★★★★★ Euclid alone...

Understanding Analysis

Stephen Abbott

According to John Derbyshire. Mathematics is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. You know what arithmetic and geometry are, and you probably have taken a high-school algebra class. "Analysis", however, is a little obscure. The word has a specialized meaning in mathematics. It is that branch of mathematics that includes calculus. More properly, analysis is the mathematics of the continuum.

The calculus was developed in the late 17th century by Isaac Newton and Gottfried Leibniz. Newton and Leibniz however, didn't quite know what they were doing and inevitably they were a little sloppy about defining things. (This is usual when a new area of mathematics is developed.) At the heart of the problem was this: calculus is all about continuous things -- in calculus space and time are continuous. What that means roughly is that we assume in calculus that every point on the line between two points A and B exists. (There is reason to believe this may not be physically true, but that is not relevant to the mathematics under discussion.) Furthermore, we assume that a number can be assigned to every one of those infinity of points.

That is not a precise definition of continuity. Defining continuity is surprisingly difficult. The ancient Greeks were aware of the problem -- this is what Zeno's paradoxes are all about. Furthermore, the Greeks knew that no number (as they understood numbers) could be assigned to the length of the diagonal of a 1 ⨉ 1 square.

In the 19th century this problem was figured out by European (mostly French and German) mathematicians. Some names to conjure with here are Weierstrass, Dedekind, Cauchy, Riemann, and Cantor. These are names every mathematician knows. Over the course of several decades they figured out how to rigorously define the continuum and to assign a number to every point on the line. These are called the real numbers, symbolized ℝ. The 19th century analysts did work of astonishing beauty, which, sadly, most people will never perceive. Analysis is now a course that every undergraduate math major is expected to take. It is generally regarded as the most difficult such math class.

In 2015, I was a professor with a 40-year career as a scientist behind me. I decided to retire and go back to school for an advanced degree in mathematics. I had never taken a course in analysis. That was a gap in my education I needed to remedy. I therefore worked my way carefully through Stephen Abbott's Understanding Analysis. This worked. In fall 2015 I took my first actual analysis course -- Functional Analysis, a postgraduate course. I don't remember my exact grade, but it was in the 90s.

So that was good -- it was why I read Abbott -- I got what I hoped from it. But I got much more than that. I was not prepared for the aesthetic experience. Math students don't talk about the beauty of analysis -- generally they are too traumatized by the effort to get through the most difficult course they have ever taken. Abbott does, though. In his preface he writes,

Yes these are challenging arguments but they are also beautiful ideas. Returning to the thesis of this text, it is my conviction that encounters with results like these make the task of learning analysis less daunting and more meaningful.

So, I will dare to challenge Edna St. Vincent Millay -- it is not Euclid alone who has seen beauty bare. Weierstrass, Dedekind, Cauchy, Riemann, and Cantor have also seen her. And thanks to Abbott, I, too am one of those fortunate ones

Who, though once only and then but far away,
Have heard her massive sandal set on stone.

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