Mathematical Experience

We often hear that Mathematics consists mainly in "proving theorems". Is a writer's job mainly in writing sentences?

A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.

To master mathematics is to master an intangible view, it is to acquire the skill of a virtuoso who cannot pin his performance on criteria.

By Gian-Carlo Rota on "The Mathematical Experience" by Philip J.Davis and Reuben Hersh

Trying to Understand the Essence of Education

Shuttleworth Foundation is trying to acclerate global innovation. Stumbled upon this very valuable site tracking some blogs on Python (programming language). Reading the goals and comments by the participants is very educational. Wonder what we, as bloggers, can do about improving education.

• Trying to understand the essence of education- what skills are gained from primary and secondary education, and to extract the core skills needed in life:
  1. Communication skills
  2. Analytical skills

Mathematics - The Science of Patterns

Mathematics is the science of patterns. Different kind of patterns give rise to different branches of mathematics. For example:

• Arithmetic and number theory study patterns of number and counting
• Geometry studies patterns of shape
• Calculus allows us to handle patterns of motion
• Logic studies patterns of reasoning
• Probability theory deals with patterns of chance
• Topology studies patterns of closeness and position

Mathematics, the science of patterns, is a way of looking at the world. The study of mathematics is ultimately a study of humanity itself. For none of the entities that form the substrate of mathematics exist in the physical world.; the numbers, the points, the lines and planes, the surfaces, the geometric figures, the functions and so forth are pure abstractions that exist only in the humanity's collective mind.

There is scarcely any aspect of our life that is not affected, to a greater or lesser extent, by mathematics, for abstract patterns are the very essence of thought, of comunication, of computation, of society, and of life itself.

From The Language of Mathematics - Making the Invisible, Visible.

How to Read Mathematics

When I was a small kid, my grandfather used to teach me how to read English. I was born in India, so we did not get to reading English till we were 6 or 7. His technique was simple. He would mark portions of a sentence with a vertical mark (in pencil). He told us that we need to pause at the vertical bar, at comma and a pause a bit more at the period.

I wish I had seen this article on How to Read Mathematics, when I was young, since it was one of my favorite subjects. This is a profound statement, a quote in the article, made me take a quick read.

“Reading Mathematics … involves a return to the thinking that went into the writing” (ibid page 16).

Here are a list of things to do (in a nutshell).

• Don’t miss the big picture
• Don’t be a passive reader
• Don’t read too fast
• Make the idea your own
• Make sure that you are the intended audience

The Example of Mathematical writing towards the end, is a clear step by step dialog on applying these techniques.

For more math blogs try http://wordpress.com/tag/math/

Mathematics and Programming

In this wonderful blog post, Ted Dziuba talks about learning programming like a mathematician. I am glad that the emphasis of mathetic thinking is coming back.

This article is a little more abstract, talking about the similarities between learning mathematics and learning a programming language. The goal of this article is to give programmers a framework by which they can effectively learn a new programming language, much in the way that a mathematician learns a new area of mathematics to the point where he or she can be effective.

In conclusion, Ted summarizes the essential similarity between math and programming.

1. Find the fundamental theorem, the sine qua non, of the language.
2. Understand how this fundamental theorem influences the structures and design decisions of the language, and how it is used to establish relationship between different parts of the language.
3. Practice and read documentation to the point where you can mentally picture how to fit the language structures together in the most efficient possible way to solve a problem.

Peter Norvig in his Teach Yourself Programming in 10 years says:

Learn at least a half dozen programming languages. Include one language that supports class abstractions (like Java or C++), one that supports functional abstraction (like Lisp or ML), one that supports syntactic abstraction (like Lisp), one that supports declarative specifications (like Prolog or C++ templates), one that supports coroutines (like Icon or Scheme), and one that supports parallelism (like Sisal).

I think the essense of programming is to build powerful mental models of the problem and the ability to deal with abstractions. A training in Math definitely helps in developing this skill.