Of Geometry And Worldviews

Believe it or not, mathematics is a gateway for some pretty wild stuff.  Oh, it’s true that every mathematical system has order to it, and, therefore, predictability.  In fact, that’s what draws folks like me to it: the orderly aesthetic of a world that makes sense.  (Now, if you saw my messy house, you might challenge my self-avowed love of order, so I should clarify that it’s mental order in which I dwell!  Physical order is indeed beautiful, too, although thus far it’s been an elusive goal for me.  But I digress…)

Anyway, if you dip your toe past the shallow waters of grade school arithmetic, you may be surprised at the other-worldly creatures you will find swimming in the deep end.  That’s why I called mathematics “wild.”

Mathematics must be built progressively, like a house.  Admittedly, I am no contractor, but even I can understand the basics: first you start by pouring a foundation.  You’ll need the right set of tools and some sturdy, consistent building materials.  You can’t just start haphazardly throwing down row after row of brick.  The angles and placement have to be just right, especially in the beginning, because that will set the shape of the structure that will result.  Mathematics is no different.

Merriam-Webster defines an axiom as “an unprovable rule or first principle accepted as true because it is self-evident or particularly useful.”  Also called a postulate, it is accepted without proof.  Once you lay down your axioms and definitions, you can then use them to prove theorems, and then you can use those theorems to prove other theorems.  And so the house is built.  Whole structures of mathematics start from these unprovable but self-evident axioms and definitions.

If you’ve never taken a math class after high school, my guess is that the mere mention of the words “theorems” and “proofs” transports your mind back to geometry class.  Proofs and theorems are actually the building blocks for all kinds of different mathematical structures, including number theory and its simplification into the arithmetic with which we’re all so familiar.  But let’s hang out in our geometry house for a while and have a little fun.

Did you know that there is more than one kind of geometry?  The kind that you probably imagine, the kind typically taught in high school, is Euclidean geometry, named after the ancient Greek mathematician, Euclid.  In his mathematical treatise called “The Elements,” written circa 300 B.C., Euclid collected 23 definitions and 5 postulates, or axioms.  Euclids “Elements” rendered him the father of the most well-known geometry of the past two thousand years.  Take a look at his first four postulates, or axioms, and see if you agree that they are pretty straightforward and self-evident:

1. A straight line may be drawn between any two points.
2. A piece of straight line may be extended indefinitely.
3. A circle may be drawn with any given radius and an arbitrary center.
4. All right angles are equal.

The fifth postulate has an interesting history.  I’ll let curious readers research it for themselves.

So what kind of geometry can be constructed with these postulates and definitions?  Well, a Euclidean plane is flat.  It exists in two dimensions.  Think of length and width, or imagine a sheet of paper and then imagine it extending out infinitely in both directions.  As we have stated, Euclidean geometry is built on a certain set of axioms and definitions.  Euclid’s Definition 23 states, “Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.”  Again, that seems pretty straightforward and familiar.  Parallel lines don’t intersect.  Okay.  Easy peasy.  Got it.

But now, haha (oh, yes!)…now let’s have a little fun.

What if we change our axioms and definitions so that “parallel” lines can intersect?

This will lead to a whole new animal.  Ready to meet one of those other-worldly creatures in the deep end?  Let’s try!  First, imagine a geometry where the shortest path between any two points is a curved line.  Imagine that there is actually more than one shortest path between those points!  Imagine a scenario where the sum of the interior angles of a triange exceeds 180 degrees.  Now, don’t get too excited (or freaked out, as the case may be).  We actually haven’t gone very far into the deep end, and we haven’t encountered a very exotic creature at all.  You see, all you have to do is imagine a globe.  Spherical geometry will illustrate my point quite sufficiently and yet still be familiar enough to feel rather comfortable.

Think of the lines of latitude and the lines of longitude.  

Would you consider lines of latitude to be parallel to each other?  How about lines of longitude—would they be parallel to one another?  Can you call these things lines at all?  It all depends on your definitions and axioms.  That’s my point.  It’s not hard to imagine parallel lines intersecting if we allow lines of longitude to qualify as parallel lines and consider that they intersect at the north and south poles.  Now, the punctilious mathematician will notice that I am not speaking strictly of standard spherical geometry, in which straight lines are replaced by geodiscs, parallel lines still cannot intersect, and there simply are no parallel lines.  Spherical geometry is indeed an interesting creature, one that turns out to be quite tamable and willing to harness its powers to benefit us with practical applications in navigation and astronomy.  But the purpose of this blog post is not to teach about spherical geometry, or even of exotic creatures like hyperbolic geometry, in which a line has two parallels and an infinite number of ultraparallels through a given point.  Rather, this blog post is about the consequences of choosing different axioms.  Your axioms will determine the entire structure that you can build using them.  Throw out Euclid’s parallel postulate and you have a whole new geometry!

So how does this relate to worldviews?

Looking at my word count, I see that this post already should take more than five minutes to read!  I’ve also run out of time to write the rest of it.  (Oops!)  So I’ll have to leave you hanging.  Please stay tuned for a continuation in the future!  But in the meantime, here’s a teaser for you, for your pondering pleasure:

On an intellectual level, what kind of axioms underlie the way you see the world?  How would your worldview change if you replaced one of those axioms?

On an emotional level, what are the axiomatic beliefs by which you live your life?

Feel free to comment below!