Just Curious

A Concise History of Mathematics

I have also finished Professor Dirk Jan Struik's A Concise History of Mathematics recently. It was a very concise account of the mathematical developments humankind has made in the past 50 centuries. There are many things in it that I was not aware of beforehand.

For instance, the ancient definitions for trigonometric functions are terribly clumsy, for sines and cosines really are defined in terms of chords and half-chords in a circle. It wasn't until the great Leonhard Euler (1707-1783) that mathematicians started to use the trigonometric proportions definitions that are used today by middle-school kids.

Another example would be that of the complex numbers.It is mildly amusing for modern readers to note that in not-so-modern times, people really had problems dealing with more abstract ideas in mathematics. For instance, in ancient Greece and China, negative solutions to a system of linear equations are called "false solutions" while complex solutions are given the name "impossible solutions!"

Aside from these factoids, it is also very interesting to observe how ideas are developed. If a modern student of mathematics were to read the original treatise on differential calculus written by Sir Isaac Newton, he or she wouldn't understand a word in it. The reason is simple: It wasn't clear to me at all that Sir Isaac himself knew exactly what he was talking about.

For instance, he did not understand the concept of infinitestimal. He also found the idea of a limit to be terribly confusing. A century and a half had to elapse before French and German mathematicians like Karl Weierstrass would lay down the foundations of analysis, when concepts like metric space, continuous mapping and derivatives, etc. were introduced.

So you see, we, as human beings, almost always acquaint ourselves first with how things work, not what things are. We learned how to use differential and integral calculus before we looked into their foundations.

Finally, I feel obliged to comment on what I would perceive as an unavoidable flaw in this book. Frankly speaking, one ought to have at least a sound undergraduate mathematics education to understand most of the concepts covered by the author. This is especiallly true in the case of the last two chapters, in which mathematical advances made in 19th and 20th centuries are covered. I call this flaw unavoidable since most ideas in 20th century cannot be explained at all without using some technical terms, etc.

At any rate, reading this book proved to be an immensely gratifying experience for me. I recommend it to anyone who's interested in learning how mathematics have progressed to the opulent art and science it is today!

posted by James Chang @ 6:30 PM