Welcome to the unique independent viewpoint of John Manimas Medeiros
How the Mathematicians Failed Civilization
Copyright 2009, John Manimas Medeiros
Ages ago, I sat in my Geometry class in Fairfield, Connecticut, alert. I paid attention to Mr. Launer because he was a good teacher. He explained things slowly and clearly when you asked a question. He told us the definition of a circle, which was just as he said at that time, with the precision of integrity, the Euclidean definition of a circle:
"A circle is an infinite set of points, having no dimensions, equally distant from a given point in the center, that center point also having no dimensions."
I immediately understood that this Euclidean circle was "ideal" or "theoretical" and was used for the purposes of pure logic, or geometric logic, or pure thought, but could not be the description of a real, physical object because its defining boundary, the circumference, was comprised of nothing (an infinite set of points having no dimensions).
Zero x N = Zero. I accepted that definition of a circle, but not long thereafter Mr. Launer asked the following question:
"Can we construct a square with exactly the same area as a given circle using only the compass and straightedge?"
He said that the Pythagoreans had asked this question. He paused. And then he gave an answer: "No. We cannot square the circle, we cannot construct a square with exactly the same area as a given circle using only the compass and straightedge." But I do not recall that he offered an explanation. I do remember, and have always remembered, that I immediately rejected his answer in my own mind. I would not openly challenge Mr. Launer on this issue in the classroom, or even discuss this with anyone at that time, but I do remember, and always have remembered, that on that day I concluded immediately that the answer to the question must be "Yes." It had to be "Yes" because the Pythagoreans were smart and they would not have asked this question if it did not have a positive answer. I told myself, privately in my own mind, that I would solve this problem some day. And I did, in the middle of the 1990's. My journey is described here on the jmanimas website and also referenced at my second website pythagorascode dot org. The best and most recent summary is A Child a Particle and Pi. My high school freshman course in Geometry is the only academic course of study in which I received an "A+."
Here is how the mathematicians have failed us: They have ignored the Euclidean or theoretical definition of a geometric "point" on the invalid grounds that it is a "philosophical" issue.
Here is where we are led by the comparison of the "geometric point" with the real world:
Construction of a physical circle: Suppose you and a friend agree to make a circle of bricks to be used for meetings at a campsite. If every brick is exactly the same size, and you cannot use any object that is smaller or shorter than a brick, is the shape a regular polygon, or is it a Euclidean circle?
The bricks are real objects, making a real, physical circle, and any reasonable person can see that the circle you have made is not a Euclidean circle because the circumference is made of "points" or objects that have dimensions. But any real object must be comprised of matter, and must be comprised of "pieces" or objects that possess dimensions. If we accept this reality about physical objects, then we will see that the definition of a circle or circular plane is inseparable from the "atomic" theory of Democritus of ancient Greece, who said that there is a "smallest possible particle" and he called it the "atom."
We have come to hear of many other particles, smaller than the atom, the earliest being the electrons, protons and neutrons. Many have been added, but if one studies physics one may wonder if there are really other particles or just instrumental detections of events or forces that physicists have named as "particles" because they want more in their cupboard. In any case, there is no sound physical evidence that matter as we know it does exist in a form smaller than the atom. The reason I make this assertion is because the Periodic Table of the Elements lists all of the "matter" that can exist independently for any appreciable duration of time. Because we have oxygen, iron, carbon, nitrogen, sulfur, calcium, gold and uranium, and a total of 92 to 96 elements, depending on how you count such elements, we can have or hold an atom (or molecule) of any one of the elements, but we cannot have or hold any piece of any element that is smaller than an atom. Although we can hear about and "see" a picture of a muon or gluon or shish-ka-blip, does not establish that any of these alleged "particles" are in fact independent forms of matter. For practical purposes, when we want to make something, we have to make it with atoms. We have no technology to make a tricycle out of muons or any other blip on the instruments that physicists have invented to substitute for brains.
So, let's get back to the point, which is the definition of a geometric "point" on the circumference of a circle. One can find academically acceptable definitions of the "point" on the "net." Here are a few, at first hinting at the definition of a point, but then getting it clear and correct:
Free Dictionary by Farlex: Definition of a Circle: A plan curve everywhere equidistant from a given fixed point, the center.
Math Open Reference: Circle: A line forming a closed loop, every point of which is a fixed distance from a center point. Tangent: A line passing a circle and touching it at just one point.
Wikipedia (The authority that ate Chicago?)
A circle is a shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the center. The common distance of the points of a circle from its center is called its radius. … The circumference of a circle is the perimeter of the circle (especially when referring to its length).
So one can see that a circle is comprised of "points" but we need more information about what a "point" is. Do you get the point?
Definitions on the web: A point is a geometric element that has position but no extension. A point is defined by its coordinates [two numbers on a graph].
Our campfire circle is now warming up.
Geometry Glossary - ThinkQuest: Point - a zero-dimensional figure; while usually left undefined, has four main representations - the dot, the node, the location, and the ordered pair of numbers [measured coordinates].
The location, and the ordered pair of numbers are consistent with the Euclidean point that has no dimensions. The dot is described as - a description of a point in which the point has a definite size. Therefore, the dot, like the atom, has dimensions or finite size.
The node, an intersection, may be the mathematical land between real and zero. I never knowed a node I didn't like.
Then the Geometry Glossary gets real: Points are zero-dimensional. That basically means that they have no height, length, or width. They are just there.
The Math League: A point is one of the basic terms in geometry. We may think of a point as a "dot" on a piece of paper. We identify this point with a number or a letter. A point has no length or width, it just specifies an exact location.
The website "roangelo" The Philosophy of Geometry, begins to discuss the logical problem presented by "points." This source includes some historical research, the statements of philosophers and mathematicians, suggesting that the mathematicians "have no interest" in the definition of a point. They just use the [ridiculous] definition that it is a location that has no dimensions. Points: "They are just there. … it just specifies an exact location." But, unlike the thoughts of mathematicians, the universe is comprised of physical matter, and physical matter always has dimensions, and -- last time I checked -- ALL physicists believe there is a "smallest possible particle." Therefore, until a physicist demonstrates how to make a tricycle out of muons, or strings, or blips, we are compelled by reason (those who do feel such compulsion) to believe that we are able to make physical objects only out of atoms, the smallest possible independent particle of matter.
And this leads to the reason for and conclusion of all my somewhat massive geometrical work on my website: Mr. Launer was right, and so was I, because if we re-interpret history, the way I am certain it should be re-interpreted, when the Pythagoreans asked the question (which I believe was originally a statement), they were honoring Democritus and amending Euclid. Here again is the ancient question, and the correct positive response:
"Can we construct a square with exactly the same area as a given circle using only the compass and straightedge?"
Of course we can construct a square with exactly the same area as a "circle" because we are referring to a real, physical circle, a plane comprised of atoms, which have dimensions, and therefore the physical circle or circular plane that exists in the real, physical universe has to be comprised of matter, and has to be comprised of atoms, and atoms possess dimensions, and being the smallest possible particle, they cannot be divided further, and therefore the physical circle has to be a regular polygon.
This is the truth. This is reality. The mathematicians have failed civilization because they insisted that their theoretical world is more important than understanding reality. This issue is NOT TRIVIAL or philosophical or semantic. The reality that a physical circle has to be a regular polygon changes the meaning and value of pi, and actually means that in the real, physical universe the value of pi is variable; and it means that the infinite decimal value of pi that is accepted and defended by mathematicians does not exist in the real, physical universe.
Precision is the essence of truth. Precision is the sign of intelligence. The mathematicians of the world have committed the unforgivable offense of imprecision on a matter of grave importance. They cannot wangle out of this. These are the same people who say "the universe is mathematical." If the real universe is mathematical, why do they worship a circle that has a circumference of zero? Do they propose a religious philosophy that the universe is made of nothing? Let's ask "Name a few familiar objects that are circular." This question could be asked of any adult but is more likely to be asked in the first grade. A pie, a quarter, my bicycle wheel, the clock, the Moon, Mrs. Murphy's candy dish. But the mathematicians have never explained publicly, and emphatically, that none of these real objects are mathematical circles. They are not the Euclidean or theoretical circles that mathematicians recognize and discuss exclusively as being circles.
I have at least sixty math books in my home and I have looked at many more. Nowhere has any mathematician taken the responsible and mandatory action to explain that the mathematical circle is not a real object, and that real objects are not and cannot be mathematical circles -- or globes. This is imprecision that discredits math and science. This is what one rightfully calls a "lie of omission." Taken together, "the universe is mathematical" and "we cannot square a circle" do not describe reality. If the original and sacred purpose of science is to have us understand reality, the mathematicians have led people away from the truth, possibly the most important truth.
In your lifetime, both mathematicians and physicists will arrive at this higher level of precision, and accept and adopt the corrected understanding of the ancient question and the positive response, and the corrected definition of pi. Because there is a smallest possible particle, the physical circle is and must be a regular polygon, and we can construct a square having exactly same area as a regular polygon using only the compass and straightedge.
To: A Child (a Particle) and pi, best synopsis to date of why the geometry.
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