The Fog

I want to start with a disclaimer. This post is NOTHING LIKE the rest of my posts. It’s literally one giant metaphor centred around mathematical understanding (I usually write about effective ways to clean your flat). In case anyone’s interested, I’m just about to finish a Ph.D. in pure maths, and so I wanted to impart some knowledge onto the world (knowledge about knowledge, how very meta!)

This post is about understanding (well, it’s technically about understanding understanding, but I already made that joke in the last paragraph). It was initially about mathematical understanding, but it can actually apply to understanding in general. In most subjects, “understanding” is a given. If someone tells you that Napoleon Bonaparte was born in 1769, you should instantly understand what that means (at least I’d hope…) But in maths (and other more technical fields), this is seldom the case. Let’s take an example: Cantor’s diagonal argument shows that there are different sized infinite sets. It shows that the set consisting of all the natural numbers i.e. {1,2,3,4,…} is smaller than the set of real numbers i.e. all infinite decimal expansions. Check out the proof if you’re interested (this website has done a really good job explaining it to non-mathematicians so seriously go check it out. Right now. I’ll wait.)

I chose this example because it’s something I’m really comfortable with, and understand well. If I were to explain this theorem and proof to someone, I’d be able to tell if they had understood it. If someone else was explaining this theorem to me, I’d be able to tell if they had understood it. If someone wrote down the theorem on a piece of paper, I’d also be able to tell if they had understood it. The Fog is the reason why.

The Fog: A Layman’s Guide To Mathematical Understanding.

Imagine you’re walking through an obstacle course with lots of other people. There are various obstacles lying around, and you need to be careful you don’t walk into a traffic cone or trip over a box on the floor or walk into a lamp post.

Here’s the catch: some people can see absolutely clearly. But some people can only see their surroundings covered in fog. They might see only a little fog, or they might be rendered almost blind. There are many different levels of fog – let’s say 0% to 100% foggy.

If someone is too far away from you, you can’t really tell what they’re doing. They could be jumping through hoops and navigating everything beautifully (like a land dolphin, if you will), or they could be walking directly into every single lamp post around them. This is equivalent to their field being so different from your own, or the work that you’re familiar with, that you simply can’t tell how much they understand. But the people in your immediate vicinity might be visible to you.

If you see no fog at all, not only can you navigate around the traffic cones and jump through the old tyres laid out on the ground with ease – you can also see what everyone else is doing. You can see exactly how much everyone else around you is stumbling, and so if someone else sees even a bit of fog you should be able to tell. If someone says something which makes absolutely no sense, you’ll be able to call them out on it.

Similarly, suppose you see 50% fog. Then the people around you who are walking into brick walls and falling over one another might look like they know where they’re going. You simply can’t tell from where you are. And if they speak with enough conviction, they might just be able to convince you that they know what they’re doing.

Another area where The Fog is blindingly obvious is understanding of languages. If someone claims to speak fluent mandarin, then whether or not this is true will be obvious to a native, and yet might be unclear to someone who has just started learning the basics.

The more technical the field, the harder the obstacle course is to navigate on the whole, and so the more clearly you can tell if someone is struggling (as even if the world seems foggy to you, someone getting absolutely battered will stand out). In many less technical fields, there might only be the occasional cowpat here and there that you should avoid stepping in. And if you walk confidently enough, you can make it look like you knew exactly where you were going and like you haven’t stepped in anything at all! This is what I like to call “bullshitting”.

Arguably the most important mathematical skill is being able to identify exactly how much fog you’re seeing at any given time. Some people live their entire lives having never experienced less than 20% fog, and so they are convinced they fully understand things they may not. But if you’ve experienced the full range, and you *know* when you’re missing 20% of your vision, you can at least be prepared and avoid stumbling too hard. To continue the metaphor, you can use your hands to guide you, be cautious, and most importantly, if someone says “Stop! You’re about to walk into a lamp post!” – you can listen. In real terms, this might mean not jumping to conclusions, listening to experts in your field, and reading up on the background material so that you can clear that fog away next time you approach the topic.

It took me over two years of studying maths to even begin to clear The Fog away. I distinctly remember being in a perpetual state of fog throughout my second year of university, and yet not being able to understand why I wasn’t getting top marks despite studying hard. I didn’t GET that I didn’t get it. After returning from my year abroad, the fog had cleared up so much that I could more or less predict my grades in any given subject (and more often than not, they’d end up higher than I predicted due to grade scaling).

It’s also obvious when marking my students’ work. If you try to bullshit your way through a question, I can tell. Actually every marker ever can tell. You may have written enough correct things to get the points, but we all know you don’t understand the material.

How to clear The Fog away.

So how can we avoid The Fog? How can we make sure our understanding of something is actually clear?

First of all, just be aware of it. Be aware that It doesn’t just apply to academic subjects such as maths and languages. Be aware that it applies in the industry as well – business, consulting, accounting, programming – any job involving technical understanding. Realise that sounding like you know what you’re talking about, and actually knowing what you’re talking about, are two very different things. And that many people around you who sound like they know what they’re doing are actually in a perpetual state of fog.

Second, you can practice the following to make sure you understand a concept:

1. Explain it simply. Without any jargon or hand-waving, explain your concept in basic terms to someone who knows nothing about it. Have you ever had a word where you’ve gone “I know what it means, I just can’t explain what it means!” I claim you don’t really know what it means. You might be able to use the word in a sentence, but the more you can explain in concise and simple terms what the word means to someone who has zero understanding of it, the better your own understanding of it is.
2. Change the parameters slightly. What happens if you alter an element of the problem – do you still understand what is going on? Maybe you understand the majority of the concept, but there’s some little detail you’re not quite sure about. How does it apply if the scenario is a little different? How can you make the concept simpler?