By N.S. Palmer

Sometimes, we don’t know what we’re talking about.

Our everyday words and ideas work just fine for getting things done and understanding our normal world. But what about for understanding God?

The ancient Greeks had it easy. They believed that Zeus had a physical body and lived on Mount Olympus. He was just like a human, except for being immortal and more powerful. When they talked about Zeus, it was mythology but it made perfect sense.

We, on the other hand, do not have it so easy. We believe that God is infinite and that He transcends our universe. He doesn’t have a physical body. Unless we resort to crude anthropomorphisms like the Greeks did with Zeus, we can’t form a concept of what He is like. When we use the word “God,” to a large degree, we don’t know what we mean.

The inevitable conclusion is troubling: In terms of our normal world, statements about God are meaningless and unprovable. But sometimes they are also true.

The same applies to any other statements about transcendent realities.

Therein lies a seeming paradox:

*If such statements are meaningless, then they don’t say anything that could be true or false.*

*If such statements are unprovable, then we can’t know if they are true or false.*

So why do most of us think that such statements mean something? Why do we think that we can know if they are true or false?

Even atheists and materialists, in order to claim that such statements are false, must believe that they are meaningful and that they assert facts.

Whether we believe or not, almost all of us have an inchoate sense that statements about God are meaningful and important. Why is that?

I have a hypothesis. It’s meaningful but unfortunately, it’s not provable. All I can do is offer it for your consideration.

### Two Statements About Euclidean Geometry

Let’s consider a relevantly similar example. Euclidean geometry is the geometry of normal space. It doesn’t apply over interstellar distances, at great speeds, or at the sub-atomic level. But it applies in our everyday lives. If you pick up a pen off the desk, you’re doing it in Euclidean space.

Euclidean geometry got its name from Euclid of Alexandria (ca 325 – 265 BCE), who wrote *The Elements* as well as numerous other books, most of which have been lost.

Though Euclid’s name is on the book, most of the geometry in *The Elements* wasn’t new: people had already been solving geometric problems for centuries. Euclid’s contribution was to consolidate and systematize earlier knowledge, add some new insights of his own, and to give *proofs* of geometric ideas that had previously been taken on faith. He also went beyond geometry into number theory, for example, giving the first known proof that there are infinitely many prime numbers.

So let’s examine a couple of true statements about Euclidean geometry:

- The sum of a triangle’s internal angles equals 180 degrees.
- Euclid of Alexandria wrote
*The Elements*.

### What Euclidean Geometry Is

Before going any further, you need to understand what Euclidean geometry actually is. We’re talking about the system defined by Euclid, not about the space it describes.*

Euclidean geometry starts with some primitive, undefined ideas (point, dimension) and some logical rules. It uses the logical rules to combine the primitive ideas, thereby generating new ideas and proving things about them.

For example, suppose you start with the idea of a point and the idea of a dimension. You have an intuitive sense of what those ideas mean, but you can’t define them because they are fundamental. There is nothing more primitive that you could use to define them (at least, without circularity). Euclid defines a point as being “without extension,” but that’s just an indirect way of using the idea of a dimension.

So you’ve got a point and you’ve got a dimension. What happens if you move along the dimension and add another point? Then those two points define a line segment. If you extend the line segment infinitely in each direction along the dimension, then you have a line.

If you add one more dimension, then you can move a little way in that direction and add another line. Those two lines allow you to define a plane. (In fact, any three points define a plane. We’ve got more points than we need here.)

If you add one more line in the plane, and none of the lines are parallel, and no point in the plane is in all three lines, then you’ve got a triangle. Each line segment is rotated a certain distance from the others in the plane. We measure that distance in “degrees,” so a complete rotation is 360 degrees. The rotational distance between two intersecting lines is the angle between the lines.

Now, we’re ready to prove our first statement: *The internal angles of a triangle add up to 180 degrees.*

Let’s add one more line segment that intersects the top of the triangle at point A. Make the new line segment parallel to the line segment BC at the bottom of the triangle.

Our proof goes like this:

- Draw a line segment DE through point A so that DE is parallel to line segment BC.
- The angle defined by line segment DE is 180 degrees. Notice that if we rotate around point A from D to E, we’ve done half of a complete rotation, so the distance is 360/2 = 180 degrees.
- Angle DAB + Angle BAC + Angle EAC = 180 degrees. Together, those three angles add up to the angle defined by the line segment DE. From (2), that’s 180 degrees.
- By (3), Angle B = Angle DAB, and Angle C = Angle EAC. Euclid previously proved that alternate interior angles of parallel lines are the same. We apply that here.
- Angle B + Angle BAC + Angle C = 180 degrees. By (4) and (3), substituting Angle B for Angle DAB, and Angle C for Angle EAC.

We’re done! The triangle was arbitrary, so our proof applies to any triangle in the Euclidean plane.

So now we know that in Euclidean geometry, the internal angles of a triangle always add up to 180 degrees. It’s true, and we have proven that it’s true.

### Considering Our Second Statement

Now, let’s consider our second statement about Euclidean geometry: *Euclid of Alexandria wrote The Elements.*

Uh … wait a second. Euclidean geometry starts with the ideas of a point and a dimension. It doesn’t say anything about people, Alexandria, writing, or books.

If our “universe” is limited to what’s defined in Euclidean geometry, then our second statement is meaningless. It doesn’t use the primitive ideas with which we started, and it doesn’t use any of the ideas that we derived from them.

A being who lived within the system defined by Euclidean geometry would say:

“Wrote”? “Alexandria”? Can you translate it into points, lines, planes, and angles? Those I understand. That other stuff is nonsense.

In addition to being meaningless within Euclidean geometry, the statement is also unprovable within Euclidean geometry.** There is no way to start with fundamental ideas such as point and dimension, then deduce the existence of a man named Euclid who lived in Alexandria and wrote a book. It’s not happening.

And yet, we know that the statement is both meaningful and true.

### Both Meaningful and True

How do we know that the statement is both meaningful and true?

*Because we don’t live inside the system defined by Euclidean geometry.*

We have access to other knowledge. We see people, books, and cities. We know what it means to write.

### Where We Do Live

But we *do* live (mostly) inside the physical universe.

The things we see, the ideas we form, and the logical principles we use are based on the physical reality around us. They are ill-adapted to talking about or understanding anything “outside” that system*** – including God, the soul, eternity, or even moral laws, to the extent that they transcend the merely physical. As Israeli physicist Gerald Schroeder put it in his book *The Science of God:*

The laws of nature exclude the possibility of seeing outside our universe even if there is an outside. It is a theory that can never be tested by observation. (Loc. 578)

When it comes to statements about God or any other transcendent reality, we’re in a position similar to the Euclidean being. Those statements talk about things that aren’t defined within our physical universe. The system of knowledge derived from the physical universe, and the principles of logic that work within the physical universe, do not apply — or at least can’t be known by us to apply.

### How Do We Know — or Do We?

Given those limitations, why do we sense that statements about God and the transcendent are meaningful? That they can be true or false? That they are vitally important?

I can’t give you an answer that’s provable in terms of this universe, but I can offer a hypothesis.

In 1964, physicists Arno Penzias and Robert Wilson at Bell Labs discovered a relatively faint cosmic microwave background radiation that is now believed to be a remnant of the “big bang” – a huge explosion about 13.8 billion years ago that created the physical universe.****

They were trying to eliminate radio interference with communications satellites, and they discovered it by accident.

The cosmic microwave background is a faint echo of the universe’s beginning. We can’t normally detect it, but if we point the right instruments in the right direction, we can.

What about our own built-in instrument: our own minds? We can’t normally detect any transcendent realities. But scientific evidence shows clearly that our minds have abilities far beyond those we currently understand. The exact nature and causes of those abilities are disputed, but that they do exist is beyond any reasonable dispute.

If we point our minds in the right direction, and through prayer, meditation, or other means we tune them to just the right spiritual frequency, do we detect a faint echo of a reality that transcends what we see around us? Does that enable us to stand partly outside of the physical universe and to perceive beyond it? Is that what Moses and other ancient prophets did?

I can’t answer those questions for you. Neither can atheists, at least not based on evidence of the kind we use to understand the physical world. The questions have no answer within our universe: Just like statements about Euclid’s authorship within Euclidean geometry, such answers are meaningless and unprovable in terms of the physical world.

It comes down to whether or not you believe there’s a kind of “spiritual background radiation” that we can detect at the extreme outer edges of our perception – and if so, what that background radiation means. Logic and normal empirical evidence can’t settle the issue.

I believe that it means the universe is created and sustained by a loving God, an infinite Consciousness, who cares about each and every one of us.

What do *you* believe?

——————————

* I’m giving the essence of Euclid’s system as a modern analytic philosopher understands it. This isn’t exactly how Euclid explained it.

** I was deliberately vague when I said that the two statements were “about” Euclidean geometry. Only the statement about Euclid is *about* Euclidean geometry. The triangle theorem is a statement *in* Euclidean geometry and is *about* Euclidean space.

*** Obviously, “outside” is a spatial metaphor for an idea that can’t be expressed literally in our language.

**** Analogies to “Let there be light” (Genesis 1:3) are obvious but, as Schroeder admits, are unverifiable.

Copyright 2013 by N.S. Palmer. May be reproduced as long as byline, copyright notice, and URL (http://www.ashesblog.com) are included.

Nicely done. And you have to know that anytime God and mathematics are used in the same discussion, I’m going to perk right up.

By:

Jim Greyon September 29, 2013at 9:17 am

Thanks, Jim! I forget who said that God was the first mathematician, but it’s a reassuring thought.🙂

By:

N.S. Palmeron September 29, 2013at 10:41 am

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With Lines and Angles – Euclid – Changed the World – One Person | One Person Changed The Worldon November 12, 2013at 6:24 am