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.

Wednesday, August 10, 2005

Websites as Notebooks

The following comments apply to both my Learning of calculus as well as to other areas of Learning that I am interested in.

I have just finished reading and making a mindmap for the book "Tomorrow's People" (2003) by the neuroscientist Susan Greenfield. In my first read through I made a small ink mark in the margin when I noticed a point that I found particularly striking or important.

On the second reading, I created a mindmap using the software MindGenius. This really helped me identify the important points and see where they fitted in the overall description of her views about the future.

As I have indicated on many posts to this blog, I am keeping my notes as a website (currently located on my hard drive) using the web authoring package Dreamweaver. As with any website, I have the capability to make links to other web pages (both my own as well as to external web resources) and perhaps more importantly, to files created by other software programs. Links to these files will work, and as long as the user has the same software that originally created the file, the file will open in that environment where one has dynamic control of the file.

In the case of this mindmap, this is preferable to seeing a static portion of the map on the screen. I can scroll through the complete map or collapse and expand nodes to suit my current area of interest. The difference is dramatic.

Similarly, when I am Learning calculus, I can use sophisticated software such as Mathematica to explore my ideas and these files can be saved and accessed from within the website that contains my narrative notes. Thus my electronic notes are much more than just a static textual presentation. They permit me to quickly move among various software packages with just a couple of mouse clicks and give me the added power that each package provides.

There is no doubt in my mind that these web-based notes are superior to anything I might try to create using more conventional materials. Each reread of the book, each paragraph that is created as part of note-making, each activity that uses a software package to play with the ideas provides numerous opportunities for review and enhancement. The cumulative effect of all these activities is to modify my neural patterns (a point made repeatedly by Greenfield) which is then manifested by my being able to engage in other activities such as a discussion (perhaps on a blog) that exhibits aspects of this new Learning experience.