By Gray Manicom.
Both Theology and Mathematics have been celebrated as “the queen of the sciences”. Perhaps the famous mathematician Gauss, who thus crowned Mathematics, was critiquing the aged words of the famous theologian Aquinas. Perhaps he was implying that the rigorous logic of mathematics enabled a scientific method that did not need God, but found its trustworthiness in human endeavour. After all, it does appear that Mathematics rules modern science with the iron fist of her own standards and predictions. Even modern psychology, considered a pseudoscience not too long ago, now finds that it stands upon the rock of mathematical rigour, the same rock upon which stand Physics, Chemistry, Biology and the other sciences.
Then again, perhaps that is not what Gauss meant. After all, when there are two queens on a chess board, it does not mean that they are in opposition to each other. Likewise, theology and mathematics can mean different things to science, and answer different questions, without being in opposition. It is not necessary to remove the rule of theology from the physical sciences, or vice versa, if we understand what each ruler contributes to science and what sort of explanations they provide. It is a common theme in the gospels that people ask Jesus bad questions, and that he responds with good questions. As Christians, it is important that we understand what types of evidence and knowledge belong to what types of questions. I hope to share some thoughts about where to look for answers to the questions that we ask.
Apologists often point out that where science asks how-questions, theology asks why-questions. Professor of Mathematics John Lennox often uses the example of making tea. Science tells us how electricity is generated, transported to a house, converted into thermal energy, how the water boils and what that means, and how the tea diffuses in hot water. Theology tells us why– because someone wanted a cup of tea. Thus theology brings meaning to why we do science. It also provides answers as to why we can do science at all, both why science can exist in our minds and why it would exist outside of our minds. What I want to focus on is what sort of questions Mathematics tries to answer. Indeed, it appears to me that a lot of the confusion in debates between John Lennox and his arch-nemesis Richard Dawkins arise from the fact that John Lennox is thinking in terms of deductive mathematical laws and Richard Dawkins is thinking in terms of inductive biological ones. It is not only that the content of the laws differs, but that their implicit epistemologies differ, that leads to further miscommunication.
To begin with, we can eliminate a number of question types. Mathematics does not answer where or when something happens, or what something is, or who someone is, since all of these questions concern particular characteristics. I should clarify that there will be types of intra-mathematical statements that say when something is or what something is. However, these are mathematical statements about mathematical objects or events. For example, I could define a function y=x+1. That tells me what that function is. However, even if y does represent something real and tangible, the function only describes how y changes as x changes, it does not actually describe the real or tangible thing. My field of dynamical systems is the study of systems that change in time, and so we say when things are expected to happen. Again, though, the events are mathematical events, not physical ones. They might correspond to some physical event, but they are, themselves, only tools to provide insight. Mathematics is concerned with abstractions, and abstractions are subtractions. No mathematical objects are to be observed in self-checkout at Pak ‘n Save. They come from subtracting specificity from the original object, so that once the crust, the gravy, the steak and the kidneys, and the pie, are removed from the steak and kidney pie, you have a mathematical object- an ellipse. As Bertrand Russel wrote, “It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so, we introduce something irrelevant and not formal”. One and one may be two, but Bob and Sue are not two. It is this lack of specificity that gives mathematics its incredible explanatory range; almost all pies (as well as planetary orbits and a great many other things) can be described as ellipses, and thus obey the mathematical laws of ellipses. Thus one can claim Mathematics describes some characteristic of specifics; it is a pie-in-the-sky description.
We can also eliminate the how-questions of physics and chemistry. The how-questions of the physical sciences are based on cause and effect, and generally find answers through inductive reasoning. Mathematics provides a type of explanation different to a causal one. Newton might wonder how the apple fell on his nose, and through inductive reasoning conclude that light objects will always fall towards heavy objects. The effect of a hurt nose was caused by the apple falling which was caused by gravitational attraction to the very heavy Earth. However, when Newton developed calculus, he was no longer asking how-questions. He did not ask what caused the Fundamental Theorem of Calculus, just like it is strange to ask what causes one and one to be two. Indeed, even his proof of the Fundamental Theorem of Calculus is not an account of the causal history of integrals or derivatives. As Mark Colyvan writes, “Numbers and other mathematical objects do not seem to be the right kind of things to be in the causal nexus of the universe.” The mathematician’s head is not in the clouds, it is above them, so that any glimpse of the Earth below only yields arbitrary shapes and colours strewn out in some pattern.
That brings us back to why-questions. Mathematical proofs seem to say why a statement must be true, based on what was already known to be true. This is not true of all mathematically valid proofs. For example, computers are now used to prove theorems when there are a finite number of cases to examine. If every case checks out, then the theorem is true. This type of proof does not say why the theorem is true, but just shows that it is true. However, the more desirable mathematical proofs (known as constructive and/or explanatory proofs) explain why. Even with a computer proof, it is still mathematical thinking that led to the hypothesis of the theorem in the first place, and that thinking led the mathematician to conclude why that theorem should be true. This is significant because there is an irrefutability to mathematical statements that is unique. Unlike in any other academic discipline, where theories are always revised and reformulated as more data becomes available, mathematics textbooks never go out of date and never need replacing, because better theories are never found (although more general ones or more insightful proofs often will be). This does not mean that equations are never replaced, just that the equations that are replaced are equations describing real, specific things, not the relationships between strictly mathematical entities.
Thus both Mathematics and Theology ask why-questions, but there is obviously some difference. The first difference one might be tempted to make is that mathematical why-questions yield only mathematical answers, whereas the scope of theology is everything. However, it is a remarkable thing that mathematics often explains why non-mathematical statements are true, in a way that non-mathematical explanations could never. For example, there are several species of cicada that have prime-numbered life cycles. Every 13 or 17 years they emerge from the ground en mass. The reason for this is mathematical, prime numbers will overlap with the least number of periodic life cycles, which predators follow. Thus if you follow prime numbered life-cycles you will have the highest chance of avoiding mass predation over time. Of course, some biological explanation is needed to make the mathematical explanation practical- cicadas don’t want to be eaten- but it is the mathematics that does the heavy lifting of why the cicadas have these life cycles. There are many other such examples ranging across almost all the sciences.
So what is the difference? After some pondering, I have come to the rather (un)profound conclusion that Theology answers why-questions to do with God, while mathematics answers why-questions that have to do with patterns. Now, since God is a rather big deal, that gives Theology a massive domain of explanation. Thus ontological, existential and epistemological questions such as “Why are we here?” and “What is the correct action to take?” fall into the category of Theology, since they are tied to what sort of creator there is (if there is one) and what sorts of beings we are. Questions like “Why are there mathematical patterns?”, “Why do we want to do Mathematics?” and “Why can we describe those patterns using Mathematics?” fall into both categories, since they require an understanding of creation and an understanding of what Mathematics is. Questions like “What is the square root of -1?” or “What is the shortest path between two points in curved space-time?” belong to Mathematics. That is not to say that there is some part of Mathematics in which God has no place, but merely that within some parts of Mathematics God is not necessary as a means of explaining things. If God is needed, then it is Theology.
Of course, there are times that Theology and Mathematics will overlap. Just like Science meets Theology when asking “Where do we come from?”, Mathematics and Theology overlap when regarding many questions. Famed mathematicians Liebnitz and Godel both used mathematical logic to try and develop understandings of and proofs for God and his existence. The prominence of infinities and imaginary numbers in modern mathematics have led many to search for metaphysical explanations. The ancient Greeks’ religious beliefs were closely linked to their mathematical ones. The question “Does God play dice?” and the fine-tuning argument is inherently mathematical, and yet is framed theologically.
This blog hopefully serves as a reminder that much confusion and miscommunication can be had by matching inappropriate types of explanation to some questions. Many of us are aware of the historical complications of ascribing theological claims to scientific questions and vice versa, but mathematical claims and questions are different to both. It is always worth taking a step back and reflecting on which types of answers are appropriate to the question. In doing so we may find that science has more than one master, and that both Mathematics and Theology have their own realms as the queens of science.
Russell, Bertrand. Introduction to mathematical philosophy. Courier Corporation, 1993.
Colyvan, Mark. “The ins and outs of mathematical explanation.” The Mathematical Intelligencer 40.4 (2018): 26-29.
Cohen, Daniel J. Equations from God: Pure mathematics and Victorian faith. JHU Press, 2007.