Part 1, The Language of Mathematics, Chapter 1: This Chapter Has No Title
By Andy Walsh
We are going to blog through the book, “Faith Across the Multiverse, Parables from Modern Science” by Andy Walsh. Today is Part 1, The Language of Mathematics, Chapter 1: This Chapter Has No Title. Walsh begins by talking about paradox. He notes many people find the idea of paradox off-putting and aesthetically irksome, to the point that they refuse to acknowledge true paradox and attempt to resolve them to one end or the other. Walsh finds paradox to be illuminating because they invite us to closely examine what it is we really mean by a statement and each of its parts. Are we making assumptions that we don’t even realize? Perhaps what we think is a paradox is really a failure to understand reality from the right perspective.
When faith is set at odds with science, it usually is defined as unquestioning acceptance of dogma, or belief in the face of contrary evidence. When that definition is set up, you have a situation where boundaries are drawn so that faith can only ever be false because anything true would automatically belong to science. Walsh says:
Dogma is a product of religion, but I don’t have faith in a religion, I practice a religion because I have faith in God. Dogma is certainly up for questioning; no less a champion of faith than Jesus challenged the dogma of his day. And Jesus did not ask us to have faith in God regardless of evidence; he presented himself as evidence of God and his nature.
And that, he says, is where math comes in. Math encourages precise definitions of abstract concepts in order to clearly reason about them and their properties. Right? But, nevertheless, we have to choose a small number of statements we assume are true rather than proving them via the rules. These assumed truths are called axioms. The intention is to choose axioms so obvious and self-evident that no one would deny them. Unfortunately, self-evident has proven to be a somewhat slippery. Arithmetic, algebra, calculus, and other widely used mathematical machinery have been developed incrementally as needed, often to solve challenges in practical fields like engineering or architecture. They were not constructed from axioms initially. Thus the goal was to find the right axioms that would bring forth all the familiar math we wanted and none of the paradoxes we didn’t. A set of axioms and the math constructed from them that met these criteria would be considered complete.
However, here is where it gets interesting, not only did the proposed basis for math fail to be proven complete, it was actually proven incomplete; some questions were demonstrably unanswerable.
Here’s Walsh’s explanation:
A simplified explanation of the proof of incompleteness is that you can create an expression whose meaning is essentially, “This expression cannot be proved.” And yet, since it is built up from proven expressions according to the rules of the language, it should be considered proved. In other words, what was proved was the unprovability of the statement. This essentially rendered that expression undecidable, meaning there is no way to know if it is true or false, given the axioms we chose. That doesn’t mean that in some absolute sense, it is neither true nor false; it just means we cannot prove it either way. Thus this system of axioms and formal language doesn’t get rid of our undecidable paradoxes.
Okay, John Barry, sit down right here and breathe into this paper bag for a few minutes. Ya alright, buddy? Let’s move on. Geometry provides another conundrum with no definitive answer. The fifth axiom of geometry defines two lines as parallel if they both intersect a third line at right angles. Picture football goalposts; the two uprights intersect the crossbar at right angles and so are parallel.
If you keep extending those goal posts uprights, they will never meet or cross; the official rules of football rely on this property. However, when we draw a globe, the lines of longitude all meet at the poles, even though they intersect the equator at right angles. Maybe parallel lines can intersect?
Where is Walsh going with all this? If one cannot prove or decide all questions of interest in rigorously and narrowly defined fields such as mathematics, maybe we shouldn’t expect to construct a complete and incontrovertible framework for understanding the entire world from first principles. Taking a cue from mathematics, we can switch instead to considerations of usefulness. If math can function without being able to prove everything perhaps other domains can as well.
What if we take belief in the God of the Bible to be axiomatic? Walsh asks the question, that instead of defining faith in terms of dogma or rejection of evidence, we instead say faith is choosing a set of assumptions, or axioms, for understanding the world. Assuming God rather than proving him might seem like a dodge to the requirement that we provide evidence. But if axioms cannot themselves be proven; as with pudding, the proof is in the tasting. Walsh is primarily interested with what conclusions follow from my belief in God and how useful they are in my real life.
The idea that God is not a provable conclusion but an axiomatic assertion, and just one possible axiom among several alternatives is obviously going to be uncomfortable for some believers, but isn’t the idea consistent with what the Bible says in many places? Psalm 34:8, “Taste and see that the LORD is good; blessed is the one who takes refuge in him.” How about the refrain in Ecclesiastes (1:2) “Meaningless! Meaningless!” says the Teacher. “Utterly meaningless! Everything is meaningless.” Rather than descend into nihilism, he ultimately choose to build a framework for understanding the world and living in it based on a belief in God, not out of the logical undeniablity of the premise, but because he found a life so constructed to be fruitful. In the parables, Jesus describes the Kingdom of God that follows from his view of the world, and invites us to be part of that kingdom. This is an appeal to the usefulness of his assumptions, not their completeness.
If thinking in terms of axioms and theorems is uncomfortable and difficult, then consider a game of chess instead. In chess, we have starting positions for all the pieces (axioms) and rules for how those pieces can be moved. Different arrangements of pieces on the board are like theorems; if you can get to a given arrangement by following the rules from the starting positions, then it is valid or true. There are many, many possible board positions, enough that you could play for your whole lifetime and not see them all. And yet they were all there in the rules, just waiting to be played out.
So when can you say you know the game of chess? Once you have learned where all the pieces start and how they move? Walsh notes you are not much of a chess player if that is the extent of your experiences with the game. The only way to acquire the skills of a grandmaster is to spend many hours playing. That is how one comes to know chess. Walsh says:
Translating that to our definition of faith as choosing God for an axiom, we see that the book of James says something very similar. “What good is it, my brothers and sisters, if someone claims to have faith but does not have works?” (James 2:14). In our terms, choosing the axiom of faith is not enough to know God; it is simply a prerequisite. Knowing God is the process of taking that axiom and figuring out what truths it contains. That means playing the game—living your life according to the theorems, the true statements that follow from belief in God. This is what James means by showing faith through works, and what separates knowing God from knowing of God, which, as James notes in verse 19, even the demons know of God.
As the old-timers would say, “that’ll preach!”