2008-09-23

lambda calculus and function

(Japanese version )

The word lambda calculus itself has ``calculus,'' so it can also calculate numbers. But, when I started to learn the lambda calculus, I want to know that what is ``calculate'' means. Then I also think about ``what is the number?'' If I want to teach ``what is the number'' to small children, I do not know how to teach it. Also I did not recall how to learn that. But, I think I know what numbers are.

I can tell that there are some properties about numbers. First, it does not matter how to read it. In the hitchhiker's guide, there was a planet called Earth. The people on the planet have a lot of languages. For example, English, German, Latin, Japanese, and so forth. Interestingly enough, every language seems to have a concept about numbers. There are many representations for numbers, for example, (1, 2, 3, ...), (one, two, three, ...), (ein, zwei, drei, ...), (ichi, ni, san, ...), and so on. But no matter how you read them, there should be a common concept underneath. In the guide, maybe (herring, sandwiches, herring sandwiches, ...) or something like that. I know the definition by negation (it does not matter how to read them) is not a good idea here, but this at least tells you one aspect of numbers. There is a concept ``numbers'' which is independent from how to read them. Then, what is the substance of numbers? You can not say, it is one, two, three, ... Inside the guide, there must be ``something,'' and they are presented in some representation according to readers. The representation is always chosen as the reader can understand. I called it ``native'' in one section before. This is the substance of guide, someone could call it as information. But, it does not matter which language is employed to store such information inside of the guide. We could investigate further the substance of the guide and also substance of numbers.

I think lambda calculus is a mathematics that talks about function. Function is a useful idea in mathematics. It's like a vending machine that if you put something in (for example, money), then something comes out (for example, a drink). Many of the cases, people imagine a box which converts from something to another thing. For instance, a converter box which tells you that one Altair dollar is how much Sirius yen. The guide must also have such a box inside.

The function is not necessary to be a box. However, I learn in such way at my school. Moreover, the character of ``function'' in my language means ``a number box.'' So, I usually write a box and put a name like f or g. I think this was a standard on earth. Anyway, both teachers and students did not think about this deeply. Especially, if the usual people know what is function, then computer scientists have a problem that they do not know what it is. Nowadays function is a popular idea, so, computer scientists needs to invent something fancy.

Most of the computer scientists use the words ``lambda calculus'' as to deceive the people. It is easy to find such people, if you ask them ``What's lambda calculus?'' then they will answer, ``fixed-point...,'' ``Combinator blah,'' ``computability...'' to deceive you.

Lambda calculus might be a deep idea and might be useful to think about function. Although I am just a Sunday wannabe mathematician, I could enjoy some superficial results. I could even finish this with ``Church-Rosser blah blah'' to deceive you. However, as you know I am now writing this article. Such a person usually wants to brag that ``I know it! I know it!'' So, I will continue this story at least until Marvin will show up. By the way, Marvin is THE genus robot in the Hitchhiker's guide. He/(or She/It?) has the most depressed mind with the best brain ever made in the universe by Sirius cybernetics corporation. This company first made a brain which has equal ability to a few hundreds bionic brains. But the result shows that a genus is crazy. So be Marvin.

Finally, this section is ended here since I could write about my favorite Marvin.

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