Not all Curves are Lines
The reality of advice that you should and shouldn't take
“Not all curves are lines”
How many hours before your flight should you arrive at the airport? 1? 2? 4?
How many people should you invite to your birthday dinner (not party)? 5? 8? 15?
People have different perspectives on these questions. But everyone can agree that there is some benefit to increasing your answer, and also some benefit to decreasing your answer.
It is beneficial to spend as little time as you need to at the airport (because they’re uncomfortable), but it is also beneficial to spend more time so you don’t miss your flight.
It is nice to be surrounded by many friends on your birthday, but it is also beneficial to invite a smaller number of friends so it feels cozier.
Assuming your moral compass was perfect,1 would you recommend that a friend should arrive earlier or later to the airport than they already are?
You don’t have an answer, right? That’s because the right choice depends on how early/late they already are.
This means there is a danger to applying some kinds of advice. If you are advised to arrive at the airport 2 hours earlier, you might actually be worse off if you already arrive 4 hours before your flight. If you are advised to be more assertive about your desires, you might actually be worse off if you tend to be pushy.
These kinds of advice are called Dichotomous Advice and you have to think twice before applying them.
I will define Dichotomous Advice in the end, but first, let’s explore the concept underlying Dichotomous Advice: Curves vs Lines.
Not all Curves are Lines
How Not to Be Wrong: The Power of Mathematical Thinking is a book by Jordan Ellenberg, a Math Professor at the University of Wisconsin-Madison2.
“Not all curves are lines“ is from this book. Ellenberg explains that “A line is one kind of curve, but not the only kind“.
A line is what you and I would call a “straight line“, but it turns out there are no (mathematical) lines that aren’t straight. All other “bent lines“ are generally called Curves.
Furthermore, Ellenberg explains that not all relationships are linear: follow a straight line. In other words, more of a good thing is not always better3.
Ellenberg illustrates this point with criticism from Daniel Mitchell, a member of the libertarian Cato Institute. Mitchell posted a blog entry titled “Why is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?“. Mitchell continues,
“If Swedes have learned from their mistakes and are now trying to reduce the size and scope of government, why are American politicians determined to repeat those mistakes“
The implication of this argument, as Ellenberg explains is as follows:
Source: Jordan Ellenberg
This is what linear thinking is. It assumes that more government is bad and less government is good. QED.
But the following is the view of people who agree with Obama:
Source: Jordan Ellenberg
A moment’s thought will make you realize the second curve is more accurate.
Note that to achieve the same outcome (prosperity), the US and Sweden would have to take opposite actions, according to this curve.
Takeaway: what to do depends on where you already are!
This means you should frequently sanity check the advice that you receive.
Remember the question I asked at the start:
How many hours before your flight should you arrive at the airport? 1? 2? 4?
Ellenberg actually discusses this airport dilemma, quoting the 1982 Nobelist in Economics, George Stigler:
“If you never miss the plane, you’re spending too much time in airports“4
Source: Jordan Ellenberg
PS: The idea behind the comment and figure above is that the chance that you miss your flight if you arrive only 15 minutes beforehand is very high. But you can’t reduce this chance to near zero unless you arrive 4+ hours before your flight. But if you fly frequently enough, “you’re spending too much time in airports“ (4 hours every time!)
Since you’re still hung up about whether you should arrive earlier or later, let’s talk about Dichotomous Advice.
How to Interpret Dichotomous Advice
First, let’s define Dichotomous Advice.
Dichotomous Advice is the kind of advice that you should either apply directly, apply the opposite, or a mix of both, depending on where you are.
Dichotomous Advice is often governed by (curves) Nonlinear relationships.
Example: Chasing shortcuts vs Focusing on consistent work.
This is pulled from James Clear’s terrific 3-2-1 newsletter. (Sign up already!)
I posted on Instagram that I disagree with the implied advice: stop wasting time searching for shortcuts and just do the hard work.
However, after reflecting on my comment, I realized that I said that I disagreed because I agree with the opposite. But the more accurate comment is: it is Dichotomous advice.
Search for shortcuts, work hard, and sometimes do both.
The key to this dichotomy
Are you an expert or a beginner?
As you get closer to being an expert, that is you have practiced your craft, you should focus on consistency and showing up to do the hard work.
When you are Lionel Messi and you have signed that deal with Barcelona, winning the Champions League is simply about showing up to practice every day and doing the hard work.
When you get tenure as a University Professor, gaining recognition for your research is simply about mastering the atomic habit of daily work on your research.
Messi and University Professors already understand the path to success. This is why I agree with James Clear. When you already understand the path to success, it is mostly futile to invest in finding shortcuts.
But what if you’re a beginner?
Not so fast. Sometimes even as a beginner you just need to get started and show up consistently. This is especially true for writing. This past summer I woke up an hour earlier to write daily because I needed to work on being consistent.
But more recently, that is when I’ve moved to the amateur stage, quality of writing has become more important. This meant researching the right techniques: Should I write as I research or after researching? How long can my sentences really be?
Here’s an instance where a shortcut is worth discovering.
When I watched an episode of The Chef Show 5, I was astonished at the Chef’s creativity. To be fair, this technique might be common knowledge to frequent bakers. Check out the illustration below:
With the Chef‘s technique, you can create thrice as many cakes!
If I wanted to be on The Chef Show in the future, I can’t just “stop wasting time on shortcuts and focus on hard work.“
“give me six hours to chop down a tree and I will spend the first four sharpening the axe“ - Abraham Lincoln
So what should you do about all these Curves and Dichotomous Advice?
For the specific example of Chasing shortcuts vs Focusing on consistent work:
Identify if you are an expert or a beginner. You need to judge how much knowledge you have. If you have a lot, then focus on consistent hard work. If you don’t have much knowledge, then focus on discovering best practices (shortcuts).
For Dichotomous Advice that is governed by curves:
There is a first problem of identifying if the advice is Dichotomous. Then there is a second problem of figuring out where you are. Awareness of these two problems goes a long way.
I will write another post that goes deeper into dealing with Dichotomous Advice in daily life. I would love recommendations (ie as a comment) on examples of Dichotomous Advice.
Did you enjoy that Deep Dive? Now don’t just keep it to yourself, gift it to a friend!
By the way
Isn’t most advice dichotomous?
Shoutout to Aliris for asking this insightful question. I am not sure right now. This is because the question is well framed. The question doesn't ask if all advice is dichotomous. This is certainly not true because of obvious examples.
For instance, if you want to gain a great deal of success, you will have to work very hard. Sorry, this is true with almost no exceptions. The opposite is especially untrue: “to gain great success you have to work less hard”.
But a lot of advice we hear seems to be dichotomous. One way of identifying dichotomous advice is that you will observe some people advocating for the opposite advice.
For instance, “Don’t compare yourself to others“ and “Discover your unique strengths and weaknesses“.
It isn’t.
He is also a two-time Math Olympiad gold medalist (with perfect scores!)
An astute reader might wonder about the converse: less of a bad thing is not always better. This is true for some ‘bad‘ or at least undesirable things like problems. Having no problems gives you no reason to exist. From Squid Game: “One thing a very rich and very poor person have in common: they are both miserable“
The reason this is true is because you should tradeoff directly between the probability of missing your flight and avoiding the pain of waiting in an airport.
not the episode above but this uses the same technique for the same cake.