Some Reasons Why We Cannot Believe In an Analog Reality by Thomas Sanford Wagner

Dear Eckard

Thank you for alerting me as to the rating system of which I was unaware. I will certainly now reread your essay (I have been fighting some deadlines here so I could not give this entire project the attention I would like. I am now going to take the time to read some of the other essays as the interplay between those of us who entered essays seems to be the most interesting part of this whole experience.

Tom Wagner

My easiest essay

http://www.fqxi.org/community/forum/topic/946

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

"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

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 to Sommerfeld on page four. When he states that no wave is reflected from infinity in finite time sounds a bit like the notion that a moving object cannot transverse an infinite number of points in a finite amount of time. Be that as it may, I am more interest in the next statement, Standing waves are strictly speaking approximations.

A standing wave is a very definable and precise physical phenomenon. It is the initiator of most and perhaps all sound. This can best be seen in a musical example. More than half a century ago, Frederick Saunders wrote an article about the physics of music for Scientific American. This article had more errors and misconceptions that I have ever seen in one article. Saunders was a noted figure in the acoustical world, but physicists are only human (at least most of them are).

One false assumption that most people make about the generation of a music as sound, and I use Saunder's example of an instrument such as a clarinet or an oboe, is that it is the movement of the traveling longitudinal wave that transverses the from the mouthpiece to either the end of the instrument or to the first open key actually creates the sound.. A conjugate is returned and wave moves back and forth through the instrument.

Saunders makes the statement that it is the air that flows in and out of the finger holes that creates the sound. He then went on to state the fundamental is the only note whose sound goes out of the end of the instrument. If this were to be true then why do they put bells on both clarinets and oboes if they only affect a single note?

The movement of the traveling wave back and forth sets up the frequency of the tone. The structure of the sound begins in the reed of either instrument. This is fed from the mouthpiece to the sides of the instrument. The movement of the air creates a classic standing wave, which is modified by the information residing on the sides. The severe impedance mismatch between the air and the materials from which an instrument is created means that the body of any instrument contributes little to the sound we hear. The primary interface that creates the sound lies across the plane of the open end of the instrument. This is why a bell increases the volume of the sound; it increases the area of interface.

This is true of most instruments. The standing wave that forms in the body of most instruments is a resonance. Perhaps the biggest obstacle in understanding sound is the lack of understanding that a vibration and a resonance are two related but decidedly different things. Any material with some elastic properties will resonate to any frequency. Only if the resonance is near to the overtone structure of the resonating material will that material vibrate. On the other hand, a vibration is necessary to create the resonance initially. Both the vibration and the resonance are digital.

Sound is not a single isolated occurrence; it is a process that ends in the Organ of Corti. The Organ of Corti is a fluid filled canal in the cochlea, which houses the hair cells that stimulate the nerves to the brain. The final argument for hearing being a discrete process is that the messages the nerves send to the brain are in the form of discrete pulses. They respond to an increase in amplitude by sending more pulses per unit time.

Since all of the nerves that send data to the brain are quite the same we have to wonder if all sensations are transmitted to brain as discrete pulses. While I agree that the brain is not necessarily just a big computer we have to be aware of the fact that the complex of nerves that address the brain do behave a bit like a computer bus and the pulses are, in effect, bit patterns.

Thanks for a very provocative essay.

Tom Wagner

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

Tom

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

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

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