Dear Sir,

Congratulation for such a brilliant essay. You deserve the best.

Though our views of numbers, infinity and zero (discussed elaborately in our essay) are different from that of Deepak Chopra, we hold that infinities are real and not a very big number. The number of infinities is not infinite, but only four. Zero is the absence of something at here-now that is known to exist elsewhere (hence real; otherwise we will not perceive its absence at all). Since all measurement/perception are done at here-now, the concepts of the objects (information about it) only are available and not the real objects. To that extent you can call it unreal. Infinities are everywhere; hence it is not like zero. You are using the term infinity for very big numbers, which is not its proper definition.

Your statement that "dx could be part of an equation describing that beautiful curve" is partially correct, as no equation can describe beauty. A smile is not the same as curvature of the lips - snide also has the same characteristic. Newton and Leibniz evolved calculus from charts prepared from the power series, based on the binomial expansion. The binomial expansion is supposed to be an infinite series expansion of a complex differential that approached zero. But this involved the problems of the tangent to the curve and the area of the quadrature. In Lemma VII in Principia, Newton states that at the limit (when the interval between two points goes to zero), the arc, the chord and the tangent are all equal. But if this is true, then both his diagonal and the versine must be zero. In that case, he is talking about a point with no spatial dimensions. In case it is a line, then they are all equal. In that case, neither the versine equation nor the Pythagorean Theorem applies. Hence it cannot be used in calculus for summing up an area with spatial dimensions.

Newton and Leibniz found the solution to the calculus while studying the "chityuttara" principle of ancient India, which is now called Pascal's differential triangle. To solve the problem of the tangent, this triangle must be made smaller and smaller. We must move from x to ﾎ"x. But can it be mathematically represented? No point on any possible graph can stand for a point in space or an instant in time. A point on a graph stands for two distances from the origin on the two axes. To graph a straight line in space, only one axes is needed. For a point in space, zero axes are needed. Either you perceive it directly without reference to any origin or it is non-existent. Only during measurement, some reference is needed.

Your point about roses was uniquely enjoyable, but we hoped you would end it in a more realistic manner. After the bill for 400 hundred roses was paid, the driver should have taken those away in the other truck and claimed another bill for the -400 (lack of 400, which used to be there) roses. Tachyon roses are somewhat imaginary. You are correct that in SR, the squared velocity yields tachyons as one of the possibilities, if v is negative. We have discussed many other aspects of SR (including wrong mathematics) in our essay.

We are reminded of an anecdote relating to a famous scientist, who once directed two of his students to precisely measure the wave-length of sodium light. The students returned with two different results - one resembling the normally accepted value and the other a different value. Upon enquiry, the other student replied that he had also come up with the same result as the accepted value, but since everything including the Earth and the scale on it is moving, for precision measurement he applied length contraction to the scale treating the star Betelgeuse as a reference point. This changed the result. The scientist told him to treat the scale and the object to be measured as moving with the same velocity and recalculate the wave-length of light again without any reference to Betelgeuse. After sometime, both the students returned to tell that the wave-length of sodium light is infinite. To a surprised scientist, they explained that since the scale is moving with light, its length would shrink to zero. Hence it will require an infinite number of scales to measure the wave-length of sodium light! Then consequences of the model are extrapolated - as you say.

Your statement that: "in observing particles that decay and experiments in super-colliders: all particles decay into energy which consists of photons moving away from the site of decay. All except neutrinos" is true. Also "Photons can randomly come together to form a particle if the area of photons is dense enough. The small number of photons gathering would be such that the particle would not persist and decay into its component photons. This is observed in the real universe". Yet, as you say, there are caveats.

Regards,

basudeba

    A beginning comment in your post is, "...we hold that infinities are real and not a very big number."

    "...we hold..." is not a mathematical nor physics form of science.

    There are many flaws in your following explanations. However, it suffices to point out that the phrase, "...infinities are real and not a very big number," indicates your unwillingness to face reality or the projection of consciousness that manifests what we call reality. Ergo, the remainder is null and void.

    I have reread your post several times. I am confused. Who is "we"? Is it the physics community or the Deepak community? Next, you have made a statement about infinity. It is presented as the ultimate fact, "...we hold..." Since the purpose of the contest is to discuss the aspects of mathematics, such a statement is inappropriate. I can understand stating your side but would it not be better to write up your opinion and submit it as I have.

    The third issue is about politeness on forums. Submitting a number of statements is inappropriate. This is a technique Deepak uses often in his discussions with others. He throws out a fistful of points or what he considers truths and the other party has no time to discuss any of them. Then he puffs his chest out as if he is the winner. It is very impolite.

    I am worried that if I respond to each of your comments you will respond to each with a multiplicity of responses. It is a cheap trick to bury someone without making progress.

    I would like to respond to your statements but not as if I am being attacked.

    Dear Sir,

    Decorum is a part of our culture. We beg to be excused if you feel otherwise. Our post did not "attack" you, but supported and expanded your views in many areas.

    You have rightly said that "the purpose of the contest is to discuss the aspects of mathematics". For this purpose, we want to keep out I from the discussion to make it a collective effort to find the truth. We would be grateful if you could educate us about our faults by justifying your statements.

    Regards,

    basudeba

    Dear Sir,

    On second thoughts we feel that our reference to smile and snide might have hurt your feelings. If so, we are deeply sorry and profusely apologize to you. It was totally unintentional and quoted from our essay. We were surprised at your reaction since we had written nothing against you and of all the essays we had rated, we had given you the highest score. We also do not know Dr. Deepak Chopra except for comments on him in the net. We have not studied his works. Hence cannot comment on it.

    Finally, before forming any opinion, we request you to kindly read our essay once.

    Regards,

    basudeba

    I would eagerly read your essay if I knew what it was.

    Second, I have essentially given up on this contest.

    It appeared that the popular authors were getting all the attention.

    I am honored that some have given me credit.

    Actually, I am ecstatic.

    If you would let me know which essay is yours, I will read it and we can continue.

    Now I am curious what you have to say.

    All the best.

      The purpose of this post is to reiterate the thesis of this essay.

      Present Operation:

      At present the physics community values real world experimental data and a mathematical explanation. It abhors mechanical models used to relate experimental data to the mathematics. This author believes that this breeds incomplete understanding.

      Desired Operation:

      Understanding can be improved by using experimental data, mechanical models, and mathematical descriptions.

      In this post, three scenarios are presented to attempt to bring this idea home.

      The first is the rose garden scenario from the essay and a post from basudeba. As pointed out, the truck drivers could have delivered the +400 roses then, to deliver the -400 roses, the +400 roses would be carted away. This is a very accurate assessment. However, this assessment is made due to the fact the truckers and gardener has a model from which to work with. That is, they can hold roses in their hand. That is, they have real world data, a model from which to work, and a mathematical understanding. The gardener did not process the model in his thought process. He thus thought there were some mysterious roses. The truck drivers understood the model and sold the gardener his mysterious roses.

      This is to mirror the way special relativity is treated. There, people have proposed tachyons. The "mathematics" proposes that tachyons are particles moving backward in time at a speed faster than the speed of light. This author proposes that this scenario is generated because the physics community abhors the idea there is a model that explains special relativity. The essay suggested (in the physics community this is considered a sin) a mechanical model that represents how special relativity works. The point is that once a model has been selected, many ramifications of the concept of special relativity can be drawn. Understandably, those ramifications can come under fire of skepticism and challenge. This is as it should be. The point is that without a model, further discussion does not exist.

      As a third attempt to explain the prime thesis of the essay, consider quantum wave collapse. Presently a quantum particle is considered a probability wave that upon observation collapses to occupy a specific position and time in space. This author believes this concept is due to no model or a false model. At present the physics community clings to the idea that particles are little hard balls that can appear randomly in locations in space adhering to quantum mathematics descriptions. Consider a different model. Perhaps particles are like oil drops that undulate as they move through space. Maybe we should consider a particle as a loop of string that undulates as it moves through space. The concept here is that the center of density of the particle is shifting up, down, left, right, forward, and backward.

      Now, the process of observation is the process of measuring the density of the particle at all points within the particle and finding the center of the density. The physicists of today will consider that is where the particle is at. As the physicists consider all these things hard little balls, they will declare that is where the ball is at, at a specific space and time. From this the concept is that quantum particles can move backward in time and experience Heisenberg Uncertainty.

      The point is that they are making a false assumption. They claim that the little ball can be mysteriously anywhere and mysteriously finds its location when observed. In reality, during observation, the averages of all parts of the particle are averaged to a specific point in space and time. The oil drop or string has no idea it has been observed, experiences no change in behavior, and just continues undulating along.

      One might ask, "OK, what do you mean we average the parts of the thing." Well, what happens when a beam is observed in a cloud chamber or we see splats of light on a piece of paper caused by some flying thing? We look at the center of the trail or locate the center of the splat and declare that is the things position in time and space. Thus we get the average of a position in time and space.

      Now one might refute the concepts illustrated here. The point is that differing models along with real measurements and descriptive mathematics will yield differing interpretations. Some will work together better than others. We then have a better chance of understanding the world around us.

      Hello Al,

      I love your conclusion section:

      Math is a language that can describe reality. When used to describe reality there should be a model accompanying the math. The model and math should be compared to reality to test the validity of the math and model. If these three things are not in harmony, we do not understand the phenomena and need more research.

      I totally agree and it is amazing that the interpretation of quantum theory has remained unresolved for so long.

      I have a great sense that Albert Einstein was outvoted and sidelined in the quantum theory debate when all along the issues that troubled him were totally valid.

      I hope you will get a chance to read 'Solving the mystery' where I have tried to provide a descriptive model that matches reality.

      Regards

      Richard

      I understand that Einstein, Schrodinger, and deBroglie believed there was something real about the equation and all were "outvoted". I suspect they were outvoted by a particular individual that had power and had his own ideas about the reality of the universe. It is incomprehensible to me that the three men that really made it all work were pushed aside. I get the feeling that if you are right you don't count. I have studied the history of physics and have found this common. I feel for people like Avagadro and Carnot. I am going to read "Solving the mystery."

        Dear Mr. Schneider,

        Your comment about Einstein, Schrodinger, and de Broglie echoes the introduction to my own essay, "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory". I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What's more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

        But questioning the foundations in this way is considered heretical, and is unpublishable in physics journals.

        Alan Kadin

        8 days later

        Dear Mr. Schneider

        Being myself involved in model theory, I read your article with great interest. And I fully agree with you that the understanding of physics is not exclusively a question of math. On the contrary, here in France and in Europe, many of us are grateful especially to Richard Feynman for his way to revolutionize the pedagogy of physics.

        However, if the understanding of physics is not exclusively question of math, the latter plays an essential role in the elucidation of physical reality, and at this level, there is a great mystery remaining until further notice.

        You say in your essay that math is a language like English. It is true that math, like English, can describe things that do not match reality, and that only experience determines if mathematical or English proposals correspond to the part of reality we try to describe. But I think that math has "something" than English or Spanish or Nipponese etc. do not have, and that this "something" plays an essential role in the relationship between math and physics.

        Let us consider cartography, a domain on the periphery of physics. Of course, we can try to replace a map by a description of the area in question in English, Spanish, Nipponese. But the semi-offshore amateur sailor I am - I do not like the term "yachtsman" - prefers a map. On a map, rather basic mathematical tools allow me to establish and correct my course, to determine my position etc. This would not be the case with an even meticulous English description of the waters where I am sailing. Being a projection of a spherical surface on a plane, a map always involves deformations. But mathematical tools depending from the adopted type of projection - in non-polar navigation it is usually the Mercator projection - allow to effectuate the necessary corrections. Here again, English or Spanish alone would not be of much help.

        On the other hand, in order to establish not a simple picture but a real map, the cartographer in turn needs mathematical tools, including differential geometry beginning with the famous ds, even if this ds does not exist in the physical reality.Let us now a bit of science fiction and imagine space-time relativistic cards destined for future and hypothetical space vessel diameter pilots traveling at a speed close to c. The establishment of these maps, necessarily computerized to be quickly accessible, requires the use of the imaginary number i, i2 = - 1, although the latter, by definition, in turn has no real existence.

        For all these reasons I think that math have "something" that other languages do not have. Regarding the nature of this "something", there is a great mystery, and this mystery is in my opinion the main motivation for the present contest.

        In my own paper "A Defense of Scientific Platonism without Metaphysical Presuppositions" (posted Feb. 25), I try - among others - to show that modeling of mathematics and modeling of physics by mathematics meet in both cases objectively given constraints and that this issue differs math from any other language and any other modeling procedure. In this connection my own paper defends a Platonic perspectiveand therefore a metaphysical position. But I try to show that any form of anti-Platonism is neither less nor more metaphysical than Platonism and that the latter is finally simpler and needs less (metaphysical and other) assumptions than its challenging theories, especially regarding the difference between existence within material reality and mathematical existence.

        Your paper and mine diametrically opposed positions express. But boths papers have a common denominator: they refer to the concept of modeling. Since the purpose of this contest is a constructive confrontation of ideas about a mystery where, in absolute terms, no one may be right, I would greatly appreciate to know your comments about my own essay.

        I would also like that this discussion could be continued for further, even after the closing of this contest.

        With best regards

        Peter Punin

          Peter Martin Punin:

          After reading this post and your essay, here is a few words that jump out at me.

          "For all these reasons I think that math have "something" that other languages do not have. Regarding the nature of this "something", there is a great mystery, and this mystery is in my opinion the main motivation for the present contest."

          I can explain this mystery. The many believe that math is the great and perfect descriptor of the all. Since it clearly fails, the many are dismayed, perceive a mystery, and search for some deeper meaning. There is none.

          In all of history, great leaders have gathered thousands of men and gone to war. Thousands on both sides in many battles have died. The great declare themselves the winner. Have those that died on the right side also won? It all depends upon one's point of view.

          All I am saying is that dependence on math does not automatically yield a correct description of reality. I am not questioning how it does it. Whether it is good or bad depends upon one's point of view.

          I have read your response and attempted to read your essay several times. Your education is superior to mine. I cannot extract understanding from the sophisticated language you use.

          Al Schneider

          Dear Al,

          You have presented the theme for this year's essay in a very entertaining and very informative way. In fact, if the remaining of the essay was as interesting and factual as the first five and a half pages, your essay would have been one of those to beat in this year's competition.

          The way you started by pointing out the scenes as the drama has unfolded was really good. Following that, your use of the parable of the "The King's Roses" to illustrate why math formula must correspond to reality makes the point clearly. This will spare us the various negative mass and negative velocities plaguing our physics today, which are the aftermath of the spell cast on our physics from Copenhagen.

          However, the euphoria I had from reading the early pages, reduced somewhat when in my opinion you went too deep into speculation in the latter part of the essay. For example, some things to ponder, if photons travel at c and are never at rest, how can they form a particle? Do they have an attraction force between them to bind them into a stable form? Your assumption does not take into consideration that light may be more of wave than particle. I will not tackle you on special relativity, which in my opinion you tried unsuccessfully to model. Special relativity cannot be modeled because it is not real, but as that is not the main focus of your essay, let me spare you the facts proving this (unless you want to discuss it on another forum on this site).

          Finally, under your Caveats, points of zero dimension are not a certainty. Indeed, in my 2013 essay I discussed the historical arguments between the Pythagoreans, Proclus and Aristotle on the one hand who held that the point had some minimum irreducible dimension and Plato who held that it was of zero dimension but not a "geometrical fiction". I continue on the same path in my essay this year, where I show that an infinite number of points on a line results in the kind of absurdity you demonstrated with your tale of the roses. I am sure you will like it because it contains things you have yourself pondered over and discuss in your essay. As a teaser, how do you propose to cut the kind of line on Deepak Chopra's hand as you mentioned, if the number of points are infinite and points are not things that are 'cuttable' into parts by definition?

          All the best in the competition.

          Akinbo

            Mr. Ojo:

            Thank you for responding to my essay. I am honored. I think you have pointed out that the beginning of my essay is on track. In viewing the second half you have pointed out what you think are flaws. In reaction to that, I would like to point out that when I talk of the things that are not modeled correctly, one would be the wave theory of light. In writing the paper I had considered addressing special relativity or the travesty of present understanding of light. I chose special relativity. I expect responses like yours and value it. Incidentally, the criticism of zero point particles was shown me by the director of a physics department of a significant university when I presented my idea to him. He didn't like my idea either.

            I will go take a look at your essay.

            Thank You

            Al Schneider

            Before I go however, I must point out that I wrote a book titled "New Age Quantum Physics" available on amazon dot com. It addresses the questions you have posed.

            20 days later

            Some time has passed. I have reread your essay. I am even more impressed. I believe your comment, "But questioning the foundations in this way is considered heretical, and is unpublishable in physics journals." is true. Then I believe, when the truth is out, there will be an earthquake in the physics community.

            12 days later

            Dear Al,

            I think it is impossible to disagree with your conclusion :) I don't think though that there exists presently any reason to rule out that reality ultimately is mathematics and the model that you speak about is too, so that we are just looking at self-similarities. I really like the focus of your argument though, it is well done. I wish you good luck with the contest,

            -- Sophia

              Dear Al,

              Your essay makes several sensible points to which I'd like to give the following comments:

              I agree that mathematics is a language which can sometimes describe things which are not real. It seems to me though that already in everyday contexts which seem rather far removed from mathematics. If you do me a favor and I say that "I owe you one", this does not prima facie have anything to do with math, yet it could be modeled as me being in possession of -1 favors from you.

              I think the real difficulty is that mathematics obscures when we transition from aspects of models that represent real things to those representation that already incorporate some abstraction because the transition is entirely seamless. To me, it is kind of like a slippery slope, in which the first abstractions are easily reconciled with our intuitions but are then followed by transitions to abstractions which are not so easily reconciled thus, Quantum mechanics being a prime example.

              In my opinion, at least part of the problem has to do with the expressive power of mathematics. The more expressive, the more likely it would be to serve as a language by means of which one can express models of reality that closely correspond to our intuitions about it. My own entry to this contest is an effort in this direction which, as far as I know, had not previously been undertaken, and the goal is precisely to find a way of expressing the standard formalism of quantum mechanics in such a way that it no longer clashes (at least as violently) with our intuitions.

              You devoted a substantial portion of your paper to a discussion of special relativity. I do not quite agree that "There is no model used to explain it. The claim is that understanding depends totally upon the math behind it."

              A book that is excellent for understanding it and which uses hardly any equations but only pictures is "Visualizing Relativity" by Lewis Epstein. I highly recommend that you peruse it. In fact, it stimulated me to write a paper on the foundations of special relativity that addresses one problem that his book unfortunately does not discuss, you might find that one interesting as well. It was my entry to the first FQXi essay contest:

              http://fqxi.org/community/forum/topic/329

              A version with minor corrections can be found here:

              http://deepblue.lib.umich.edu/handle/2027.42/83152

              Finally, I would like to mention a couple of corrections:

              "Given that a particle in this photon universe is a cluster of photons, an observer in this

              universe could identify a particle as his frame of reference."

              In a universe in which there are only photons, there is nothing to which a rest frame can be attached i.e. there is no physical object that corresponds to an "observer" (The problem is that observers are associated with a non-zero spacetime intervals i.e. they "age" while photons do not, so you cannot consistently attach a spacetime observer frame to a photon frame). So, while you can argue about the ability to mathematically represent such frames ( which I think is what you mean by "stationary states") it is not physically realizable, just like -400 roses. Different physicists seem to interpret this fact differently (an indication that this is not yet settled science). My view is that a universe with only photons in it is not a 3+1 Minkowski spacetime. Regardless of how you interpret this, I think any argument based on rest frame observations in a universe in which there are only photons is highly suspect, to say the least.

              "Then, present science today does not contain the concept of colliding photons."

              No, it does. The higher the energy of a photon, the more particle-like its properties. For gamma rays, the energy is sufficient that you can model photon-photon collisions. See, for example,

              http://en.wikipedia.org/wiki/Two-photon_physics

              What I see in your work is a call for taking a common-sense approach to modeling reality, in the sense of the opposite of what might be called a "blindly mathematical" approach, and with that I wholeheartedly agree. I hope you found my comments useful.

              Best wishes,

              Armin

              PS. I am a fellow alum of WSU (pharmacy) but I graduated quite a bit later than you did, and now I'm at U of M, completely immersed in physics, philosophy and mathematics.

                Dear Armin,

                Sorry for the length of this. I wrote a letter that occurs at the end of this post. Then, I went over your post line by line. Here are my thoughts.

                I am rewarded by your initial comments.

                [If you do me a favor and I say that "I owe you one", this does not prima facie have anything to do with math, yet it could be modeled as me being in possession of -1 favors from you."]

                Are we keeping score?

                [You devoted a substantial portion of your paper to a discussion of special relativity. I do not quite agree that "There is no model used to explain it. The claim is that understanding depends totally upon the math behind it."]

                This is not my claim. It is the claim of the literature I have read. That may well be limited. It is also the opinion of the Ph. D. physicists I have talked to about this subject.

                [A book that is excellent for understanding it and which uses hardly any equations but only pictures is "Visualizing Relativity" by Lewis Epstein. I highly recommend that you peruse it.]

                I did a search on it. Lot of confusion on the web about it. Finally found it for sale on amazon. Now I need to decide if I want to buy it. New, it is pricy.

                [So, while you can argue about the ability to mathematically represent such frames ( which I think is what you mean by "stationary states") it is not physically realizable, just like -400 roses. Different physicists seem to interpret this fact differently (an indication that this is not yet settled science). My view is that a universe with only photons in it is not a 3+1 Minkowski spacetime. ]

                I object to this. Bad English using something the reader is probably not aware of. Good writing would require naming the source as you have but demonstrate how the material applies to the subject at hand. I have successfully written a hundred papers on a wide variety of subjects that have been accepted by the appropriate audiences around the world. I am renowned in those industries. I do not use ambiguities such as this to impress my audience.

                [Regardless of how you interpret this, I think any argument based on rest frame observations in a universe in which there are only photons is highly suspect, to say the least.]

                This subject depends on how you define observer. In the context you select, (note that you select the frame of discussion then say, regardless) the observer appears omniscient. This is a drastic error. Note that I propose that the only material in the universe is photons. That includes the observer as well. In that context I propose an observer would be a particle consisting of whirling photons. Then, observation consists of photons moving between whirling clouds of photons. One observes using photons only at the position where one resides. To be able to observe the motion of photons at a distance is to be an omniscient observer. Any desire to be omniscient blows it all apart. Understanding this took me years to understand. I don't expect you to grasp it in a moment.

                ["Then, present science today does not contain the concept of colliding photons."

                No, it does. The higher the energy of a photon, the more particle-like its properties. For gamma rays, the energy is sufficient that you can model photon-photon collisions. See, for example,

                http://en.wikipedia.org/wiki/Two-photon_physics]

                Here I believe you are right. I did not want to open the discussion about photon collisions in the little bit I presented in my essay. I did not believe that suggestion would be acceptable. I did not want to ruffle too many feathers. I expected my presentation to be a hard sell as it was. Perhaps I was wrong.

                And now my initial letter.

                Dear Sir:

                I truly appreciate your post. Time will be required to study it. I scanned one of your documents quickly and noticed a reference to general relativity. I hope that is not used heavily in your arguments for I find the theory wanting. I find the concept of spacetime confusing and cannot wrap my mind around it. I suspect it is one of the victims of misapplying math models. While I believe special relativity is one of the most solid theories in existence, I find a flaw in general relativity. I see this when applying Feynman's QED to the meaning of light as it passes through space. Understanding his concepts crushes the expansion of space and warped space. I have published a video on YouTube addressing this issue. https://youtu.be/p0E5AWnYjys "Space is not Warped." The point is that photons move in straight lines from electron to electron. While a photon is interacting with the electron, gravity affects the electron and thus the path of the photon through it. However, when the photon is not interacting with an electron of matter, it is unaffected by gravity. Hence, general relativity is not wrong but its interpretation is slightly off.

                I must add my feelings to the presentation of special relativity in my essay. It only touches the significance of its meaning. This was not an idle study. I spent about two years examining the bit presented in the essay. I spent about three years attempting to explain my conclusions about that. With what is offered there, the math of special relativity unfolds. I believe if one follows the thread one encounters an explanation of what time and distance are. Then, a definition of energy and mass follows. From that the equation E = mcc falls out with ninth grade math.

                At one point I got out my Modem Physics book from college days and, beginning with chapter one, I paged through the book. With the turn of each page, the concepts that unfolded in my mind explained what I saw on the pages. As mentioned elsewhere, I have produced a book titled, "New Age Quantum Physics." It sells on amazon dot com. It details my path through this subject.

                Well, I guess I got carried away.

                I sincerely appreciate what you have offered and promise to study it.

                All the best.

                Al Schneider

                Dear Sophia,

                Thank you very much for your kind comments. However, I do not expect to place in this contest. The material presented by those here is of such a caliber and from such distinguished individuals, I feel small by comparison. I am very happy to present my ideas with the simple idea that people like you will read it and contemplate it a bit. In this, I am very honored.

                Al Schneider