2008-10-13

Defining natural numbers 2

Japanese version

This is continued from the last article. Please refer the Peano's axiom in the last article.

The second definition means that we can create a next natural number from the current one. This function creates a successor number from the current number, therefore it called ``successor'' function. This defines ``plus one'' function. We have already the first natural number, zero, then we can make a successor number from zero. This successor number is ``something'' of zero. It is usually called one, but not necessary. This definition just said, it is something different from zero. We have now:

1. there is the first number,
2. we can make a successor number from a number.

Out of these two definitions/rules, it seems we can make the whole natural numbers, but this is not enough for that.

The third definition said that there is no loop of successor function, i.e., if you repeat the successor function staring with x, the result of them will never return to the number x. This seems natural since x + 1 + 1 ... + 1 != x. There is such mathematics (modulo), however, Peano's natural number does not think about that. For example, there are such system, like 12 + 1 = 1, or 31 + 1 = 1. If you think this is odd, think about your calender. Amazingly, the next day of December 31st (31.12) is January 1st (1.1). If you look at the month part, 12 + 1 = 1 and days part is 31 + 1 = 1. Time has also similar system, next of 23:59 is 00:00, means 59 + 1 = 0. Some might say that is not a calculation, but, there is such calculation in your computer. Someone may think 1 + 1 = 2 is the simplest mathematics. I think it is not so simple.

The fourth definition said that the same number's successors are always the same. In other word, different two numbers's successors are never the same. The calender example also does not follow this rule. 31 + 1 = 1 for January, 28 + 1 = 1 for February, 30 + 1 = 1 for April. 31, 28, 30 are not the same number, but the + 1's result are all 1. This definition tells you that the calender system never happens in the natural number.

I think now Marvin want to complain my explanation since these are so obvious and not need to explain. The last definition said all the natural numbers follows this rule are also the natural number. (This is the base of mathematical induction.)

Marvin: ``By the way, this explanation is quite similar to the explanation from Kouji Shiga's book. You should refer that.'' That's true. This explanation comes from ``A story of growing mathematics.'' Unfortunately, I do not have this book now. I left them Japan... I like all the Shiga's book. When I found out that he had a open seminar at Yokohama, I visited his class. But it took four hours to go and back the class by train. It was fantastic classes. When I left Japan, I regret that I could not take his class anymore.

Peano's axiom reminds me two things: Lao-tsu and Wittgenstein. The coincidence works again for the guide. The section 42 of Lao-tsu starts with ``Tao is created by one. One bears two, two bears three, and three bears everything...'' Excellent.

I have no idea about what Wittgenstein try to explain, however, his word: the words are just projection of the world. I think it sounds right. Lambda calculus projects back this words to a machine. I.e., logic (logos) is projected to the world again. So far, I just describe calculation with words. But no matter what words I use, this can be run on a machine, which belong to the world. I am satisfied when I see the logic really runs on a machine. Then I feel this is not just meaningless words.

Next time, let me describe the natural numbers by lambda.