To Infinity... and Beyond!

"Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity—distance is incomprehensible and therefore meaningless." -Douglas Adams, The Hitchhiker's Guide to the Galaxy

It has been my experience that very few people spend time thinking about infinity.  They are hardly to blame; as a mathematical concept it is relatively new and mostly ignored at levels lower than calculus.  And this idea barely exists outside of the theoretical realm.  I mean, let's face it, most of the things we describe as "unlimited" are anything but: just look at the recent law suits surrounding cell phone companies and their so-called unlimited data plans.

I believe that this is largely due to the fact that everything we see around us is just the opposite of infinite.  The physical world seems bound by the immutable law of finiteness.  Our money, our time, the population of earth, the number of stars in the galaxy, even the number of atoms in the universe.  All are limited, all quantifiable, all finite.

And in all our experience, we start to think that this must be all there is, that everything in the entire universe must be bounded.  We start to doubt the infinite, and all of a sudden existence starts to seem, well, rather small.  But there are still some out there who claim that infinity is not just a whimsical artifact of mathematical theory.  The Bible itself rails against the notion of existential delimitation: that is to say, the notion that a thing must be finite in order to exist.  In order to describe the deity pertinent to its narrative, this book throws around words like eternal and everlasting.  These words are also used to describe the kingdom that is coming to this earth, and which is inextricably tied to God Himself.

Missing the Forest for the Trees

But how can this be?  Surely anything that claims to be infinite, even in duration, must be completely imaginary.  Only in theory can something truly be boundless.  In fact, I am writing this blog post because I was asked a question very similar to this.  How can someone logically believe in eternal life, or eternity at all?  Now, I do not claim to have a perfect answer to this question, but I would begin my answer by pointing out the practicality of infinity.

We are given to think that nothing around is is infinite, but in that we are mostly incorrect.  It is perhaps true that mankind can never fully understand the concept of infinity, but we can certainly do a little better than this.  Think about all the examples I stated above and you will begin to see the error in our thinking.  What is the one thing that money, population (I'll get to time shortly), stars, and atoms all have in common?  They are all discrete.  Now, do not confuse my meaning.  The mathematicians among us will recognize that this is not discreet, meaning "modestly unobtrusive" (Dictionary.com), but rather the idea of individual parts of a whole.  Each of these examples is one piece of many, and of course there can only be a finite number of them in existence.

Such a digital world we live in!  This way of thinking has even pervaded our sense of time, such that we are only able to conceive of it in terms of seconds and hours and centuries.  And to all these there is surely a limit.  But to a certain group among us, there is another aspect of time which entrances us, which drives us mad for our lack of full comprehension.  This quality which so intrigues and so vexes is what I can most accurately describe as continuity.

Infinity Contained

In my time as a university student, I have taken several different classes which involved not a small amount of mathematical modeling.  And if there is one thing such modeling will impress upon a student, it is how nearly futile the idea itself is.  As George E.P. Box said, "All models are wrong, but some are useful."  Certainly, some processes, like the effect of a single force on a single body, can very nearly be approximated by simple mathematical prestidigitation, but these cases are rare, and they even contain a marginal amount of error.

The reasons for this inaccuracy are many, but I wish to press one in particular upon the reader.  This is the continuity of certain phenomena.  If time were really, in essence, made of tiny, discrete pieces, then there would be a perfect, divisible (i.e. digital) mathematical model for nearly every system and process on earth (or at least a great many more than there are now).  To avoid going too in-depth suffice it to say that this has much to do with the properties of differential equations, which are central to mathematical modeling but are usually impossible to solve with the information provided by nature.  What is done in practice many times is that the whole of something is broken up into smaller pieces, which can be individually analyzed.  This "discretized" model is often adequate for many applications in control theory and physics.  Thus we have studies such as Finite Element Analysis and Discrete Mathematics.

At any rate, the point is that time and other phenomena are not truly discrete, and therefore they can cause problems in algorithms based on discretization.  This can also cause major problems in digital instrumentation: digital machines are capable of reading a signal at certain time intervals—whatever happens in between two different readings is lost.  And because of their discrete natures, these models and machines are fundamentally incapable of ever providing a perfect picture of nature.

What I am attempting to explain (in a far more circuitous way than I first intended), is that time, when considered as an entity, is infinite.  Consider the space of a second: there are infinitely many individual moments contained in this "finite" set of time.  Or think about a ruler.  You can divide an inch into two half-inches, and each half inch into two quarter-inches, and each quarter inch in half again.  The space which resides inside an inch on your ruler is, in a very real sense, infinite.  Instead of all the atoms in the universe, let us consider the mass of only one.  No matter how small an atom might look, it is conceivable that its "mass" (whatever that means) can be divided into infinitely many "pieces."  Even in light of the size of the universe, the difference between empty space and a single atom is infinite, in almost every meaning of the word.  How vain of us to package up the infinite in our tiny, comprehensible units so that we can pretend it does not exist!

Ad Infinitum

I was going to spend some time discussing how the sheer practicality of infinity in the world of mathematics must lend it some significance, but I feel that this argument would be superfluous and hardly impactful after the discussion above.  Suffice it to say that the concept of infinity is nearly as useful to mathematics (especially at and above the level of calculus) as is the concept of zero (which, coincidentally, could be called infinity's multiplicative inverse).

Now, this argument does very little to prove the existence of something which is innumerable (or perhaps indivisible), but I believe it goes a long way in suggesting that infinity can and does exist in the physical universe, especially with regard to unity and continuity.  In this respect, one of the most important concepts in all of Scripture is the idea that God is One. (Deuteronomy 6:4, Mark 12:29)  In this sense, God at the very least contains infinity, even if it doesn't say that He takes up an infinite amount of space and/or time.

Perhaps I will strive to show in another post that these ideas too are true (or at least possible) for God.  For now, however, I think this discussion should suffice.  I hope I did not bore my readers; at any rate, this was an important idea for me to put into writing.

"How great you are, Sovereign Lord!  There is no one like you, and there is no God but you, as we have heard with our own ears....  You have established your people Israel as your very own forever, and you, Lord, have become their God. 

"And now, Lord God, keep forever the promise you have made concerning your servant and his house.  Do as you promised, so that your name will be great forever.  Then people will say, 'The Lord Almighty is God over Israel!'  And the house of your servant David will be established in your sight. 

"...Now be pleased to bless the house of your servant, that it may continue forever in your sight; for you, Sovereign Lord, have spoken, and with your blessing the house of your servant will be blessed forever." -2 Samuel 7:22, 24-26, 29