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FQXi FORUM

March 1, 2021

CATEGORY:
Is Reality Digital or Analog? Essay Contest (2010-2011)
[back]

TOPIC: Some Reasons Why We Cannot Believe In an Analog Reality by Thomas Sanford Wagner [refresh]

TOPIC: Some Reasons Why We Cannot Believe In an Analog Reality by Thomas Sanford Wagner [refresh]

Abstract The concepts of analog vs digital cannot be answered by mathematics. It is, in fact, the underlying philosophies of our mathematics that clouds our thinking. This essay is an attempt to sort it out utilizing, among other things, simple logic.

Foxi Bio Professional composer from 1953 to 1975. Formed and directed The Wagner Renaissance Opera Company, Inc. in 1970 which toured for four years 1960 received a joint patent for a music engraving system 1978 wrote the Structural Resonance papers 1996 created a novel medical billing program 2005 received a grant from the Richard Lounsbery Foundation to prove the tenet of the Structural Resonance papers 2008 created the world's first perfect music tuning system Since 2005 developing a conditioned violin, condition listening environs, development of uses of the middle ear for use in both music and hearing

Excellent paper. One additional argument you might have used would be from Brukner's coarse-graining. To my knowledge he doesn't specifically apply that approach to the discrete-versus-continuous issue as such, but he's argued elsewhere (along with Zeilinger) for weeding infinities and continua out of a final QM formulation. Anyway, in my own opinion, we're clunky, imprecise measuring instruments who try to finesse our ignorance with high-sounding ideational junk (e.g., continua, infinities, infinitesimals).

You shouldn't underplay the infelicity of "Digital or Analog" when "Discrete or Continuous" would have demonstrated a much greater savoir-faire. Or, on the other hand, since you're a contestant maybe you should.

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You shouldn't underplay the infelicity of "Digital or Analog" when "Discrete or Continuous" would have demonstrated a much greater savoir-faire. Or, on the other hand, since you're a contestant maybe you should.

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If asked, most would say that motion is an analog function, but consider this. In order for an object to be said to have moved it must be in a different place than it was originally. This implies a distance as without a distance the object could not be said to have moved. We can take half that distance and then half again for as many times as we wish but the distance can never become infinite. Otherwise the object would not have moved.

This is another way of saying that an object cannot move by taking infinitely small steps which, if motion is truly analog, it would have to do. Since infinitely small steps are impossible there must be an initial step, however small this might be. The next step must be finite as well so motion is digital because it cannot be otherwise.

Dear Thomas,

I am of the opinion that an object can move by taking infinitely small steps. It’s the speed of the object that makes a difference.

Let us assume a small bird is flying from one point P1 located in New York to another point P2 located in Tokyo. At the same time, a jet liner is moving from P1 to P2. It could be said that the small bird would take infinitely small steps per time unit as it moves forward. On the other hand, the plane is propelled at a speed say 1000 times greater than the bird’s speed. So the plane would cover steps 1000 larger than the bird’s small steps per time unit simply because of differences in speed. It could be said that the bird would never reach Tokyo if it was to face deadlines similar to business passengers in the plane.

It is very important to stress that we deal here with relative concepts, thus it is always possible to set an acceptable error. If the bird takes a very small step per time unit, then the plane could take 1/1000th time unit to cover the same small distance. Thus time also could be infinitesimal, thus the continuity.

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This is another way of saying that an object cannot move by taking infinitely small steps which, if motion is truly analog, it would have to do. Since infinitely small steps are impossible there must be an initial step, however small this might be. The next step must be finite as well so motion is digital because it cannot be otherwise.

Dear Thomas,

I am of the opinion that an object can move by taking infinitely small steps. It’s the speed of the object that makes a difference.

Let us assume a small bird is flying from one point P1 located in New York to another point P2 located in Tokyo. At the same time, a jet liner is moving from P1 to P2. It could be said that the small bird would take infinitely small steps per time unit as it moves forward. On the other hand, the plane is propelled at a speed say 1000 times greater than the bird’s speed. So the plane would cover steps 1000 larger than the bird’s small steps per time unit simply because of differences in speed. It could be said that the bird would never reach Tokyo if it was to face deadlines similar to business passengers in the plane.

It is very important to stress that we deal here with relative concepts, thus it is always possible to set an acceptable error. If the bird takes a very small step per time unit, then the plane could take 1/1000th time unit to cover the same small distance. Thus time also could be infinitesimal, thus the continuity.

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Simply put, how can an object take an infinitely small step? The continuum of a line requires an infinite number of points. Assume a line that is one centimeter long; it has an infinite number of infinitely small points which are infinitely close together. Now consider another line that reaches from here to the Andromeda Galaxy. It too has an infinity of infinitely small points that are infinitely...

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Hi to both of you,

Relevant dear Mr Wagner, rational all that.Well said also the words about elites in opera, it's funny.The world doesn't turn correctly dear Mr Wagner, the opulences above the poors, sad reality of our big cities,sad reality of human stupidities.They listen arts and ART but they do not understand his heart!

Steve

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Relevant dear Mr Wagner, rational all that.Well said also the words about elites in opera, it's funny.The world doesn't turn correctly dear Mr Wagner, the opulences above the poors, sad reality of our big cities,sad reality of human stupidities.They listen arts and ART but they do not understand his heart!

Steve

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Hello Thomas,

Please read my reply to Antonio Leon. It applies to you too.

You wrote: "Mathematicians tell us that our math has become sufficiently sophisticated that we now know that the sum of an infinite series ... equals 1. If that be true, what is the penultimate fraction? If such fraction does not exist then the result of an infinite series simply cannot equal 1."

non sequitur

Eckard

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Please read my reply to Antonio Leon. It applies to you too.

You wrote: "Mathematicians tell us that our math has become sufficiently sophisticated that we now know that the sum of an infinite series ... equals 1. If that be true, what is the penultimate fraction? If such fraction does not exist then the result of an infinite series simply cannot equal 1."

non sequitur

Eckard

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The series I referred to in my initial posting is an endless series. No matter how far we extend it you can always divide the existing fraction by 2. Both the theory of infinitesimals and the theory of limits are useful mathematical tools but they have no basis in physical reality.

If we sum the fraction as we progress we find that each sum produces (n-1)⁄n regardless of how far we extend the series. If the series does, in fact, equal one at what point do we experience n⁄n ? To say that it does appear exposes us to pure math and philosophy and immerses us into a sort of transcendental fog. In the real world n⁄n simply and absolutely never appears.

In the real world there are no abstractions. Natural events, even when they appear indecipherable always have an ultimate definable explanation, even if we don't as yet understand what it is.

Some twenty years or so ago there was a huge gathering of pure mathematicians. They all gathered and read their latest papers to each other. A review of the proceedings appeared on the front page of the New York Times. The reviewer noted that all of the papers he heard seemed to be divided into three parts; a first part where just about everyone could understand; a second part which only those very conversant with the logic and concepts of pure math could understand and finally a third part which only the author could understand. The reviewer then went on to state that perhaps it was to save time but he noticed that all of the papers jumped immediately to the third part.

We must remember when dealing with abstractions that they are indeed abstract.

Tom

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If we sum the fraction as we progress we find that each sum produces (n-1)⁄n regardless of how far we extend the series. If the series does, in fact, equal one at what point do we experience n⁄n ? To say that it does appear exposes us to pure math and philosophy and immerses us into a sort of transcendental fog. In the real world n⁄n simply and absolutely never appears.

In the real world there are no abstractions. Natural events, even when they appear indecipherable always have an ultimate definable explanation, even if we don't as yet understand what it is.

Some twenty years or so ago there was a huge gathering of pure mathematicians. They all gathered and read their latest papers to each other. A review of the proceedings appeared on the front page of the New York Times. The reviewer noted that all of the papers he heard seemed to be divided into three parts; a first part where just about everyone could understand; a second part which only those very conversant with the logic and concepts of pure math could understand and finally a third part which only the author could understand. The reviewer then went on to state that perhaps it was to save time but he noticed that all of the papers jumped immediately to the third part.

We must remember when dealing with abstractions that they are indeed abstract.

Tom

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Hi dear scientists,it's me the crazzy.

The words of the day....the rationality is the sister of the proportionalities, the series are like a music, they can be finites or infinites, that depends of what we want calculate or what we want create. Reality sings with objectivity, the subjectivities can be synchronized by pure reals only.It's a real objective rationality of things simply.

You can repeat still and still your games with a good method as Hannon.These notes are precises,the speed as a personal choice, it's objectif and subjectif when the creativity appears with its series of harmony.The superpositions of creations as a torch of sounds.Now of course can you insert 3 diesis for the Ré major for example, no evidently as for the sol, 1 is sufficient and essential even.The superimposings in the series of harmony seem showing a beautiful road.

Regards

Steve

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The words of the day....the rationality is the sister of the proportionalities, the series are like a music, they can be finites or infinites, that depends of what we want calculate or what we want create. Reality sings with objectivity, the subjectivities can be synchronized by pure reals only.It's a real objective rationality of things simply.

You can repeat still and still your games with a good method as Hannon.These notes are precises,the speed as a personal choice, it's objectif and subjectif when the creativity appears with its series of harmony.The superpositions of creations as a torch of sounds.Now of course can you insert 3 diesis for the Ré major for example, no evidently as for the sol, 1 is sufficient and essential even.The superimposings in the series of harmony seem showing a beautiful road.

Regards

Steve

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Thomas Wagner,

You wrote:"The series I referred to in my initial posting is an endless series."

Do you consider me naive or did you not read my essay? I have to face rejection by those who feel hurt by my unwelcome irrefutable arguments. If you prefer simple logic you should be able to understand, accept, and vote for my arguments. Engineers like me have to correctly use mathematics.

Eckard

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You wrote:"The series I referred to in my initial posting is an endless series."

Do you consider me naive or did you not read my essay? I have to face rejection by those who feel hurt by my unwelcome irrefutable arguments. If you prefer simple logic you should be able to understand, accept, and vote for my arguments. Engineers like me have to correctly use mathematics.

Eckard

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Simply put, how can an object take an infinitely small step?

Dear Thomas,

Your question is thought-provoking, and I acknowledge I do not have enough knowledge in music at present to give any comment on that line of reasoning.

Nevertheless, I basically think the universe is continuous because continuity is rich. There are so many possibilities that we are left with food for thought for eternity.

Let us assume we want an object to move through a 1 cm distance. An object could take it by halves: 1/2 cm, then 1/2 + 1/4 and so forth.

Another object would take it differently: 1/10 cm, then 1/100 cm, then 1/1000 cm and so forth. Each object thus enters an infinity and they will not progress at the same rate because their rate is drastically reduced at each step.

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Dear Thomas,

Your question is thought-provoking, and I acknowledge I do not have enough knowledge in music at present to give any comment on that line of reasoning.

Nevertheless, I basically think the universe is continuous because continuity is rich. There are so many possibilities that we are left with food for thought for eternity.

Let us assume we want an object to move through a 1 cm distance. An object could take it by halves: 1/2 cm, then 1/2 + 1/4 and so forth.

Another object would take it differently: 1/10 cm, then 1/100 cm, then 1/1000 cm and so forth. Each object thus enters an infinity and they will not progress at the same rate because their rate is drastically reduced at each step.

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Yes the rate is drastically reduced but it cannot become infinite as then it would cease to exist altogether,

Tom

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Tom

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Thomas - I enjoyed your paper. I would not have thought that the numerical notions of ancient Greece would apply to our modern understanding of physics, but you make a compelling argument. I may express it in another way - Nature evolved in simple steps- a step implies a bifurcation, a digital item. Pythagoras and his followers understood the importance of digits because they did not have other notions to distract them. Now we human beings, who also evolved within Nature step by step, have become too clever for our own good. We invent wonderfully baroque mathematics and expect Nature to follow it as we imagined it. String Theory may be a good example of that. That is why it was refreshing to read what you said. Physics, as I have explained in my paper, needs to be reconstructed, and not necessarily from the concepts that have come to be adopted over the past few centuries.

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Vladimir

The Ancient Hellenes were remarkably aware of both mathematics and their environment. One reason they had for not believing that the earth orbited the sun was that when they observed the sky at one time and the viewed the same part of the sky six months later is that could not observe a parallactic shift, which they correctly assumed should they should see. What they were not aware of was the vast distances involved. Even the nearest star crates a parallactic shift of less than one arc second.

Tom

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The Ancient Hellenes were remarkably aware of both mathematics and their environment. One reason they had for not believing that the earth orbited the sun was that when they observed the sky at one time and the viewed the same part of the sky six months later is that could not observe a parallactic shift, which they correctly assumed should they should see. What they were not aware of was the vast distances involved. Even the nearest star crates a parallactic shift of less than one arc second.

Tom

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I think you have it exactly right. Math finally caught up with the physics and became very predictive, and so then somehow we started to believe that physics was a crystallization of the math because the representations seemed so accurate. Great job. My essay is a spoof of a similar idea, obviously for beginners.

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Ddear Thomas Sanford Wagner,

Nice to know you.,I love music, I play piano and guitar .It's wonderful the musics.Could you tell me more about your researchs about the sounds, acoustics and musics,please?

Good luck also for the contest.

Regards

Steve

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Nice to know you.,I love music, I play piano and guitar .It's wonderful the musics.Could you tell me more about your researchs about the sounds, acoustics and musics,please?

Good luck also for the contest.

Regards

Steve

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I have been away from the music scene for quite a few years so I have not really written anything about music. There is a short description of the Structural Resonance paper on my website (www.wropera.com)

I am debating as to whether to put the Structural Resonance papers on the website.

When I get some good examples of my tuning system I will post then as well.

Tom

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I am debating as to whether to put the Structural Resonance papers on the website.

When I get some good examples of my tuning system I will post then as well.

Tom

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Thomas Sanford Wagner,

I just printed off your essay. I still need to read it. One thing I will be looking for is support for this statement:

"Simply put, how can an object take an infinitely small step? The continuum of a line requires an infinite number of points."

So far as I know, a line does not require steps or points. I assume that your essay will reveal why you think a line involves steps.

James

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I just printed off your essay. I still need to read it. One thing I will be looking for is support for this statement:

"Simply put, how can an object take an infinitely small step? The continuum of a line requires an infinite number of points."

So far as I know, a line does not require steps or points. I assume that your essay will reveal why you think a line involves steps.

James

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Hi Tom,James(Master Yoda),all,

Dear Tom, thanks,it's nice.

Dear Master Yoda, be the force with you .

Regards

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Dear Tom, thanks,it's nice.

Dear Master Yoda, be the force with you .

Regards

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Thanks Thomas for some common sense thinking. I totally agree with this early statement:

"This cannot be resolved by mathematics. In fact it

is our mathematics, miraculous as it is, that is one of the chief contributors to the dilemma."

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"This cannot be resolved by mathematics. In fact it

is our mathematics, miraculous as it is, that is one of the chief contributors to the dilemma."

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Thomas

Thank You. Brilliant essay, I entirely agree, particularly about maths and infinities. To the point and well written, definitely worth a high score. I've identified assumptions about maths as the root of many problems with physics, and pointed to overcoming them. Indeed going beyond that, we use motion and vectors within geometry, when motion invalidates any geometry.

Einstein specified Descartes x.y. and z axis as being 'attached to a body'. Mathematicians immediately discarded the body and treated the abstracted points and lines as real. No wonder physics has been in a rut for so long!

I really hope you'll read and score my essay (I really need the abstract points!) and most importantly I think you may be conceptually able (only about 1 in 5 are!) to see the real dynamic solution that arises by removing abstraction. Just cleaning up SR a bit with Logic to derive it, and GR with a Quantum Mechanism. the process is also falsifiable (I'm an Architect and have to build reality not illogical theories).

The mechanism is all based on wave harmonics and interaction. I couldn't get much into that in the Essay, but Christian Corda is currently considering a paper on refraction including the paralell between wavebands (with 'absorption bans of reversed refraction between) and octaves which I'd quite like your views on later if possible. Did you know rainbows reverse immediately they are out of the visible band?

Do please give me your views. But don't try to 'scan' over it quickly or you probably won't be able to follow the visualisation and dynamic relationships.

Very best wishes.

Peter

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Thank You. Brilliant essay, I entirely agree, particularly about maths and infinities. To the point and well written, definitely worth a high score. I've identified assumptions about maths as the root of many problems with physics, and pointed to overcoming them. Indeed going beyond that, we use motion and vectors within geometry, when motion invalidates any geometry.

Einstein specified Descartes x.y. and z axis as being 'attached to a body'. Mathematicians immediately discarded the body and treated the abstracted points and lines as real. No wonder physics has been in a rut for so long!

I really hope you'll read and score my essay (I really need the abstract points!) and most importantly I think you may be conceptually able (only about 1 in 5 are!) to see the real dynamic solution that arises by removing abstraction. Just cleaning up SR a bit with Logic to derive it, and GR with a Quantum Mechanism. the process is also falsifiable (I'm an Architect and have to build reality not illogical theories).

The mechanism is all based on wave harmonics and interaction. I couldn't get much into that in the Essay, but Christian Corda is currently considering a paper on refraction including the paralell between wavebands (with 'absorption bans of reversed refraction between) and octaves which I'd quite like your views on later if possible. Did you know rainbows reverse immediately they are out of the visible band?

Do please give me your views. But don't try to 'scan' over it quickly or you probably won't be able to follow the visualisation and dynamic relationships.

Very best wishes.

Peter

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The premise upon which this essay is based is that there is no such thing as infinity, not even as a concept.

Thomas,

Well argued, but does a recycled universe meet the criteria of infinity or recycled galaxies? My argument is for analogue and I do rely on model concepts in supporting it, including string theory and its offshoots.

Jim Hoover

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Thomas,

Well argued, but does a recycled universe meet the criteria of infinity or recycled galaxies? My argument is for analogue and I do rely on model concepts in supporting it, including string theory and its offshoots.

Jim Hoover

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Jim

I assume you are referring to the notion of the 'big crunch' where the universe eventually reverses the expansion and collapses once again to a very small structure and then the big bang happens again. The recycling universes itself is discrete. If there exists analog occurrences they must be part of discrete events.

Tom

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I assume you are referring to the notion of the 'big crunch' where the universe eventually reverses the expansion and collapses once again to a very small structure and then the big bang happens again. The recycling universes itself is discrete. If there exists analog occurrences they must be part of discrete events.

Tom

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Dear Yuri Danoyan,

"Since infinitely small steps are impossible there must be an initial step, however small this might be. The next step must be finite as well so motion is digital because it cannot be otherwise. Some might say that we have studied motion with exquisitely fine instruments and such a step has never been measured. We have never measured a Planck Length either but no one doubts its existence."

I don't think that you have shown that motion is digital. What you may have shown is that your method of measuring is digital. By the way, I doubt Planck Length and the other Planck units. However, I am not a physicist so that is just my opinion.

"I think we forget sometimes that mathematical abstraction goes back at least to Euclid. Look up the definition of a point in any dictionary and you will read that a point is thought of as having a location in space but having no dimensions. Thus a point, by definition, does not exist."

The point may be an abstraction but the math is not. You mention infinitely samll steps. Is that your understanding of what an infinite number of points is? Or, are you imagining an infinitely small 'step'?

I am reading through your essay and am trying to acquaint myself with your logical basis. Any clarification would be welcome. Thank you.

James

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"Since infinitely small steps are impossible there must be an initial step, however small this might be. The next step must be finite as well so motion is digital because it cannot be otherwise. Some might say that we have studied motion with exquisitely fine instruments and such a step has never been measured. We have never measured a Planck Length either but no one doubts its existence."

I don't think that you have shown that motion is digital. What you may have shown is that your method of measuring is digital. By the way, I doubt Planck Length and the other Planck units. However, I am not a physicist so that is just my opinion.

"I think we forget sometimes that mathematical abstraction goes back at least to Euclid. Look up the definition of a point in any dictionary and you will read that a point is thought of as having a location in space but having no dimensions. Thus a point, by definition, does not exist."

The point may be an abstraction but the math is not. You mention infinitely samll steps. Is that your understanding of what an infinite number of points is? Or, are you imagining an infinitely small 'step'?

I am reading through your essay and am trying to acquaint myself with your logical basis. Any clarification would be welcome. Thank you.

James

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"Einstein often said in various ways that out mathematics does not reflect reality. The things we have created with the various flavors of the calculus is truly remarkable when you consider the calculus is based quite literally upon a grand oxymoron; instantaneous speed. From this of course we got the derivative of a real value function which led to the whole magic of the calculus."

What reference, or your own explanation, can you give to show that calculus is based upon instantaneous speed? It is not clear to me that your statement is true.

James

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What reference, or your own explanation, can you give to show that calculus is based upon instantaneous speed? It is not clear to me that your statement is true.

James

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James

The Plank length came from Plank's constant, which is part of the cgs system. It arose as a viable explanation to the ultraviolet catastrophe. We cannot prove the existence of the Plank length as it is really small - how small? There are more cubic Plank lengths in a cubic centimeter than there are stars in the visible universe.

If we define a square in geometry and state that each side is one inch long, we can accurately say that the perimeter is four inches and the area is one square inch. This uses only integers and is, of course not abstract. Now if we draw a diagonal of that square the length should be the square root of 2. It is physically impossible to draw such a diagonal, as the square root of two has no definable value. In this case, the math is abstract.

In speaking of points on a line, we must remember that the line itself does not exist, so referring to points on a line is an absurd statement to begin with. Much of our notions of continuity come from our approach to a graph. If we have a number of points on a graph, they represent a digital structure. If we draw a line through these points we have something else altogether. When we linearize such a structure by this method, the data becomes abstract.

In the late seventeenth century, Frederick Leibnitz was pondering the notion of instantaneous speed. Speed is the result of two finite quantities, that of time and distance. He realized that in order to create a valid equation or this time must have a positive vale regardless of how small this value is. Calculus deals primarily with changing quantities so Leibniz created what may be calculus' first bit of notation. If time is t he wrote that the change in time is dt. He did the same for distance, which he notated as y, and then he had dy. A change in speed is then the result of dt/dy.

He needed a small change in either value so he came up with the notion of a number that is smaller than any other number but not equal to zero. It is here that the calculus was really born. He now had the infinitesimal. This sounds strange but it worked even though they did not why.

This caused quite a disturbance in the mathematical community, as Leibnitz was a reputable figure in seventeenth century mathematics. Bishop Berkley called the infinitesimal the 'ghost of departed quantities'. It turns out that the solution to instantaneous speed is quite like that of area under a curve.

Some decades later Cauchy came up with notion of limits and calculus took more or less the form in which it exists today. I realize this is a cursory definition but at least it establishes the roll of instantaneous speed as initiator of the calculus.

Tom

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The Plank length came from Plank's constant, which is part of the cgs system. It arose as a viable explanation to the ultraviolet catastrophe. We cannot prove the existence of the Plank length as it is really small - how small? There are more cubic Plank lengths in a cubic centimeter than there are stars in the visible universe.

If we define a square in geometry and state that each side is one inch long, we can accurately say that the perimeter is four inches and the area is one square inch. This uses only integers and is, of course not abstract. Now if we draw a diagonal of that square the length should be the square root of 2. It is physically impossible to draw such a diagonal, as the square root of two has no definable value. In this case, the math is abstract.

In speaking of points on a line, we must remember that the line itself does not exist, so referring to points on a line is an absurd statement to begin with. Much of our notions of continuity come from our approach to a graph. If we have a number of points on a graph, they represent a digital structure. If we draw a line through these points we have something else altogether. When we linearize such a structure by this method, the data becomes abstract.

In the late seventeenth century, Frederick Leibnitz was pondering the notion of instantaneous speed. Speed is the result of two finite quantities, that of time and distance. He realized that in order to create a valid equation or this time must have a positive vale regardless of how small this value is. Calculus deals primarily with changing quantities so Leibniz created what may be calculus' first bit of notation. If time is t he wrote that the change in time is dt. He did the same for distance, which he notated as y, and then he had dy. A change in speed is then the result of dt/dy.

He needed a small change in either value so he came up with the notion of a number that is smaller than any other number but not equal to zero. It is here that the calculus was really born. He now had the infinitesimal. This sounds strange but it worked even though they did not why.

This caused quite a disturbance in the mathematical community, as Leibnitz was a reputable figure in seventeenth century mathematics. Bishop Berkley called the infinitesimal the 'ghost of departed quantities'. It turns out that the solution to instantaneous speed is quite like that of area under a curve.

Some decades later Cauchy came up with notion of limits and calculus took more or less the form in which it exists today. I realize this is a cursory definition but at least it establishes the roll of instantaneous speed as initiator of the calculus.

Tom

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Dear Eckard

I have reread your essay and I do have a better picture of what you are saying. While I do have a fair layman's grasp of the study sub-atomic particles I do not possess the sophistication nor the experience in such matters to allow me to engage in a meaningful debate. I plan to read it again as there is much included that stimulates ideas.

I was struck by your reference...

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I have reread your essay and I do have a better picture of what you are saying. While I do have a fair layman's grasp of the study sub-atomic particles I do not possess the sophistication nor the experience in such matters to allow me to engage in a meaningful debate. I plan to read it again as there is much included that stimulates ideas.

I was struck by your reference...

view entire post

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