Sometimes, Simple Mathematics is Better than Our Experiences

humble pi simple math

“If a new system is implemented, humans can be very resourceful when finding new ways to make mistakes. It can be very dangerous when humans get complacent and think they know better than the maths.

[Humble PI: A Comedy of Maths Errors, Matt Parker]

If someone has accumulated knowledge based on a lot of experiences in a particular subject, we call her/him an expert. When a paradigm is shifted or a new knowledge system comes, we often follow existing experts’ opinions without a doubt. I don’t underestimate the important role of existing experts to show a vivid and clear vision for the new age, stemming from appropriate heuristics. However, sometimes their knowledge and experiences are obsolete. For example, the experts about the Ptolemaic system lost their power in the age of the Copernican system. When a new system of knowledge is coming, what can we do? Should we stick to old ways of doing things?

Our model based on previous experiences is suitable for predicting similar tasks but vulnerable to predict rare events. The model (or knowledge) based on mathematics, however, is still robust to an extreme case or a rapid change. Moreover, our knowledge from experiences seems to be fragmentary and unconnected for understanding a big complex system while mathematics can describe it more effectively and clearly. Hence, when you do something totally new, please think one more time before act on instinct or previous experience. Human keeps on making the same mistakes over and over again. Also, previous experience may not predict something new effectively. Instead, find evidence (or facts), do mathematical thinking (or making mathematical model) with them, and take the simplest way to understand without logic fallacy. I cannot say such mathematical thinking is the best (or the only) way to be right. Rather, mathematical thinking is the way not to be wrong.

[Wrap up] Book Review: How Not to Be Wrong: The Power of Mathematical Thinking

We need to focus on the book title: How not to be wrong. Why did the author, Jordan Ellenberg, not say like: How to be right? This is because mathematical thinking is not the fruit of the Tree of Knowledge. Even though we equipped ourselves with concrete mathematical thinking, we cannot get the right answer to some problems we faced in the world. However, mathematical thinking helps us to correct our view based on a popular misconception and prejudice and to understand the structure of the world more clearly.

In this book, the author presents several mathematical misconceptions (more focused on statistics) that make the wrong decision and prediction, and show how mathematical thinking can overcome such kinds of obstacles. Since mathematical thinking is the extension of common sense by other means, the author said that we need more math majors for non-mathematician such as more math majors for non-mathematician such as math major doctors, high school teachers, CEOs, and politicians.

The following links are some quotations from the book with my thoughts.

(1) Do You Want to Be a Nonlinear Thinker?

(2) The Past is in the Past: the Law of Large Numbers

(3) Improbable Things Happen All the Time

(4) Can We Predict our Future in Chaos?

(5) Make Your Problem Harder!

(6) The Triumph of Mediocrity: Do not Stumble on Your Success

(7) Everything is Connected but Not Correlated

(8) When You Meet a Mathematical Genius

When You Meet a Mathematical Genius

how not to be wrong

“Athletes don’t quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was ahead of them.”

[How not to be wrong, Jordan Ellenberg]

I know this is a little bit off the topic (and the style) of this blog but I would like to write this post for kids/students who want to be a future mathematician. In my life, I have met several mathematical geniuses equipped with complete mathematical skill sets, intuition, reasoning, and creativity. Many people may think that geniuses are not willing to work hard but all the geniuses I met before put their whole energy into solving some mathematical problems always. Hence, when I had met them, I had felt that there is NO chance to defeat such kinds of geniuses and had felt depressed every single day. Many prodigious kids/students give up chasing their dream to become a great mathematician like this way.

However, doing mathematics is not a race and competition to choose the only one winner. It is more like team sports. For example, in Football, the Quarterback looks like the one and only hero to win the game but it is not true. There are many unsung heroes to try to get a score and win the game. Likewise, the development of mathematics is not the exclusive property of the math geniuses. I don’t want to underestimate the role of math geniuses; they always give us a new point of view about mathematical thinking. But rather, I would like to redound to many roles of other mathematicians such as building rigorous mathematical formulation from brilliant ideas and applying this mathematical concept to various problems in the real world. Hence, “Genius” may represent not a person but a team (or generation).

So, here is my humble advice when you meet mathematical geniuses in your life:
(1) Do not compare yourself to them (everybody has an important role in developing mathematics).
(2) Learn everything from them as much as you can.
(3) Put your whole energy into developing, extending, and applying their brilliant idea.
(4) Do not give up.
(5) Be grateful for being contemporaneous with the great geniuses.

Everything is Connected but Not Correlated

How not to be wrong

“Correlation is not transitive. … The non-transitivity of correlation is somehow obvious and mysterious at the same time.”

[How not to be wrong, Jordan Ellenberg]

In Hollywood, the Bacon Number of an actress/actor represents the closest connectivity to the actor, Keven Bacon through movies. Surprisingly, we observed that almost all the actresses/actors can be connected to Keven Bacon within six steps, called this: “Six Degrees of Separation” or “Small World.” This concept originally stems from “Erdős Number” in mathematics and science research, representing a collaborative distance to the mathematician, Paul Erdős. (My Erdős number is 4 by-the-way). What a small world and we feel that everybody is connected!

Sometimes, we confuse a correlation with a connection (or relation). A correlation is not transitive. Even though A and B are strongly correlated and B and C are also correlated, nobody can guarantee that A and C are correlated. However, we often think that there should be a correlation between A and C because we get used to syllogistic reasoning. Moreover, when we mixed up with causality, correlation, and relation, it’s a disaster. So, please do not make any transitivity for mutually correlated data. Also, we keep in mind that uncorrelated data can have a relationship with each other. We, you and I, are connected in the small world but we may not (or may) be correlated with each other.

The Triumph of Mediocrity: Do not Stumble on Your Success

Triumph of Mediocrity

“That’s what causes regression to the mean: not a mysterious mediocrity-loving force, but the simple working of heredity intermingled with chance.”

[How not to be wrong, Jordan Ellenberg]

At the beginning of the month, I check the number of visitors and views on my blog and say: “What? Too many people come in! Then, my blog is ON PACE to break my monthly record!!” I am really excited about this shock rise. At the end of the month, my eyes widen in surprise because the average number of people visited, no new record (Sigh). This shows “The Triumph of Mediocrity.” Some data intertwined with deterministic factors and uncertainties show a tendency to regress to the mean.

This simple mathematical observation gives a lesson about how to live. There is no (deterministic) equation of success. Even if it exists, it has too many uncertainties so we cannot solve this equation. When you achieved something that you want, this success does not only stem from your skills, abilities, intelligence, and effort. Rather, uncertainties (many people call this “luck”) may drive your way to success. Just when you think that you find the equation of the success, your next try may fail and you will be back to the mean – we call this “Sophomore Slump.” So please be humble. please do not stumble on your success. Also, if you did your best but failed, please try one more, the triumph of mediocrity may take you to the success.

Make Your Problem Harder!

How not to be wrong

“Instead, we turn to the other strategy, which is the one Birbier used: make the problem harder. That doesn’t sound promising. But when it works, it works like a charm.”

[How not to be wrong, Jordan Ellenberg]

When your friend was struggling with a difficult problem, we often said: “Don’t make it complex, just start with a simple problem”. This is because we have experienced that this simplification provides some clues for solving the difficult problem. This is what mathematicians actually do every day. When proving some statements, they start from the simplest case and expand it to the target problem. However, sometimes, making the problem harder suggests a simple alternative way to solve your real problems effectively.

Many data scientists have focused only on reducing the number of features to make a data-driven model simper. However, this approach does not always give the simplest model. The projection onto the low-dimension (fewer features) may make the data structure more complicated, leading to a failure of spotting the hidden pattern. Hence, sometimes, they need to increase features to make a model simpler (because of more data, more simple). This alternative thinking (adding more features) embodies the trade-off between a simpler model with many features and a complicated model with few features.

Can We Predict our Future in Chaos?

“For human action we have no such model and may never have one. That makes the prediction problem massively harder.”

[How not to be wrong, Jordan Ellenberg]

In the weather, the very tiny scale of energy at a certain location can change the global outcome dramatically – we called this chaos. Edward Lorenz discovered this and wrote: “if the theory were correct, one flap of a sea gull’s wing would be enough to alter the course of the weather forever”. Even though we have an accurate mathematical model (or a data-driven model) for the weather forecast with tons of measured data, we can make only a short-range prediction.

Our behaviors in society are much more chaotic than the weather, leading to a failure of prediction of future outcomes. Moreover, we have no mathematical model to describe our behaviors effectively. Hence, it is really hard (or impossible) to find “right” causation from the massive data. In this chaotic system, we should keep in mind the followings: (1) don’t make any causation from your success (rather, say, just “lucky”); (2) don’t follow others’ successes (a tiny different condition makes a totally different outcome); (3) don’t prejudge the situation using “common sense” (no one can predict the outcome).