No, The Infinite Book isn’t one of those magical books that never has a final page and you can read it forever. But I must say that it feels like I’ve been reading it for an eternity.
The Infinite Book is, simply, a book about the concept of infinity (∞), its history, and some of the fields which are currently making use (or misuse) of it. The first half of the book is about the concept in abstract and the attitudes of philosophers, mathematicians, and scientists toward it over the years. I found those sections quite interesting while the later sections—considering ideas like “if there are an infinite number of parallel universes then everything that can happen has happened and will happen again and then there is no point to existence”—well, those later sections seemed like mind games completely unrelated to why I find infinity fascinating.
Because I do. I find infinity fascinating. Take the concept of Set Theory, which describes how one can have an infinite set which has infinite subsets inside of it, none of which are smaller than the entire set. For example, the integers, or whole numbers (<— -3 -2 -1 0 1 2 3 —>), are an infinite set. From zero, you can go infinitely in either direction (remember adding the arrows on the ends of your number lines in algebra?). You’ll never reach a highest number beyond which you cannot go. Yet, if I took all the odd numbers in that sequence and made a set from them (<— -3 -1 1 3 5 7 —>), that set would also be infinite. So would a set of even numbers. Yet one would be tempted to say that a set of all the even and odd numbers would be twice as large as the set of even or odd numbers alone. Yet, since all sets extend infinitely, there is no largest (or even larger) set. And you can even unendingly spiral down into the infinities between whole numbers (1.1, 1.01, 1.001, 1.0001, etc.). The set of whole numbers may have a larger decimal total than all the fractional numbers between 1 and 2, but there aren’t more numbers in the whole number set than nest in the fractions between 1 and 2.
If you start letting this worm its way into your mind, a question should eventually arise (at least, it does in my quixotic brain): where do these infinities come from? Or, rather, where do they go? Or where are they now? If, on a mathematical level, they describe our world—they live in it, so to speak—then where do they live? What part of our world are they describing? I mean, a tree isn’t infinite, nor a sandwich or a desk. Or are they? What is infinite in our world? Are these infinities just human limitations—evidences that our observations are inaccurate or that our mathematics aren’t advanced enough?
Okay, I know I’ve lost some people. So let me take this from another angle: say you have two mirrors. You stand the mirrors facing each other and then step between them. In the mirror, you see your face, but you also see a reflection of the mirror behind you, which is also reflecting the mirror in front of you, which is reflecting the mirror behind you reflecting the one in front, and on and on forever. Well, not really. See, due to imperfections in our mirroring technology, we can’t actually make a mirror which reflects light perfectly without scattering some of it. (If you’ve ever tried the above trick with two mirrors, you may have noticed that your image gets blurrier the further back in the reflection you look.) But also, due to the finite speed of light, it would take an infinite amount of time for the light to bounce back and forth to make an infinite sequence of reflections. But say light was instantaneous. And say our mirrors were perfect. Then an infinity would have entered our world in a real, concrete way. There would be something in a limited, finite universe which was infinite. How could that be possible?
Talk about mind games. Who’s playing them now, right? Well, here’s where I think infinity provides some wonderful insights into the nature of God. Relating infinity to God is something that’s been done for thousands of years, so I’m certainly not original in my thought. But imagine those whole numbers again, containing infinite sets (positive and negative numbers) inside of the infinite set of whole numbers. Since we can understand that a set of numbers can be both infinite and distinct, yet part of another set which contains that set and yet is no larger, that gives us one metaphor for understanding the Trinity. God is all the sets, yet there are distinct sets within Him that are not the whole, but yet are no less than the whole.
Another fascinating thought is how an infinite God could incarnate Himself, containing the infinite inside a finite. Or how can a soul, which God has given eternal properties (everlasting existence) be contained in a finite universe? It seems related to the question How could infinite numbers exist in our universe? We use infinities in calculations, we can describe infinity in great detail, yet our universe cannot contain infinity. It’s like Thoreau’s line, extending from beneath the water, to a second line rising into the sky. Infinity passes right through our universe and extends into Something Else. It both resides in our universe and should draw our minds right out of this world and into eternity.
Or take the irrational numbers, like Pi (π). It’s an eternally permutating number: 3.14159~ (and on to infinity, never ending and never repeating). Pi helps describe the circle, something we know exists in our world, yet something whose construction requires a number that never repeats and never ends. How does any perfect circle ever get drawn? You may say that a perfect one doesn’t exist, but that we just approximate. Fine. But since the concept exists, and we know it describes something real, the reality must exist as well. And how beautiful is it that God put a concept humans will never fully understand into the simple elegance of the circle? Doesn’t that also give a glimpse into what eternity will be like? We’ll be constantly learning about God, yet never exhaust Who He is. He is infinite. If we understand that we’ll never, even with the most sophisticated computing programs devisable, know the final digit of the number π, why is it difficult to imagine the inexhaustible character God? He created Pi as a single letter in the immense poem of the universe—and that on a Monday morning before breakfast.
Infinite sets inside infinite sets, indeed.
∞
I got a new CD recently and have been enjoying it more than I have enjoyed any music in quite a while. It’s called The King’s Consort Collection by, obviously, The King’s Consort. Most of their music is late medieval & baroque & classical and done on period instruments with a keen attentiveness to good soloists and a deep love for the music they’re performing. The “Collection” is just that: a sampler of a wide variety of pieces from their over 90 CDs.
