“Maths is the true language that the universe is written in – the key to understanding the world around us.”
– Marcus du Sautoy, in the conclusion of this series
In America, we often shorten the word “mathematics” to just “math.” In Britain, they retain the pluralization of “mathematics” to make it “maths,” even when shortening it in this way. Thus, no one from Britain would ever be likely to say just “math,” and would probably consider it an Americanism that would sound a little strange to them. But whatever you call it, I've been tutoring people in the subject since 2012, as a professional “math” tutor (and I am an American, as you may have guessed from my spelling of this word).
Disclaimer: My degree was in business, rather than math
I should give a disclaimer to say that I am not really a true mathematician, since my degree was in something other than “math” (namely, business). Nonetheless, I know enough about it to tutor the lower levels just fine; and I'm interested enough in its history that I decided to watch this documentary just for kicks. I figured that if I was going to be a professional “math” tutor, I ought to know something about the history of the subject. It was certainly enlightening to me, and it helped me to better appreciate our heritage of great mathematical ideas. Perhaps it will even help me as a tutor somehow – although I would have considered it to be worthwhile for me, even if this program had not done so.
Episode One: “The Language of the Universe” (Egypt, Babylon, Greece)
The series is about four hours long, and it is divided into four episodes of roughly an hour each. The first episode is called “The Language of the Universe,” and it starts at the very beginning of the history of mathematics. The Ancient Egyptians were the first to pioneer a “base-10 numbering system,” which is sometimes referred to as a “decimal system.” The word “decimal” actually has an additional meaning from its being part of the phrase “decimal point,” and it is sometimes used to mean digits to the right of a decimal point. But the origin of the word is from Latin “decimus” (meaning “tenth”), which comes from the Latin word for ten. In this context, it means a base-10 numbering system, which is definitely a legacy of the Ancient Egyptians. It was an important invention, partly because it was built on the fact that a human being has ten fingers. This made counting in such a system easier for most humans, and allowed for “place value” columns of ones, tens, hundreds, thousands, and so forth. They also invented a “base-2 numbering system,” which is also known as a binary number system. Obviously, this would be important for the history of computers later on, where “on” and “off” switches were a major part of their functioning.
Ancient Egyptian pyramids at Giza
But the greatest Egyptian invention in mathematics may have been another innovation, which was the fraction. This is still used today in many contexts, and modern calculators have made it even easier to use them. The documentary also makes some note of the Sumerian and Babylonian “base-60 numbering system,” which was another legacy of the ancient world. We don't use base-60 numbers much in the contemporary Western world, but we do have 60 seconds in a minute and 60 minutes in an hour (not to mention 60 minutes in a degree for some angle applications). Thus, we may have been more influenced by this number system than we realize. The documentary also notes Ancient Greek mathematics which were more theoretical. Among other things, the Ancient Greeks discovered the existence of irrational numbers; but they also developed the proof as one of the logical tools of mathematicians, and made mathematics into the analytical subject that we know today.
Pythagoras, the person for whom the “Pythagorean theorem” was named
Euclid of Alexandria, who developed Euclidian geometry
Episode Two: “The Genius of the East” (India, China, the Middle East)
The second episode is called “The Genius of the East,” and it focuses on major mathematical ideas from outside the West. For example, Chinese mathematics for engineering are mentioned. But the most important development mentioned in this episode may have been what are called the “Hindu-Arabic numerals.” They are often erroneously referred to as just the “Arabic numerals,” but they were actually a Hindu innovation. They acquired the name of “Arabic numerals” mainly because they were popularized in the West through the Middle East, who had a lot of great mathematicians of their own (I acknowledge with some admiration). For those who don't know, the “Hindu-Arabic numerals” are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. They are obviously the building blocks of all larger numbers (such as the number 250), as well as numbers after the decimal point (such as the last two digits of 0.75). More importantly, they included a symbol for “zero,” which was one of the most important breakthroughs in the history of mathematics. The Babylonians had included a blank space for the number zero, but no separate symbol for this mathematical concept. Although the prior languages of human history had included words for “nothing” and “nothingness,” a separate mathematical symbol had never really existed for zero before. Thus, it was definitely the invention of the Hindus, and adds to the glory of India in many ways.
Gwalior Fort in central India, where one of the oldest mentions of zero is found
Like the “Hindu-Arabic numerals” themselves, the idea of “zero” is often erroneously credited to the Arabic world, but it was the Hindu world that really discovered it. The Arabs, however, were truly more important in developing the discipline of algebra than anyone else – the word “algebra” being an Arabic borrowing, translating to “the reunion of broken parts.” The word's origins may thus hint at a kind of rearranging, and this kind of rearranging is fundamental to many kinds of algebra. When I tell my students to “do a little algebra,” I'm typically referring to rearranging the equation so that the variable is by itself. This allows you to say what the variable equals, and thus gives you the solutions to many different kinds of problems in the process. The Hindus also invented negative numbers, infinity, and trigonometry – plus calculating a relatively exact value of π, or “pi.” (I would comment on this episode's coverage of the Fibonacci number sequence, but I admit that I don't really understand this sequence, even if I do recognize its importance for some things.)
Leonardo Fibonacci, famous for the Fibonacci number sequence
Episode Three: “The Frontiers of Space” (Europe)
The next episode is called “The Frontiers of Space.” It focuses mainly on European mathematics and its various breakthroughs. Some have actually criticized this episode for being “Euro-centric” in some unknown way; but considering how much the previous episodes are actually focused on the East, I find this criticism somewhat hard to support. There actually isn't much coverage of American mathematicians in this series, either; but as an American, I don't feel at all shortchanged by this. The United States actually hasn't been around for as long as some of these other nations, and so I thus feel that we will gain more triumphs given the adequate time. Among other things, the presenter also discusses René Descartes's merger of algebra and geometry, which allowed us to graph equations on what we still today call the “Cartesian coordinate plane” (named after Mr. Descartes).
René Descartes, who merged algebra and geometry via the “Cartesian coordinate plane” (a. k. a. graphing)
Another breakthrough was in the simultaneous invention of calculus by both Isaac Newton and Gottfried Wilhelm Leibniz. These two men actually hated each other intensely, and each tried to minimize the contributions of the other. It turns out that they both invented calculus independently from one another, but that Newton's way of writing it was not easy to follow (to put it mildly). Leibniz's way made much more sense, so it was the version that eventually triumphed among Western mathematicians. It is still the standard way of writing calculus today, even in non-Western nations. This is actually one of the greatest controversies in Western intellectual history, and it is thus covered in some depth in this episode. The presenter's view is that Isaac Newton may have been the greater physicist, but that Gottfried Wilhelm Leibniz was still the greater mathematician. (The presenter does acknowledge, though, that Newton had some major contributions of his own in mathematics; and that students today are still taught “Newton's method” for obtaining roots and “zeroes” of some things in calculus – a testament to the mathematical accomplishments of Mr. Newton.)
Isaac Newton, co-inventor of calculus
Gottfried Wilhelm Leibniz, the other co-inventor of calculus
Episode Four: “To Infinity and Beyond” (the twentieth century)
The last episode is called “To Infinity and Beyond,” and it focuses mainly on mathematics in the twentieth century. This episode may be a little theoretical for me to really appreciate it, because it seems to be mostly about complicated proofs of various modern hypotheses. I suspect that its logical validity is actually quite good, but I lack the mathematical background to find this out for myself. More to the point, I don't really want to acquire the background necessary to do so; and would much prefer to stick to tutoring the lower levels of math in my job. It just sounds like way too much work for someone of my interests. Thus, I may not be qualified to write a good review of this episode, although I did enjoy it nonetheless.
Bernhard Riemann, the originator of what is today known as the “Riemann hypothesis”
Conclusion: The greatest television history of mathematics that you're likely to find
I have not really done justice here to the history of mathematics in a blog post of this length. Even this four-hour BBC program doesn't really have the time to do it justice, as it turns out. But I hope I have given my readers a brief overview of a few major developments in the history of mathematics. If you want to learn more, feel free to watch this series in your spare time. It might be a good investment of time, if you spend any of your professional life or your free time doing mathematics. You might also enjoy it if you just like to hear about history, and wouldn't mind hearing about how human beings have thought and reasoned in the past, and how they continue to do so today based upon past discoveries in this area.
“Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers.”
- Marcus du Sautoy, as quoted by The Guardian
DVD at Amazon
If you liked this post, you might also like:
Some thoughts on mathematics education
The difference between skepticism and closed-mindedness
Why is my stats class so focused on bell curves?
How do the economic laws of supply and demand work?
How fractions are used in the United States Constitution
See also the audiobook series
Science & Discovery
Others to be covered later
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