Math Every Morning

For fifty consecutive mornings, I have enjoyed math. Specifically, self-study of Stephen Abbott’s Understanding Analysis has given me exercise and enjoyment.

This is, or ought to be, a review for me. I took Calculus and Analysis in High School, 34 years ago, from a wondrous teacher. My appreciation for this man, the late Mr. Jack Dobelbower, has grown over time, and more so in the last 50 days as the cobwebs are cleared from my memory. I can see clearly now how much I missed by not studying more deeply then. Since High School, I have taken more advanced math classes, getting C’s and B’s sometimes A’s. but I never felt that I really mastered math, except for parts of Computer Science.

Math is full of mysteries. Being able to “do” math is not the same as understanding it fully. For example, how can it be that the infinite sum of the reciprocal of the squares of the natural numbers is equal to Pi squared over six? What does this mean? Connected to the natural numbers is the most “circular” of all numbers — but why?

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Math is deep; I want to understand it deeply. In the past, I did enough homework to pass a class. In a typical college class, one might be assigned one fourth of the exercises as homework. Now, I am doing every exercise in the book. I can’t stand to leave one undone. Perhaps a more confident student, or a student hungrier to reach more advanced material faster, would not need to do this. I have done 55 problems (usually proofs) in 50 days at about 35 minutes each morning.

My friend John K. Gibbons and I similarly work problems from Introduction to the Theory of Computation by Michael Sipser on a weekly basis; but we are not doing every exercise in the book.

I have learned:

  • Abbot has written a great undergraduate book that combines good explanations with excellent exercises coherently, enlivened by historical insights.
  • It remains unclear to me what constitutes a really good proof. I have always struggled with this; I tend to mistrust proofs that are not very formal.
  • Not having a solution book, nor anyone to grade my work, I don’t know for sure that I am solving the problems correctly. However, except for occasional self-delusion and confusion, I subjectively “get it” after enough time, and so am probably correct most of the time.
  • My curiosity for even the simple mathematics introduced in Chapter 2 on sequences and series, is quickened, not quenched. Question arise; I usually can’t tell if these are deep questions or silly misunderstanding on my part.
  • I would remind all students, as Abbott does, that even early chapters of a modern course took humanity centuries to figure out.

I cannot prove that this is making me smarter. I have no interest in competition; I don’t want to compare myself to others. Nor can I even assert that I know much more than when I started this morning self-discipline. At the current rate, I may require an entire year to finish Understanding Analysis. I have faith it will be valuable; I have certainty it will be fun.