kurt Gõdel.

Dear Steve, thanks for your interesting comment. Along the lines of your words, there was a time when I was wondering whether all fundamental physical constants are irrational numbers, since up to the precision we know them, they are surely not rational. I also think that we can not give a definitive answer to my question. However we are free to speculate about the implications of a yes or no answer to it.

Andre, you might like to see this:

https://www.researchgate.net/post/Is_Plancks_constant_a_rational_number

Thanks a lot for showing me this. Kåre Olaussen's answer is just awesome!

Hi Guys:

Please allow me to join your discussion group - so late in the game.

Thank Wulf for leading me here; you all seem to have similar interests to me in fundamental questions.

Allow me to introduce myself via a recent essay I just finished that Wulf privately reviewed.

Dear Herb Spencer,

Thanks for your message and interest in this discussion. I believe it would be very beneficial if you could share your thoughts on this discussion ahead of us reading your document.

Best Regards

Dear Andre:

Thanks for the acceptance

but summaries are for the lazy.

Those who stop reading are cutting

off their most useful sources.

Remember: Tweets are for the Stupid.

As it is, I had to squeeze the ideas

down to 20 pages and was criticized

for not giving more detail.

As a wise man once said:

"You can't please everyone all the time."

Well, I have a different perspective on this. It is my opinion that a book without preface and introduction would hardly leave the bookshelfs

Dear Herb,

I have read about half of your essay and find it fascinating. I hope to read the remainder. It encompasses many ideas that have developed over time. It is clear that a lot of work and thought has gone into this. I appreciate that ideas not well accepted today are presented anyway. I would recommend that others read this, not necessarily to accept everything in it but to consider and evaluate.

Dear Andre:

You are being an intellectual snob (par for the course for most mathematicians). Some of us have chosen to ignore the historical approach for better communicating with the general public rather than the privileged in their Ivory Towers. Academia.Edu is accessible to everyone.

Dear Herb,

I think what Andre was getting at is that few people will read the essay (regrettably) and shorter statements on the subject at hand would be welcomed. My experience with him is that he is very interested in a dialog. He is certainly not an intellectual snob.

Thank you, Thomas:

For explaining on behalf of Andre.

I have no interest in communicating with the lazy: no effort = no reward.

People who do not bother to read deserve to remain ignorant.

They will fail to influence others, as many have failed in the past.

Dear all,

Thank you Thomas for your kind words. I really appreciate that. Herb, I apologize for sounding like a snob. As Thomas putted it, I believe that short statements on the subject are indeed welcomed. What I had in mind with my reply to your comment is that, at least for me, reading a summary of a large text ahead of reading the text itself is very helpful. It helps me putting things into perspective before reading the entire work. Moreover, it would help your text reaching a broader audience then just a handful of the people following this post. Also, I certainly didn't mean that I would not read your document without a summary.

Best Regards,

Andre

For an ongoing project concerning the set-up of a more general concept of "numbers" which are appropriate for physics, in particular for physical systems which cannot be tackled numerically by attaching to the system an "ordinary" number in the "ordinary" unique way, see

https://www.researchgate.net/project/Mathematical-Concept-of-B-numbers-and-their-Representations-in-Rigged-Hilbert-Space

Dear Karl:

The final extensive models of the hydrogen atom (Wave Mechanics and Dirac's) illustrate the flaw of a math-based physics. They simply calculate a number that agrees with experiments (so cannot be too wrong) but they offer no insight into Nature, which is MY motivation for 'doing physics'. This is also true for Lagrangian or Hamiltonian techniques: all math; zero insights. That's why we still have the very expensive nonsense of CERN.

Dear Andre:

Too many people stop at the abstract, others stop after section one; so there's no easy way for lazy people - why bother? Such lazy people will never put in the effort (10 years minimum) to make an original contribution.

Dear Herb,

"Numbers" are in principle abstract artefacts, which might be convenient and useful for various distinct purposes. To use "numbers" or any math. theory involved with "numbers" or any other math. artefacts often serves as a tool to express things much more unambiguously and precisely as it can be done in common speech without any use of mathematical expressions.

The main flaw in physics, in particular in theoretical physics, is to identify "nature" or "reality" or parts of it with the math. theories which are used to formulate phys. theories, and similarly, to identify any math. feature of an imperfect physical theory with some feature of reality.

Even in the world of physics, it does no harm to have some extensive skills in advanced mathematical theories. What really matters to recognize which branch of math. is appropriate or most appropriate or misleading in order to describe "reality" or parts of it in a --with regard to logical and ontological consistency and verifiable predictions-- successful manner.

Best regards

Karl

Dear Karl:

I agree 100% with your view of the role of symbolism in math, as I wrote in an old essay criticizing Descartes' simplistic views.

I would like to add some comments about numbers that are based on an elaboration of an example found in Herb's essay.

The idea of counting could have begun by shepherds trying to keep up with their sheep to see if any had been stolen. They would associate a pebble with each sheep and observe the pebbles at the end of each day. Recognizing the various patterns in the pebble colleection, eventually counting of pebbles developed.

Modern mathematicians would say that the number was neither the flock of sheep nor the pile of pebbles but was closer to the idea of the bijection between them.

Not being completely satisfied with this definition, the idea of cardinality was born. The pebbles and sheep were both replaced by sets. Numbers were then defined as the equivalence classes of these sets, each number being all sets in bijection with a certain type of set. Finally we had a definition of number that was completely free from nature! I will grant you that the whole idea of number was originally based on observations of nature but we mathematicians take great joy in obscuring our motivation and intuition in presenting our final product!

Dear Ari Ben-Judah,

In answer to your question, an open interval about pi, however small, has the same cardinality as the set of all real numbers. If the open interval consists of rationals only, then it has the cardinality of the set of all rationals. I am not aware of any application of these to statistical mechanics.

Dear Ari,

The results of one of the rare attempts to clarify the relationship between "physics" and mathematical theories in a rigorous manner, can be found in G. Ludwig's

"Die Grundstrukturen einer physikalischen Theorie" (Springer 1978, 1990)

and

"A New Foundation of Physical Theories" (Springer 2007).

https://www.springer.com/gp/book/9783540308324

Ludwig's ideas offer greatest help for not to get lost in the jungle of various physical theories and of physical facts and theoretical fantasies and of all kinds of mixtures thereof.

Best regards

Karl

Dear Ari,

"i", i.e. "sqrt(-1)" should not be considered as a "value" like "3" or "-5" or "123.456789" or "pi", because the word "value" usually reminds to something that can be quantified by means of an ordering relation. But in the set of complex numbers there is no ordering relation. "i" should be considered basically as an "algebraic entity", combined with some rules for handling square-root expressions in such cases where the argument is a negative real/rational/integer number.

Complex numbers and the "i" are useful in various branches of physics.

For example, calculations which are involved with alternating electric currents can be performed easily. In addition to complex numbers, which actually are pairs (x,y) of real numbers, it is sometimes convenient to use "quaternions" (a,b,c,d), which are related to specific complex 2x2 matrices, in order to handle numerical calculations involved with geometrical operations in a 3-dim. affine Euclidean space more elegantly (or efficiently) or to simplify or beautify the formal appearance of various math. formulas concerning relationships for geometric objects and geometric operations (translations, rotations, etc.). In field theory complex numbers occur because the Lagrangian-Noether machinery with complex field amplitudes seems to be the best recipe to combine transformation properties of the phys. object "field" with it's conservation properties, and thereby can serve for definitions of which math. terms of a field theory can be reasonably regarded as expressions representing the physical quantities "energy", "momentum", "angular momentum", "charge", current", etc. Contemporary phys. theories concerning elem. particles and their interactions contain even more number-involved ingredients. Looking at QFT one encounters Grassmann numbers and Caley numbers ("octonions"). Again, all these objects have not the character of a "value", in the sense sketched above, but of an "algebraic entity".

Hence, the more general and less mystic question to be put reads:

"What algebraic entities are useful and successful for describing physical phenomena?"

Best regards

Karl

how to circumvent the contemporary RG problem with accepting answers:

(1) ignore the warning box

and just

(2) RELOAD the page

( works for Firefox web browser , JavaScript enabled )

According to Kronecker, all number systems can be built from the positive integers. As I indicated earlier these can be built from sets without regard to nature. The complex numbers are ordered pairs of real numbers in which the arithmetic operations are defined a certain way. In this field 'i' is just a convenient symbol to represent the pair (0,1).

Dear Karl:

Thanks for the referral to Ludwig's book - seems worth reading but it's price is way beyond most independent researcher's means.

Do have access to a cheaper English version (preferably free pdf?).

Ari,

For those who are interested in the very nature of a "photon" this RG publication and RG project might be of interest :

Research SECRETS BEHIND THE MACH-ZEHNDER PHENOMENON

and

https://www.researchgate.net/project/Hypotron-Theory

However, even in "Hypotron Theory" there are merely "photons", because "anti-" is understood only as "opposite charge". And the "photon" in "Hypotron Theory" has zero charge ( and zero supercharge ), although the 6 constituents of the photon are objects having non-zero charge.

Dear Ari,

You are right about i. It depends on the context. It can be viewed as a vector in R^2 or as a scalar in a vector space over the field of complex numbers.

Dear Ari,

I am impressed with the challanging problems in physics that you are undertaking as well as the impressive physics of Karl. I also appreciate Herb's background in history of science and many other related areas. Steve also presented some great ideas in physics, although we haven't heard from him in a while.These backgrounds can be useful to resolve or at least contribute to the question at hand. My viewpoint is that mathematicians have developed the number systems from set theory and logic. The more difficult part is to characterize "nature" and I see that the recent discussions about photons and related ideas in physics are headed in that direction. When I find time I would like to raise some questions about the nature of nature. These concepts are near the borderline between physics and metaphysics.

Wishing you much success,

Tom

Ola Andre,

You wrote

"...neither quantum fields nor classical fields can be either perceived or observed. Since the ontological status of quantum fields is still unknown,..."

I agree with this opinion, but not with your conclusion to leave out or to abandon the concept of a "field", in particluar "field amplitudes".

In conjunction with the Lagrange-Noether formalism the field concept allows to calculate the 4x4-tensor of energy-momentum-density as well as the 4-vector of charge-current-density from a SINGLE quantity, i.e. the Lagrangian. Moreover, transformation properties of field amplitudes can be related to conservation properties (Noether's theorem). Hence, the concept of "field amplitudes" represents a convenient mathematical tool for describing a material object, which can be characterized by a value for energy, momentum, charge etc., in the case where the material object is considered to be a (large or tiny) spatially extended object. However, this math. feature does NOT imply that "field amplitudes" must have a physical meaning too. The latter might be inferred from quite different physical as well as mathematical arguments, provided that this is reasonable and possible at all.

It should be noted that the concept of a "field" can be introduced in branches of science and engineering which are more or less different from physics, just because the Lagrange-Noether formalism is basically a mathematical machinery which can be useful whenever there is a system under consideration which shows up properties which can be mathematically described by real numbers, continuous functions, differential equations, continuous groups, etc.

Best regards

Karl