Essays on Mathematics.

Welcome to Descriptive Mathematics. I hope you find these essays informative and enjoyable. The two core articles in this web are are: Descriptive Mathematics and The Calculus in Perspective. I will add essays on the prime numbers, system architecture, the diagonal method, and other philosophical issues.

Please note, I am paying for this web from my own resources. I would like to add databases, java programs and more, buts these are expensive. So, please feel free to check out the sponsors for this site. They pay for the content. 

Descriptive Mathematics

Most math classes emphasize problem solving. They treat the study of mathematics as a series of increasingly complex problems. Unfortunately, this approach gives students an unbalanced view of the subject, and has the negative effect of alienating those students whose verbal skills are stronger than their “problem solving” skills.

Descriptive mathematics takes a more holistic approach to the subject. In this work, I concentrate on how to use mathematics to create and describe complex mathematical models.

The goal of descriptive mathematics is to help students develop the ability to explore and discuss issues using mathematical models. This method encourages students to develop a strong mathematical vocabulary in conjunction with their problem solving skills. Students who study descriptive mathematics should not only be able to solve problems, but should be able to express their ideas. more…

The Calculus in Perspective

The Calculus in Perspective is a prime example of how I would use descriptive mathematics to develop a math class. In this work, I present a unique approach to teaching the Calculus. I begin by creating a model of visual perspective, then expand this model into a larger theory of the calculus.

There is a natural progression between visual perspective and the calculus. Visual perspective is a mathematical description of space. Calculus is a mathematical description of change.

Perspective gives the students the skills needed to project a three dimensional shape onto a two dimensional surface. Adding calculus to perspective gives us the skills needed to animate the image.

To avoid the philosophical problems inherent in the use of limits, infinitesimals and other infinite constructs, I have chosen to present the fundamental theorem of the calculus with simple primary school algebra. By avoiding the complexities of the infinite, I believe that we will be able to teach students the vocabulary of the calculus at an earlier age. My ultimate hope would be for students to learn the basic model of calculus in junior high school, or high school at the latest, then to take a rigorous two year course in Calculus in their first years of college.

This work is substantially easy to comprehend than the standard two year college course. The method builds on standard high school geometry and algebra. Students should be able to read this work if they know to following:

• how to describe a line with an equation,
• how to chart a graph on the Cartesian plane,
• how to solve a binomial (y = ax² – c) using either the quadratic equation, or by completing the square.

Although this work is easier to master than the standard college level course in Calculus, I would not recommend using it as a replacement of the  rigorous two year study of the Cauchy method–taught in most colleges. There are some very important things that students should learn about function theory, limits, and solving complex differential equations. The Cauchy method does a better job of at presenting these advanced topics.

However, reading through this web site before taking a college level course will make the course more enjoyable. Most importantly, studying the Calculus in Perspective is better than never taking Calculus at all. (read on)

Prime Territory

6/4/2003 Sorry about taking so long. I just started the upload of this article. Hopefully, I can get it finished soon. I thank you for your patience.

A Critique of the Diagonal Method

Unfortunately my work, The Calculus in Perspective, has pulled me head long into the debate about the nature of the calculus. The Cauchy method teaches that integral and derivatives are limits. In my work, I view the calculus as a model that is independent of the tools used to derive the model. Limits, infinitesimals, and standard high school algebra are all valid tools for exploring the calculus. I agree that, once you have mastered the concept limits, limits are the best means for finding integrals. But the integral itself is independent of the tool used to find it.

Because, my work is involved in this on going controversy, I found it necessary to study different views of the foundation of mathematics. One of the most prominent views is called transfinite theory. This theory was developed by Georg Cantor at the end of the 19th century. Since its discovery, transfinite theory has been the subject of much heated debate, and have caused major schisms in the mathematical community.

Unfortunately, I have found that most introductory text in transfinite theory do a poor job of describing the theory and the controversies surrounding the theory. In A Critique of the Diagonal Method, 

System Architecture

I must confess. I find the world of computer programming, database modeling and object oriented design to be more interesting and fast paced that the than the world of pure mathematics.

Computer programmers face many of the same challenges as descriptive mathematicians. So, as I develop my works on descriptive mathematics, I will jam all of my observations and discoveries about system architecture into this section of the web.

The Roots of Sound Rational Thinking

This is a full length book by Virginia Vallee Delaney concerning the foundation of logic, epistemology, and more.

Writings

I am busily polluting the net with many other projects. Most notably, The Ghost of Alma Matterson is my first attempt at full length fiction. It is a story of a young man raised by computers who breaks to the debugger. I will put amateur art at the location www.rgreetings.com, and amateur photography at www.protophoto.com.

I also have an extensive collection of book reviews which I hope to expand.