This post is a re-run of a post that originally ran in August, 2018.
This post is part of a series of posts on Pi, infinity, and other things mathematical. Click here for the series.
Illustration from an argument on infinity in Johann Georg Locher’s 1614 book, Mathematical Disquisitions.Why might a person see great intrinsic value in mathematical knowledge? Why might a person be a “math nerd”? Consider this lengthy quotation, from a piece written by Fr. Ron Rolheiser that was published in The Record (the newspaper of the Archdiocese of Louisville, Kentucky) some time ago,* in which Fr. Rolheiser reflects upon the words of a woman who had gone through a profound spiritual experience:
What she remembers most and most wants to share with others is this: “I learned that God is very close. We have no idea how close God is to us. God is closer to us than we ever imagine!” Her experience has left her forever branded with a sense of God’s warmth, love, and welcome, but what’s left the deepest brand of all inside her is the sense of God’s closeness. I was struck by this because, like millions of others, I generally don’t feel that closeness, or at least don’t feel it very affectively or imaginatively. God can seem pretty far away, abstract and impersonal, a Deity with millions of things to worry about without having to worry about the minutiae of my small life. Moreover, as Christians, we believe that God is infinite and ineffable. This means that while we can know God, we can never imagine God. Given that truth, it makes it even harder for us to imagine that the infinite Creator and Sustainer of all things is intimately and personally present inside us, worrying with, sharing our heartaches, and knowing our most guarded feelings. Compounding this is the fact that whenever we do try to imagine God’s person our imaginations come up against the unimaginable. For example, try to imagine this: There are billions of persons on this earth and billions more have lived on this earth before us. At this very minute, thousands of people are being born, thousands are dying, thousands are sinning, thousands are doing virtuous acts, thousands are making love, thousands are experiencing violence, thousands are feeling their hearts swelling with joy, all of this part of trillions upon trillions of phenomena. How can one heart, one mind, one person be consciously on top of all of this and so fully aware and empathetic that no hair falls from our heads or sparrow from the sky without this person taking notice? It’s impossible to imagine, pure and simple, and that’s part of the very definition of God. How can God be as close to us as we are to ourselves?
The woman had approached Rolheiser spontaneously, wanting to communicate God’s closeness. Rolheiser, in contrast, finds it hard “to imagine that the infinite Creator and Sustainer of all things is intimately and personally present”.
Infinity is a mathematical concept of sorts. It cannot be directly expressed, but it can be studied by a certain process of imagining a progression of simple calculations. For example, consider simple division, say 100 divided by 2: 100/2 = 50. Now let us divide 100 by a larger number, such as 5: 100/5 = 20. Dividing by a larger number (5 rather than 2) yields a smaller result (20 rather than 50). Now let us extend this to dividing 100 by progressively larger and larger numbers: 100/10 = 10; 100/50 = 2; 100/100 = 1; 100/500 = 0.5; 100/1000 = 0.1; 100/10,000 = 0.01. Dividing by progressively larger and larger numbers yields progressively smaller and smaller results. So while we cannot actually divide 100 by infinity, we can imagine that as we increase the dividing number more and more, the result will be smaller and smaller still; thus we can say that as the dividing number heads up toward infinity, the number resulting from the division heads down toward zero. And thus in some sense, 100 divided by infinity is zero.
Now let us go the other direction, and divide 100 into some other number. Let’s divide 100 into 20: 20/100 = 0.2. Now let’s divide 100 into something larger, like 400: 400/100 = 4. Let’s keep dividing 100 into larger and larger numbers: 1000/100 = 10; 40,000/100 = 400; 10,000,000/100 = 100,000. Dividing 100 into progressively larger and larger numbers yields progressively larger and larger results. Thus while we cannot actually divide 100 into infinity, we can imagine that as we divide 100 into increasingly larger numbers, the result will be larger and larger, so that we can say that as the number being divided heads up toward infinity, the number resulting from the division also heads up toward infinity. And thus in some sense, infinity divided by 100 is infinity. And this does not change if, instead of dividing our various numbers by 100, we instead divide by 200, or by 10,000, or by a billion, or by an umpteen gazillion. Our “math nerd” fact for the day: Infinity divided by any number is infinity.
(This is not something new in mathematics. Johann Georg Locher and his mentor, Fr. Christoph Scheiner, S.J.—both math nerds to some extent—wrote about infinity at some length in their 1614 book Mathematical Disquisitions. It was not new stuff then, either.)
So, if there is an infinite Creator and Sustainer of all things, it makes no difference whether there are billions of persons on this earth, with thousands of people being born, dying, sinning, etc. every minute. It makes no difference that there are trillions upon trillions of phenomena (indeed, far more than that). We human beings each have a finite amount of time and attention to divide among the things we have to do or keep track of, so that the more things we have the less we can focus on each. However, an infinite Creator and Sustainer of all things has an infinity to divide, and our brief study of mathematics has shown us that infinity divided by any number, is infinity. Therefore that infinite Creator and Sustainer of all things can focus an infinity of attention on a hair that falls from your head, or a sparrow that falls from the sky. Our knowledge of mathematics helps us to understand that the infinite Creator and Sustainer of all things really will be intimately and personally present. If there is an infinite Creator and Sustainer of all things, that infinite Creator and Sustainer of all things knows each of us, has more time for us, and is closer to us, than we know and have time for and are close to ourselves—infinitely more. For the person who believes that an infinite Creator and Sustainer of all things exists, a “nerdy” knowledge of mathematics can help him or her to understand how that infinite Creator and Sustainer of all things can know and care about a hair or sparrow.
It is socially acceptable, even among educated people, to be ignorant of mathematics, and even to dismiss it as being beyond explanation and beyond interest—as being the province of the nerds who can wrap their minds around such things. However, knowledge of math is valuable and worthwhile in so many ways, including, as we see here, in ways related to faith. This is important even to those who might not believe that an infinite Creator and Sustainer of all things exists—at least if they care about math and science and related “STEM” areas of knowledge, and about knowledge and education in general. In the U.S., at least, students go into debt for education, and there is increasing pressure within the world of education to engage students and to do everything possible to ensure that students are successful and that they do not accumulate debt yet no degrees. So what in fact will engage people? Some will be engaged by math because they love math; others because they believe knowledge of math will help in getting a good job. But to engage a diverse audience of people means to consider what a diverse audience finds engaging and wants to really know. So often, the subject of knowledge that really engages a broad swath of people—that makes people want to spontaneously come forward to share what they know with others—is faith. As I have discussed in the past, the future is full of people of faith; full of people who have a strong reason to see a great intrinsic value in mathematical knowledge, to be attracted to nerdy mathematical thought because through it they can understand how God can be close.
*November 30, 2017 edition of the Record, page 5—click here. Also available on Fr. Rolheiser’s website—click here for the website.
