Thursday June 22, 2006 7:30 am Lethbridge Alberta

A. Morning Musings

7:30 am The temperature is + 10 C with a high forecast of + 23. There is a 30% chance of showers.

I will wait a day or two and let the sand dry out a bit before laying down the patio bricks. But I may go over and pick up about 50 bricks. That should give me something to play with and see if my ideas are going to work.

The 2 hour Learning block for today will be devoted to learning to use the MacBook Pro. I will also try to spend a little time on "The Computational Beauty of Nature".

B. Plan

Immediate
Chores	Clean mortar from cement blocks in rear fence	2 hr
Health	Walk & exercise	1 hr
Model Trains	Add blue backdrop to layout	2 hr
History	Continue reading "Citizens"	2 hr
Literature	Begin reading "The Eagles' Brood"	1 hr
Mathematics	Read "The Computational Beauty of Nature" Chap 2	1 hr
Later
Chores	Install patio bricks
	Buy sealant for join between garage floor and wall
	Take 5th wheel in for maintenance	2 hr
	Investigate water softeners for home	2 hr
	Paint door frame white	1 hr
Technology	Read manual for cell phone	1 hr
Technology	digital photography
	try using extender lens and monopod	2 hr
Mathematics	Larson "Calculus"
	Gardner "The Colossal Book of Short Puzzles"
History	Watson "Ideas"
Model Trains	Review layout for under the table turnouts
	Visit Lethbridge Model Train club at 7 PM on Tues	2 hr
	Wire lower mainline track for a power block
	Fasten lower mainline track to layout
	Draw schematic diagram of track layout	2 hr

C. Actual/Notes

7:20 am Garbage pickup using the new (for Lethbridge) bin system worked just fine. I had a brief chat with the operator - he was delighted with it. No more picking up heavy bags.

2:30 PM Much of today has been spent on the MacBook laptop. The major difficulty was getting the wireless system to recognize my home wireless base station. It kept asking for a password and when I gave it the password from my Windows computer it kept rejecting it as invalid. After verifying that I was using the proper password and running all of the diagnostic windows on the Mac that I could (many times) I finally found a message that suggested that the problem might have something to do with how the security encoding was set up on the base station. I eventually simply tried all possibilities and the very last one was the correct one. I now am able to surf the Web using Mac OS. I have yet to try the Windows version on the Mac.

Meanwhile something has gone wrong with email. Thunderbird simply goes into a loop and repeatedly says that is is unable to connect with the uni mail server. I have been able to verify that it can connect to the telus mail server so that seems to indicate that the uni is having difficulty. I will try again a bit later.

6:40 PM There is definitely something wrong with the uni mail server. I finally went into the campus and tried logging in from one of the email stations. Same problem. Fascinating how this happened right when I was in the middle of setting up email on a new computer.

Previous	Mathematics Session 2	Next
Mathematics Chronology
6:45 PM I began reading "The Computational Beauty of Nature" by Gary Flake yesterday. I am sufficiently excited to stay with this book for awhile and see what notes I can make. Much like Spivak's book on calculus, this book also says that the topic is largely about numbers, and how little we understand them. I would like to begin by noting a few quotes and my reaction to them. Preface [p. xiii - xviii] "The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful." [p. xiii] substitute experimentalist, theorist, and simulationist for engineer, mathematician and computer scientist [p. xiii] "A simulationist is a relatively new breed of scientist who attempts to understand how the world works by studying computer simulations of phenomena." [p. xiii] "... simple recurrent rules can produce extremely rich and complicated behaviors." [p. xiv] A superb example would be our number system. The natural numbers are generated by the rule that each number has a successor. We then add a couple of rules defining what we mean by addition and multiplication and we have almost all of mathematics. "If a particular equation confuses you, the first thing you should do is see if the surrounding text gives you a little more insight into what the symbols mean. If that doesn't work, then look to see if there is a picture that corresponds to the equation. Finally, if all else fails, you should do what I do when I don't understand an equation: mentally substitute the sounds 'blah blah blah' for the equation and keep going. You would be surprised how often this works." [p. xvi] Lovely! This same strategy works for not just for equations but for any material that one does not understand. Chapter 1 Introduction "traditional scientists study two types of phenomena: agents (molecules, cells, ducks, and species) and interactions of agents (chemical reactions, immune system responses, duck mating, and evolution). [p. 2] "... agents that exist on one level of understanding are very different from agents on another level: cells are not organs, organs are not animals, and animals are not species. Yet surprisingly the interactions on one level of understanding are often very similar to the interactions on other levels." [p. 3] "Complex systems with emergent properties are often highly parallel collections of similar units." [p. 4] e.g. humans, or almost any living organism. "Parallel systems that are redundant have fault tolerance." e.g. humans, or almost any living organism. "While animals must react according to their surroundings, they can also change this environment, which means that future actions by an animal will have to take these environmental changes into account." e.g. sheep and over-grazing; invasion of Afghanistan and Iraq Section I "Systems that compute can represent powerful mappings from one set of numbers to another. Moreover, any program on any computer is equivalent to a number mapping." [p. 9] I have never seen the statement in red before. That is a very powerful insight. That is saying that ALL programs may be thought of as a form of mathematics. This may also mean that any statement about programs may have an equivalent statement in mathematics, and vice-versa. The actual process of computing can be defined in terms of a very small number of primitive operations, with recursion and/or iteration comprising the most fundamental pieces of a computing device." [p. 9] Chapter 2 Number Systems and Infinity Flake begins this chapter with a discussion of Zeno's famous paradox of Achilles and the Tortoise. He then shows that this story is mathematically equivalent to summing a series of terms, each of which is half the size of the preceding term (e.g. 1/2 + 1/4 + 1/8 + 1/16 + ... ). If one continues this series forever what is the total? Flake then uses a diagram where he successively divides a unit square into halves and then sums all of the little squares. Clearly the result is the complete square (i. e. 1) [p. 12 - 14] Beautiful! Using a rough sketch one can see at a glance what the sum must be. This may not be a tight algebraic proof, but it makes sense. This is also a neat demonstration that an infinite sum may have a finite answer. For many people this is counter-intuitive: adding up an infinite number of little things should result in an infinite amount. Not so, if the little things are sufficiently small and getter smaller with each term. The example, as Flake points out, involved looking at the properties of an infinite number of fractions (i.e. rational numbers). Flake then quickly demonstrates another counter-intuitive idea: the number of even numbers is the same as the number of counting numbers. Intuitively one might think that there are half as many. But using the idea that two sets have the same number of elements if one can find a mapping from each element of one set to each element of the other and vice-versa. From this one can easily imagine many more such examples. All such sets of numbers are called "countably infinite". In a single page Flake begins with the counting numbers (positive integers) and quickly comes up with an advanced idea in mathematics having to do with the size of infinite collections. What kind of an idea is infinity? 7:50 PM

 

D. Reflection