Why we should stop saying "explain it to me like a five year old"

A disquisition into what learning actually is, how to do it, and why you shouldn't feel bad if you don't understand something complex...

The insidious and derisive claim that “if you can't explain it to a five-year-old, you don't understand it,” is adopted by people who claim to be parroting Einstein (it must be true because everything Einstein said was true, right?). But this claim is false on so many levels that it's difficult to know where to start.

The Simplification Trap

Well, firstly, let's start with attribution: Einstein didn’t say this! In fact, Einstein implicitly conveyed the opposite when he stated that he couldn’t enter the 1920s competition to explain relativity in 5,000 words to a general audience. Note, that that’s a general audience—not even a five-year-old, but rather an educated, say, 18-year-old. Further note that you get 5,000 words. Now, you try saying anything uninterrupted for 50 words to a five-year-old, let alone 500, let alone 5,000!

Secondly, even Feynman said to a reporter, who was asking him to explain QED: “if I could explain it to you, it wouldn’t be worth the Nobel prize…”

Thirdly, what someone calls “simple” is based on one’s own familiarity with the terms, and not the inherent simplicity of the concepts. Indeed, something as “simple” as “logarithm” requires you to know what multiplication is, which itself required months of drills and hammering home in elementary school, which itself requires the concept of addition, and that, too, required months of drills. It’s only after this process of usage and boot camp that you think it’s a “simple” concept. But there’s nothing congenitally “simple” about it!

Fourthly, it’s harrowingly often the case that understanding a subject deeply and being able to explain it are only mildly correlated phenomena. For instance, in my alma mater of the University of Toronto, it’s (in)famous for being a research university first and foremost. This means they hire based on a professor’s knowledge of the field, and not on their ability to teach. Being a great explainer and a great understander are different skillsets.

Fifthly, why the arbitrary cut off at five years old? Why not fifteen years old? Why not twenty-five? Why not two?

(Next up: explaining second-countable Harsdorf spaces to cellular division.)

The Succinctness, Simplicity, Accuracy Trade-off

There's an adage in business that you can only have two of the following:

1. Speed

2. Quality

3. Cost

That is, you can't have something quickly with quality unless it's expensive, and you can't have something inexpensive and quality without it being slow, etc.

I think something similar is true for explanations. When asking someone to explain something to you (or for you to explain something to someone else), you only get two of these, and not all three:

1. Succinctness

2. Simplicity

3. Accuracy

If you want something accurate but succinct, the explanation will not be “simple.”

For instance, let's tackle the question of “What is a classifying space?” The succinct and accurate answer is: “A classifying space B(G) for a group G is a topological space such that its principal G-bundles over any space X are classified by the homotopy classes of maps [ X , B(G) ].”

After you recover from the aneurysm, you'd realize this is not simple in the least. However, it is compendious and true!

So, what if you want an explanation of what a classifying space is that's simple yet succinct? Well, it's “A classifying space shows all possible ways some objects (or groups) can be organized or arranged.” Okay… not informative in the least, nor is it entirely accurate, but ah well, we did select simplicity and succinctness.

How about if instead we select a simple explanation that is accurate? Well, in some sense, that's what Hatcher's Algebraic Topology or Bredon's Topology and Geometry are. They're simple (starting with basics) and absolutely accurate, but that's decidedly not succinct!

TL;DR: Good luck.

Compression as Distortion

If you actually listened to the explanations given to a five-year-old, you'd see that what's being said is such a poor, 10,000-meter-in-the-air explanation that sometimes it's worse than saying nothing.

In fact, some spiritual teachers believe this to be the case of language in general. That is, language so fails to capture the nature of consciousness or whatever else are the “deepest” questions we want answered, that to speak about it is to instantly debauch and fabricate it; in some schools, it’s best to be silent.

It takes months (if not years) of consistent practice of meditation, and wrestling with koans, to start to understand what it means to grasp the notion of emptiness. If you were to explain it to a five-year-old, your concept of “empty” would indeed be empty!

In other words, compression doesn’t whittle away the message to its essential core but instead modifies it.

Take a photograph of my home town Toronto. Remove the CN Tower. Slowly remove more and more until it’s just trees. Sure, you’ll have some idea of Toronto, but it will be a misleading one.

However, what’s “non-essential” can sometimes be removed. Taking the first few terms of a Taylor expansion, or doing what Picasso did by finding only the "relevant" curves that need saving to capture some essence of the original image, are examples but they’re not always a straightforward process. Infamously, Picasso took decades to get to that point, and we take for granted how much others have contributed to our sense of the “gist” in other scenarios (like the skyline of a city), that we think it’s effortless to convey the “gist” in every scenario. Further, there exists such functions as non-analytic ones!

(…By the way, you can summarize my entire article as “not all concepts are analytic.” This statement is succinct and accurate… but not simple.) ;)

From Set Theory to Storks

“Explain it to a five-year-old” is one of those unexamined phrases that people echo often because it prevents their ego from being bruised, since it’s no picnic to not understand something. It’s better if the problem lies with the speaker than with the receiver, lest the receiver have to learn more to grasp the underlying ingredients to the explanation.

“But what about these videos where you explain something at 5 different levels?”

Hogwash.

If you watch this video about “explaining infinity at 5 levels” with the brilliant Emily Rheil, you’ll see the issues.

Firstly, the opener is a nine-year-old, not a “five-year-old.”

Secondly, if you listen carefully, you’d notice the title is misleading. Riehl is NOT explaining “infinity at 5 levels!” Instead she’s explaining different ideas related to infinity at five different levels. She starts with:

1. [Nine year old] Explaining that infinity is what’s unbounded

2. [Thirteen year old] Explaining a paradox about infinity (Hilbert’s hotel)

3. [Undergraduate] Explaining cardinality

4. [PhD student] Explaining the axiom of choice and its conclusions, as well as equivalences (in ZFC)

5. [Professor] This last one wasn’t even “explaining” anything but rather “talking to another person casually about what’s fascinating about infinity.” If you want to pull out a “lesson” from here, it would be about the “Continuum Hypothesis and how proofs are constructing a function where the domain is the hypothesis and the target is some moduli space universe.”

While you can argue that what Riehl did was “explain” Hilbert’s hotel (though that’s a stretch), it would be implausible to argue that she explained the “axiom of choice and its equivalences in ZFC to a nine-year-old,” let alone that she “explained” the “Continuum Hypothesis and how proofs are constructing a function where the domain is the hypothesis and the target is some moduli space universe” to a nine-year-old.

Oh, and let alone to a five-year-old.

Even researchers would be hard-pressed to do so without ChatGPT as your aid.

PS: For fun, try getting the five-year-old to explain it all back to you so that you can ensure your explanation indeed landed.

Once you've finished explaining to your therapist why you tried explaining ZFC to a kindergartener, you'll realize that explaining something in such a manner that a "child will understand" may lead the child to think they understand it. But the inquisitive child would realize that explanation simply can’t be the way the world works.

For instance, the simple explanation is that babies come from storks, rather than sexual reproduction. Is that accurate in the least? Even if you were to say a simplified truth, which is that "daddy puts his penis in mommy," you're still left wondering—what the heck does that have to do with having a kid?!

Explaining the Punctured Plane to a Five-Year-Old Misses the Point

Now, the person who doesn’t understand some explanation doesn’t need to feel bad and resort to the retort of “you must not understand because you can't explain it to a five-year-old. Checkmate.”

Learning anything technical is formidable and strenuous. This is okay.

Often our misapprehension comes from unconscious questions not being asked. For instance, in general relativity, explaining it nonchalantly as a bowling ball on a mattress, unconscious questions arise: “why is it constrained and why does it have to move at all, why not stay still? In our world, you can jump up and down and be motionless if you like.”

These questions aren’t obvious to you, as they lie at the undeclared level.

Not realizing these are questions lurking underneath prevents you from asking these questions. This evinces itself as an implicit feeling that something is “left unexplained.”

This means that as a student, or an interviewer, the more you can become in touch with that feeling of knowing something is left unexplained, and the more you can ask by dredging what’s unarticulated, then the more quickly you learn.

In other words, you have sticking points, but you're unaware of precisely what they are. That is, you have some misunderstanding, and you don’t know that your trouble understanding a derivative concept is because of having the wrong intuition about a sub-concept.

For instance, some people may be stuck not understanding how the ring of integers forms a group because g⁻¹ isn't usually an integer. You then realize that 2⁻¹ isn't to be thought of as “two to the power -1,” but rather “the inverse with respect to the operation at hand” (in this case, addition). Only then you can move on.

However, because you don’t know that your misunderstanding is there, you can’t tell a teacher “here’s my sticking point.” Thus, your teacher throws plenty of examples and watches to see where you falter so that they can triangulate your sticking point for you. When you don’t have a teacher, you have to simply look up concepts and examples, over and over, often getting more confused until you have a realization that you misunderstood some concept and then finally you can move forward.

Your ignorance about your ignorance is the issue. And that’s fine. The sooner you can realize this, the better you feel and the more motivated you become to tumble your way through the learning process instead of fault-finding in someone else.

Keep in mind that there's also something physical (neurological) about learning and knowledge. It would be like going to the gym and some bodybuilder Brandon tells you to deadlift 450 lbs. You can’t, and you then say, “Well, Brandon, you must not understand the 450 lbs. My lack of being able to lift must be because your lifting is false somehow!” This robs you of the opportunity to sharpen yourself by placing culpability elsewhere.

Pro tip: No amount of whey isolate will help you digest quantum mechanics.

Just Get Wet: Becoming Comfortable with Complexity

Despite this, there is something to be said about watching and listening to what is far “above” your pay grade. John von Neumann once remarked, "You don't understand mathematics—you just become used to it." While this is a facetious quip, there's some truth to it. Why?

Part of what holds someone back from understanding complex ideas is the intimidation and anxiety that come with more advanced concepts. This intimidation diminishes as you become more familiar with the terms, allowing you to grasp the spirit of the concepts, even if you don't fully understand the text.

In other words, there are three levels of explanation:

1. one that provides just a “gist”

2. another that offers rigor

3. and a third where you can smoothly transition between the two—understanding how to start from one and derive the other, and even invent new concepts. This comes with familiarity with the concept (as well as copious calculations!).

The point isn't to drink from the firehose, but instead to just get wet.

I think what’s doing the heavy lifting in this entire article is the word “explain.”

What does it mean to “explain?” How do you know when you have had something actually explained to you? Is it the same as “understanding” it? What is “understanding”?

It’s those these latter questions which are at the heart of what it means to give an account of the “nature of reality” or the “nature of existence.” In some sense, you can think of the questions of “what does it mean to explain and understand?” as what we explore here in this labyrinth, and on the Theories of Everything YouTube channel.

- Curt Jaimungal

PS: A shining example of explaining concepts simply yet accurately is Scott Alexander of Astral Codex.