When we debate and dialogue with dissenting views, both sides always enter the discussion with unstated assumptions. These unstated assumptions color the discussion in a variety of ways, sometimes helpful, and sometimes unhelpful. But often, our unstated assumptions remain unquestioned, even in our own minds. Is this ok? Can’t we just assume that things that we don’t even feel like we have to say don’t need to be questioned?

A lot of people might answer yes – but I answer no. My name is Will Craig. I am a graduate student studying mathematics at the University of Virginia. I have also studied philosophy, history, theology, and apologetics in my spare time since becoming a Christian a few years ago. I have been graciously added as a guest author on Adherent Apologetics, and I write my own blog called Mathematical Apologetics – where I write about theology, apologetics, philosophy, mathematics, and more in an accessible way, aimed to help anyone who reads the blog develop sharper critical thinking skills, a broader knowledge base, and an awareness of when we need to question our assumptions.

Why do I say that unquestioned assumptions always must be questioned? It is because I am a mathematician. Questioning my unquestioned assumptions is a huge part of my job. And it is an important part of my job because sometimes numbers behave very strangely! I want to give just one example of how things can become very strange indeed in the world of mathematics.

Suppose you want to calculate 1+2+3+4. The commutative property of addition tells us we can add up these numbers in any order we want. So, 1+2+3+4, 2+3+1+4, and 4+2+1+3 all give the same final answer. We learn this rule as a universal truth that we are always allowed to use. But is that really right? Actually, no. While it is universally true for basic sums like these, things get weird when you do an infinite sum. One famous example of infinite sums that mathematicians look at is

$1 - \dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{4} + \dfrac{1}{5} - \dfrac{1}{6} + \dots$,

with the same repeating pattern going on towards infinity. Without going into too much detail, there is a subtopic within calculus that helps us assign an actual number to infinite sums like this one – and this sum is exactly equal to about 0.6931… ($\ln(2)$ for those of you who are mathematically inclined). If we look closer, using some mathematical tricks we can realize that the positive fractions $1 + \dfrac{1}{3} + \dfrac{1}{5} + \dots$ actually add up to infinity, and the negative fractions $- \dfrac{1}{2} - \dfrac{1}{4} - \dfrac{-1}{6} - \dots$ add up to minus infinity. But any mathematician will tell you that $\infty - \infty$ is nonsensical – you can literally make $\infty - \infty$ into any number you feel like making it.

What does this mean? What it means is that we cannot use the commutative property for infinite sums. It only works for finite sums. As a mathematician, this makes me wary of my presuppositions when I go into discussions about God, Jesus, and Christianity. If something as basic as the fact that a+b = b+a sometimes goes wrong if you stretch it too far, then I should be humble and willing to ask whether I am making any assumptions that I am stretching too far. I am still firmly convinced of the Gospel of Jesus Christ, and on the basis of very good reasons, but that doesn’t mean I understand all of reality correctly. Because of my careful training in logic and mathematics, I try to be much more careful about what I do and do not assume when I go into discussions.

For more thoughts from a young Christian mathematician’s perspective, check out Mathematical Apologetics (https://mathematicalapologist.com/).