T

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

February 2006 Mathematics Notebook
Introduction      
Goals
   
     

An Example of a "Learning Process" Journal
Wednesday February 1, 2006
Learning Log Number 6
6:50 am Ballina NSW Australia

The intent is to continue making notes for Coincidences, Chaos, and All That Math Jazz.

I will begin with a mind map to sketch out the overall structure.

The first three chapters are about uncertainty. These are my notes for chapter 2 Chaos Reigns.

The first chapter emphasized that mathematics was about ideas. I will review my yellow highlighted passages in the book and identify the important ideas that merit repetition on this page. As before, I will insert a tan box with my personal commentary when I feel that I want to emphasize a point.

  • Much to our surprise, there are many (most) situations that are impossible to predict, no matter how hard we try. This is not because we do not understand the situation well enough, nor because we are unable to measure variables accurately enough. Rather it is impossible, in principle, to predict the future for these situations. Examples include:
    • weather forecasting
    • computation (!)
    • connected pendulums.

  • Mathematical chaos refers to situations involving repeated mathematical processes (i.e. iteration) that have a sensitivity to initial conditions.

  • Almost all computation by computers gives the wrong answer!

This example did catch me by surprise. As soon as one has a situation that involves some rounding then the machine (e.g. calculator, spreadsheet, Mathematica, ...) must truncate and if this result is then used in subsequent calculations, then the error becomes compounded. This is true in principle, it is not a technical issue. The "correct" answer is one with an infinite number of digits. When a computation involves an appoximation then it must, after a suitable number of iterations, differ wildly from this correct answer, but how wildly we have no idea as we are unable to compute the correct answer.

There is an entire field of mathematics called numerical analysis that deals with this issue. However there is no real cause for concern. The answers that various routines give satisfy certain criteria that give us reason to believe that the answer is either acceptably close or that it is genuinely random.

I do wonder about statistical analysis software packages that involve millions of computations such as multivariate statistics.

I will try to pursue this a little on google and see what else I can find:

  • "But perhaps chaos in classical mechanics limits our potential to predict the future of physical systems even more than quantum mechanics does." [p. 40]

Stunning! I have never seen this idea before. My head is genuinely spinning. In the last few minutes I have experienced two new ideas. The book is already worth its price.

I want to google this as well: "mathematical chaos", prediction, "physical systems"

  • Here are some examples of iterated systems in reality:
    • weather forecasting
    • population dynamics
    • fluid dynamics
    • economics
    • stock market
    • oscillating chemical reactions
    • electrical networks
    • heart rhythms
    • brain waves

7:50 am This has been one of the most stimulating hours in a very long time. Two new ideas in one hour! Fantastic. Now to do some googling. This is where the Web begins to show its value. When one hits the boundary of the original resource one now has recourse to another source.

This is a profound change in our Learning environment.

Total time for this session: 1 hour.