Lev,

"All our mathematics has evolved under the assumption of continuity"

The natural numbers and aritmetics does not assume continuity but discreteness. It took a long time for mathematicians, the history of the invention of the real numbers, to find a way to express continuity in mathematics.

Lev,

"All our mathematics has evolved under the assumption of continuity"

The natural numbers and aritmetics does not assume continuity but discreteness. It took a long time for mathematicians, the history of the invention of the real numbers, to find a way to express continuity in mathematics.

Louis, it is true that natural numbers, as the more immediate / direct product of our temporal perception, are indeed of a discrete nature. However, as far as the modern science is concerned---even going back to the ancient geometric practices, and then to the discovery of the irrationality of the sqrt (2), and particularly to the development of differential and integral calculus---I just have to quote the great Bernhard Riemann:

"As is well known, physics became a science only after the invention of differential calculus. It was after realizing [rather postulating] that natural phenomena are continuous that attempts to construct abstract models were successful. …

True basic [physical] laws can only hold in the small and must be formulated as partial differential equations. Their integration provides the laws for extended parts of time and space."

So we *are* talking about the continuous formalism.

"This is a problem which haunts me every day..."

Ramon, you are in good company. For example, Einstein was obsessed with the question most of his professional life (see "The Other Einstein: Einstein Contra Field Theory" by John Stachel at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1457116 )

Lev, you state

"while all basic processes in Nature are discrete".

Do you share the naive view that quantum theory says 'everything consists of quanta, and thus is discrete'? If you have better founded reasons for this statement, please let us know them.

Ulrich, although you sound quite strange, ;--) you may prefer to hear the above point from the Schrödinger's mouth:

"If you envisage the development of physics in the last half-century, you get the impression that the discontinuous aspect of nature has been forced upon us very much against our will. We seemed to feel quite happy with the continuum. Max Planck was seriously frightened by the idea of a discontinuous exchange of energy ... Twenty-five years later the inventors of wave mechanics indulged for some time in the fond hope that they have paved the way of return to a classical continuous description, but again the hope was deceptive. Nature herself seemed to reject continuous description …

The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time. ... So the facts of observation are irreconcilable with a continuous description in space and time …"

Lev,

Do you interpret Schrödinger in that quote as saying that reality is fundamentally discrete?

Louis,

How else can you interpret this?

" Nature herself seemed to reject continuous description … "

"So the facts of observation are irreconcilable with a continuous description in space and time …"

Actually I think it's all simpler than that.

Discreteness in nature fundamentally comes from limits - from "endpoints", which often lead to the establishment of a stationary wave system of whatever relevant phenomenon or attributes obtain within the bounded domain - leading to attributes' peaks and troughs (owing to the reflecting (back travelling) waves' which interfere with the outgoing waves.) Just picture moving up and down a rope tied at one end to a wall, or moving the same rope when it extends to infinity.

If the universe were unbounded aka infinite, its very spacetime would be a continuum.

I'm not sure I agree with the assertion that our mathematical formalism is predicated on continua. Calculus, for instance, works with infinitesimal but nonzero dx 's. A lot of continua in math are idealizations 'at the limit' , of something that tends to infinity asymptotically, without ever truly reaching it. So I'd say that a lot of math reflects in fact a discrete space (in the broadest sense of space).

Issues of continuum, and hence related issues of infinity, such as the continuum hypothesis, issues in number theory, and so on, have a way of often being sorely difficult (There is a pretty good BBC documentary on that at http://www.youtube.com/watch?v=Cw-zNRNcF90 )

Lev,

David Bohm and Basil Hiley seems try to explain the quantum world with a process based approach which is non-local in space-time. In their approach, space-time is not the ground of reality but emerge from more fundamental processes.

Chris, the "limits" have been an integral, possibly the central, part of the continuous model. If you remove limits, there is no "continuous" formalism. ;--)

Lev, I take it you mean limits as in my phrase "idealizations at the limit " above, not as a synonym for 'endpoint'.

You are right with the two provisos that 1- it is clearly an idealization and 2- at times - rather unseldom at that - there are discontinuities and/or singularities when you extrapolate to infinity

Luiz,

I'm sure that *now*, when the continuous formalism has been fully developed and you have zillion of mathematicians (I was educated as a mathematician myself), you can propose all kinds of fancy new models.

The point is that the very idea of differential geometry would have been unthinkable without the continuous formalism.

So the issue is not to invent a clever model that looks like the old one in a new guise.

The main issue is to invent a radically new formalism that is *not* going to look like anything we knew and will suggest to us how to think about discreteness in Nature. Because, currently, we use the term "discrete" as a *negation* of "continuous". We need a formalism that presents a fundamentally new idea of what the discreteness means.

Lev,

I don't feel that I can agree with the statement that

"All our mathematics has evolved under the assumption of continuity..."

Certainly all of quantum mechanics is developed around the concept of discretization. The correspondence principle tells us that at the macroscopic scale classical mechanics is an excellent approximation to QM, e.g., if you're building a bridge or sending a lander to Mars then you don't need to worry about lack of continuity.

Luiz,

I'm afraid you are missing one very pig point.

The main scientific objective (and utility) of mathematics---or whatever it will be called in the future---is to supply 'useful' forms of data representation together with the corresponding apparatus for "data processing".

Due to the well known historical tradition going back many thousands of years, we have relied on various numeric forms of data representation and, eventually, on the continuous formalism, or apparatus.

It appears that now---for the first time in our history---we have reached the point in the development of science when we may need to change this basic numeric form of data representation by the structural forms of data representation that have very little to do with the numeric forms. These new, structural, form of data representation is supposed to capture previously inaccessible side of reality and, in particular, to clarify the "quantum" nature of reality..

Lev,

the Schroedinger citation you gave, refers to the measurement process as an (in a sense) non-continuous element of the quantum description of the world. With a partial understanding of decoherence (Zurek et al.) this is no longer so hard a point. It never was an argument for discreteness in Nature, though. That said, I would like to get from you an idea what kind of "ubiquitous discreteness in Nature" you have in mind when diagnosing an "central inconsistency in physics".

Ulrich,

Since I'm not a professional physicist by education / training, as far as the "ubiquitous discreteness in Nature" is concerned, I have to rely on the opinions of physicists I respect. If, for example, both Schrödinger and especially Einstein (who was very seriously concerned about it throughout his life) say so, I accept it, especially since I have my reasons coming from my own experience in pattern recognition. Einstein's concerns are documented in the above paper by John Stachel.

Now, as far "partial understanding of decoherence" is concerned (about which I'm aware), *I* don't see anything which would make me change my opinion: it appears to be one of those convenient cop-outs, which many physicists today are looking for in order *not* to face the "ubiquitous discreteness in Nature". I can understand this tactics as self-preservational rather than a scientific one.

Of course, I also understand that most physicists are loyal to their basic formalisms to a much greater extent than they are to any of their close relatives (which, to a point, is quite commendable). ;--)

Lev,

physics is an empirical science. With respect to the actual behavior of quanta today experiments are possible which Einstein and Schrödinger could hardly have foreseen despite their undisputed physical insight. The picture that merges from such modern experiments (a very accessible account is Serge Haroche and Jean Michel Raimond: Exploring the Quantum; Atoms, Cavities, and Photons) surprisingly differs not too much from what physicists had learned from Bor, Heisenberg, Schrödinger, Feynman, ... but, you may believe it or not, this is not the result of an unreflected adherence to "their basic formalisms".

Ulrich, please forgive me, but . . . as I already mentioned, such (quite pervasive and understandable) attitude, the Ostrich Syndrome, appears to me to be self-appeasing, when one cannot or does not want to face the "unpleasant" scientific reality: such behavior is in our genes. ;--)

Although you are quite sincere, I believe you are kidding yourself if you think that Einstein or Schrödinger would change their opinions if they were alive today: as Dirac also noted, we have made no theoretical progress to change the situation to such extent. At this point, the experimental advances do not give an answer to the question we are discussing.

However, I'm not saying that the experimental advances *cannot*, in the future, give such answers: I'm sure they can. But for this to happen we need to put on their table a new, 'discrete', concept of a physical process.

Is there an inconsitency in physics between continuity and discreteness. Typically general relativity is absolutely continuous whereas quantum mechanics has aspects of discreteness within it. Interestingly the Schroedinger equation is a partial differential equation which is defined with continuous space and time variables, now upon its solution we obtain so-called discrete values which represent energy levels within say a hydrogen atom which describe the orbits of electrons about the nucleas. So, it is possible to obtain purely discrete data from an equation which is described as purely continuous.

I believe that the problem here is not a fundamental inconsistency in nature but rather one of the way human beings conceptualise nature. It has been shown in quantum mechanics experiments that the particle-like nature of phenomena and the continuous (wave-like) nature of reality can be actualised through the kind of experiment we perform upon nature. One experiment appears to show that nature is continuous, another shows it is discrete. So, in reality it may be that nature is not any of these two strictly separate ways of looking at it. Rather it is capable of showing itself in one or another way. This means that it is neither wave-like nor particle-like but far more subtle than this. Henri Bergson made the comment that in reality time is both continuous and qualitatively heterogeneous which gives one an idea of what nature is capable of.

"What you said about the need of a change in the representation of structures of our knowledge is exactly what the work I cited is about. It changes the focus in the mainstream constructions to a categorial one (by means of an axiomatic theory)."

Luiz,

I'm afraid you missed the point completely.

If you have to represent an image or a Google query in computer, what concrete help can the "categorical constructions" you mentioned give us?

Frank,

"So, in reality it may be that nature is not any of these two strictly separate ways of looking at it. Rather it is capable of showing itself in one or another way."

My point: is that we need a more consistent formalism, in which we will not face such fundamentally unsatisfactory situations, where the discreteness will be clarified in a positive manner rather than as a negation of the principal (continuous) formalism.

It appears that for the vast majority of people (not trained in pure math.) such inconsistencies are not very troubling. Keep in mind that the formal language is the only thing we can rely on when studying Nature. There comes a point in the history of science (today in particular) when the formal language needs to be changed radically.

I'll attempt to answer here point 1 of Avik Dutt 's reply to me (one day ago) at

https://www.researchgate.net/post/Is_there_a_general_physical_not_mathematical_reason_behind_the_conjugacy_of_a_pair_of_variables_in_quantum_mechanics?ev=tp_feed_post_xview

when I suggested that "We will see such evidence [for the inadequacy of QM] as we begin to pursue the alternative formalisms." (His other points were already addressed there.)

=======================================================

Avik Dutt:

"Lev,

In reply to your last answer:

1. 'We will see such evidence as we begin to pursue the alternative formalisms.'

For that to happen, you either need experiments to suggest something that can't be explained by QM, or you need to come up with predictions from the alternative formalisms that you suggest which can violate QM. Till then, I'm placing all my bets on the present formalism of the theory."

=======================================================

I'm replying here since this thread is more appropriate.

As we know, quantum mechanics does not deal, i.e. *does not model*, the ubiquitous mutual transformation of 'particles', since in classical mathematics no such transformations exist. As far as I know, QM does the only thing classical math. 'allows' it to do: it simply 'stitches' together the corresponding wave functions.

However, the processes of mutual transformation are not peculiar ones at all: the very concept of a 'particle' includes such processes. There are no particles without the transformations, since the transformations form the inalienable part of any particle. Put differently, the structure of a 'particle' is *defined* by such transformations.

On the other hand, in our (ETS) formalism, we have put this fact at the very basis of the this formalism for structural representation. We postulated the idea of structured events (at which transformations occur) as the basis of structural (or discrete) representation.

So here is a very basic prediction. According to ETS formalism, *all* particles are sequences of structured events, and for each particle the *structure* of the corresponding several events involved (as defined in ETS) is to be determined experimentally. There are no particles, just sequences of events.

@Lev,

'Reality' consists of 'events' (which have neglectable spatiotemporal extension) and particles are causal links between events. This view (reproduced here out of my memory) is nicely explained in the influential book by Rudolf Haag: Local Quantum Physics, Fields, Particles, Algebras, Springer. particularly chapter VII.3 'The Evolutionary Picture'. This is less simplistic than your view: 'particles are sequences of events'.

Luiz: "The model adopted today is constructed on continuity assumptions, but the concepts behind the model are independent of those assumptions."

This does not make any scientific sense!

My dear Luiz, now I'm really surprised! (Although I should have known better.)

Your answer is really puzzling, to say the least, especially considering the above Riemann's quote

==================================================

"As is well known, physics became a science only after the invention of differential calculus. It was after realizing [rather postulating] that natural phenomena are continuous that attempts to construct abstract models were successful. …

True basic [physical] laws can only hold in the small and must be formulated as partial differential equations. Their integration provides the laws for extended parts of time and space."

==================================================

and the following Von Neumann quote

==================================================

“The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations describes observed phenomena.”

==================================================

As Riemann observed, in the continuous model, "true basic [physical] laws can only hold in the small and must be formulated as partial differential equations. Their integration provides the laws for extended parts of time and space." But now we know that they do not "hold in the small", and the logic of the continuous model is broken.

Of course, I understand that---as was the case with Ptolemaic epicycles---one can continue to patch up the inconsistencies for a while, but in the end we need "to smell the roses" and produce a new model of reality. And I am trying to "keep open mind to get answers". However, I'm afraid, somehow, you continue to ignore the presented evidence I am referring to regarding the big picture. It is a separate issue why so many scientific workers are quite insensitive to the above inconsistencies.

@Lev,

To a physicist your statement

========================================================

But now we know that they do not "hold in the small", and the whole logic of the continuous model is broken.

========================================================

looks ridiculous: Does the fact that water is made of molecules break the logic of Navier Stokes equations?

Luiz,

Why do you keep pushing the 'party line'?

What do all these postwar developments (about which I'm quite aware, including the non-commutative geometry) have to do with what Riemann said?

Ulrich:

"Does the fact that water is made of molecules break the logic of Navier Stokes equations?"

In a sense, yes, but as the numeric and relatively crude approximation, no.

Again, as Von Neumann hinted, formal models are the only models of reality we have. Obviously we want to have the model not for the "numeric purposes" only, but we need it to tell us what the reality is structurally (or qualitatively) like.

By the way:

"Although these equations were written down in the 19th Century, our understanding of them remains minimal."

http://www.claymath.org/millennium/Navier-Stokes_Equations/

I see Luiz: apparently, somehow, we now gained a 'deeper' understanding of the conventional mathematical and physical models than Riemann, Einstein, and Schrödinger did. ;--)

Don't forget we are talking about the fundamental features of those models, which no formal niceties can change.

I also hope that you haven't forgotten that the general relativity---and hence our understanding of the large scale structure of the Universe---are based on (somewhat advanced by Einstein) idea of Riemann.

Luiz, I asked the question because, being involved in the field of Pattern Recognition, I gradually came to realize the fundamental limitations of the vector space as the representational medium. I.e. I realize that the numeric forms of representation are extremely limiting from a structural point of view, where by "structural" I mean the interrelationships among various object "parts".

Gradually, as the development of our ETS formalism proceeded, I also realize that the structural side of reality cannot be separated from its temporal side, i.e. an object's structure emerges temporally, during its evolution, or formation. So it became clear that this might be the essence of the discrete (event-based) view of reality. in contrast to the conventional space-based, or spatial, view.

Hi Lev, sorry for joining the discussion that late-you are partially right with some of your remarks on calculus, vector spaces, etc. However, I'm feeling chilly with some implicit identifications/connections of calculus or some of its aspects and physics as well as with your remarks on vector spaces. Of course, some of these techniques are what you denote by 'party line', and there are lots of advertizers of the 'party line', too. However, nobody prevents you from going beyond vector spaces or differential calculus :-) just take Grassmann and Plücker or Clebsch, for example :-) and I disagree with claiming that these descriptions have to be associated with an understanding of 'physics'. At its best, the mathematical apparatus of QM describes certain restricted physical aspects, and one always has to respect the limitations of the Ansatz (which nowadays some people of the 'party line' sometimes forget or suppress :-) ) The same holds in a lot of situations where differential calculus is used in physics because in physics we are usually talking about observable ('integrated') phenomena...Solving differential equations is only one way to find appropriate/suitable state reps, although nowadays a lot of text books and people claim this to be the 'holy' approach... ;-)

Rolf,

The issue I'm referring to is much more pervasive than you mentioned.

Actually, what is at stake here is not just calculus and geometry but the entire *numeric*, in contrast to a structural (not in the conventional sense), orientation of science.

Lev, I don't understand your comment. As was mentioned above, it took a long way to continuity and numbers, and the results didn't drop from heaven. You can go back to Hilbert's, Klein's etc. discussions of geometry and numbers and their interrelation, or you can go just to a (discrete) geometry, define your coordinate system geometrically and introduce your numbers e.g. by taking three points on a (real) line and using the harmonic ratio. So your question already bases on some (to my opinion) artificial assumptions or classifications which are not really related to physics or to any physical problem... maybe they constitue a problem for some of the descriptions used in physics or IT nowadays...

Sure, Luiz.

1."The category theory allows to solve (I guess) the same kind of issues that you are concerned. In what way (or points) your approach is better?"

Not at all! Category theory does not address the issue of representation at all. I.e. in physics, we represent the state of a system by a point in some space, most often in a vector space (even if it is infinite-dimensional).

I'm suggesting that to elucidate the nature of 'discrete'---a better term, 'structural'---representation we need to initiate a completely new to mathematics (event-based) form of object representation, which should allow us to see a *completely new*, structural, side of reality. To understand why category theory does not address this issue, I asked you above to think about how would it help us to represent *concretely* an object, for example a tree, or a 'particle' trajectory.

2. "Why you separate space and time so drastically? If I understood correctly, your point is centered in evolution systems. However, space and time cannot be separated in general problems. Thus, we need an approach in which they are treated on equal foot."

At present, the separation of space and time appears to be a consequence of the non-spatial nature of the ETS representation ("struct").

More accurately, the spatial object appears to be instantiated on the basis of this structural representation (which logically precedes the spatial representation). This structural representation appears to be the primary one.

I should note that, in many ways, this is quite consistent with much work in quantum gravity (causal set theory).

3. "Can I see pattern recognition as an entanglement (a strong correlation) between space and time? If so, why making time privileged on space?"

I'm not sure what you mean. ;--)

Pattern recognition is about object classes and object classification. It is directly related to induction.

Luiz,

1. "You can't represent a tree in a reasonable way in category theory because you have to say what are morphisms between two (or several in the case of multicategories) trees. But you can consider the (a bit useless) discrete category of trees."

There is nothing wrong with trees: the whole point of a general formal language is its universal applicability. And this universal representation should not depend on the morphisms. It should be the other way around the morphisms should depend on the objects' representations.

To see the incredible artificiality of the (standard categorical) constructions you describe for the trajectories, compare these highly artificial "representations" with the biological ones. Note that one should not doubt the authenticity of the latter, since, during the evolution, they must have been adopted *directly* from the more primary ('physical') representations in Nature.

2. "If separation of space and time is a consequence, then it cannot be consistent with any work on general relativity."

That's true but as I mentioned, there is an extensive work by quantum gravity researchers who made progress in this direction.

3. I have written sufficient criticism of the Neural Networks approach to pattern recognition (PR). The main point is that they are advanced function approximation machines, nothing more and nothing less. But the main problem of PR is the concept of class and its representation, hence connection with induction.

Induction and PR is, mainly, about the capability to construct class representation (e.g. class of cats) on the basis of a small training set of examples from that class (several cats). Without having a class representation we can't do classification: if I don't have a representation of the class of cats I will not be able to recognize many cats.

Read about it in section 1.3 and chapter 2 of my book.

Rolf,

I'm afraid you completely missed the point.

So let me try to put things in different perspective.

Physics has emerged as a science--- not counting some of the truly great Hellenistic scientists such as Archimedes---after the work of a number of scientists during the 17th century, including Galileo and Newton. It emerged from their work as a science studying *motion in space*. The concept of space here must be understood as it emerged during several millennia. The associated formalism is what we call the "continuous" formalism. This formalism has its integrity which cannot be preserved if we attempt to "discretize" it.

My (and it appears also Schrödinger's and Einstein's) point is that we must seek a fundamentally new way of approaching processes in Nature which would be consistent with the discovered ubiquitous "granularity". The new formalism, from the very beginning, should clarify the nature of this "granularity".

It goes without saying that such transition to a new scientific language cannot be a business as usual affair: we must begin building physics anew, mainly because we cannot rely on the conventional concept of motion (and on the associated formal apparatus).

from what i have read,it is important for us to define what a space is in a way that fits both physics and mathematics. For a start,let time be a space on its own right then let us assume that there is a difference between TIME and AGE!

we may not have any separation problem at all if we redefine time to meet the needs of all sciences!

@Lev: It would probably help if you could give an example of your "ubiquitous granularity". For me (a theoretical physicist who became a computational physicist) the experience is quite contrary to what you seem to feel. I work with computational, thus digital, granular, models of physical quantities (e.g. many-particle wave functions) and would like to see a digital nature or behavior in these quantities themselves. Without any success so far. Meanwhile I share the oppinion of most physicists to expect granularity not earlier than at Planck's scale.

"It would probably help if you could give an example of your "ubiquitous granularity""

Dear Ulrich,

Why do you pretend (do you?) to be ignorant (even after I quoted Schrodinger)? ;--)

This is *very* annoying, especially since you do "share the oppinion of most physicists to expect granularity not earlier than at Planck's scale."

In any case, here is some more 'help' for you.

From the penultimate paragraph of the book "the Evolution of Physics" by Einstein & Infeld:

"WE SUMMARIZE:

Again the rich variety of facts in the realm of atomic phenomena forces us to invent new physical concepts. Matter has a granular structure; it is composed of elementary particles, the elementary quanta of matter. Thus, the electric charge has a granular structure and most important from the point of view of the quantum theory so has energy. Photons are the energy quanta of which light is composed."

Also, from Feynman's QED:

"Newton thought that light was made up of particles---he called them "corpuscles"---and he was right . . . We know that light is made of particles because we can take a very sensitive instrument that makes clicks when light shines on it, and if the light gets dimmer, the clicks remain just as loud---there are just fewer of them. Thus light is something like raindrops---each little lump of light is called a photon---and if the light is all one color, all the "raindrops" are the same size.

I want to emphasize that light comes in this form---particles. It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told something about light behaving like waves. I'm telling you the way it *does* behave---like particles."

@Lev, I insisted ( not to annoy you but to make you think, unfortunately I only triggered reflexes) since I could hardly believe that unhappiness with atomism should be the only impetus for your attempts to put thinking about Nature on a new basis. I hoped to learn from you some specific aspect that striked you and told you that a new approach is -- if not needed -- so at least helpful. Instead, you seem to build exclusively on misunderstood utterances of some of great old physicists.

"I could hardly believe that unhappiness with atomism should be the only impetus for your attempts to put thinking about Nature on a new basis"

That's a completely different story: I did describe in my developing book some reasons for the development of the new formalism, but they came from pattern recognition and the problem of induction (i.e. information processing). What I still will address in one of the future chapters is the limitations of the conventional numeric formalisms w.r.t. the demands of a *structural* object representation.

However, I'm still amazed at your last characterization----"misunderstood utterances of some of great old physicists". Again, how can one "misunderstand" them? The only way I can see that is if one *ignores* or *dismisses* them completely!

Lev,

when discussing scattering of particles, Feynman, writes on p. 137 of 'Quantum Mechanics and Path Integrals' "since we are dealing with a wave phenomenon, there is diffraction into the shadow". In physics, words are only the music in the background, what counts is the formalized analysis. So understanding, misunderstanding, ignoring, or dismissing are mental activities that remain quite pointless when applied to purely verbal utterances of physicists when the corresponding physical theories are not taken into account.

I may remind you that it is the understanding of Nature (which I narrowed down to physics) for which you promote your ideas here and not pattern recognition and information processing.

So, as I see it, you should better not center on 'understanding Nature' when it comes to describing the motivation for your thoughts.

Dear Ulrich,

I'm quite surprised that, as an applied physicist, you are so indifferent to the physical 'reality' beyond the formal apparatus, even as it is registered by the measurement apparatus, . Although, given the dominance of the prevailing Bohr's computational and quite 'convenient' attitude, I'm not *that* surprised.

I'm also surprised that, given your indifference to the *nature* of reality, you, at the same time, are hinting at some professional expertise in this respect. (when you hint at the lack of my expertise in this respect). ;--) By the way, I have been relying on the expertise of some of the leading physicists of the last century, whose opinion you so easily dismiss. I begin to see now that, as you said yourself, you really don't care that much about their 'informal' opinions.

Frankly, I can't really understand why you would be interested at all in the present discussion: as you stated, for you, the 'reality' *is* the continuous apparatus.

@Lev: I participated in this discussion to make you aware of the weakness of the physics-related part of your motivation. I made no statements concerning physical 'reality' and your inferences regarding this could hardly be more wrong.

@Ulrich, OK, let's talk about "the weakness of the physics-related part of your [my] motivation", including "misunderstood utterances of some of great old physicists". (By the way, so far, I have not seen a *single* hint of that in what you said.)

Lets go back to what Feynman said:

"We know that light is made of particles because we can take a very sensitive instrument that makes clicks when light shines on it, and if the light gets dimmer, the clicks remain just as loud---there are just fewer of them. Thus light is something like raindrops---each little lump of light is called a photon---and if the light is all one color, all the "raindrops" are the same size."

I took this quote from a useful pedagogical paper by the late applied physicist (who studied phase phenomena in quantum optics) John H. Marburger, III "What is a photon?" http://tpt.aapt.org/resource/1/phteah/v34/i8/p482_s1?isAuthorized=no&isAuthorized=yes

I'm asking you as an applied physicist:

Is this not an evidence of the "discreteness of Nature"?

(Of course, from a *theoretical* point of view, the view of a photon as that which is detected by a click of the detector is unsatisfactory.)

@Lev:

If we shift Feynman's 'very sensitive instrument' the frequency of clicks will vary as a continuous function of the shift vector. Is this not an evidence of the "continuous behavior of Nature"? Things are not that simple that a single aspect of an experiment allows to infer properties of Nature.

So far no attempt to base the understanding of quantum physics on a discrete basis was successful. Interestingly Heisenberg's matrix theory looked quite discrete and Heisenberg's motivation came from the discrete nature of spectral lines. However, matrix quantum mechanics turned out to be mathematically equivalent to Schroedingers 'continuous' wave mechanics. Any decent textbook on quantum physics (e.g. the famous Feynman lectures) shows that there is nothing artificial in the way 'discrete quantum events' are handled on the basis of classical calculus. If your methodology claims to improve the situation, much hard work waits for it.

"If we shift Feynman's 'very sensitive instrument' the frequency of clicks will vary as a continuous function of the shift vector. Is this not an evidence of the "continuous behavior of Nature"? "

How can you verify that it "will vary as a continuous function of the shift vector"?

You can't claim that, since we don't have the means to verify it.

@Ulrich,

When I said that “you are so indifferent to the physical 'reality' beyond the formal apparatus”, I relied on your disinterest in what some leading physicists think about 'physical reality' and on some of your statements, e.g.:

"In physics, words are only the music in the background, what counts is the formalized analysis."

Later, you also said:

"For me (a theoretical physicist who became a computational physicist) the experience is quite contrary to what you seem to feel" (i.e. contrary to my view of 'reality').

=============================================

I just noticed that you referred to yourself as a "theoretical physicist", while I ---reading that you "Retired from industrial R&D"---thought of you as an applied physicist. Of course, this explains some things.

But let's focus on the topic: now, I'm waiting to see if there is any justification for the "continuous behavior of Nature", as you put it.

@Lev: Of course I use 'continuous' here as it is used in empirical sciences: no steps detectable. I trained students several years in topology and know what continuous means in mathematics. If you are scholastically minded you have a problem with klick frequency to define a function. Here I consider such problems as ignorable, although I sometimes feel that it would be nice to have a widely accepted solution for them.

PS: This is not an answer to your last note but to the one before. For an answer to the last note it is too late today (for me).

By the way, I have no problems with topology: my diploma work was in topology, and it was for quite a while my big love. ;--)

Lev, I'm sorry, but the fact that I don't follow your personal definitions or citations doesn't mean that I 'miss the point's. If you restrict physics to the point of view of 'motions in space' (and some accompanying philosophy you've mentioned), then of course you are going to run into trouble and this type of discussions. But that's an issue of an ill-posed problem, not of physics.

As Jean answered, you have to talk on the spaces, I'd like to enhance this topic - the representation spaces. And what you learn e.g. in spontaneously broken symmetries is the idea that the transformations generate their rep spaces themselves. However, that's only part of the story and that's mainly based on Klein's Erlanger program and the discussions following afterwards, especially with respect to differential calculus and Weyl's ideas.

However, that's far from physics and your original question. For me, your discussion is purely based on more or less suited formalisms which you identify with (to my opinion a special understanding of) physics. And yes, those formalisms introduce problems and contradictions. So where is the beef?

The important issue which got (once more) focus by 'quantum' theories is the fact that one should include the observer into the description. This necessity has also been known in classical physics, however, in nowadays discussions and formalisms this requirement is often neglected. And although people neglect the observer, they claim to be talking about physics... funny.

So this is no discussion of 'continuous' vs. 'discrete' but a discussion of appropriate representations. And it is really not necessary to claim with every poor representation theory that thereis another problem in 'physics'.

Rolf,

I'm afraid, I didn't quite follow your point.

We are discussing, in Riemann's language, the implicit physical postulate "that natural phenomena are continuous", on which all present physics stand, including all field theories.

In some sense, this implicit physical postulate was inevitable given the fact that all our millennia old geometrical knowledge needed it. But I'm asking: Where is the physical basis of this postulate? If anything, as Schrödinger put it

"The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time".

In fact, it is even hard to imagine how would one go about the verification of such "continuity" postulate.

Lev,

I don't see this 'implicite physical postulate' nor is it necessary to assert that 'all present physics stands on it, including all field theories' nor is this 'postulate' 'inevitable given the fact that all our millennia old geometrical knowledge needed it' nor is it (to my personal opinion) helpful to cite Schrödinger without the appropriate context... There is a plenty of literature (maybe not along the 'party line'...) addressing some of these issues...

Rolf,

The only thing I can clearly see now is that *you* prefer not to see this 'problem' (which is, of course, your choice). ;--)

maybe there is a 'problem'..., and yes: seems for me, too, I do not really see a physical problem :-)

Rolf,

I don't 'blame' you: most of those working in science (especially in physics) are married to the basic models in a much more permanent way than they are to their spouses. ;--)

Science does requires that kind of dedication, but then, in a vast majority of cases, hardly any mental energy is left for the doubt (because it is very 'subversive').

Right, and there are other cases where almost all 'mental energy' is left for 'doubt' even if only the wording has some overlap with 'science'...

I see, I didn't realize that you are also a self-appointed custodian of science.;--)

@Lev. Let me come to these of your words:

================================================================

But let's focus on the topic: now, I'm waiting to see if there is any justification for the "continuous behavior of Nature", as you put it.

================================================================

I assume that you buy 'natura non facit saltus' as far as classical physics is concerned. Then we had to deal with quantum physics exclusively.

Let us begin with Feynman's clicks that accompany the absorbition of a photon in a suitably designed device. Here the 'discrete' event is made by the device which is characterized by an avalanche-based amplification process. The primary disengagement of an electron by a photon which starts the avalanche is a continuous process that so far is only amenable to our mental eyes if we look how quantum mechanical scattering theory describes this process or when we model this quantum mechanical scattering process by a computer program.

This example shows, what all good quantum mechanics texts write, that the characteristics of quantum behavior are the sensitivity to perturbations and not a graininess. If one inspects molecules and atoms under the raster tunnel microscope or the atomic force microscope one finds continuously distributed soft entities with no granularity built in.

The discontinuity which according to many (bad) textbook presentations of quantum mechanics is built into the formalism under the name of 'collapse of the wave packet' (or R-process, or objectivation process) is based on erroneously forgetting the nature of the wave function as a descriptor of statistical properties. The presently wide-spread idea is to replace the R-process by 'decoherence' in a way that all discontinuity disappears. This is, however a field that is open to some debate. At least, claiming a fundamental discontinuity in Natur's behavior based on properties of the R process is a weak position.

A last (trivial) remark: spectral lines are discrete only superficially, as are state changes in phase transitions, currents in quantum Hall effect, ...

Ulrich,

OK, finally we get something to discuss.

1."I assume that you buy 'natura non facit saltus' as far as classical physics is concerned. Then we had to deal with quantum physics exclusively."

No I do not: in the original question, I'm referring to the opposition between the 'observed reality' and our formal apparatus.

2. "The primary disengagement of an electron by a photon which starts the avalanche is a continuous process that so far is only amenable to our mental eyes if we look how quantum mechanical scattering theory describes this process or when we model this quantum mechanical scattering process by a computer program."

Again you refer to the "theory" and not to the 'observed reality'.

(By the way, I'm still *very* surprised, to say the least, that you so much underestimate Feynman's ability to apprehend, for example, the nature of photon as opposed to its theoretical treatment. He wrote QED in 1985.)

===================================================

In general, as I mentioned earlier, we don't have the means to verify Leibniz's "Law of Continuity" (http://en.wikipedia.org/wiki/Law_of_Continuity), since we don't have, and will never have, the continuous, or even 'rational'---i.e. based on the rational numbers only---measurement devices. So the postulate of continuity of Nature probably cannot be verified. However, as Schrödinger suggested, all the observed "jumps" point in the opposite direction.

Lev, I did not expect that you would come to childish topics like measurements based on rational numbers. We obviously lack a sufficient ammount of common understanding of the topic. Changing this would be tedious, too tedious for me to try. So, I'll quit the discussion and wish you the very best for the physics-unrelated parts of your ideas.

Childish??

Again, you missed the basic point:, which I repeat for the *third* time: we don't have the experimental means to verify the continuity postulate. This fact already says something about the postulate.

Another underlying question is this:

How *far* can we go along the present path, i.e. using the continuous postulate as an approximation of reality? Given the observed facts (the discrete jumps), the mathematical wisdom is suggesting that as far as our understanding of the *structure* of reality is concerned we need to look for a fundamentally different formal language, and the reason for the mystery why the numeric *approximations* have worked so well will be clarified much later.

Currently, the 'weight', or structural unity, of our physical models is quite unsatisfactory. We need structurally much simpler, more transparent, and more unified picture of reality.

I guess, what has been missing from the above discussion---and from physics and mathematics in general---is the realization that there might be a completely other than numeric, "structural" (but not in the conventional sense), form of data representation, which might clarify the nature of the observed "discreteness" of Nature.

It is quite possible that the observed discrete "jumps", the wave-particle duality, and the entanglement are the *numeric* manifestations of the more appropriate non-numeric form of representation.

Eberhard,

Please note that the question is much more 'subversive' than implied by you. ;--)

Dear Eberhard,

I don't want at all "to enclose the discussion to a smaller group excluding" you.

Also, I didn't mean at all to suggest the irrelevance of you answers.

When I said that "the question is much more 'subversive' than implied by you", I simply meant to *add* to your point, that taking into the consideration what you and I said, there are reasons to believe that we may need as yet unprecedented reconsideration of physical concepts.

In other words, conventional math structures, including functional spaces (e.g. Hilbert space) and categories (from category theory), don't provide a satisfactory formal language for explicating the nature of the observed quantum 'discreteness". "Discreteness" is a 'dirty' term in science, since we have to deal with it indirectly, via the conventional continuous models, while we need a fundamentally new formal language to address it.

Eberhard, you might be disappointed to hear that, by education, I'm not a physicist but a mathematician. The general introduction to the proposal ( "entry point") can be found at http://www.cs.unb.ca/~goldfarb/BOOK.pdf and a complete formal exposition (with some minor misprints), at http://www.cs.unb.ca/~goldfarb/ETSbook/ETS6.pdf .

If you wish to be motivated by Feynman diagrams, take the more abstract formal (non-spatial but informational) view of them, where each node is an event that has a particular structure. In general, there are only a small finite number of basic events (which can compose macro-events).

We postulated that the underlying, or informational (pre-spatial), ‘reality’ consist of stream of such events. So ‘objects first emerge and evolve as such informational streams, but it should not be that difficult to see how *spatial instantiations* of such streams can be constructed.

So the new formal language is associated with such streams (no points, spaces, etc.) and operations on them.

You may have noticed some similarity with the “causal sets” approach to quantum gravity, except that their underlying formal part (causal sets) is quite undeveloped compared to our ETS formalism. On the other hand, the physical part for the ETS formalism will have to be developed, and that is why I’m trying to find physicists interested in the collaborative work. ;--)

József,

Again, as I mentioned several times, the point is that the meaning of adjective "discrete" comes from its negative connotation. Here is a quote from one of my papers:

"We are facing a very peculiar situation concerning the term “discrete”. Although it is quite popular in mathematics and natural sciences, its familiarity is quite deceptive, since we use it simply as a name by which we designate all non-continuous models—e.g. graphs (with nodes and edges), strings over a finite alphabet—or models obtained from a continuous model by simply “discretizing” it. So, “discrete” means anything that is not continuous, i.e. we are dealing with the negation of a particular formalism rather than with another, clearly delineated formalism . . . ."

And I'm suggesting that it is high time to begin to approach the issue more constructively, i.e. independently of the "continuous": What is the "discrete" model as a *fundamentally new* model of reality? Besides, the discretization of a continuous model is, in fact, an oxymoron, i.e.a decontinuousation of a continuous model. ;--)

Lev, if I may contribute my 2 cents' worth, what is wrong or objectionable in defining discreteness as opposed to 'continuous' ? You could equally define 'continuous' as 'non discrete' and define discrete directly.

In mathematics, one of the most powerful and fruitful modes of proof has historically been by contradiction, i.e. by opposition - by so-called 'reductio ad absurdum'. Some of the greatest proofs in math have been established this way, and are hardly amenable by any other method. That does not give those proofs a 'negative' connotation or in any way invalidates them..

Same in Physics - theories are falsified by just one counter experimental result.

You seem to be mostly playing with words, your whole argument since the beginning seems to be in part a game of semantics, and in part gratuitous assertions - for instance, the discretization of a continuous model is absolutely NOT an oxymoron - by that definition, 'to realize a dream' , for instance, etc., would also be an oxymoron, which of course it isn't.

Discretization is of course perfectly legitimate, and useful in engineering (finite elements analysis is based on this, for instance), in physics (you keep talking of about a 'continuous postulate or formalism' which is a bit of a baffle ) and so on.

The study of whether any aspect of reality is continuous or discrete can absolutely be conducted using the current tools at our disposal - i.e. current number theory, calculus, and the theorems of physics, such as Heisenberg, and so on.

You seem to want to replace number theory with a new formalism, but it seems to me that any valid new formalism will inevitably have to map onto some aspect of number theory, and may well be invalid if it does not. Your whole approach seems to be based, again, on unfruitful wordplay. Your original question itself is IMHO just wordplay - there is no central inconsistency where you allege there is, it looks like you are making an issue out of whole cloth where there is none, and the 'old formalism', as evidenced by such tools as e.g. calculus, etc., is not exclusively 'continuous' , but can be put to proper use in continuous as well as in discrete environments.

It would most definitely behoove you to cite papers from authors other than yourself - in general terms, citing one's own work exclusively is not very conducive.

I'm sure you won't like this answer, but rest assured that I am wholly uninterested in discussing this further

@Chris and all: Since Lev writes:

=====================

On the other hand, the physical part for the ETS formalism will have to be developed, and that is why I’m trying to find physicists interested in the collaborative work.

====================

it is certainly legitimate that he cites his ETS papers.

Yesterday I spent 3 hours trying to understand the basics of this method/theory from the second of these cited sources. I found many ideas that look natural ( that reminded me of own ideas, to be precise). One of the basic intentions seems to be to find (here simply to posulate) a formal system that promises to describe processes that do not require space as an environment (surprisingly 'time' is assumed from the very beginning without discussion) and that make up a 'reality' (Bell would have said: are the 'beables' of the theory). Similar intentions are worked out in some detail in 'A New Kind of Science' by Stephen Wolfram. In contrast to most beable-based theories ETS is non-deterministic and decribes systems that learn from the systems they interact with (and which are expected to be close to them in a space that hopefully can be constructed based on these learning interactions). This reminds me of the relational view of quantum theory by Carlo Rovelli.

However, when it comes to a formalized presentation I was lost with the third formula. It looks like mathematics but I was not able to decide whether it actually is. What I understood the least was why the author insists to need a new representational system when it is evident that any modern object oriented programming language can be used to provide a faithful model of all these finite-set-based structures (as far as they are defined at all).

To reason (or speculate) about a 'real' structure below understood physics is quite natural for a physicist but only the naive will estimate the chances of success larger than winning the jackpot of a lottery. The chances of success for a particular such endeavor do not grow if one insists in the preliminarity or even inconsistency of prior knowledge.

Chris:

"Lev, if I may contribute my 2 cents' worth, what is wrong or objectionable in defining discreteness as opposed to 'continuous' ? You could equally define 'continuous' as 'non discrete' and define discrete directly. "

I don't really understand how you can---after my explanations---still say such thing.

We do not have even a single "discrete" model in the same sense that we have "continuous" models, simply because physics (e.g. the concept of field) was built on the "continuous" foundation.

What has been called "discrete" in mathematics has never had a status even remotely approaching that of "continuous", simply because all the main math. structures have evolved out of the "continuous". More importantly, even if one points to some "purely" algebraic structures (e.g. groups), from the applied point of view they cannot play the foundational role because if you remove the concept of (continuous) field, you need to start building physics anew, as mentioned by Einstein. And group theory is not going to help with this undertaking.

=============================================

Ulrich,

Neither Wolfram nor Rovelli nor anyone else (to my knowledge) address the issue of object / process representation to the full extent. Unfortunately, "representation" has in mathematics and physics many meanings only one of which is relevant here: this is the meaning associated with the process of representing objects / processes in the form of data. (In science we collect "data", which so far has been of numeric nature and which we propose to change.)

Contrary to what you think about ETS representation ("struct"), time is not "assumed from the very beginning"! In fact time emerges form ETS representation, or from the stream of the corresponding events.

I presented a very simple illustration in section 2.9 of http://www.cs.unb.ca/~goldfarb/BOOK.pdf .

Now, let's address your point:

"To reason (or speculate) about a 'real' structure below understood physics is quite natural for a physicist but only the naive will estimate the chances of success larger than winning the jackpot of a lottery. The chances of success for a particular such endeavor do not grow if one insists in the preliminarity or even inconsistency of prior knowledge."

Why are the chances much better than you think? ETS is the first serious attempt to address the issue of *structural*, or relational, representation of 'reality' (as opposed to the numeric representation). If, in this undertaking, it captured / postulated the basic ingredient---e.g. the structured event---correctly in general outlines, then we can begin to rebuilt science on this new "structural" foundation, where structures are not derived from the conventional mathematical structures, but are built into the data representation from the very beginning. We then begin to see the "physical" reality in a completely new light.

Moreover, as you know, the importance of the "relational' side of physical reality has been recognized for some time by a number of prominent physicists.

József, it's nice to know that since my PhD at Waterloo the quality of education there has considerably 'improved'. ;--))

(By the way, since then I have lectured several courses on numerical methods.)

Having read all these clear ideas as written above, we can all agree that science cannot be divided though it has many streams as people say ! This is just my little opinion ! we may not tell where general relativity may take us ! Bertrand Rusell did argue that social and human sciences depend considerably on biology while biology needs chemistry as chemistry needs physics. The main challenge may be that Bacon's general relativity can not be different from einstein's general relativity.

My sincere apologies to anyone concerning the expression "social and human sciences" since I am not sure whether it is permitted or not!

@Luiz: Thank you very much for the link to the Tom Leinster article. Great stuff!

1. We have to be cautious when making statements about Nature. There is a dominant discourse about "ubiquitous discretness of Nature", it does not imply that Nature is discreet (or continuous). The best that scientists can achieve is to provide models (representations) of phenomena which are efficient (with reasonable predictive capabilities) and explicative (easy to understand and to use for the workers in the field). This is not to say that some things are beyond our understanding, it is just that we always strive to do better.

2. Probably the most efficient model in "hard sciences" is the atomic model of chemistry : with a single line such as H2+1/2O2->H2O + 286 kJ/Mol you have almost everything that you want to understand chemical reactions. It is efficient, explicative, and has strong theoretical foundations.

3. For a long time Mathematics have been developed in the discrete picture, continuity, limits are fairly recent (late XIX ° century). And the introduction of infinity has raised many issues, some of them still puzzling.

4. In physics we have to deal with measures (figures) in one hand, and some formal representation of the phenomena on the other hand. Whatever the theory, there is a mandatory step : define variables (meaning objects with some mathematical properties) in relation with measures (figures) : a model is the organization of raw data. To validate a theory by an experiment we collect data and check, afterwards, that the assumed relations hold. This process is not without limitations. The number of measures that can be done is always finite, meanwhile most, if not all the models, involve maps acting on infinite sets, and the figures that we can get for our measures are rational numbers (up to some scale), meanwhile we generally assume that they represent real numbers. Thus the need for statistical methods. And the rise of probability and the "axioms" of Quantum Mechanics.

So, our measures, in one hand, are necessarily discreet, but it does not imply that the use of continuous model is not legitimate, and even less so that "Nature" is discreet.

5. The issue which appears with Quantum Physics is the duality particle / wave. Duality, meaning that it goes both ways. And there is no simple mathematical representation of a phenomenon which has both properties : being continuous and discontinuous. The solution which has been used so far - taking random variables - can be efficient, but is not the most satisfying. And, from this, to state that "Nature" is not determinist seems a bit of an overstatement.

@jean claude: Saying that nature is deterministic has just as shaky a meaning as saying it is discrete: Even if the contrary would be evident to common sense, one would be able to transform any continuous model into a discrete one (by numerical methods) with the same predictive power and each stochasic model in a deterministic one (by replacing the evaluation of random variables by calls to pseudo-random generators) - also of the same predictive power. Taken into account that pseudo-random generators can be controlled by binary strings (as 'seeds'), one version of such a 'derterminified' stochastic model could be fed with the binary string obtained from digitizing some meaningful human text, and the deterministic model thus obtained would agree with all empirical facts with which the original stochastical model agrees. For people not familiar with the fallacies concerning assessing the significance of empirical tests, this could be seen as a sign for the truth of the meaning of our fed-in text.

Even in the well-defined framework of non-relativistic quantum mechanics the question of determinism remains shaky. Whether radioactive nuclei decay inherently stochasticly or as determined by the dynamics of Bohmian point particles seems to be undecidable.

Jean Claude,

Two points:

1. The path from the data to the its interpretation has become too long, indirect, and very cumbersome (recall Ptolemaic epicycles, which also did a decent predictive job) .

2. We need a *direct* structural form of data representation---which we never had before---rather than to rely on the current (indirect) mathematical structures. This is where the new scientific story should begin.

@Lev: One point: Do you envision a new generation of measuring devices which output directly 'structural data' (of what kind ever)? If yes, can you give a semi-concrete example that shows in what direction your pertinent ideas go ?

Yes, Ulrich!

Quite a while ago we wrote two related papers (one with Deshpande and one with Golubitsky, see my RG papers) but they are outdated now.

To get the basic intuitive idea, it's enough to look at the popular bubble chamber photos: to simplify, what is missing are the structures of the actual mico-events. So as far experimental physics is concerned, we need to begin the gradual process of discovering the structure of the basic events for photons, electrons, etc. Actually I'm not the first to point in this direction. Here is the quote from my developing book (p. 17):

"In 1949–1950, Ya. Il. Frenkel [the great Soviet physicist] suggested viewing a particle’s motion as a series of regenerations: transformations of this particle into a different particle and the subsequent reverse transformations."

So what I'm suggesting is not that heretical, i.e. that the basic physical reality is that of structured (interconnected) events.

@Ulrich

OK, my point of view is clear : what we do is build representations of physical phenomena, using the tools at our disposal. So we must be humble in pretending that the "reality" is conform to our models. But for me it is simpler, as a model, to use a determinist one than a probabilist one.

@Lev

I think that I see your point. It makes sense in a relativistic picture : a system of interacting particles can be seen as a graph, any Noether current generates a potential segment of the graph. But it is still necessary to build a theoretical model to explain the vertices and the creation -annihiliation of particles. And to establish the link with figures...

From the Zeno paradox, Democritus thought out its atomistic model. Though, in successful theories like Maxwell's one, continuity kept creeping in. In one of his articles of 1905, Einstein pointed out the asymmetry between discrete systems like elementary particles, and electromagnetism, and that was indeed the start of quantum mechanics, where the very world "quantum" was borrowed to the discrete charge as it was then called. Thus was born the photon. But in quantum field theory, we see that the discreteness of the photon actually emerges from a continuous theory. Indeed, quantization is now the metamorphosis of discrete system to a continuous wave function. Similarly in the atom, energy levels and integer angular momentum, initially postulated as discrete by Bohr, are described by the continuous Dirac equation. As we see, discreteness and continuity have a parallel history to wave and corpuscles.

Perhaps the error is precisely trying and introduce discreteness where it doesn't belong. For example for space-time, while space-time probably has no physical existence, and is only our way to reason and represent things. Even the Planck scale doesn't involve it. It is however true that discreteness can arise from continuity, and not the other way round. There are various different indexes in topology that give integers, the orbital angular momentum being such an instance. The integer number of particles might possibly be derived in such a way.

Claude:

" It is however true that discreteness can arise from continuity, and not the other way round. "

The reason for this has nothing to do with the underlying model of 'reality', but rather with the fact I mentioned above several times: so far we have had to rely only on the 'continuous' formal models. We have not had any general and fundamentally new 'discrete' model from which the continuous one can be derived.