Background

I've just picked up a book called "How Not To Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg" that I'd started reading a while back, but never had the time or patience to get past the first chapter. It's an interesting book that highlights how approaching everyday thought mathematically can add a layer of rigour and logic to the way we think. In this post, I discuss a specific part of the book on linearity, and touch on the first chapter to reflect on what's been learnt.

Textual example of linear thinking

In the very first chapter, he points out a common flaw or hole in many arguments: the false assumption of linearity. He does so by introducing a political issue back in the 2000s. The Obama administration wanted to make the U.S. "more like Sweden" by raising taxes, while Sweden reformed to lower tax, aiming to become less like itself. This was a critique against the Obama administration brought up by Daniel J. Mitchell of the libertarian Cato Institute as a blog entry, questioning why the U.S. wanted to be more like Sweden if Sweden and other nations alike had learnt from their own mistakes—that's just stupid.

However, Jordan notes that the Cato institute had looked at the relationship between tax rates and prosperity as a linear graph. The more "Swedish" you are, the lower your prosperity. The less "Swedish" you are, the higher the prosperity. They unknowingly assume that the relationship between the two variables ought to be a linear graph, with some inverse relationship.

Jordan refutes this argument by introducing a Laffer Curve, a nonlinear graph showing how tax rates and tax revenue are related. According to the curve, at a tax rate of 0%, the government has no revenue. At the tax rate of 100%, people no longer have the incentive to work and so the government has no revenue yet again. It tells us that the sweet spot is somewhere between the two extremes, forming some n-shaped parabola. Hence, optimisation is about finding that perfect middleground. Understanding this idea of potentially having too much or too little of something allows us to see the pitfall in linearity, and is the first step to breaking free of it.

This rebuttal isn't as strong to me because there was no statement supporting the claim that Mitchell had assumed a linear relationship. The case may be that from his point of view, the U.S. was in a position that required a step towards decreased tax rates in order to reach the sweet spot. However, I haven't actually read the article so this is just a little remark to take with a grain of salt.

Effect of escaping linearity

I believe this is a false mode of thinking unintentionally adopted by many of us in our daily lives. Although I believe most are in fact, aware of the conceptual dangers of linearity, we unknowingly choose not to apply nonlinear thinking because it involves more work by forcing us to question ourselves. Below are some examples.

Example 1

"The more work I do, the more success I have"

• Linear thinking:
  • "If I am not successful, that just means I have to work harder until I reach my condition of success."
• Nonlinear thinking:
  • "If we think nonlinearly, there is a sweet spot between hard work and no work. What is that spot? How do I reach it?"
  • Once we break free from linearity, we can consider other variables and add a wholeness to our analytical lens to gain a more holistic view. In reality, there is no strict correlation between hard work and success. More often than not, working smarter instead of harder is a key part of success. So we need to explore and find ways in which we can personally optimise ourselves and our habits that drive us towards the shortest path to success.

Example 2

"I have to sleep longer hours to recover from a lack of sleep"

• Linear thinking:
  • "By working longer hours and sleeping later this week, I have built up a sleep debt of 1 hour per day. Throughout the weekend, I have to sleep 5 extra hours to recover.
• Nonlinear thinking:
  • We have falsely assigned a simple linear relationship between lost hours of sleep and compensatory hours for recovery. Modern research shows that the biology of sleep debt recovery (which is very relevant in today's digital age) is more complex than that. By accumulating the compensatory hours of sleep and attempting to recover it over the weekend, we develop an inconsistent sleep-wake rhythm, which affects the quality of subsequent sessions of sleep. Experiments also show that certain physiological aspects of the body fail to "re-regulate" themselves throughout the weekend recovery period.

Hence, we can see that by developing a habit of nonlinear thinking, we get to view many subjects on a grayscale spectrum instead of black and white. This in turn allows us to think critically and approach a subject from multiple angles, developing our ability to think critically.