Olivia Cannon | 17 and Sam Swain | 18
As math majors who follow Christ, at some point we had to ask ourselves: does mathematics exclude or reveal God? Can God speak truth to us through math?
Throughout this semester, we have considered how to look at the world and find truth embedded where there is not an initially apparent image of God. Even such a seemingly cold, unfeeling field as mathematics can give us understanding into an aspect of God’s nature. For its common label as “the language of the universe,” math sometimes may be considered as excluding God—but this is God’s universe, and even though mathematics as a language is a human science, it provides us with some useful tools for describing the abstract, with concepts not available to us in other languages.
The Fundamental Theorem of Algebra tells us that for a polynomial with complex coefficients, all its roots are complex numbers. Complex numbers are those of the form a+bi, where a and b are real numbers, and i, the imaginary unit, is the square root of -1. The real numbers are those we usually consider, like 22, π, 1.5, -6. But the real numbers are even insufficient for describing the roots of polynomials with real coefficients; we still need the complex. For example, we cannot solve x2 +4=0 with real numbers; we need 0+2i or 0-2i.
Some implications of such an idea struck one of our fellow math majors, who penned an opinion piece in the Orient entitled, “What Math Can Teach Us about Religious Ideology.” They suggest that just as the reals are incomplete next to the complex, a secular society dismissive of particular ideologies also has an incomplete understanding of itself. This was the inspiration for our own consideration of kingdom truth in relation to complex numbers.
For Christians, it is dissatisfying to say that our faith is some “imaginary” supplement to our otherwise “real” understanding. In mathematics, we do not use the complex numbers as a supplement to the reals; the set is not a completion of the reals in that sense. The real numbers lie within the complex set, but the number we know as 7 is actually 7+0i by its identification as complex. Every real number is redefined because it is known as part of this more complete set, all of whose elements are single numbers, though the limits of mathematical language necessitates the representation of these by their real and imaginary components. A single set gives us the whole scope of these polynomial roots.
The idea that all the roots actually exist in this other set of numbers (rather than just the “extra” ones we are trying to find) is indicative of an important kingdom truth—we all live in God’s kingdom and created world whether we know it or not. Life with Jesus is not an addendum. It changes everything, down to our very foundations.
The author then begins to talk about order. “Ordering” numbers is to put them in a list, from smallest to largest. We cannot do this for complex numbers because the measure of “big” or “small” is not unique—there can be multiple numbers that are the same “size.” The author continues, “Without even realizing it, for a long time I took it for granted that numbers had to have order. However, by thinking about it, I could understand how a more complete conception of numbers wouldn’t have an order at all.”
But do the complex numbers lack order completely? We still have some sort of measure of size that can allow us to classify numbers; it is just not unique. We call it the norm, and it is equal for any two numbers equidistant from the origin—points on a circle. So our human concept of the word ‘order’ does not fall apart completely. The complex numbers do have structure, just with some equality. This equality becomes even stronger when we consider the language of ideals. In rings, an ideal is a subset of a set of numbers with the property that any element can be multiplied by any number in the bigger set and still be in the ideal. In both the real and complex numbers, there is exactly one maximal ideal—that is, exactly one ideal that contains all others, but is not the whole set. (This ideal in both cases is the set containing only the number zero.) But since the complex numbers are a field, the only other ideal is actually the whole set. No matter which element you choose to “define” this other, non-maximal, ideal—as long as it is not zero—it is the same. Ideals impose a surprising equality over the set of nonzero complex (and real) numbers. This equality, too, is fundamental in our world: no worldly thing can give us an advantage in God’s eyes, or define us. We are all equally loved by Him. Equality can coexist with order and even within a structure of seeming disorder.
And, interestingly, the number zero is in the ideal generated by any other number, but its ideal is the maximal ideal. If that isn’t Christlike, I don’t know what is.
Olivia Cannon, Bowdoin Class of 2017. Mathematics, Physics. Sherborn MA.
Olivia has the unique ability to inspire those around her to wonder at the beauty of the world almost as much as she does.
Sam Swain, Bowdoin Class of 2018. Mathematics. Ashland OH.
Though slow to speak, Sam addresses the elephant in the room and unites communities.