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

August 2005 Mathematics Learning Log


An Example of a "Learning Process" Journal

Thursday August 25, 2005
Learning Log Number 1

6:20 am Lethbridge, Alberta

This is my latest attempt to create a web site that captures both my Learning and my reflections while engaging in a series of activities that I hope will improve my understanding of calculus and number theory. I began with the idea of Learning some calculus almost a year ago and have played with various formats for a web site as well as with different ideas of exactly what and how I would Learn. Where am I this morning?

First, I consider this sequence of Learning Process Journal web pages to be the heart of the process. I will not try to make a separate notebook that summarizes my understanding of calculus or of number. Instead I will create one set of pages that incorporates my comments as I proceed. The following paragraphs are copied from a page I made yesterday while still working within an older framework.

Here are two quotes that I have noted in the last few weeks. The first was from the Michael Spivak book on calculus: "A great deal of this book is devoted to elucidating the concept of numbers, and by the end of the book we will have become quite well acquainted with them. ... It is therefore reasonable to admit frankly that we do not yet thoroughly understand numbers. ... we will have to consider a little more carefully what we mean by 'numbers'. " [p. 12 - 13]

Then this morning, while collecting my books I came across a textbook from my last year of undergraduate study, "Principles of Mathematical Analysis" by Walter Rudin (1964). The first sentence is "A satisfactory discussion of the main concepts of analysis (e.g., convergence, continuity, differentiation, and integration) must be based on an accurately defined number concept." [p. 1]

Thus I now feel that I should embark on a two-prong approach to Learning calculus. In addition to focusing on calculus, I now want to also spend a similar amount of time on studying numbers themselves. If nothing else I remain flexible. In passing I notice that the undergraduate curriculum where one takes a few courses at the same time, but where the courses have some overlap, is consistent with what I am now endeavoring to undertake. The real issues are time, and intellectual background. Since almost all of the actual mathematics has not been used in over 40 years it seems fair to assume that the actual detail is forgotten. Fortunately the positive aura surrounding the detail is still intact, which should be a valuable support to my future Learning.

I have a substantial number of books on number. It is time I spent some time with them. The first task is to collect them into one area on my book shelves.

Here is a list, ordered by date of publication:

1979 Douglas R. Hofstadter Godel, Escher, Bach: An Eternal Golden Braid
1983 Graham Flegg Numbers: Their History and Meaning
1987 Eli Maor To Infinity and Beyond: A Cultural History of the Infinite
1988 Ivars Peterson The Mathematical Tourist: Snapshots of Modern Mathematics
1994 Calvin C. Clawson The Mathematical Traveler: Exploring the Grand History of Numbers
  Eli Maor e: The Story of a Number
1996 Calvin C. Clawson Mathematical Mysteries: The Beauty and Magic of Numbers
  John H. Conway & Richard K. Guy The Book of Numbers
1997 Stanislas Dehaene The Number Sense: How the Mind Creates Mathematics
  Hans Magnus Enzensberger The Number Devil: A Mathematical Adventure
  Jan Gullberg Mathematics: From the Birth of Numbers
1998 Keith Devlin The Language of Mathematics: Making the Invisible Visible
  Paul J. Nahin An Imaginary Tale: The Story of
1999 Donald C. Benson The Moment of Proof: Mathematical Epiphanies
2000 Amir D. Aczel The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
  Charles Seife Zero: The Biography of a Dangerous Idea
2001 Clifford A. Pickover Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning
2002 Mario Livio The Golden Ratio: The Story of Phi, The World's Most Astonishing Number
  Clifford A. Pickover The Zen of Magic Squares, Circles, and Stars
2003 Robert Kaplan & Ellen Kaplan The Art of the Infinite: Our Lost Language of Numbers
  Barry Mazur Imagining Numbers (Particularly the Square Root of Minus Fifteen)
  Alfred S. Posamentier Math Charmers: Tantalizing Tidbits for the Mind
  Marcus du Sautoy The Music of the Primes: Searchng to Solve the Greatest Mystery in Mathematics
2004 Bulent Atalay Math and the Mona Lisa: The Art and Science of Leonardo da Vinci
2005 Dan Rockmore Stalking the Riemann Hypothesis: A Quest to Find the Hidden Law of Prime Numbers

Perhaps most noteworthy is my preference for books. I like to surround myself with them and to move from one to another as I try to come to grips with the material.

And finally, here is a copy from a posting that I made to my blog earlier this morning.

There are moments when I feel that I am collecting butterflies. There are many kinds of numbers: the natural numbers, the integers, prime numbers, odd numbers, even numbers, rational numbers, irrational numbers, imaginary numbers, complex numbers, transcendental numbers, cardinal numbers, ordinal numbers, congruent numbers, ... I have also seen a quote ascribed to the German mathematician Kronecker in a number of my books, "God created the natural numbers, the rest is the work of man."

I find it fascinating to look at the books I have on number. There are a total of 25 that have a major emphasis on numbers. I have about a dozen on calculus. All of these books describe some aspect of the topic with some having a fairly well-defined sequaence and structure while others are more like a collection of interesting facts and problems.

However none of the books describe the actual process of Learning the material, not do any of them attempt to describe the actual mathematical knowledge that the author possesses about the topic. In fairness some of the books do attempt to convey a sense of the excitement of playing with the topic, but I have not seen a book that tries to set out how the author conceptualizes the topic and how she explores ideas within the topic. This blog has been a preliminary attempt to do this, and my future web site will be a more detailed description of my personal journey over the next few years.

However the real goal is not this blog, nor my web site. The real goal is the modification of my mind to better incorporate a number of mathematical ideas and to improve my understanding of mathematics and hence of science and philosophy. The blog and web site are only cloud chambers of an underlying mental transformation.


I have created a new web site consisting of just a home page (an index to the Learning Log web pages) and this first Learning Log page.

My next step will be to make a few notes about the natural numbers, which begin with the number one and the idea that each number has a successor. The natural numbers are also called the positive integers. A very important subset of the positive integers are the prime numbers, numbers that have only the number one and itself as factors.

7:30 am

Total elapsed time: 1 hr 10 min.

10:20 am Second session of the day

I read the first 12 pages of "The Art of the Infinite: Our Lost Language of Numbers" by Robert and Ellen Kaplan (2003) this morning.

The real challenge is one of deciding on a format for these notes. The notes should contain both some description of the essential mathematics ideas as well as my comments on the process.

Looking at what I have created so far, I have this tan background for general process statements and a bright yellow background for content and a bright aqua background for reflection. Now to give it a try.


The Natural numbers consist of the positive integers beginning with the number one, and for each number there is a successor, created by adding one to that value.

Theorem: There are an infinite number of Natural numbers.
Proof: Assume the theorem is false. That is, there are a finite number of Natural numbers. Let this number be n. But I can then create a new number, n + 1. This contradicts the statement that n is the last number.

Notation: Since there are an infinite number of natural numbers, we need a way of writing them that involves something less than an infinite number of different symbols. There is actually no pressing need for this if one does not have a written system to begin with, nor is there a need if the only reason for having the numbers is to identify how many objects are in a small collection. But if one wishes to use large numbers and if one wants to engage in some form of computation, then some form of written system is needed. There have been numerous such systems created by different civilizations over the last few thousand years. The Roman numeral system is perhaps the most familiar to us, although there were many others including the ancient Sumerian cuneiform system and the Mayan system, to name just two. Our present system uses only 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) but makes use of the idea of position to further elaborate on the nature of the value. (e.g. 11 means 10x1 + 1).

In addition to a notational system based on arbitrary symbols it is also possible to create a variety of pictorial systems that capture the essence of some properties of these Natural numbers. These are usually based on some arrangement of an appropriate number of dots, but once the dots have been placed on the surface other properties may become apparent. For example, one might categorize numbers by their "shape". Thus there are rectangular numbers, triangular numbers, square numbers, pentagonal numbers, n-polygon numbers, as well as other shapes such as notched numbers. One might even consider adding color to some numbers in order to help the eye see various patterns. One example might be a standard checkerboard where the even-numbered squares are white and the odd-numbered squares are black. One of the advantages of playing with such representations is to make explicit the idea of pattern as well as the idea of a relationship between certain kinds of numbers. Although this might be interesting, in general it has turned out that such explorations have had limited applicability outside of the specific exploration.

So far I have only discussed the natural numbers and one operation, addition.

However the exploration of patterns of numbers based on the operations of addition, subtraction, multiplication and division have yielded rich rewards in helping us look for regularities in the physical world (i.e. science).

Another special topic within mathematics is that of Number Theory, and within that is the special place for prime numbers. Prime numbers are those that have only the number one and itself as factors (e.g. 2, 3, 5, 7, 11, 13, 17, ...).

Here are a few quotes from "The Art of the Infinite: Our Lost Language of Numbers" that I enjoyed:
"Its initiates [Mathematics] speak of playfulness and freedom, but all we come up against in school are boredom and fear, wedged between iron rules memorized without reason." [p. 1] I agree with this statement, but why must it be like this? When will we change our curriculum and our pedagogy to emphasize playfulness and reason?
"Its symbols store worlds of meaning for them [mathematicians], its sleek equations leap continents and centuries. But these sparks can jump to everyone, because each of us has a mind built to grasp the structure of things." [p. 2] Lovely. This is more like poetry than prose.
"Diagrams printed out from computers communicate a second and subtler falsehood: they lead the reader to think he is seeing the things themselves rather than pixellated approximations to them." [p. 2] This is an important point. On the other hand such diagrams may permit us to visualize complexities that are not readily apparent from the symbols themselves.
"Of all the arts, mathematics most puts into question the distinction between creation and discovery." [p. 8]

I have rarely seen mathematics referred to as one of the arts, but I like it.

The dichotomy of creation and discovery may be another example of a false dichotomy. I prefer to think of the terms as being in a symbiotic relationship where each grows from the influence of the other.

"Idly messing about - the way so many insights burst conventional bounds ..." [p. 9]  
"... a model of how mathematics happens: a faith in pattern, a taste for experiment, an easiness with delay, and a readiness to see askew. How many directions now this insight may carry you off in: toward other polygonal shapes such as pentagons and hexagons, toward solid structures of pyramids and cubes, or to new ways of dividing up arrays." [p. 10] I like the open-endedness of their approach. The second sentence is important as it illustrates how one might extend an idea.

I like the tone of the Kaplans' book. It is informal and relaxed where the emphasis, at least initially, is on the essence of an idea rather than on the formality of the approach.

12:00 PM

Total elapsed time: 1 hr 40 min.
Total elapsed time for the day: 2 hr 50 min

9:35 PM I have just completed reading and yellow highlighting the second chapter of "The Art of the Infinite: Our Lost Language of Numbers" [pp. 29 - 54]. I thoroughly enjoyed it, and felt that I understood the content. Now to make some notes about this material. ... another day.