What is Einstein's Moon?

In the course of this blog, I have been speaking about this contention of mine, that the split between science and philosophy is focused around who addresses the “how” questions and who deals with the “why” ones. I’ve shown that for the most part, science sticks to “how” and leaves the “why” to philosophy. Lately, in particular, I have been speaking about the scientific method, as to what it contains or allows, as well as what it does not. One of the key elements of the method, is the use of mathematics. Now as we know one of the most basic tools in science, is mathematics. In fact many mathematicians would claim that it is the most fundamental of the sciences, rather then physics. There are others that would say that mathematics is not a science at all, but rather a hybrid of logic. However, the purpose of this post is not to enter this debate. What I would like to talk about is how mathematics itself, to a large extend, mirrors the rest of science in the context of how it forms our attitudes in regards to the discovery of truth in the natural world. To be fair I should say our common view. The example I will use to expand on all this is a mathematically defined shape that all of us are familiar with and that is a circle.

Now to begin, if I asked you to describe a circle, what comes to mind? This would seem to be a easy question, yet if I were to take to the street and solicit answers from a cross section of the populous, I suspect I would get a variety of replies. Some might say, it’s the shape we see when we look at a wheel. Others might say it’s the shape we see when we look at the moon or the sun. The more scientific might give a explanation that is somewhat more technical. In general though, I would suspect that most people would draw on things found in nature or constructed by man to describe it. This said , I would also wager that when you think of a circle, you see it in the abstract, as to what it is without reverence to any physical object. Now if you were to look for a formal definition, what would you find? Well I looked to many sources, including several dictionaries. For instance in all dictionaries I referred to, the one given was almost identical to the following, stated in Webster’s online dictionary as:

"a closed plane curve every point of which is equidistant from a fixed point within the curve"

For a more professional definition one might refer to Wolfram’s Math World (a site I highly recommend), yet in this case it states much the same:

"A circle is the set of points in a plane that are equidistant from a given point."

Now what is the difference between what many of you describe or only hold in your mind and what we find here as a definition. The difference is, in your mind, you hold it as a preformed thing, in its completion or “what” it is. In the definition, what is being described is “how” it is, as to how you would go about constructing a circle. Like the task performed when you use a compass to create one. This definition also, in a indirect way,both implies and relies on another abstract concept and that is the one of infinity. For the points described must be both infinite in number and have no space between them, otherwise one ends up with something that although circle like, would still not be a true circle. So is there another way to describe a circle, which does not rely on a reference to other objects or ill defined abstract concepts?

Before I reveal this other way to describe a circle, let’s first review what I told you were the key “hunches” that science uses to find truth in the natural world. There were two. One was economy and the other was symmetry. Now let’s suppose that we are nature. In this case I will limit us to be nature of a two dimensional world. That is, one that is restricted to a plane or flat land. Now let’s suppose, we as nature, constructed this world to comply with these hunches I pointed out. So if we imposed these hunches as our rules to make real, a basic and fundamental shape for this universe, what could be one that we would end up with? Okay, let’s imagine we say we want a two dimensional form, that always is in proportion, a line of least length to enclose the greatest amount of area . For as I said, we are nature and so therefore we want our basic structures economical in both form and explanation. So what form would we have thus just demanded? Well as I think you will have guessed, if you didn’t know already, it is the circle. Now what about the symmetry part? Remember we defined symmetry as when one takes a thing and does something to it in regards and respect to the allowed levels of freedom (dimensions) and it there after remains unchanged. What can we do to the circle to see if it has symmetry? First, we could just move it around our two dimensional universe. When we do this, ( to no great surprise) no matter to what distance moved, it still remains the same in both form and function. Now what if we rotated it as allowed in this universe, will it change? No it will not. So now we have found by invoking this rule of economy, we have in turn ended up with something that also demonstrates symmetry. Now you might retort, that’s fine for this mystical flat land, but what of the real world. In response, I would remind you that the three dimensional projection or analogue of the circle is the sphere. The sphere described in the way I have just shown, would be a form that always is in proportion the surface of least area enclosing the greatest amount of volume. The economy here is also evident and the tests we imposed to confirm symmetry, would hold the same.

Now you say, so what, for the objects just described are simply abstractions. You continue by insisting they are therefore forms only created in our minds. You could thus feel this should end it all. That would be true, if the circle and its somewhat distorted cousin the ellipse were not so prominent in both the form and action of our natural world. That it is to say, it has proven to thereby have purpose. The other way to state it is, “why” is the circle (or the sphere)? The answer of course is to have a entity(s) that can act in purpose that is both economical and symmetrical in form and function. I thus find it unfortunate, that although science looks to these hunches to discover truth of the world, that it at the same time denies the “why”, that also lends insight into not only the means of it’s construction, but also of its utility.

In summation today, I would bet in the beginning, when I mentioned mathematics many of you thought, oh no, here come the formulas and equations. In contrast to this expectation, as demonstrated above, many might be surprised to find that if they explored both mathematics and science, at the more fundamental level, you may discover it to be simple, beautiful and yet excitingly mysterious. I would ask now, the next time you are given to describe a circle, what will be your reply?

As a foot note to this, let me leave you with a quote from Milton’s “Paradise Lost” that I find projects the spirit of what I have attempted to convey:

From man or angel the great Architect
Did wisely to conceal, and not divulge
His secrets to be scanned by them who ought
Rather admire; or if they list to try
Conjecture, he his fabric of the heav'ns
Hath left to their disputes, perhaps to move
His laughter at their quaint opinions wide
Hereafter, when they come to model heav'n
And calculate the stars, how they will wield
The mighty frame, how build, unbuild, contrive
To save appearances, how gird the sphere
With centric and eccentric scribbled o'er,
Cycle and epicycle, orb in orb.