Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind

September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Wednesday September 14, 2005
Learning Log Number 10

6:35 am Lethbridge, Alberta

Today is a travel day as we plan to drive to Calgary to see a Paul Anka concert.

There is no doubt that I am still changing my mind on a regular basis. I have, at the moment, 3 key books:

  • Prime Obsession (2003) by John Derbyshire
  • Calculus (1994) by Michael Spivak
  • What is Mathematics? (1941) by Richard Courant & Herbert Robbins,

plus 3 more in reserve

  • Calculus (1998) by Larson et al.
  • Calculus (1977) by Morris Kline
  • The Nature of Mathematics (2004) by Karl J. Smith.

Finally, I have the book that started this review: The Road to Reality (2004) by Roger Penrose.

Each of these books is thick and comprehensive. Together they provide a rich resource as I struggle to form a synthesis. After all the goal is not to say that I have read one or more of these books, but rather to say that I understand the mathematics that they all discuss.

The goal is to understand the Penrose book. This means that I need to review, to my own satisfaction, the mathematics that he mentions. This quickly leads to a discussion of complex variables and functions. Now working backwards, I need to understand functions of a real variable and this leads to a clear understanding of calculus. So far so good. I have some excellent books on calculus. However both Spivak and Courant point out that calculus is closely tied to our understanding of numbers. And Derbyshire provides an excellent overview of prime number theory, which happens to involve a strong use of calculus and real functions. Everything connects. Again, no problem. The task is simply to map out these connections.

The trick is to find a balance between the rigor of mathematics but without becoming enmeshed in the formality.

Spivak's chapter 1 "Basic Properties of Numbers" [p. 3 - 20] is a superb presentation of the axiomatic basis of the natural numbers and the integers. His second chapter "Numbers of Various Sorts" briefly extends the concept of number to include rational and irrational numbers and real numbers. His next four chapters discuss functions (and their graphs) and the concept of limit, leading to the idea of continuous functions. This all forms the foundation for a discussion of calculus. I feel quite comfortable with all of these ideas, but feel that I am lacking specific skills for handling most questions and problems at this level. I want to rectify this situation.

The Courant book is the other book that looks most promising at the moment. Chapter 1 "The Natural Numbers" is about the positive integers with a supplement on prime numbers and congruences. Chapter 2 "The Number System of Mathematics" discusses rational and irrational numbers as well as real and complex numbers. Excellent! I then want to skip three chapters that focus on geometry, and begin again with chapter 6 "Functions and Limits". This lead to chapter 7 "Maxima and Minima" and chapter 8 "The Calculus".

In summary, both books adopt a similar sequence:

  • Numbers
  • Functions
  • Limit
  • Calculus

Each of the words represents a topic in its own right. What are the key concepts and ideas within each topic? Similarly, what are the key techniques and skills that one needs to play with these ideas? One important idea that Derbyshire mentions is that of having an automatic reflex when one sees a statement to ask how that statement might be made more general, or more specific.

Looking at the four topics I have identified, it is also clear to me that the real heart of the matter lies with the concept of function. At its most general level, the idea is straight forward: a function is a rule or procedure for taking one number (of any kind) and arriving at another number (also of any kind). However the devil is in the details. There are a phenomenal number of types of functions, and I should become intimately familiar with many of them before proceeding to look at calculus. In addition to algebraic functions, logarithmic functions, exponential functions and trigonometric functions should become my friends. Interestingly these functions cycle back to reveal new numbers (notably p, e and i). As I approach the topic of limits, I will enter the world of infinity as well as sequences and series.


Reviewing my books, I feel that Courant provides the best coverage of numbers, although Spivak's treatment of the axiomatic basis of the integers is first rate. The important point from Spivak is that from a small set of axioms one may rigorously derive most of the properties of numbers that are so familiar to us, for example that a negative number multiplied by a negative number must always be a positive number. This is simply a logical consequence of the axioms.

I am now about to read chapter 1 of Courant & Robbins, "The Natural Numbers" [p. 1 - 51]

7:45 am I am delighted with what I have written this morning on this web page. I feel that I now have a road map to help me on my journey. Unfortunately today promises to be very busy and I am not sure when I will be able to get back to this. Not likely today, and perhaps not for a few days.


Next day, when I begin chapter 1.

Here are a few quotes from "What is Mathematics?" by Richard Courant & Herbert Robbins (revised by Ian Stewart).
"What is Mathematics? is one of the great classics, ... " [Stewart]

This is not just another math book.

"Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either. " [Stewart] This is the essence of the Penrose book "The Road to Reality", which was my initial incentive to return to mathematics.
"Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. ... The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally." [Stewart] This web site might be thought of as weak journalism, but I should give some thought to trying to develop it into weak literature.
"The teaching of mathematics has sometimes degenerated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual independence." [Courant] Surely my goal is real understanding and intellectual independence.
The goal is genuine comprehension of mathematics as an organic whole."[Courant] I would say I have that sense of mathematics, but I lack the fine detail.
"Actual contact with the content of living mathematics is necessary." [Courant] This is a paraphrase of the common saying that mathematics is not a spectator sport.
"For almost two thousand years the weight of Greek geometrical tradition retarded the inevitable evolution of the number concept and of algebraic manipulation." [Courant]

Fascinating. I had never thought of the Greek geometry as being an impediment.

I wonder if we are at another point in history where we are being held back by our reluctance to embrace technology. What would happen if we had adopted Logo in the elementary grades as an important topic, and if we extended this to Geometric Supposer and Geometer's Sketchpad, as well as to Maple and Mathematica?


A nice beginning.

8:10 am

Total elapsed time: 1 hr. 35 min

Total elapsed time for the day: 1 hr. 35 min