March 21, 2006

From MathWiki

  • I played a radio segment from NPR (http://www.npr.org/templates/story/story.php?storyId=5279787) about an interesting entrepeneur who uses mathematics to build the latest technology (that wasn't how NPR cast the segment, but it is my take on the interview). There is a segment of video on this page where they demonstrate the technology which I found particularly fascinating.
  • I showed the headline from todays Metro page. "TTC mulls Google trip plan" - the point of the article is that the TTC was considering partnering with Google in order to build an on line trip planner which will tell you how to get from point A to point B on public transportation. If Google were just storing the information of how to get from any point in the city to any other point then this is an information retreival problem, instead what Google would (if it ever does this project) keep in its computer is the information of all the routes that the TTC runs, then when you ask how to get from point A to point B the computer 'figures out' all possible means to travel between those points and determines which is shortest. The 'figures out' step is usually involves some mathematics and computer science, but depending on how organized the information that you start with this problem has been done before. (http://carbon.cudenver.edu/~hgreenbe/sessions/dijkstra/DijkstraApplet.html)

Image:Shortestpath.jpg

  • The reason that I found these particular stories was because I found a blog by someone who uses computers for a living (http://steve-yegge.blogspot.com/2006/03/math-for-programmers.html). I want to quote a couple of lines from it which prompted me to bring these last two stories to class.
  1. Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.
  2. They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer.
  3. Knowing even a little of the right kinds of math can enable you do write some pretty interesting programs that would otherwise be too hard. In other words, math is something you can pick up a little at a time, whenever you have free time.
  4. Nobody knows all of math, not even the best mathematicians. The field is constantly expanding, as people invent new formalisms to solve their own problems. And with any given math problem, just like in programming, there's more than one way to do it. You can pick the one you like best.
  5. Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)
Anonymous: Very well put! I used to hate math, then I started playing with Flash and become interested in math as art, it really is beautiful. Also reading Bart Kosko's books were very intriguing, bridging, discussing philosophy and art...for the first time I finally understood some of it.
I wish they would have talked about some of this stuff when I was in high school, I would have paid much more attention. When you start understanding the history and the actual problems that are solved and more importantly what mathematics attempts to explain...oh my...I would have been a super geek in school.
  • We talked about minimizing the surface area of a milk carton. We looked at equations for area of cardboard and then graphed them and found on a 1 litre carton of milk we should see the height slightly larger than twice the base (if our assumptions about the rest of the cardboard cuttings was correct) because that shape is the most efficient.
    • Consider the volume of a cube is fixed at 1 so that if lwh = 1, then the surface area is = 2\ell w + 2hw + 2h\ell = 2lw + \frac{2}{\ell} + \frac{2}{w}. The surface area can be graphed with respect to the length and the width (the height is determined because h = 1 / lw) and we see:

Image:GraphzSA.jpg

    • Notice that the minimum point must occur in the 'middle' of this graph where l=w because it is symmetric in the length and width (look at the formula for the surface area) and so the lowest point must be in the center.
    • Because of this if we want to find the lowest point then it has to be the lowest point on SA

= 2w2 + 2 / w + 2 / w (because \ell=w is where the lowest point is). Now that graph looks like the picture below. Notice that the lowest point is where w = 1. This implies that w = \ell = h = 1. Image:GraphySA.jpg


  • We then talked about logic and logic puzzles. I brought in a book called "What is the Name of This Book?" Considering that I didn't get to spend much time on this we might come back to it next time.
  • I mentioned that the idea of formal logic (like most branches of science) is an approximation of the way that things really are: Assume that all statements are either true or false and then try to figure out what we can about this model of the language. Most statements that we make are not either true or false, but somewhere in between and so everything that we figure out about logic is only an appoximation for the way language really works.
  • Logic has found applications in all areas of technology because robots dream in 1's and 0's.
  • The first thing that we notice is that compound statements depend on the truth value of the parts, e. g. the truth value of the sentence "the sky is blue and the grass is green" depends on the truth value of both of the statements "the sky is blue" and "the grass is green." AND and OR are two common connectives in English and probably most languages since language is constructed to express thoughts and logical statements are things that humans would like to express.
  • There are others "IF it rains tomorrow THEN I will take my umbrella" can be abstracted to the statement "If A then B" where A = "it rains tomorrow" and B = "I will take my umbrella" We use the convention that if A happens to be false then the statement "If A then B" will be true because if it does not rain tomorrow then the statement "IF it rains tomorrow THEN I will take my umbrella" is true even if I don't take my umbrella.
  • The following is called a 'truth table' since it records the possible truth values of the statements and the truth values of the compound statements
A B A and B A or B if A then B
T T T T T
T F F T F
F T F T T
F F F F T
  • Given this we solved the following logic questions from the book that I mentioned above "What is the name of this book?" by Raymond Smullyan:
    • A man was being tried for participation in a robbery. The prosecutor and the defense attorney made the following statements:
      Prosecutor: If the defendant is guilty, then he had an accomplice
Defense Attorney: That's not true!

Why was this the worst thing that the defense attorney could have said?

    • There is an island consisting of knights and knaves. Kights always tell the truth, knaves always lie. Three of the inhabitants-- A, B and C-- were standing together in a garden. A stranger passed by and asked A, "Are you a knight or a knave?" A answered, but rather indsitinctly, so the stranger could not make out what he said. The stranger then assked B, "What did A say?" B replied, "A said that he is a knave." At this poin the third man, C, said "Don't believe B; he is lying!" The question is, what are B and C?
    • Suppose the stranger instead of asking A what he is, asked A, "How many knights are there among you?" Again A answers indistinctly. Do the stranger asks B, "What did A say?" B replies, "A said that there is one knight among us." Then C says, "Don't believe B; he is lying!"
  • Finally we discussed the homework (at least the polygonal number problem).

Previous class : March 14, 2006

Main class page : Mathematics 1590 Nature of Mathematics II