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2016-08-22
Continuum - a bibliographic assistance
Hi,I'm looking for a good book/article that analyzes the concept of continuum (not just in space and time but on the general level, including properties, numbers etc.) and surveys its definitions.
I'll be grateful for your references.
Thanks,
Benny

2016-08-29
Continuum - a bibliographic assistance
Benny, your question is a bit too open-ended for me. However, I've written one article and have submitted another that perhaps does as you suggest, although you may well not find it appealing. I redefine "matter", not as tangible actualities or entities, but as ontic probabilities. I make the argument that the value of an ontic probability is grounded by proximate actualities, and an actuality is simply the probability value that is maximally probable in reference to proximate actualities. The circularity here is addressed in terms of a spontaneous symmetry break.

These probability distributions are continua. Spatio-temporal localization, properties, tangibility or causal action  (observability) are simply an extrinsic value in that continuum.

Haines Brown 

2016-08-29
Continuum - a bibliographic assistance
Not sure it'll be what you want, but Hermann Weyl seems an obvious suggestion. 

2016-08-30
Continuum - a bibliographic assistance
Reply to Haines Brown
Dear Haines,Thank you very much for your reply. For one thing, does your article DEFINE continuum or cope with existing definitions?
In any case, I'll be grateful if you send me a copy of your article to:
Benjamin.brown@mail.huji.ac.il
Thanks again!
Benny

2016-08-30
Continuum - a bibliographic assistance
Dear Peter,Thank you for your reply. Weyl will have to be an important chapter in the book I'm looking for, but he's only one of many names who dealt with it. What I need is a comprehensive survey of the definitions of continuum and the theories related to them .
Talking about Weyl, is there any English translation for his systematic work on continuum? Or even a French one? My German is too poor and to everyday-life-centered to enable me to read Weyl in the original.
Thanks again,
Benny

2016-08-30
Continuum - a bibliographic assistance
Sure. Das Kontinuum is available in English. It is mostly maths but there is some useful discussion.Other works by him are also in English. Not sure he will quite answer your question though.

Out of interest I trawled through a few definitions and realised that they vary considerably. Not all the definitions that I read are correct, in my opinion, but quite often define a series of points instead.

This is one of Weyl's proposals, that what we often call a continuum (the real numbers, say) is not one.

My own view would be that a true continuum cannot have parts, cannot be extended and is a unity. But this is not a common useage.    

2016-08-31
Continuum - a bibliographic assistance
Peter,

It seems to me that Weyl offered a variety of approaches, and I wonder if you wouldn't mind being more specific either as to source or about which of his ideas you have in mind.

For example in Space, Time, Matter, he employs the idea of a tensor calculus. I'd like to take a moment to look at tensors and ask myself if they are a continuum. 

First of all, a tensor is a mathematical entity, and it strikes me that anything called an "entity" is closed rather than continuous. Jessica Wilson and Jeremy Butterfield have done some work on this. An entity is defined by its intrinsic properties, whether they be essential or accidental. This means the unobservables beyond that enclosure are not relevant, so an entity by definition cannot be not part of a continuum.

A tensor or vector presumes a point in Cartesian space-time that anchors a magnitude pointing elsewhere in Cartesian space. To "point elsewhere" is not a continuum, but a specific value. Points on a pseudo-continuum, as in calculus, remain points, albeit infinitesimally small. They only approximate a continua, just as movie stills approximate motion. More broadly any presumption of a Cartesian space {an Aristotelian "substance" having a location defined by the three conventional dimensions and possessing a set of properties) seems to presume an ontological closure right from the start, for the entity is defined by what is local, whatever the origin of its properties may be. 

This does not apply to extrinsic properties such as ontic probability values. Without arguing the case, I believe a representation of a continuum can only be done through extrinsic properties.

Even more broadly and boldly, it seems the modern Western world view is incompatible with continua. One might look at Buddhism or Daoism for an alternative. Rather than start by presuming a sacrosanct self at a spatio-temporal point and lending the world meaning relation to it (as advocaed by Leon Battista Alberti) the world is a flux, and he self is secondary, an entity that imposes limits on that flux (kind of an explicit symmetry break)..

A tensor is a mathematical representation of the relation of mathematical entities, whether they be scalars, vectors or other tensors. Is this a structuralism that would have relations exist independently of their nodes? If so, it seems to imply an objective idealism or reification that I suspect is contrary to the intent of the OP. That is, if we reify relations, then of course we have continua, but the price we pay is superstition. In other words, reified abstractions by definition transcend particulars.

I suspect a continuum is beyond the reach of folk psychology/science. It seems the consensus in neuroscience that the signal from sensory organs is analog in origin but converted to a digital signal so that it can be cognitively processed. Mathematics goes beyond folk psychology, but I doubt that a mathematical representation of a continuum is itself a continuum. I also suspect that any representation of a continuum must, perhaps to borrow from Leibiz, continuum must essentially joint both the whole and its parts. .

2016-08-31
Continuum - a bibliographic assistance
Dear Peter,  Short after I wrote the message I searched the web, found Weyl's "Continuum" in English and ordered it. Should have done this before and not trust my memory. Thanks, anyway.
The problem is that most of the definitions relate to space, time and the number line, or at least that is what the definers had in mind. Everyone is aware of property continua, but somehow many of the definitions do not seem relevant to it.
Take for example a patch of colour that begins with light grey, then gradually becomes darker and darker until it ends up black. That patch is a continuum. Now take the definition you suggested. A unity? maybe (it's an elusive term); cannot be extended? - it definitely is extended; cannot have parts? - yes, it can; we can divide it to any number of slices (equal or non-equal; each giving another range of darkness), just as our brain divides the colour spectrum, for example. So this definition doesn't work. 
I somehow have the feeling that the idea of continuum in properties should be the basis for the idea of continuum in numbers, space and time, and not the other way 'round. You may call it a logicist bent (though I'm not committed to logicism on the whole). 
Benny

2016-08-31
Continuum - a bibliographic assistance
Benny, I'll toss the article your way shortly. However, it is about universal history and does not focus on the continuum issue.

It seems to me we have to distinguish two questions: a) in ontological terms do continua exist? b) if so, how do we represent them in thought? Put simply, I perceive the clouds moving across the sky. This seems to be a process, a continuum of states rather than a set of discrete states of being. But can I represent that through non-poetic language or in rational thought?

On the ontological question, in mathematical terms I suppose a variable is a continuum. But does a variable without value have meaning? x=2y refers to two variables, but seems to have meaning in reference to the world only when we plug in discrete values, at which point they cease to be continua.

Another aspect of this I have trouble wrapping my head around is "vague objects", which have drawn much interest in recent years. A common view, although I don't know that it is the consensus, is that we have an intuitive sense that vague objects do exist, but they are not subject to philosophical analysis. Saint Augustine in Confessions made this point concerning time: we have an intuitive sense that time exists but cannot articulate it in language to convey it to others.

To repeat my earlier suggestion, I suspect that any representation of a continuum must do two things: a) specify spatio-temporal location or state of being so that it refers to the world, b) specify that it at the same time lacks locale and is therefore universal, a becoming that refers to possibilities. Ontic probabilities do this. Do Leibiz' monads (albeit in idealist terms)? Does David Bohm's implicate order (albeit in terms of quantum physics)? A good example is John Bell, "Beables for Quantum Field Theory" in Speakable and Unspeakable in Quantum Mechanics (2004). 

Does action satisfy the conditions of both being and becoming? Of course, this begs the question. But if so, an action theory would have to replace the Cartesian primacy of thought characteristic of the West. It seems to me that rationality and logic are artifacts of cognitive closure and therefore are incompatible with continua. Action, on the other hand, discovers possibilities that reference a state of being but do not reduce to it .

2016-08-31
Continuum - a bibliographic assistance
Hmm, yes, this is tricky.

I would say that what we call a continuum is never a true continuum. When we see a range of values between two fixed values we call it a continuum, but really it is just an arbitrary series of notional points or measurements. If it is a continuum those points and measurements are approximate and refer to ranges, not points. The real numbers cannot be associated with points since there are no discrete points in a continuum by definition. 

But this is not a necessary problem for me. I reserve 'continuum' for time and space since that is where the idea is most important. If space-time is a continuum then there are no moments in time or points in space. For these we have to create what Weyl calls the 'arithmetical' continuum, which is a digitised model of the real analogue thing.

But time, when we take away the past and future, is simply NOW, and always is. No extension required.

Pardon me, this is a bit woolly. I haven't had much practice with this argument.  

..."Mathematics goes beyond folk psychology, but I doubt that a mathematical representation of a continuum is itself a continuum. I also suspect that any representation of a continuum must, perhaps to borrow from Leibiz, continuum must essentially joint both the whole and its parts."

Yes! .  .

2016-09-05
Continuum - a bibliographic assistance
Reply to Haines Brown
Hello Haines.

Thank you for a wonderful post, which I largely missed first time around. I'm not a mathematician but my guess would be that the Weyl of The Continuum and The Open World would have agreed with everything you say in it. I certainly would. I enjoyed reading it properly and learned much. You make some excellent points and I'll copy it for future reference and plagiarism.   

I would agree when you say, 'I suspect a continuum is beyond the reach of folk psychology/science.' I would rather say bluntly that it is. It is cheering to hear someone else arguing this point. In his book on the subject Weyl proposes just this, that the 'arithmetical' continuum on which mathematics depends is not a continuum. 

As for representing the continuum, I do not believe it can be done. This would be for the same reason that Tao cannot represented. The parts can be reduced to the whole but the result is beyond the reach of language and concepts. This takes us into 'nonduality' and the idea that time and space are not truly real. I recently watched a TED talk by Chris Field in which he disposes of the parentheses we place around 'objects' and 'subjects' in space-time in just the same way Kant disposes of the categories of thought. Physics and philosophy were bound to coincide in the end.  

An extended continuum can only makes sense if we assume that its extension is conceptual, such that for any ultimate analysis there would be no such thing. Buddhist philosophers speak of space in terms of 'non-obstruction' and do not reify spatial extension. This allows space-time to be reduced.

How can space be extended? I mean, where would we put it?  

2016-09-06
Continuum - a bibliographic assistance
Reply to Haines Brown
Thank you, Haines. What I understand from your reply is that if I nevertheless insist on integrating the concept of continuum into some formal system (I understand that you're against, but suppose I insist...) - I have to take it as a primitive. Is that correct?Benny

2016-09-06
Continuum - a bibliographic assistance
Dear Peter,   Thank you. I agree, and this is one of the issues in which I'm doubtful about Dedekind's argument. If I remember well, he coped with someone else's definition (forgot whose) who said that A and B are continuous if you can always find an object C  between them. Dedekind said that according to this the line of the rationals is continuous, while we know it is the irrationals break its continuum. To that I would say in the spirit of your comment: the irrationals are points on the line, and points do not have magnitudes so they are not objects fir this purposes, but only boundaries of objects. 
I wrote hastily and hope I was clear.
Apologize,
Benny

2016-09-06
Continuum - a bibliographic assistance
Peter, I'm trying to arrive at a better understanding of your points.

What do you mean by a "true" continuum? You seem to offer two contradictory definitions. One is a range of values or notational points and another is a space-time continuum. Are you saying the first is a sequence, at best the simulacrum or approximation of a continuum, and the second is a continuum that is "true" in the sense that it does not rely on closed units or points?.

Would you agree that we experience true continua? If I watch the clouds move across the sky, am I not experiencing a true continuum? This you appear to say is indeed a continuum in terms of space. But I'm a little nervous limiting a continuum to these two dimensions. How about "its getting warm in here" and "mounting anger" and "atmospheric pressure decreases as we climb"?

I'd suggest, and wonder if you agree, that the distinction is that between ontology and epistemology. That is, we experience the world as contsinua and thus know it exists as continua, but we can only describe or think about them in terms of a sequence of closed entities such as qualities at points or units of measure.

Mathematics I can't get my head around. For example, differential calculus is conceived in Liebniz's terms as infinitely small differences, but in in using it we seem to deal with true continua, not infinitesimals. Does differential calculus lead a double existence?

Discussions of time are a philosophical swamp. My own intuition is that the past and future do not exist except as traces on one hand (probable relations slow present rates of change) and the other as ontic probabilities in the present that have not been made actual. I don't understand why folks want to reify the non-existent past and future. And yet my simple-minded outlook finds little support among saner folks. No doubt it is practical to speak of the past and future as being real, but that is not the issue. "The US went to war" is also fictional reification of a totality, but it is a useful statement nonetheless.

2016-09-06
Continuum - a bibliographic assistance
Hello Benjamin, 
When I was studying logic, set theory, and the foundations of mathematics, the continuum was defined for me using Dedekind cuts. [ https://en.m.wikipedia.org/wiki/Dedekind_cut ]. The proximate starting point for the cuts is the set of rational numbers, which are themselves obtained from the integers (or from the natural numbers if you're interested only in the non-negative real numbers, which are also a continuum).

Other approaches to the continuum use the set of all infinite series of 1s and 0s, with some identifying of certain notionally different series such as .01000.. and .00111.. .

Perhaps I'm not getting at the core of what your question is. Are these traditional definitions of a particular continuum, the real numbers, inadequate to answer deeper questions of interest to you?

Bill Courtney

2016-09-06
Continuum - a bibliographic assistance
Hello Benny,
I think some care is going to be needed here. The rational numbers (a countable set) have many of the properties we ordinarily think of as being important to the notion of the continuum. 

The most important of these properties is that the rationals are dense in themselves. I.e., between any two rationals there is another. Thus, the infinite divisibility of a finite span is not emblematic of the continuum. The color patch you describe could as easily be a span of rationals as a continuum.

The color patch could also be not a span of rationals but instead be a finite, but extensible number of gradations. The notion of a completed infinity has troubled mathematicians (and others!) for quite a while. Finitist notions of uncompleted infinity are sometimes sufficient.

For example, your color patch can be thought of as an excedingly fine discrete set of jumps between darker and lighter. How fine? As fine as your current ability to discriminate. When you ability improves, the refinement of the jumps increases. The color patch is always finite.  

If these dynamics are troubling, we can instead say that the patch is more refined than we have been able to measure so far and whether there is truly an infinite number of levels of light is unknown to us.

In the physical universe, since light is a collection of discrete and indivisible photons which themselves have intensities taken from a discrete set of values, there is no continuum. There is not even a span of rationals.

So not only do we need to take care to avoid confusing the continuum with the rationals, we need to determine whether anything in our world other than our ideals can be infinite or infinitesimal.  

Regards, Bill Courtney

2016-09-06
Continuum - a bibliographic assistance
Reply to Haines Brown
Haines and Bill, thank you for these useful posts. I'm a little out of my depths but will try to keep up. I'll take you at your word, Haines, and attempt to clarify my view.   

---"What do you mean by a "true" continuum? You seem to offer two contradictory definitions. One is a range of values or notational points and another is a space-time continuum."

I probably shouldn't have have used 'true'. I would normally follow Weyl and call the former the arithmetical continuum, which would be a fiction, and the latter the intuitive or empirical continuum, the continuum of experience. I may have referred to the latter as 'true'.

The complication would be that the intuitive continuum would also be a fiction, just not such an obvious one. If we take the view that mental and corporeal phenomena are not metaphysically real then we are left with a single moment of awareness that is not extended. There is no continuum if there is no extension. This would take us beyond the flux of change to the unchanging phenomenon required for its manifestation, thus to a fundamental theory that reduces not just time and space, and thus the continuum.  

One reason why time seems such a complex issue is that rarely do our notions of it make sense. In turn, one reason for this may be that it is far from obvious that the unreality of time makes any sense.     
 
---"Would you agree that we experience true continua?"

In the sense that I think you mean it I would. We do not hear see or hear any joins even if there are any. Metaphysically-speaking, however, this would not be a yes or no question but one for a careful essay. If we go beyond time and space we have to go beyond experience and this idea is a can of worms. This view goes even beyond solipsism.

The key word in the question would be 'we'. For the Buddhist view of time to work we would have to say that there is no such thing as 'we'. As in physics, if I understand it correctly, the boundaries we place around objects and subjects would be conceptual.  

---"How about "its getting warm in here" and "mounting anger" and "atmospheric pressure decreases as we climb"?"

These would be experiential continua with the provisos added above. What they actually are would be another question, one that confuses me. Bill has some good thoughts.
 
---"I'd suggest, and wonder if you agree, that the distinction is that between ontology and epistemology. That is, we experience the world as continua and thus know it exists as continua, but we can only describe or think about them in terms of a sequence of closed entities such as qualities at points or units of measure."

The view I am endorsing is that all distinctions are metaphysically false, false for an ultimate analysis. This would include the distinction between epistemology and ontology. When we dig deep enough we would find that they are the same topic, that what is explains what we know.    

I'd agree that we can only conceive of a continuum in an incoherent way, as something that is extended but yet is not a series of discrete locations, and that we must describe and think about it in these terms, but I would not agree we know that the world exists as continua. I don't think this idea makes sense. Rather, I'd say that to the extent it appears to exists it appears to exist as continua.     

---"Mathematics I can't get my head around. For example, differential calculus is conceived in Liebniz's terms as infinitely small differences, but in in using it we seem to deal with true continua, not infinitesimals. Does differential calculus lead a double existence? "

Nor me. I see it as a useful fudge with no metaphysical implications, but I'm not a mathematician.
 

---"Discussions of time are a philosophical swamp. My own intuition is that the past and future do not exist except as traces on one hand (probable relations slow present rates of change) and the other as ontic probabilities in the present that have not been made actual"

Memories and expectations. We are only ever here and now.

---"I don't understand why folks want to reify the non-existent past and future...."

The past and future are necessary to the self. It is reported by explorers that if we see what lies beyond or behind the self, and thus see that the self is unreal, we will have travelled beyond time and space. This would be the Grail experience that bestows not immortality, but the knowledge that one always was immortal, just like everyone else. Hence it is a modest cup. To be mortal would require the passing of time.

Having got up a head of steam I'll enter dangerous territory and go on to mention that for this view of time the Holy Cross of Chistianity would represent the meeting of two worlds or aspects of the world, the horizontal as the world of time and space, the vertical as Eckhart's 'Perennial Now' or Divine world of timeless awareness, with Jesus, just like the rest of us, suffering at the intersection. Or seemingly so.

Thus for this view it would be impossible to separate the topics of time and religion, which can cause some problems on philosophy forums.  
 

 


2016-09-07
Continuum - a bibliographic assistance
Dear Editor, I have been having some trouble typing my post and I fear that I have submitted partial posts by accident. Below is the post I would like to submit. Thank you, Bill Courtney
Hello Peter,
On the real line there are points. For example 1 is a point. It is true that the point 1 is not discrete in the mathematical sense. That is, any interval of non-zero length around 1 contains other real numbers as well as 1.
But a point does not have to be discrete in order to be a point. For example, a line L in plane geometry is considered to be a continuum. An different, but intersecting line would pass through a single point of L. That point of intersection exists even though it is not discrete in either of the intersecting lines or in the underlying plane space.
I do not understand why you say that points on a continuum and real numbers cannot be associated. The usual association is that one picks a point arbitrarily and calls it 0, picks a distance and calls it 1, and picks a direction along the line and calls it positive. Then each point on the line is associated with its distance, positive or negative, from 0. That is a one-to-one association between the reals and the points on the line.
Whether moments of time exist and, if they do, whether they form a continuum is a conundrum, but the mathematics of points and continua cannot resolve it. We need to look to physics and metaphysics for that. Mathematics can model continuous time or time as the rational numbers inside a notional continuum or time as atomic intervals, and can model other notions of time as well. While mathematics can help us understand what the consequences of each type of existence would be, it cannot pick out which one is our circumstances.
Personally, I find it very difficult to imagine that we can be both conscious and physically constrained to exist only in an instant of NOW and a point of space (if such points even exist). Instead, we can try to model (mathematically or less formally) what conscious existence must be like if we were so constrained and see whether that existence would be similar to our experience or whether a model of existence in some larger piece of spacetime is necessary.
Regards, Bill

2016-09-07
Continuum - a bibliographic assistance
Benny, I would not presume to tell you what to do. Sorry it sounded that way.

It just seemed to me, to put it simply, that epistemology  (cognitive processing) is not up the the world of actual experience. We experience contniua, but don't know how to express it through ordinary language.

Nevertheless, science must somehow manage, for the world consists entirely of processes or continua. I believe that any project to represent a process must explicitly reach beyond the language and cognitive a prioris of folk psychology.I would go further, without defending the point here, that all understanding, as distinguished from mere narration or description, requires that we do so. Causality, as a predictable correlation of empirical change, is a description and thus remains empty and mysterious. 

So in fact, we can represent processes. As I suggested, differential calculus engages continua, but can only be explained in terms of fictitious infinitesimals. 

Haines.  

.



2016-09-07
Continuum - a bibliographic assistance
Peter,

At the risk of derailing the thread, I wonder if you wouldn't mind expanding a bit on your bringing Buddhism in.

Buddhism is highly diverse, but I have a take on it that may be relevant to the points raised here. I 'd appreciate a critique.

The world is in flux. It consists entirely of continua or processes.  By acting on it out of desire or instrumentally, we impose limits of self on this natural flux. This brings frustration and pain. The way to avoid it is  with wú wéi, a "mindless" action without desire or goal. This allows our actions to accord with the flux of the world about us, and so alone provides satisfaction.

Is this a complete misunderstanding?

Haines

2016-09-07
Continuum - a bibliographic assistance
Reply to Bill Courtney
Bill

---"Personally, I find it very difficult to imagine that we can be both conscious and physically constrained to exist only in an instant of NOW and a point of space (if such points even exist). Instead, we can try to model (mathematically or less formally) what conscious existence must be like if we were so constrained and see whether that existence would be similar to our experience or whether a model of existence in some larger piece of spacetime is necessary.
Regards, Bill"

It would be the case that we are constrained to exist only NOW and HERE. How could it be otherwise? But there is something about this idea that doesn't quite work since neither NOW or HERE would have extension and therefore would have no existence. Any theory for which time and space are metaphysically real will have these problems.

A further complication for the view I'm endorsing would be the evanescent nature of the 'thing-events' that make up our world. They would not by persisting 'things'. There would be no persisting things.

This and other ideas would have to be combined for this view of time to make overall sense.


2016-09-07
Continuum - a bibliographic assistance
Reply to Haines Brown
Haines

Your summary ...

"The world is in flux. It consists entirely of continua or processes.  By acting on it out of desire or instrumentally, we impose limits of self on this natural flux. This brings frustration and pain. The way to avoid it is  with wú wéi, a "mindless" action without desire or goal. This allows our actions to accord with the flux of the world about us, and so alone provides satisfaction."

Not way off but I'd rephrase it.

The space-time world would be forever in flux. Attachment to it will lead to suffering because there is nothing permanent anywhere. In particular, our attachment to the ego and identification with it brings suffering. The ego would be impermanent and would not really exist. In short, we will experience suffering if we do not see the true nature of Reality.

The way to avoid this suffering would be to transcend the ego and its desires (as represented mythologically by the Crucifixion of Jesus) and to be able to stand back from ones suffering and, as it were, not take it personally.

Many people who have suffered great pain have found that there is a place in consciousness where the pain doe not reach, or, rather, does reach but only as a phenomenon to be noted, not as something that one would want to participate in or own. This is what Buddhism points us towards, this disassociation with 'me' and 'my' suffering. (I had this learning experience as a teenager - and remain very grateful for it). 

For an ultimate view there would be no suffering and nobody who could suffer. This would, of course,  be a purely conjectural theory for a non-practitioner and of little use to them. It would be fact of life for a 'realised' individual. It would be to some extent an experienced truth for the average person who does a bit of practice and perseveres. Practice would 'make perfect'. 

Wu wei would be a natural response to the situation once we know what the situation is, but may not be directly relevant here. As Ramesh Balsekar writes, 'Why carry your luggage when you're being carried in a vehicle?' (His book The Ultimate Understanding is  good on this). 

This is a poor post. My head is not yet in gear. I might go check my database for a clear statement from a Buddhist.


2016-09-07
Continuum - a bibliographic assistance
Reply to Haines Brown
Dear Haines,Your words sounded 100% OK, and there is definitely no reason to apologize. If I sounded offended - am the one who should apologize. 
Historical processes are a different type of continuum, bordering on the metaphorical sense of the word. This, I think, is stage 2 or even 3. But I'll think about it. 
Benny

2016-09-07
Continuum - a bibliographic assistance
Reply to Bill Courtney
Dear Bill,Thank you for your reply. (BTW, there is some mess in this website regarding the replies. The system puts the messages online long after the person writes them (I can't get the point of this procedure, but never mind), and then you can't really trace back who answered whom and when. My message that mentioned Dedekind was probably written before your message, but appeared later, etc. etc.).
What you send about the photons is actually a quantification of qualities (or properties). This is one way to solve the problem. I personally doubt if it works. Can Beethoven's Ninth be reduced to the number of frequencies and atom motions that the instruments and the singers exert? 
In my patch, the atoms are countable, but the subjective notion of the quality of grey-to-black colour remains "without skips" in our consciousness. But I have to think abut your suggestion.


2016-09-08
Continuum - a bibliographic assistance
Benjamin - I don't think this delay should be occurring. Are you signed up as a member, or are your
posts being re-routed for moderation?

It has belatedly come to me that I have an essay on the continuum that discusses Weyl and other things.

 http://philpapers.org/archive/JONTCE.pdf


2016-09-08
Continuum - a bibliographic assistance
Dear Peter,Thank you for the essay! I will certainly read it soon. 
So far I thought I am a member, but I'll check...
Thanks,
Benny

2016-09-23
Continuum - a bibliographic assistance
Are you still around, Benny? I was hoping you'd comment on that essay and whether you felt it addressed your question.  

2016-10-24
Continuum - a bibliographic assistance
Dear Peter,  I now see I missed your message, and apologize for that. The article is indeed interesting, but does not at all meet my needs. In the mean time I found what I was looking for: John Lane Bells' book, The Continuous and the Infinitesimal in Mathematics and Philosophy  (the whole book is online!: http://publish.uwo.ca/~jbell/The%20Continuous.pdf) and its summary in the SEP:  https://leibniz.stanford.edu/friends/members/view/continuity/sc/ This exactly what I was looking for. Thank you all for your help and insights.
Benny