Sunday, July 31, 2011 Edmonton

6:50 am

The temperature is +14 C, with a high predicted of +25 C.

From the Environment Canada website: Today Mainly sunny. Increasing cloudiness this afternoon then showers with a few thunderstorms. High 25. UV index 6 or high. Tonight Showers with a few thunderstorms ending near midnight then clearing. Wind northwest 20 km/h gusting to 40 becoming light near midnight. Low 12. Normals Max: 23°C Min: 11°C.

Today's forecast for Jasper is holding good. Both Monday and Tuesday are supposed to be sunny. The plan is to drive there early tomorrow morning and have the geocaching for finds #999 and #1000 completed by early afternoon. This will be the last item on our list of "to do's" for this trip. We will stay in Jasper tomorrow night and drive back to Lethbridge on Tuesday.

6:00 PM Mathematics

I have read the first three articles in "The Best Writing on Mathematics 2010", a book I bought a couple of days ago.

Now to set up a format for note making.

1. Foreward - William P. Thurston

• "... the goal of mathematics is to develop enhanced ways for humans to see and think about the world." [p. xi]
  
  
  • This is a better statement than the one I have usually used which is "the study of patterns". This is much more applied and is more generic with respect to Thinking.
• "... mathematics takes a highly symbolic, algebraic, and technical form. Few people listening to a technical discourse are hearing a story. Most readers of mathematics (if they happen not to be totally baffled) register only technical details - which are essentially different from the original thoughts we put into mathematical discourse. The meaning, the poetry, the music, and the beauty of mathematics are generally lost. It's as if an audience were to attend a concert where the musicians, unable to perform in a way the audience could appreciate, just handed out copies of the score." [p. xi]
  
  
• I wish I had developed the skill of reading beneath the surface much earlier. As I read, I stop and ask, What's the author trying to say? What is the author really thinking (if I suppose it is different from what he put in the mathematical text)? What do I think of this? I talk to myself back and forth while reading somebody else's writing. But the main thing is to give myself time, to close my eyes, to give myself space, to reflect and allow my thoughts to form on their own in order to shape my ideas." [p. xii]
  
  
  • This is very close to the two-color approach I use on this web site when reading a non-fiction article.
• Mathematical ideas can be transcribed into symbols organized into precise descriptions, equations, and logical deductions - but such a transcription is typically far removed from the mind process that generates the ideas." [p. xiii]

2. The Role of the Untrue in Mathematics - Chandler Davis

• "... the 20th century brought us to an acknowledgement that truth may be of various strengths. ... modal logic, many-valued logic, fuzzy logic, probability" [p. 6 - 8]
  
  
  • This is well worth remembering in a time of black-and-white. The world is even more than gray - it is colorful.
• "... that what we employ, whether in reasoning or in observational science, should be not mere association but the structure of the association." [p. 9]
  
  
  • Excellent! Let's push the envelope.
• "Even if we call on high technology to explore a graph of connections between items, it will be natural to refine it to be a directed graph, a colored graph, and surely much more. Only connect! - but there is such a wealth of ways that two nodes may be connected." [p. 9]

3. Desperately Seeking Mathematical Proof - Melvyn B. Nathanson

• "... Elementary proofs are not better than other proofs. ... I've changed my mind. In this paper I argue that elementary (at least in the sense of easy to check) proofs really are better." [p. 15]
  
  
  • This is great - I have rarely seen an article where the author admits to changing his mind.
• "Occasionally speakers respond to a question from the audiencewith a look that the conveys the message that the questioner is an idiot. That's why most mathematicians sit quietly through seminars, understanding very little after the introductory remarks, and applauding politely at the end of a mostly wasted hour." [p. 16]
  
  
  • Yep!
• "Different theorems can be proven from different assumptions. The compendium of mathematical knowledge, that is, the collection of theorems, becomes a social system with various substructures, analogous to clans and kinship systems, and a newly discovered theorem has to find its place in this social network. To the extent that a new discovery fits into an established community of mathematical truths, we believe it and tend to accept its proof. A theorem that is an 'outsider' - a kind of social outlaw - requires more rigorous proof, and finds acceptance more difficult." [p. 16]
  • Nicely put.
• "Perhaps we should discard the myth that mathematics is a rigorously deductive enterprise. It may be more deductive than other sciences, but hand-waving is intrinsic. We try to minimize it and we can sometimes escape it, but not always, if we want to discover new theorems." [p. 17]
  • Lovely! We need flexibility to think.

4. An Enduring Error - Branko Grunbaum

• This paper describes in some detail a number of mathematical errors that have been repeated numerous times. Mathematicians do make errors.