A Closer Look at Hyper-Geometry..Post #8

In 1962, I purchased a mathematics book at a small bookstore called The Book Nook just outside the campus of Indiana University. It is titled Geometry of Four Dimensions by Henry Parker Manning (Dover Publications, 1956).

In its introduction, Mr. Manning made this statement:

Although it is doubtful whether we can ever picture to ourselves the figures of hyperspace in the sense that we can picture to ourselves the figures of ordinary space, we can nonetheless reason about them. And knowing that the validity of our geometry depends only on the logical accuracy of our reasoning, we can proceed to build upon any figure within our geometric system without waiting for a realization of it. Subsequently, we may in time acquire such facility in handling the geometrical proofs of the theorems. Thereby we are able to state precisely the forms and properties of the figures such that it is almost as if we could see them.


Now I admit, at the time these words had little impact on me. My motivation for buying the book was just because I wanted to better understand what little I had read about Einstein's Special Theory of Relativity. I was confused at the time. Some might argue that I was insane to even want to buy such a book. My confusion was that I was trying to figure out how time could be a 4th dimension in terms of space. I had not yet learned the difference between a spatial dimension and a temporal dimension.

What Manning said in these few sentences is that math, specifically hyper-geometry, allows one to deal with shapes that we cannot see. We can describe the shapes of the hyperspace just as we describe the shapes of plane and solid geometry. We can additionally state properties concerning these shapes, and prove our statements using deductive reasoning built upon some initial definitions and axioms. As an analogy, a blind man cannot see the world in which he must ultimately interact. But through descriptions fed to his brain by others, and through his surviving senses, he is able to almost see the three-dimensional world in which he lives.

In a similar manner, math is our way of almost seeing the higher dimension worlds, including the shape of our Universe. It may even be the key to figuring out how this Universe came about in the first place.

Some people are uncomfortable in dealing with the abstractions of math as our only means of “sensing” that something exists or does not exist. As I mentioned, I had the same problem during my own earlier exposure to abstract math. To help us with these uncomfortable abstractions, we can aid our thought processes by riding on our imaginations. I borrow here Webster’s definition of imagination: “the act or power of forming a mental image of something not present to the senses or never before wholly perceived in reality”.

Imagination is a powerful tool for opening our perceptions and widening our realms of understanding. Einstein had revealed during an interview one time that much of his reasoning and theory (behind his general and special theories of relativity) stemmed from imagining what it might be like if you could “catch up with a light beam”. Of course, we know you cannot, but by allowing himself to mentally achieve such a feat, he was able to provide some of the most insightful reasoning ever introduced. Taking this mental journey enabled him to look at space and time with a whole new perspective and then to express his “observations” with math equations. Later most of his equations were tested and verified against real world observations. One must not lose sight of the fact that it was imagination that provided that fresh perspective.

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