2008-10-08

Defining natural numbers 1

Japanese version

Peano defined natural numbers.

He actually described properties of natural number, not seems to try to define the natural numbers. But these are somehow the same. The following five definitions are called Peano's axiom which defines the natural numbers. If you are not familiar with mathematical notation, it might be hard to get what they said. But, the basics are not so difficult. These are copied from Mathworld.
  1. Zero is a number.
  2. If a is a number, the successor of a is a number.
  3. zero is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.
The first definition said, there is the first number called Zero. Here it said Zero, but it does not matter which number is. It should be a ``something.'' However, you may ask ``What is something?'' It is really just ``something.'' Why we need to say the first number as Zero? It is fine as One, 42, -1, or x... You can write in Japanese ``Rei'', or in German, ``Null.'' In short, this is ``something.'' But, one thing I would like to make this clear, this ``Zero'' is not the number 0. Because we want to define numbers, so we do not know any numbers, even 0. It is just something the first number and we can just call it Zero. Personally, I prefer to write x since it seems more ``something.''

Definitions are similar to rules of a game. Therefore, we should not think about why this is defined. That is just a starting point of the discussion. It is easy to imagine that this is hard for especially someone who is not familiar with mathematics. This is the same as rules of some game, like soccer game, ``A player is not allowed to touch a ball by hand except goalkeepers.'' If you ask ``Why a player can not use his/her hands?'' Then one can only answer that ``That's a rule of the soccer game.'' This is also true as the rule of chess, shougi, or go... If there are ten kind of games, there are ten kind of rules. Mathematics is the same, there are many rules of mathematics and each rule makes different mathematics. We can make arbitrary kind of mathematics, only necessary condition is such mathematical system must be consistent. But, most of the arbitrary rules can not make interesting mathematics. ``Interesting'' is quite subjective word and it seems not so fit to mathematics. But, many can feel that. I sometimes encountered that some people believe that the mathematics is a kind of truth in the universe. But, mathematics is nothing related with how the universe is. That's the physics' area. However, well established mathematics can describe our universe well. I am fascinated this interesting point of mathematics. It is like a game/sport that has a well established rule are so fun and interesting. As there are many kind of games and sports, there are many mathematics and each mathematics may have different rules. Although, many rules can be shared in mathematics. One can make up new rules and can create a new sports. But it is difficult to create a new interesting sports. It is the same in mathematics, you can create own mathematics easily by making up several definitions, but it is very difficult to make an interesting new mathematics.


By the way, one day for one small thema is still too long for blog for me, I will keep an article short from now on.

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