[Wrap up] Book Review: Humble PI: A Comedy of Maths Errors

In our life, mathematics is very important for logical thinking based on evidence-based knowledge through rigorous mathematical analysis. Especially, when we predict something new, the power of mathematics overwhelms our instinct or heuristics. However, when using mathematics improperly, catastrophic results are waiting for us. In this book, the author, Matt Parker, said such an important role of mathematics and showed examples of disasters stemming from mathematical errors through exhilarating stories he has experienced. 

Then, what is the role of a human in mathematics? We try to use mathematics when deciding something important. And then, we should check all the types of mathematical errors to avoid the disaster. I would like to introduce his last paragraph. “Our modern world depends on mathematics and, when things go wrong, it should serve as a sobering reminder that we need to keep an eye on the hot cheese but also remind us of all the maths which works faultlessly around us.”

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

(1) What Number Is a Really Big Number?

(2) Please Give Math More Time to Pick up the Pieces

(3) I Don’t Count on You When You Count Numbers

(4) More Approximations, More Problems in Your Life

(5) Probably, We Are Not Independent

(6) Searching for Average Man

(7) Sometimes, Simple Mathematics is Better than Our Experiences

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.

Searching for Average Man

humble pi average

“How many of the 4,063 people in the survey could wear such an approximately average uniform? The answer is zero.”

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

An average is the simplest statistic to measure the characteristics of data; just add up all the values of data and divide it by the number of data. This simple calculation has been widely used to set a standard for assessment such as average earning, average height, or average life expectancy. Statistically, the average is vulnerable to an outlier. Moreover, the average does not give us information about the distribution of data. That’s why statistical analysis usually provides the average and its standard deviation for sample data. Still, these two values are not enough to represent data due to a large variability (e.g. using any values of average and of standard deviation, we can draw a dinosaur silhouette). Hence, you should choose the right statistic for a clear understanding of the data.

Then, can we say the average is the best number for representing our group or society? The answer is NO. The average man is not a real person and also not a standard for us. In the book “The End of Average“, the author, Todd Rose, showed the failure of our education based on the statistical average. An average is just a number calculated without any context. So, we don’t need to set the average as our standard for success or failure. The annual performance rate cannot measure your success rate in your life. The academic GPA cannot give you the grade of your life. Please don’t follow a phantom of the average. Keep pace with your own plan in your life. After being an outlier in success, please tell us your amazing story, not just give a number.

Probably, We Are Not Independent

humble pi joint probabitlity

“Getting our head around probabilities is very hard for humans. But in high-stakes cases like this, we have to get it right.”

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

Since a human is born to be a prediction machine, we naturally estimate the probability of everything. When amazing things are happening in a row, we think it is improbable. This is because we usually estimate the probability that each thing happens and multiply all these probabilities in our head, leading to the tiny probability (close to zero); this is a simple rule to calculate a joint probability. However, to use this rule, there is a hidden assumption that all the events are independent (the probability that one event occurs does not affect the probability that another event occurs). But all the events in our life are not always independent. Some events are correlated, have a cause-and-effect relationship, or occur in the same environment. So, amazing things may happen in a row with a high probability.

If you want to predict some extreme results using such a calculation of joint probability, you consider the independence of each event first. Only the right calculation of the probability is a guardian against a catastrophic result. For example, on the street, you met ten persons whose height is over 6′ 4” in a row. Then, you think that a strange thing happens today. But if you met the same persons in front of a basketball court, then you feel that this situation is somewhat reasonable. The same persons but the different places can change your probability (sometimes we consider a conditional probability). Hence, a careful calculation of a joint probability changes looks-like improbable things to probable things (or vise versa). Even though you calculate the joint probability and the result is very small, you should keep in mind that improbable things happen all the time.

More Approximations, More Problems in Your Life

humble pi approximation

“When the time value is small, the error is also small. But the problem with a percentage error is that, as the value gets bigger, the error grows with it.”

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

Rounding a number is a quite simple method to approximate some complicated numbers, leading to easy calculation and good memory. However, such approximation makes a percentage error which makes a big problem when the value is big. Moreover, error grows bigger when multiplying this approximated number many times. To reduce a percentage error for mass production, in the late 20th century, many companies have focused on quality management such as Six Sigma. Specifically, companies have made continuous efforts to reduce a (percentage) error, leading to successful and predictable process results. In data science, to obtain high-fidelity prediction with big numbers, they have kept a significant digit as many as they can.

What about individuals? we often think that we made a 100% effort for my task but we actually made a 95% effort and rounded off. This approximation will make a problem in the future. What? you can say that the 95% effort is high enough to say 100%. Is it true? Let’s take an example, you usually complete a task with a 95% effort (here, say, a success rate) and you have 20 tasks now. Then, the probability that you complete all 20 tasks successfully is 35.8%. However, if you make a 99% effort for each task, then the probability will be 81.8% (and 98% with a 99.9% effort). This example shows that a small different percentage makes a big difference, and we agree that we should complete more than 20 tasks to achieve success in your life. That’s why we should do our best (close to a 100% effort) every time to reach your goal in life.

I Don’t Count on You When You Count Numbers

Humble PI

“The only downside is that you break the link between the number you are using to keep track of your counting with the number of things you are counting.”

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

Since Adam was a boy, counting has long been recognized as the most important skill for humans to survive in the world. So, I believe that you (even if you are a toddler) can count numbers well. Let’s check it out. how many numbers you can count on your fingers? The answer is eleven (not ten). This is because you can also count zero with all folded fingers (the more correct answer is 1024, please search for “finger binary” on google). Next, how many natural numbers from 10 to 99? The answer is 90 (not 89). Hooray, you got the right answers, I count on you!

We are more getting into trouble when counting large numbers in efficient ways. For example, when we count the total number of events for calculating probability, we use some math skills such as permutation and combination. However, these are too tricky to use simply. So, when you make a decision based on probability (e.g. Bayesian approach), miscounting the number of events results in a totally different probability, leading to a wrong decision. Please don’t count on yourself when you count numbers (specifically, counting sheep to sleep or counting cards to win the blackjack).

Please Give Math More Time to Pick up the Pieces

humble pi

“We make things beyond what we understand, and we always have done. … When theory lags behind application, there will always be mathematical surprises lying in wait.”

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

In human history (specifically engineering and technology), many successful achievements have shown their effectiveness without proven scientific theories or rigorous mathematics. For example, first flying to the moon in 1969 was achieved with little knowledge of rocket science, aerodynamics, and astrophysics. Sometimes, we called such achievements “the greatest challenge for humans.” However, the word “challenge” here implies that we do not know a mechanism or theory well. I don’t want to underestimate such a challenge but we have experienced that applications without coherent theories may lead to a catastrophic disaster.

Then, in data science, when is the right time to adopt a new model? Should we wait until we understand all theories and mechanisms and prove them all by mathematics? It may be too late. So we should decide the right time by ourselves but we always keep in mind the negative effect when the introduced model fails. So, we try to quantify uncertainties of the model (or our decision) and estimate a probable disaster as insurance companies do. It is much robust thinking rather than efficient thinking. It may lead to slow progress and more cost but it can give us a second chance to correct the model when the model fails. So, if you don’t have rigorous mathematical support, please think uncertainty and make your model robust. Moreover, when you decide something without evidence in your life, please make your decision robust, too.