October 2003 Archives

A good mathematical notation is in my opinion a necessity when solving mathematical problems. It allows to capture and communicate mathematical reasoning in a concise, efficient and unambiguous way. Anybody doubting this should just try to read old mathematical texts from before the time when modern mathematical notation was common. Mathematical historians usually quote the old masters using up a whole book page worth of text and then translate that for us in modern mathematical notation using an equation in one line or so.

Choosing the right notation for the job is also very important. Expressions in a notation should be easy to manipulate, expressions should stay small but the notation should not hide properties of the mathematical objects involved. My dad always gives this example of a bad notation choice: In some schools EE students learn to use + for "or" and "*" for "and" in boolean expressions. This is bad because in boolean algebra "or" distributes over "and" and "and" distributes over "or". But we the chosen notation one would write A (B + C) = A B + A C and A + (B C) = (A + B) (A + C). The second one looks unnatural because it doesn't work in integer algebra so students are reluctant to use it for boolean expressions even though it's perfectly fine there.

I tried to make all of this lead to this paper by E.W. Dijkstra where he explains the notational conventions that he used and why. He is one of the first to take notation very seriously and he pushed for corrections to the modern mathematical notation (For example he criticized the familiar use of the classical summation symbol as sloppy and misleading, he repeatedly argued for numbering starting from zero, ...).

I'm going to read Why Not? How to Use Everyday Ingenuity to Solve Problems Big and Small, by Barry J. Nalebuff, Ian Ayres.

This article made me curious, it sounds like an interesting book.

On the flight back from Europe I was flipping through the audio channels and I got hooked on some eurobeat tracks. In fact I liked them so much that later I went hunting for them in the iTunes Music Store:

tunnel - dining rooms

la citta nuda - dining rooms

afterwards@the bar - caia

magic love - bent

so long without you - bent

blue monday - flunk

moonbeams - bent

I don't know how many parts there will be about ewd666. It's a wicked puzzle, tough to crack. I found it on the website that collects Edsger W. Dijkstra's notes (a wonderful collection btw).

Let's state the problem again:

Of two unknown integer numbers in the range [2..99] inclusive Paul is told the product P and Sam is told the sum S. When asked whether they know the numbers, the following dialogue takes place between Paul and Sam:
Paul: "I don't know them."
Sam: "I already knew that."
Paul: "Then I now know the numbers."
Sam: "Then I now know them too."
The problem is to determine the two numbers from the above data and to prove that the solution is unique.

Spoiler alert: don't continue to read if you want to find your own solution (also don't read more than page one from the pdf file above). I don't have a solution yet, these are pretty raw ideas. Like I said, it's a tough one for me.

We went to see the Marc Chagall Exhibit at SF Moma today. The guy is good, I really like him. You can tell he enjoyed his life, he enjoyed being married, he enjoyed being with his wife. He liked the circus, parties and celebrations. His paintings have a very distinct style, escaping the trap of too much abstraction, making them accessible to people like me.

The collection is unique, it's the first time so many of his paintings are together in an exhibition and it will end Nov. 4th. I recommend buying the tickets online to avoid long lines.

Apple never ceases to amaze me. They just "get" things, they know how to do it right. iTunes and the iTunes Music Store are the latest example. One cool thing with the second generation of the Store is that music becomes addressable. You can refer to it by URL just like books became addressable when Amazon showed up.

Try for example "First Cool Hive" by Moby

When you drag a song from iTMS into a text area you get its URL and you can also use iTunes link maker to make more sophisticated links that do the smart thing when iTunes is not present.