## Math, Thinking Outside the Box-- Post #7

Geometry, similar to my initial feelings about algebra in high school, seemed like such a waste of time for me. It was only later that I realized that it had taught me one of the most valuable intellectual tools ever developed by humans. It taught me how to prove things in life that are often not readily apparent. Geometry taught me the rigors of formal proof. Math, in general, taught me how to think outside the box--outside my often biased perceptions bogged down with limitations of my ability to visualize and relate everything to my sense of reality.

Soon after I graduated from high school, I recall that I was exposed to non-Euclidean geometry and realized the Euclidean verifications were flawed, as were the initial axioms when we try to apply those same empirical measurements on either a cosmic or subatomic scale. My earlier measurements and constructions had only seemed to verify the proofs. The more I had read about the Universe, the more I came to realize that It is the non-Euclidean geometry that is so important for understanding its geometry.

Along the way too, I found Newtonian laws of physics that I had trusted so blindly in high school were also flawed. Cosmology discoveries have shown that the axioms of Euclid and the laws of Newton no longer can explain the geometry and events of the Universe within the cosmic boundaries for which we are now taking measurements. I have to be careful here even using this word “boundaries”, because it is not for certain that our Universe is bounded. To the contrary, it may be unbounded. Furthermore, we don’t know if its contents are finite or infinite either.

In reality, I don’t think many people care how many objects are in the Universe, but we do yearn to know how far out space reaches. My point here is really to express the need for us as humans to assign number to our observations and to assign boundaries to things we can’t even see or measure. It has been said that no science is established on a firm basis unless its generalizations can be expressed in terms of number. Math has in its vocabulary the word “infinite”, but we don’t really feel comfortable with the possibility of its existence as a specific magnitude. That is, it is impossible for us to conceive anything being infinitely small or large. If it were possible to retain either of these quantities before the mind for a moment, it would be just as easy to think of a smaller or a greater magnitude as the case might be.

So you see the intellectual enigmas here. We struggle with the concepts of something being infinitely small (the size of this Universe at the moment of the Big Bang) or something being infinitely large (an unbounded Universe some 15 billion years later). Ironically, we are just as uncomfortable in trying to build fences around these concepts by our attempt at expressing the cosmos in terms of boundaries. Doing so, requires us to think about the existence of something on the other side of these fences. Is this outer world the province of God? Some say this is so. Are there other Universes? Some believe this is a possibility also.

If there is a 4th Dimension (4th direction in space outside our normal 3), could there be a 5th, a 6th and so on? If there are multiple Universes, what is the upper limit of these multiple Universes? Are there ten? Are there a billion? Are there a trillion? At some point, we feel an intellectual need to say we reached the final outer layer. That is the nature of human thinking and it probably stems from the fact that everything in our three-dimensional world has an outer layer that we can visualize.

There is an intellectual need for us to want to assign a particular shape to bounded objects. We say something looks like a sphere, or a cylinder, etc. And if the universe could be expressed in terms of a bounded three-dimensional object, we could say it looked like some familiar object too. If your mind could zoom out far enough, what would be the shape of our Universe?

If the Universe is indeed bounded, most scientists believe the Universe is not a three-dimensional shape. The prevailing thought among scientists and mathematicians (believing our Universe has boundaries) is that the Universe is a higher dimensional shape. Such shapes can only be described in terms of a less familiar branch of mathematics we call

To think "outside the box" far enough to better understand the Universe, we have to understand clearly the hyperspace of which our Universe is merely a subset. My next blog entry will deal with this a little more in-depth.

Soon after I graduated from high school, I recall that I was exposed to non-Euclidean geometry and realized the Euclidean verifications were flawed, as were the initial axioms when we try to apply those same empirical measurements on either a cosmic or subatomic scale. My earlier measurements and constructions had only seemed to verify the proofs. The more I had read about the Universe, the more I came to realize that It is the non-Euclidean geometry that is so important for understanding its geometry.

Along the way too, I found Newtonian laws of physics that I had trusted so blindly in high school were also flawed. Cosmology discoveries have shown that the axioms of Euclid and the laws of Newton no longer can explain the geometry and events of the Universe within the cosmic boundaries for which we are now taking measurements. I have to be careful here even using this word “boundaries”, because it is not for certain that our Universe is bounded. To the contrary, it may be unbounded. Furthermore, we don’t know if its contents are finite or infinite either.

In reality, I don’t think many people care how many objects are in the Universe, but we do yearn to know how far out space reaches. My point here is really to express the need for us as humans to assign number to our observations and to assign boundaries to things we can’t even see or measure. It has been said that no science is established on a firm basis unless its generalizations can be expressed in terms of number. Math has in its vocabulary the word “infinite”, but we don’t really feel comfortable with the possibility of its existence as a specific magnitude. That is, it is impossible for us to conceive anything being infinitely small or large. If it were possible to retain either of these quantities before the mind for a moment, it would be just as easy to think of a smaller or a greater magnitude as the case might be.

So you see the intellectual enigmas here. We struggle with the concepts of something being infinitely small (the size of this Universe at the moment of the Big Bang) or something being infinitely large (an unbounded Universe some 15 billion years later). Ironically, we are just as uncomfortable in trying to build fences around these concepts by our attempt at expressing the cosmos in terms of boundaries. Doing so, requires us to think about the existence of something on the other side of these fences. Is this outer world the province of God? Some say this is so. Are there other Universes? Some believe this is a possibility also.

If there is a 4th Dimension (4th direction in space outside our normal 3), could there be a 5th, a 6th and so on? If there are multiple Universes, what is the upper limit of these multiple Universes? Are there ten? Are there a billion? Are there a trillion? At some point, we feel an intellectual need to say we reached the final outer layer. That is the nature of human thinking and it probably stems from the fact that everything in our three-dimensional world has an outer layer that we can visualize.

There is an intellectual need for us to want to assign a particular shape to bounded objects. We say something looks like a sphere, or a cylinder, etc. And if the universe could be expressed in terms of a bounded three-dimensional object, we could say it looked like some familiar object too. If your mind could zoom out far enough, what would be the shape of our Universe?

If the Universe is indeed bounded, most scientists believe the Universe is not a three-dimensional shape. The prevailing thought among scientists and mathematicians (believing our Universe has boundaries) is that the Universe is a higher dimensional shape. Such shapes can only be described in terms of a less familiar branch of mathematics we call

**hyper-geometry**. If you are like most people, you probably don’t know much about hyper-geometry. If you have had geometry classes, most likely you were taught the definitions, axioms, postulates and theorems of Euclid only. I know, at least, my high school education only included Euclidean geometry.To think "outside the box" far enough to better understand the Universe, we have to understand clearly the hyperspace of which our Universe is merely a subset. My next blog entry will deal with this a little more in-depth.

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## Comments (1)

katiedeamer

Mathematics is the world most exciting subject and it allows you to think out of box. The solution of math problems also available in best assignment writing services by adopting different methods. We learn about many things by solving these tinny puzzles.

May 27, 2017