Thursday, August 25, 2005

Calculus and Number

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 yesterday, while collecting my books, I came across a textbook from my last year of undergraduate study, "Principles of Mathematical Analysis" by Walter Rudin (1964). Here is the first sentence of the book: "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 spend a similar amount of time on studying numbers themselves. If nothing else my approach to Learning remains 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 that I once studied 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.

The blog I am currently using does not have the power to capture my notes so I will begin a new web site that chronicles my efforts. I will post the link on this blog when I have it ready.

I have accumulated 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.

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.

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