2009-02-19

Abstraction --- Infinite can live in finite (1)

Japanese version

Here I would like to remind you about abstraction. Abstraction derives a common idea that is not associated with any specific instance from many ideas associated with instances. Because this blog is oriented to ``getting a feeling of understanding,'' I need to explain the motivation, ``why we should need to mention about abstraction?'' We introduced the idea, abstraction, because concrete instances are not sufficient. If someone can understand about what is the addition, s/he can add a pair of numbers out from infinite combination of numbers. Because any memory devices have a limit in size, there is a limit to add the numbers in finite combination. It is a strong idea that understanding we can add any number.

We could design a machine that seems compute numbers by memorize the answers. This is a different approach to make machine to compute. We could teach 1 + 1 equals 2, 1 + 2 equals 3. Then if you ask this machine, what is 1 + 2? Then you can ask to the machine, what is 2 + 3. The machine can not answer the question since the machine has not memorized that answer for a long time. This is why remembering each instance is not sufficient.

We could abstract 1, 2, 3, ... as a concept ``numbers.'' We could design a machine which process ``numbers.'' This means a machine can treat infinite kind of number instances.

Here infinity means, we can add ``any number'' which is quite normal. 1+10 is possible, but it should not happens that 1 + 128 is not possible. Sometimes, I encounter a person who think that ``nothing can do on infinity.'' However, you can chose one number from infinite numbers and you can add it with one. It might be possible your choice is too large and you can not say that number in your life time. However, you know you can add any two numbers.

I personally think that the most important subject in the high school mathematics is differentiation and integration. It has an idea, ``Infinite can live in finite.'' You can see this through the concept called convergence. For example, there are infinite number of numbers between zero to one. We can project and create one to one mapping from an infinite plane to hemi-sphere with radius one except one point. Pythagoras thought infinite can not live in finite, Zenon made a famous paradox to indicate Pythagoras's mistake.

We could think about infinite numbers by an abstraction of number itself. This is a strong idea. In lambda calculus, we abstract functions with three definitions. All three definitions are about functions. Thanks to abstraction, we can handle any functions in lambda calculus --- infinite type of functions.

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