Archive for September, 2006

To Bear Thought

Saturday, September 23rd, 2006

“Why understand? What does it mean to understand?”. Two posts ago, I talked about compact expression, and that is part of the answer. Today I try a different angle.

Certain things “bear thinking about”. Which things those are would depend on who you are. How funny/recursive though, to have thoughts about the utility of further thinking.

The answer to a “where?” question is the name of a place, while “who?”s are answered by giving the names of people. I was thinking about “why” and “how” when this became apparent:



Thursday, September 14th, 2006

The rain splattered in vain on the glass, insignificant and ignored – my eyes following well-crafted sentences, my mind filled with images of Trantor and thoughts of interstellar politics.

Lost in another world while hurtling home on the bus in rainy weather… it makes me feel really smug.

Compact Expression

Thursday, September 14th, 2006

This post was inspired by “Who Can Name the Bigger Number?“.

“oooooooooooo”, “1100”, “12” and “1.2e1” are different ways of representing the same number.

In coming up with a new representation, the first step is to invent a process by which all members of the old representation can be written in the new one. For example, to convert the “ooo” form to “11” (binary), one would do the following procedure:

1) “ooo” – “”
2) “oo” – “1”
3) “o” – “10”
4) “” – “11”

Addition and Multiplication are defined for the old rep, and to do the same on the new rep, one can convert to the old rep, do the operation, and then convert back. However, most useful reps also have direct ways of doing the same operation. In fact, the usefulness of a rep is often determined by how efficient the operations in it are. (Transforms are used to the same effect.)

Also, in principle the different reps describe the same objects, but this is assuming we have infinite memory and time. For example, to write “1e100” in the “o”-form is impossible, since that would require more “o”s than there are particles in the universe (~1e79).

Representations neglect information that we do not need. Scientific notation, for example, lists only the leading digits of a number and allows us to discard inaccurate digits early in a computation and thus reap the results of error propagation by compacting the expression.

The equivalence of an operation in a higher and lower rep must be proven using mathematics, and not by going through all possibilities, since the higher rep is capable of expressing values that the lower rep is not.

In music, (more…)


Monday, September 11th, 2006

It’s been a while since I last wrote. In the meantime, I’ve had to restore zeno from backup after the harddisk died, and the fall term started. Taking a class this semester by Gerald Sussman, inventor of Scheme.

That’s a dialect of LISP, for those of you who don’t know. Not that you would know what LISP was. I’m not even sure I do.

Anyways, it is a new year, and have been very excited meeting all the young people coming in. I’ve said this before, I think – I find their enthusiasm infectious.

On account of all this infectious enthusiasm, I shall pull an all-nighter and get some things done tonight!