Trinity of Truth: Version Pi

In the final chapter on How to Bake Pi, mathematician Eugenia Cheng delves into her field of mathematics, Category Theory, having dropped hints and examples throughout the second half of the book. Early in the book (reviewed here), she asserted that the purpose of mathematics is to make hard things easier. But since “category theory is the mathematics of mathematics… [it] is there to make difficult mathematics easy.”

The final chapter purposes to discuss the role of category theory. Cheng thinks that mathematics can illuminate the search for truth. Her Trinity of Truth has three aspects: Belief, Understanding, Knowledge. She compares them to the three-dome-structure of St. Paul’s Cathedral. (I learned that the visible outer and inner domes are held together by the crucial but unseen middle dome.) Knowledge is akin to the outside dome – what the world sees. Belief is akin to the inside dome – what we feel inside ourselves. Understanding is what holds the two together. At least most of the time. In mathematical fashion, Cheng maps out a Venn diagram. There is an intersection of Knowledge and Belief that does not include Understanding. As examples, she writes: “I don’t really understand how gravity works, but I know and believe it works. I know and believe that the earth is round, but I don’t understand why.” But this doesn’t preclude understanding at a later time how gravity works or why the earth is round.

Cheng sees “mathematical activity in terms of moving around between these three kinds of truth.” In particular, knowledge comes from proof in mathematics. The aim is not so much to prove math theorems (which is an important activity) but more to “move things from the proved area to the believed area.” According to Cheng, “proof has a sociological role; illumination has a personal role. Proof is what convinces society; illumination is what convinces us. In a way, mathematics is like an emotion, which can’t ever be described precisely in words – it’s something that happens inside an individual.” But defining illumination is very tricky. How it leads to belief is somewhat mysterious. That’s where proof comes in. It enables me to move from my realm of belief to that of another.

The procedure is illustrated by the diagram above. Here’s how it works (quoting Cheng):

·      I start with a truth that I believe and that I wish to communicate to person X.

·      I find a reason for it to be true.

·      I turn that reason into a rigorous proof.

·      I send the proof to X.

·      X reads the proof and turns it into a convincing reason.

·      X then accepts the truth into his realm of believed truth.

Why this drawn-out procedure? Because “attempting to fly directly from belief to belief is inadvisable. We’ve all seen people try to transmit beliefs directly, by yelling.” What about just sending the reason? “The answer is that a reason is harder to communicate than a proof… The key characteristic of proof is not its infallibility, but its sturdiness in transit. Proof is the best medium for communicating [to avoid] danger of ambiguity, misunderstanding, or distortion.”

This made me think about the complexity of teaching chemistry. As an expert, I have a powerful interlocking conceptual map that allows me to travel fluidly throughout the realm of chemistry. But novice students do not have this, and the scaffolding has to be built layer by layer, tearing down old misconceptions while attempting to introduce more accurate concepts, all the while trying to avoid further confusion. Johnstone’s Triangle does not make it easy, and I’m constantly reminded when students ask questions, that the chasm is very wide between my chemistry “beliefs” and theirs. This is why it is a good thing to continuously probe student understanding and the more questions they ask, the better. (It gives insight into what they are thinking.) Chemistry however has no fool-proof Proof for transmitting knowledge. The dangers of ambiguity, misunderstanding and distortion are constantly present.

Cheng has a sobering view of why “many people grow up feeling great antipathy towards math, probably because [it was taught] as a set of facts you’re supposed to believe and a set of rules you have to follow… The important stage in between the belief and the rules has been omitted: the illuminating reasons… Schoolchildren try to follow the rules but are sometimes abruptly told that they have broken a rule. They didn’t do it deliberately… they really thought they had the right answer… being marked wrong feels like a punishment to them… Perhaps it was not explained to them in an illuminating way that could actually make sense to them. As a result, they don’t know when they will next be found to have broken a rule, and creep around in fear. Eventually they’ll simply want to escape to a more democratic place, a subject in which many different views are valid.”

That last phrase may partly explain why many students who start off enjoying math and science in the early years end up being fearful and hating it later, sometimes in college, sometimes before they even get to college. I suspect that some of my students feel that about chemistry in the way described by Cheng. I should look out for this, and continue to encourage my students to ask “Why?” questions. I’m not getting enough of those suggesting that I’m not using in-class time in the best possible way – for the asking and answering of questions through the lens of learning as a relationship. Questions can come from both teacher and student. Answers can come from both student and teacher. It’s a journey through knowledge together!