Overview
For the past year, I’ve been thinking about nothing but English…
English is great! But I’ve had enough! I’m good for now! I want to get into math! But I also want to read world history! Right when I was feeling that way, I found a book that’s like math + world history, so I read it. Here’s a summary and my thoughts.
Summary
It’s about how mathematicians have influenced world history.
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BC287 - Archimedes of Syracuse
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The “Eureka!” guy. This is what he shouted when he came up with the principle of buoyancy, after being given the task by the king: “Check whether the crown is made of pure gold without breaking it.”
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When Syracuse fought Rome, his inventions like the “Claw of Archimedes,” “catapults,” and “heat rays” protected Syracuse from the massive Roman army for two years.
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7th century - Arabic numerals completed in India
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Until then, people used Roman numerals, which were extremely hard for calculations.
- Calculations were done using a tool called the abacus, but it doesn’t leave any record of the calculation process, so it’s not ideal.
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The numbers we use today were completed in India, passed through Arabia, and reached Europe in the 13th century. That’s why they are also called Indo-Arabic numerals.
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13th century - Fibonacci of Italy
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Introduced Arabic numerals in Liber Abaci.
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Much easier to calculate! The calculation process can be recorded! But they are easy to tamper with in bookkeeping (you can just add a 0 at the end and increase the digits), which is a downside. So there was a transitional period where calculations used Arabic numerals, while documents used Roman numerals.
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15th century - Pacioli of Italy
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He was an abacus teacher (a job teaching math to merchants’ children). While teaching, he also created his own textbook.
- That became the famous 600-page work Summa de Arithmetica, Geometria, Proportioni et Proportionalità. 36 chapters. It covers “double-entry bookkeeping” and the “Rule of 72.”
- Double-entry bookkeeping is a system where the total debit and credit match. Surprisingly, it has barely changed even after 500 years.
- Because of this, he is called the “Father of Modern Accounting.” Makes sense.
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The Rule of 72 is often used in investment: the number of years it takes for the principal to double is shown by
72 / interest rate (%). -
16th century - Viète of France
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He studied algebra and contributed to deciphering Spanish codes.
- At the time, Protestant King Henry IV of France and Catholic King Philip II of Spain were in religious conflict.
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Henry IV issued the “Edict of Nantes” to protect his throne and allowed freedom of religion. Later, in the 17th century, Louis XIV revoked it.
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17th century - Jacob Bernoulli of France
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Moved to London because the Edict of Nantes was revoked.
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He came up with the “Law of Large Numbers.” As the number of data points increases, the difference between theoretical and actual probability becomes smaller. Like flipping a coin many times and getting close to 50% heads.
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17th century - de Moivre of France
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Life insurance already existed, but pricing was very rough.
- In Annuities upon Lives, he introduced the “de Moivre’s law” and made life insurance consider that mortality increases with age.
- He assumed the maximum lifespan to be 86—and he actually died at 86. (Too impressive.)
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Newton even said, “If you don’t understand something in Principia, ask de Moivre.” That’s how much of a Principia master he was.
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17th century - Daniel Bernoulli (nephew of Jacob Bernoulli)
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He came up with the “Law of Diminishing Marginal Utility,” which influenced later decision theory.
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Even with the same quantity, the first glass of beer tastes the best. Also, the joy of going from 1,000 yen to 2,000 yen is different from going from 200,000 yen to 201,000 yen.
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18th century - Condorcet of France
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In Essay on the Application of Analysis to the Probability of Majority Decisions, he introduced the “voting paradox” and the “jury theorem.” One of the early examples of applying mathematics to sociology.
- This was just before the French Revolution, when interest in democratic decision-making was rising.
- The voting paradox shows that round-robin comparisons best reflect public opinion.
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The jury theorem shows that as the number of voters increases, the probability of a correct decision by majority vote increases.
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19th century - Quetelet of Belgium
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Founded the Royal Observatory of Belgium, but it was occupied during the Belgian Revolution. This led him to become interested in human behavior.
- At that time, sociology did not use averages much and relied on subjective experience.
- He wrote On the Development of Human Faculties and introduced the concept of the “average man” using the idea of the “normal distribution.” He also created BMI.
- Normal distribution is a bell-shaped distribution where the average has the highest frequency.
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He is called the “Father of Modern Statistics.”
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19th century - Nightingale of England
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Inspired by the ideas of the “Father of Modern Statistics,” she applied statistics in field hospitals during the Crimean War.
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By visualizing data, she discovered that deaths caused by poor hygiene were more common than those caused by injuries. By improving sanitation, she reduced the death rate from 60% to 2% in six months.
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19th century - Pareto of Italy
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He noticed that 80% of his garden’s pea harvest came from 20% of the peas, and that 80% of Italy’s land was owned by 20% of the population. He called this the “Pareto principle.”
- Also, using the “Pareto distribution,” he showed that income inequality would not worsen drastically based on past data.
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This led him to develop “elite theory,” which argues that a small number of elites should make decisions and rule the majority. Mussolini applied this idea to fascism, a form of dictatorship.
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20th century - Lanchester of England
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In his report Aircraft in Warfare: The Dawn of the Fourth Arm, he proposed “Lanchester’s Linear Law” and “Square Law.”
- The linear law shows that in close combat, combat power is proportional to the number of soldiers.
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The square law shows that in long-range combat, combat power is proportional to the square of the number of soldiers. In other words, the side with more troops will have more survivors if they fight at long range.
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20th century - Benford of the United States
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The “Benford’s law” shows that in natural data, the leading digit is most often 1.
- Sounds weird, right? But natural phenomena tend to scale by ratios (like doubling every year), so 1 appears most often, and 9 the least.
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This law is used as one tool to detect fraud in financial statements and as evidence in claims of election fraud. But apparently, there are very few real cases where fraud was actually proven. (What is that.)
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20th century - von Neumann (Hungary) and Morgenstern (Germany)
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Together, they applied game theory to economics and wrote Theory of Games and Economic Behavior.
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Von Neumann was involved in developing atomic bombs during WWII and reportedly said, “Why drop them on Hiroshima or Nagasaki? Drop them on Kyoto.”
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20th century - Nash of the United States
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After reading Theory of Games and Economic Behavior, he wrote his doctoral thesis Non-Cooperative Games, where he introduced the “Nash equilibrium.”
- Nash equilibrium is like when stores keep lowering prices saying “We’re cheaper than others!” and eventually the price competition stabilizes. In other words, it’s a situation where changing your own strategy alone would make you worse off.
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This theory influenced the Cold War, where the U.S. and the Soviet Union chose to maintain nuclear development without actually fighting.
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20th century - Arrow of the United States
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Through the “Arrow’s impossibility theorem,” he proved that there is no perfect voting system that everyone agrees on and finds fair.
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20th century - Gale and Shapley of the United States
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The “Gale–Shapley algorithm” solves the “stable marriage problem.”
- Imagine an N:N group dating event… First, each man proposes to the woman he prefers most. Each woman keeps only the most preferred man among those who proposed to her. Rejected men propose to their next choice. Women do the same. Repeating this creates a stable matching.
- This is applied to job matching and donor matching.
Thoughts
It felt more like sociology than mathematics… That’s my impression. I thought “history of mathematics” would explain how functions, geometry, and various theorems were created, but this wasn’t that kind of book. It felt more like “18 cases where mathematicians were involved in historical events.” Isn’t this not really a history of mathematics?
- Actually, I used to confuse “Roman numerals” and “Arabic numerals” (like, which is which?), but thanks to the phrase “Arabic numerals that came to Europe through India and Arabia and are easy to calculate with,” I think I can remember them properly now. History is useful like this.
- But if they were “completed in India and passed through Arabia to Europe,” shouldn’t we just call them “Indian numerals”?
- I kind of liked Summa. It’s the culmination of a textbook Pacioli created for his own classes. Also, it’s interesting that double-entry bookkeeping hasn’t changed for 500 years.
- I have a personal embarrassing story about the “Law of Large Numbers.” When a teacher said, “You’ve been coming to the staff room to ask questions a lot lately—that’s great,” I replied, “Not really. It’s just the Law of Large Numbers.” “Huh? What’s that?” “(I just said it without really knowing…) You know, when you rarely see something but remember it strongly and think it happens often.” That was totally wrong. Of course, the meaning is completely different. So yeah, black history. Sorry, teacher. I was just embarrassed by the compliment.
- The jury theorem (the probability of a correct decision increases as the number of voters increases) also means that people who want to pass incorrect decisions tend to prefer closed voting. That was interesting.
- Nightingale’s “data visualization” feels kind of obvious now… Is that really statistics? But maybe that just means statistics has become part of everyday life in modern times.
- I didn’t quite get why Benford’s law is used as evidence for election fraud. It applies to natural data, right? But vote counts aren’t exactly natural, are they?
