The Trinity of Truth
There is something else that plays the role of convincing mathematicians
that something is true. I think of it as an illumination.
I’m going to talk about three aspects of truth:
1. Belief
2. Understanding
3. Knowledge
This is a bit like the three domes of St. Paul’s Cathedral. We have Knowledge, which is what the outside world sees, Belief, which is what we feel inside ourselves, and Understanding, which holds them together.
I’ve marked the different areas of overlap, so we have:
KUB: Things we know, believe, and understand. The most secure of truths.
KB: Things we know and believe, but do not understand. This includes scientific facts that are
certainly true, even if we don’t understand them. For example, 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.
B: Things we believe, but do not understand or know. These are our axioms, where everything else
begins—the things we can’t justify using anything else. For example, for me, there are things
like love and the preciousness of life. I believe that love is the most important thing of all. I can’t
explain why, and I can’t say I know for sure it is true—because what does that even mean?
After this things get a bit trickier.
K: Things we know, but do not understand or believe. Is this at all possible? I think if you’ve ever
experienced sudden grief or heartbreak you might have a sense of what this is like. Those numb
days after the event when you know, rationally, that it really has happened, but you simply can’t
believe it, you can’t feel it to be true in your stomach. And you certainly don’t understand it.
Perhaps extremes of good emotions feel like this too. Perhaps if I won the lottery I would, for a
while, know that it had happened without understanding or believing it. Winning the lottery of
love feels like that too, at least at the height of its ecstasy.
KU: Things we know and understand, but do not believe. Perhaps this is where we get to the next stage
of grieving, when we have come to understand that this terrible thing really has happened, but we
still don’t believe it. But if you’re in this state you’re probably in some state of denial, because
usually knowing and understanding something would make you really believe it’s true.
Finally we have the following sections, which I suspect are empty.
U: Things we understand, but do not know or believe.
UB: Things we understand and believe, but do not know.
I don’t think it’s possible (or rather, reasonable) to understand something without knowing it. In this way, understanding is different from the other two forms of truth, which do seem to be able to exist by themselves. Truth flows through this diagram in one direction only—from understanding flows everything else. Of course, it all depends somewhat on exactly how we define these things, but just try thinking for a second about some things you believe.Here are some things you might believe.
Understanding is a mediator between knowledge and belief. In the end
the aim is to get as many things as possible into the central part of the
picture, where knowledge, understanding, and belief all meet.
Here’s a mathematical example of the difference between knowledge and understanding. Suppose you are trying to solve an equation like this:
x + 3 = 5
Perhaps you remember that you can “take the 3 to the other side and
switch the sign.” So the next step is
x = 5-3
and we see that x is 2.
However, knowing that this works is not the same as understanding it. Why does it work? It’s because we have an equality between the left-hand side and the right-hand side, and so we can do the same thing to both sides and they’ll still be equal. Now, we want to get the x isolated by itself on one side, which means we want to get rid of the 3 on the left. How do we do that? We subtract 3. But if we do that on the left we have to do that on the right as well. So what we’re really doing is:
x +3 = 5
x+ 3 -3 = 5- 3
x = 2
Understanding this principle rather than merely knowing the rule makes the knowledge more transferable to other situations.
Proof vs. Illumination
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. What we write down is merely a language for communicating those ideas to others, in the hope that they will be able to reconstruct the feeling within their own mind.
When I’m doing math I often feel like I have to do it twice—once in my head, and then a second time to translate it into a form that can actually be communicated to anyone else. It’s like when you have something you want to say to someone, and it seems perfectly clear in your head, but then but then you find you can’t quite put it into words. The translation is not a trivial process; why do we try to do it at all? Why do we not just stick to the things that are illuminating? The thing is, illumination is very difficult to define. And moreover, different people can have different notions of what is illuminating. So illumination by itself doesn’t make a very good organizational tool for mathematics. In the end, doing mathematics is not just about individuals convincing themselves that things are true; the point is to advance the knowledge of the world around us, not just the knowledge inside our own head.
But understanding is still kept a secret, at least in mathematics. Students of all levels are shown the rules but kept in the dark about the reasons. We encourage children to ask the question “Why?” but only up to a point, because beyond that point we might not understand it ourselves. So we stifle their quest for illumination to match our own inability to provide it. Instead of being afraid of that darkness, we should bring everyone to the edge of it and say: Look! Here is an area that needs illumination. Bring fire, torches, candles—anything you can think of that will cast light. Then we can lay down our foundations and build our great buildings, cure diseases, invent fabulous new machines, and whatever else we think the human race should be doing. But first of all we need some light. -- end
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