2008-10-14

Church numerals

Japanese version

Last time we were talking about how Peano defined the natural number. Because Lambda calculus defines the numbers based on that. Mathematical formulation makes the discussion (proof) more exact, this ``exact'' is important for mathematician. But, the formulation causes increasing the exactness, which means, there are no such thing, like ``You know about the numbers, just do something like calculation in appropriate way.'' Because even every single obvious issue should be defined in formulation. As the side effect, we could execute these rules on a machine --- we can make a computer! That's the interesting point for me.

There are many ways to how to implement a computer as a machine. Pascaline and Charles Babbage's differential engine are gear based. Here our base is Peano's axiom.

Before Marvin points out, this explanation is Masahiko Sato and Takafumi Sakurai's ``Basic theory of programming.'' (By the way, It is not so convenient that I left all my books in Japan. In Japanese version of my blog is written by SKK input method, which was first implemented by Masahiko Sato.) Prof. Sato's class was tough. At least I had totally no idea if I only attended. However, when I asked questions, even though it is extremely stupid questions, he answered the question until I said ``thanks I understand.'' The problem is the class was so tough, therefore I do not have any questions, except ``What can I ask?'' It is very difficult to figure out what is I don't figure out.

Here, I use a circle as a sign of ``This is a number.'' I use a square as Peano's ``zero,'' and a successor number is represented by reputation of ``zero''. See Figure 1.


As you see, there is no one or two. We have only a sign of ``number'' and ``zero.'' Our number system is depends on base 10 system. We could also use Roman number system, Babylonian system, or Chinese characters. These are all for human's convenience, and all of these number representations are not substance. Zero is sufficient to represent the Peano's system. I use zero as a square, since I would like to stress that ``zero'' is also a name for something. It does not matter if it is ``hoge,'' or "Petrosiliuszwackelmann." Then I use a square as following the Prof. Sato's book.

When you see one square, then I understand you want to call it one. I agree with that. But, we need to represent Zero. The usual number 0 means ``There is nothing here'' -- nothing exists here. If we want to start the number zero, then I think this is the way. Although, Peano's system can start any number of natural number, one, two, or 42. It does not matter. But still I would agree with Figure 1's zero looks like one.

Next time let's start with why we need a circle instead of using only Zero.

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