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Trick or Truth: The Mysterious Connection Between Physics and Mathematics
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How Should Humanity Steer the Future?
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It From Bit or Bit From It
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Lev Goldfarb: on 2/12/12 at 17:23pm UTC, wrote Thank you very much!! However, although I find it interesting, it appears...

Anonymous: on 1/31/12 at 6:18am UTC, wrote You may find this interesting.

Lev Goldfarb: on 10/24/11 at 13:28pm UTC, wrote Thank you for the references! In light of his views expressed in the book...

Schrodinger: on 10/20/11 at 7:36am UTC, wrote Hi Lev, I would like to help you understand my views on the problem...

Lev Goldfarb: on 5/25/11 at 1:17am UTC, wrote Dear Vladimir, Thank you for your interest in my essay! I am, by far, not...

Paul Reed: on 5/23/11 at 12:00pm UTC, wrote One of the fundamental problems in continuous and discrete revolves around...

Vladimir Tamari: on 5/22/11 at 3:56am UTC, wrote Dear Lev I am honored to be your neighbor on this week's top essays...

Lev Goldfarb: on 3/30/11 at 21:33pm UTC, wrote Hi Yuri, I don't see his view as "radical", just not typical. ;-)


kurt stocklmeir: "neutrinos are tachyons - neutrinos flying around the universe would have an..." in Alternative Models of...

Gary Simpson: "All, Is there any empirical evidence that the electron orbitals of an..." in Real-Time Physics

Georgina Woodward: "Hi William, Thanks for your answer. The motivation for the vibration..." in Alternative Models of...

Ken Seto: "I endorse the idea of Newton’s “absolute time”. However, we have no..." in Real-Time Physics

kurt stocklmeir: "if space is expanding and if this makes positive energy particles have a..." in Alternative Models of...

nimit theeraleekul: "Dear friends, In my early post, I said that we could see detail of..." in Weinberg: Why quantum...

nimit theeraleekul: "Dear Administrator, I have tried to make several posts with an attachment,..." in Weinberg: Why quantum...

Gary Simpson: "Pentcho, I did not need the postulates of SR to propose the mechanism. In..." in Alternative Models of...

click titles to read articles

Bohemian Reality: Searching for a Quantum Connection to Consciousness
Is there are sweet spot where artificial intelligence systems could have the maximum amount of consciousness while retaining powerful quantum properties?

Quantum Replicants: Should future androids dream of quantum sheep?
To build the ultimate artificial mimics of real life systems, we may need to use quantum memory.

Painting a QBist Picture of Reality
A radical interpretation of physics makes quantum theory more personal.

The Spacetime Revolutionary
Carlo Rovelli describes how black holes may transition to "white holes," according to loop quantum gravity, a radical rewrite of fundamental physics.

Riding the Rogue Quantum Waves
Could giant sea swells help explain how the macroscopic world emerges from the quantum microworld? (Image credit: MIT News)

May 27, 2017

CATEGORY: Is Reality Digital or Analog? Essay Contest (2010-2011) [back]
TOPIC: Nature Is Fundamentally Discrete But Our Basic Formalism Is Not by Lev Goldfarb [refresh]
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Author Lev Goldfarb wrote on Dec. 23, 2010 @ 15:54 GMT
Essay Abstract

In physics, we have been seriously confronted with the question of continuous vs. discrete since the beginning of the last century. Why is it still with us, and increasingly so in the last sixty years? The title of the essay suggests the reason: so far, we have relied on the "continuous", or vector space, mathematics (of spatial origin), the only one we have, while the experiments suggest that, at the bottom, the nature is non-continuous, or discrete, albeit in a sense unfamiliar to us. We have tried to save the situation by "discretizing" our conventional models, but for the reasons I discuss here such desperate attempts to transform our basic formalism (by destroying its integrity) are not meaningful. We may have no other choice than to set aside for a while the millennia-old numeric, or spatial, forms of representation and the associated measurement processes and to begin completely anew, by shifting to a non-numeric-- relational, or temporal--representational formalism, which should give the meaning to the nebulous concept of discreteness and, even more importantly, should remove the enormous present gap between the physical and the mental.

Author Bio

Lev Goldfarb obtained Diploma in Mathematics (St.-Petersburg University) and Ph.D. in Systems Design Engineering (University of Waterloo). For twenty five years he worked as an Assistant and Associate Professor in the Faculty of Computer Science, University of New Brunswick, Canada. Now he conducts research, development, and consulting through his company IIS. He has served on the editorial boards of Pattern Recognition, Pattern Recognition Letters, and now Cognitive Neurodynamics. Trained as a mathematician, he realized quite early the inadequacy of the conventional numeric models and has been working on the development of a fundamentally new formalism for structural representation.

Download Essay PDF File

Cristinel Stoica wrote on Jan. 22, 2011 @ 15:32 GMT
Hi Lev,

I like how you explained the historical roots of the dominance of the continuous structures in our mathematical description of the world. Your conclusion "The reason why we have not been satisfied with this answer is simple: having excluded the "standard" (continuous) formalism, we are left with no concrete formalism to take its place, hence no adequate understanding of the "discreteness"" is very reasonable. The shift you propose, "to a nonnumeric-relational, or temporal-representational formalism, which should give the meaning to the nebulous concept of discreteness and, even more importantly, should remove the enormous present gap between the physical and the mental", the "transition from the point-based representation to the structural representation", is very needed in many domains, in particular in physics. I think that something like this would be very useful, for several reasons: It is able to capture relations and transformations in an intuitive manner. It can be used at least for the taxonomy and understanding of various processes. It provides a syntax isomorphic to the semantics. Even if the structures like those you proposed would turn out not to be fundamental in physics, but rather emergent, I think that they still may provide an important view, complementary to what we have.

Best regards,


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Author Lev Goldfarb replied on Jan. 22, 2011 @ 17:37 GMT
Hello Cristi,

Good to see you participating in this contest!

You got the main point.

However, the unfortunate thing about the period of time we live in, I believe, is that the fast pace of life does not allow for the scientific transformation that is needed. Many of us do understand the transitional nature of this time period but are not mentally prepared for the...

view entire post

Robert Spoljaric wrote on Jan. 24, 2011 @ 07:01 GMT
Dear Lev,

You identify the problem as: We have tried to save the situation by "discretizing" our conventional models, but for the reasons I discuss here such desperate attempts to transform our basic formalism (by destroying its integrity) are not meaningful.

In my essay I show an inevitable "discretising" of relativistic momentum-energy, which you may, or may not, be able to relate to your thesis.



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Member Tommaso Bolognesi wrote on Feb. 8, 2011 @ 11:50 GMT
Hi Lev,

Figure 4 in your essay rings a bell: on the right you have the essential, partial order representation of the pattern of events, with their causality interconnections explicited. On the left, you have a particular, totally ordered realization of that partial order (there may be other such realizations). Are you familiar with Causal Sets, as proposed by Levin (as models of computation) and by Bombelli, Sorkin, Reid, Rideout, Wolfram, and others, as discrete models of spacetime? I understand you add some more structure to your events, but the general idea appears to be the same...

Best regards. Tommaso

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Lev Goldfarb replied on Feb. 8, 2011 @ 13:03 GMT
Hi Tommaso,

Obviously, there is some superficial resemblance.

However, if you read beyond the figures, you will note that the difference is more radical than appears at first sight: the main point of our proposal is to change the form of object representation from the ubiquitous point form (entrenched by set theory) to the proposed structural form, the struct.

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Alan Lowey wrote on Feb. 10, 2011 @ 13:48 GMT
Hi Lev and thanks for your essay. I like the way you think with regard to basic underlying failure of the science community in the last 100 years or so. I summed it up with a response to Jarmo's essay where he imagines talking to Newton himself about the latest developments:

An excellent and entertaining entrance to your essay Jarmo, congratulations on your imagination and ingeniuty. I have a burning question which I've always wanted to ask Newton though, which is this:

Q: Since he equated the ancient greek philosophy of the smallest irreducible particle, called an atom, with the motions of the planets as observed by Galileo Galilei, does he want to know what his very large unspoken logical assumption was, which has now meant that humanity has been led down the wrong scientific path?

Ans: He assumed that the cores of the planets and sun are composed of the same everyday matter which is found on the external crust. (It's not necessarily the case and so invalidates the whole of Einstein's space-time concept imo and also invalidates the results of the Cavendish experiment to 'weigh' the Earth).

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Don Limuti ( wrote on Feb. 13, 2011 @ 04:36 GMT
Hi Lev,

1. Glad to see you in the contest.

2. I believe your proposal to "view and represent all “objects” in nature as (irreversible) temporal processes comprised of temporally related events." is the key to progress in how we understand the world.

3. I wish I could "full get" how the detailed block diagram modeled the hydrogen atom, but I think I would need to take your course in "Reality Modeling 101" which I hope you will offer soon.

4. Your blocks and connections reminded me of "Labview" a visual programming language. Labview works on a data flow scheme which may not be that difficult to make into an event driven scheme. I can see various blocks of code "drivers": a. Low level primitives. b. Photons c. Particles d. Atoms e. Molecules f. gasses. g. solids. h. Polymers i. DNA. etc That can be assembled as the user chooses.

This is a very good essay.

Best of Luck.

Don L.

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Author Lev Goldfarb replied on Feb. 17, 2011 @ 21:49 GMT
Hi Don,

I wanted to welcome you to the contest first, but you bit me to it. ;-) [I'm having a very hard time with my renovations, now a year and half in the making]

Regarding 3: there is really nothing new there (think of a more precisely depicted Feynman diagram.

Regarding 4: that's the idea.

Thanks for you comments!

Author Lev Goldfarb wrote on Feb. 20, 2011 @ 22:35 GMT
Last night I remembered that several days ago I recommended to Ian Durham (on his essay page) R.G. Collingwood's book "The Idea of Nature" (1945), in particular the end of the first chapter. So today I decided to reread this part of the book, and suddenly one important point came to my mind. Before making this point, I would like to share with you the portion of the text that triggered this...

view entire post

James Lee Hoover wrote on Feb. 21, 2011 @ 23:57 GMT

Your summary seemed to look at reality posited in models rather than the universe itself. Can we separate them?

Jim Hoover

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Author Lev Goldfarb replied on Feb. 22, 2011 @ 02:03 GMT
Hello Jim,

The reason we cannot deal with the "reality", at least so far, has to do with our inability to approach it directly, without any special "language", or formalism, designed for that purpose.

In my essay, I suggested that our present difficulties have to do with the fact that the numeric formalism, which has served us for several millennia, has exhausted its usefulness and needs to be replaced by a new formalism which is supposed to capture reality in a much more direct manner.

James Lee Hoover replied on Feb. 22, 2011 @ 21:02 GMT

One could never say that another's approach is wrong. Certainly mine depends on observed characteristics that models pose or reveal.


Jim Hoover

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Author Lev Goldfarb replied on Feb. 22, 2011 @ 21:50 GMT

I'm sorry but I'm not quite following you.

Peter Jackson wrote on Feb. 25, 2011 @ 21:35 GMT

Another excellent essay, and still too good a concept to expect any headway against the paradigm, but a top score from me. Perhaps a spectacular success in application is needed. I do have something in mind which originally used a different representational and conceptual structure to the mathematical, and hasn't yet 'hit the streets'. We can discuss if you feel happy with the concept.

My own essay will provide a hint, the conception also based on moving entities and relationships not numbers. I believe you will also find it rewarding to explore a number of other excellent and consistent essays, such as Edwins, Georginas, Rafael Castel, Constantinos, Robert Spoljaric, and others!

I'd be honoured for any thoughts on mine, which is mentally testing! but I hope you also find worth a good mark. Do ask about any doubts.

Best wishes


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Author Lev Goldfarb replied on Feb. 27, 2011 @ 03:12 GMT
Dear Peter,

I sincerely appreciate your interest in my essay!!

I already read your essay and would have commented on it if I felt confident/competent that those will be useful to you.

I believe your remark that "Perhaps a spectacular success in application is needed." is absolutely correct, and the main reason I participate (for the second time) in these contests is directly related to my hope that some people might be motivated to develop with or without me some "physical" applications of the proposed new form of data representation.

In fact, *all* I'm proposing is a fundamentally new (structural) form of data representation in science.

Best wishes,


Yuri Danoyan wrote on Feb. 26, 2011 @ 20:47 GMT

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Author Lev Goldfarb replied on Feb. 27, 2011 @ 03:28 GMT
Thank you, Yuri, for pointing to the quotes by Feynman (in your essay page) and 't Hooft (in the link) supporting my doubts regarding the applicability of the logic of calculus to physics (the second paragraph of section 3 in my essay).

Peter Jackson wrote on Feb. 28, 2011 @ 11:39 GMT

Thanks. I see your work as a little more important fundamentally, as a tool that can avoid the limitations of other tools.

As someone who can think non mathematically, can you tell me this ref my essay, (if you can remember or have time to check.) If we say the 'problem' is to explain how a constant speed of light is measured by all moving observers, does it understandably explain how this can be achieved with a quantum mechanism?

If successful, this should lead to the success I referred to.


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T H Ray wrote on Mar. 10, 2011 @ 18:42 GMT

Okay, here goes. No pressure about the ratings -- I rated you a long time ago and you know I marked you high. Time for the commentary I promised.

Having spent several years now studying your work, and corresponding with you, I remain convinced that this is very important work -- and unfortunately, probably far ahead of its time. I wish that pattern recognition algorithms were sufficiently advanced for a proof of concept, but I don't think they are.

All I can say at the moment is that I still agree in principle with the identity between time and information, and the independence of semantics and syntax. I think this all converges on my view of general self organization. My head is still into classical computation, though, and into the possibilities that quantum computing might bring.

Be assured that I will follow the progress of your group as I am able, and I hope you put me on the mailing list for any significant breakthrough!

(House remodeled to your satisfaction now? :-) )

All best,


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Author Lev Goldfarb replied on Mar. 10, 2011 @ 23:29 GMT

Thank you for your interest!

By the way, I now believe that "the independence of semantics and syntax" in a formalism might be the main source of our scientific troubles. To avoid fulling ourselves, the formalism must be absolutely transparent: syntax = semantics.

My renovations are still ongoing. ;-(

I'm glad you are doing reasonably well in the rating.

My best wishes to you

Author Lev Goldfarb replied on Mar. 11, 2011 @ 05:23 GMT
Sorry, obviously, I meant "fooling" instead of "fulling". ;-)

This is what twelve hours a day of renovations can do to my English. ;-)

T H Ray replied on Mar. 11, 2011 @ 13:48 GMT
Hi Lev,

I knew what you meant. :-) I haven't been out to Canada's east coast since 1979, where in Halifax I took immediately to the taste of the local Ten Penny Ale. Mmmmm. Do they still brew it? The rocky coast is majestic and beautiful.

Hope you're not working yourself into exhaustion. I know that renovation is one of those things that feel so much better when it's over.

Re semantics = syntax, I will look closer and hopefully be able to convince myself that it's not a return philosophically to the Logical Positivism school.

All best,


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John Benavides wrote on Mar. 13, 2011 @ 08:05 GMT
Dear Lev

I have enjoyed your essay very much. You point a very interesting fact and is how our conception of the continuum through points is based on the notion of limit. On my essay I try to explain how this conception of limit and others are just the tools that allow classical logic work on our conception of reality. Once we remove the excluded middle from our logic, concepts like limits, closed sets etc. loose their meaning and our continuum recover its relational character. We can keep then a lot of the richness of our classical formalism but getting the powerful relational character that you propose. I wonder what kind of logic governs the models you proposed, I think my ideas can be useful to you, I find particularly interesting your new conception of the Peano axioms I think it could have a closed relation with the models I propose. I would like to hear your opinions about it.


J. Benavides

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Author Lev Goldfarb replied on Mar. 13, 2011 @ 21:23 GMT
Hi John,

Thank you for your interst in the essay!

First, I should mention that I believe in the priority of (object or data) representation in science over any logical considerations: the latter should follow the "logic" of the chosen representation, i.e.logic should emerge during our "manipulation" of data.

Regrettably, the concept of representation emerged basically outside natural sciences (and was driven by CS applications), mainly because mathematics has not yet addressed it (and that is why it has been developed properly).

It took me some time to gradually realize that when replacing "point" representation by a structural representation, you get a much richer "geometry", because of the much richer variety of various path between two entities (just look at the variety of paths between two strings over a finite alphabet under deletion, insertion, and substitution operations).

I am familiar with category theory, but I felt that the concept of category is useful under the conventional mathematical (point) representation. What would be necessary in the "structural" mathematics is not yet clear to me: in the conventional math one has to seek (and build) various math. structure "outside" the (point) representation itself, since such numeric representations are too "simple". So the "structural" math. in view of very rich natural data structure, there is much less pressure to seek it in a manner similar to the conventional math.

I will comment on your essay shortly, in your essay's page.

Paul Halpern wrote on Mar. 14, 2011 @ 01:15 GMT
Dear Lev,

Your essay offers a fascinating analysis of why science has historically viewed nature as spatial and continuous. I very much enjoyed your methodical account of how spatial measurement has served as a core of mathematics, and how we may need to re-examine our assumptions. Well written and developed!

Best wishes,


Paul Halpern, The Discreet Charm of the Discrete

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Author Lev Goldfarb replied on Mar. 14, 2011 @ 02:26 GMT
Thank you Paul!

Alan Lowey wrote on Mar. 19, 2011 @ 11:12 GMT
Dear Lev,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,


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Author Lev Goldfarb wrote on Mar. 19, 2011 @ 14:40 GMT
Dear Alan,

I'm sorry, I'm by far not the best person to address your question.

My best wishes to you!

Alan Lowey replied on Mar. 23, 2011 @ 13:11 GMT
Okay, no problem. All the best


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Peter Jackson wrote on Mar. 21, 2011 @ 16:03 GMT

You have found a solution I've spent some years looking for. I've tried many words but they have normally failed me and been inadequate. I have to congratulate you on your inspired use of the word 'cataclysmic'. Well done for finally pinning the right word down so precisely.

I've recently added a couple more analogies and explanations to my string which, if you have time, I'd greatly appreciate your looking over. The last in in response to a tome by Basudeba.

The solution has come through departure from the abstractions of points and lines and reversion to simply the 3D 'body' Einstein originally specified. This is so close to your own thesis I do hope you can perceive the enormous result, well able to extract us from cataclysm. You will need all your dynamic visualisation skills and a tiny bit more. An earlier post breaks the variables down and deals with each observer and path 'condition'.

Very best wishes


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Peter Jackson wrote on Mar. 21, 2011 @ 19:21 GMT

Thank for message on mine. Shame, but it seems we're all different for good reason.

The problem with the DFM is the construct has to be in motion. Did I post you this video, with two observer frames involved.;

It still takes a lot of thought and imagination!

Best of luck.


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Author Yuri Danoyan+ wrote on Mar. 30, 2011 @ 12:29 GMT

I wonder why you did not notice or do not want to notice the radical view that an independent investigator.Remember this name: name,Friedwardt Winterberg


Yuri Danoyan

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Author Lev Goldfarb replied on Mar. 30, 2011 @ 21:33 GMT
Hi Yuri,

I don't see his view as "radical", just not typical. ;-)

Vladimir F. Tamari wrote on May. 22, 2011 @ 03:56 GMT
Dear Lev

I am honored to be your neighbor on this week's top essays web-page. I have read your interesting essay twice, and cannot honestly say I fully understand your concepts completely yet, but I did get the general idea that we need to use a new conceptual language when dealing with nature. The reason for this is due more to my own rather too-focused ideas on my own specific approach to reconstructing physics, rather than to any lack in your lucid and learned presentation. Your graphics rather reminded me of Penrose's suggestion to represent tensor operations in what he called diagrammatic notation (Roger Penrose "The Road to Reality" Figs 12.17 and 12.18) but of course the specifics and intentions of the diagrams are quite different.

From your obviously well-defined point of view and set of ideas I wonder what you would think of my scheme to reconstruct physics in my 2005 Beautiful Universe paper on which my present fqxi paper is based?

The only way to go forward for me now seems to be by computer simulation. Any ideas how I can start?

With best wishes from Vladimir

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Author Lev Goldfarb replied on May. 25, 2011 @ 01:17 GMT
Dear Vladimir,

Thank you for your interest in my essay!

I am, by far, not the best person to judge about or comment on your attempt "to reconstruct physics".

You see, I am coming to physics from the point of view *quite* different from a physicist (see my latest book proposal). The reason for the latter is that I came to a tentative but very radical conclusion for the necessity to rethink the entire science of physics as it exists today (as a science of "motion in space").

So, I'm afraid I cannot be of much help to you.

My best wishes to you!

Paul Reed wrote on May. 23, 2011 @ 12:00 GMT
One of the fundamental problems in continuous and discrete revolves around what actually happens in reality vrs the limitations of the information medium that conveys it to us. The problem is as follows:

Everything that constitutes reality is undergoing a process of change. Every sequence of change has its own intrinsic rate of change. [This is time, incidentally, but not the point I am making here].

Logically, both the maximum number of states and the fastest rate of change potentially experienceable, is a function of the maximum frequency with which the medium conveying the information is able to differentiate them. This could differ from what actually exists, which should be inferable.

So, the attributes of the medium (the obvious one being light) needs to be disentangled from the resulting experience of reality, in order to extrapolate what was reality.

© Paul Reed

April 2011

Extracted from Theory of Reality and Time posted on Re Ality (Facebook, look for the boy with his cat)

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Schrodinger wrote on Oct. 20, 2011 @ 07:36 GMT
Hi Lev,

I would like to help you understand my views on the problem continuous vs. discrete. Please take a look at these papers:

I will quote some paragraphs:

"I believe one is allowed to regard microscopic interaction as a continuous phenomenon without losing either the precious results of Planck and Einstein on the equilibrium of (macroscopic) energy between radiation and matter, or any other understanding of phenomena that the parcel-theory affords."

"I wish to consider some typical experiments that ostensibly force the energy parcel view upon us, and I wish to show that this is an illusion."

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Author Lev Goldfarb replied on Oct. 24, 2011 @ 13:28 GMT
Thank you for the references!

In light of his views expressed in the book "Science and Humanism" (published one year prior), I find his views expressed in this 2-part paper a bit puzzling. In that paper, it appears, he is leaning towards a less radical solution than he was advocating in the book. One can appreciate this: no one, at that time, could think of a fundamentally new, non-continuous, formalism.

Anonymous wrote on Jan. 31, 2012 @ 06:18 GMT
You may find this interesting.

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Author Lev Goldfarb replied on Feb. 12, 2012 @ 17:23 GMT
Thank you very much!!

However, although I find it interesting, it appears that, on the formal (and partly on the informal) side, this work is still in a relatively hazy state. ;-)

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