Human has an ability to abstract each concrete numbers, like, 1, 2, 3, ... to ``(general) numbers'' by name. And we can understand that they (numbers) can include all possible numbers. This is outstanding ability of human. It is enough interesting for me that any people have this counting ability and ability of speak except in case of diseases. But a machine seems not have such ability (yet).
Human can easily perform this abstraction. For example, human can have a concept of ``chairs.'' It is still a difficult problem to search pictures of chairs. Human can see a catalog book of IKEA (a famous furniture shop now in 2009) and can recognize chairs. How can we use it is also easy to be understood. A word ``chair'' includes infinite type of chairs. But, a human can recognize a new design chair of this year which never existed until this year.
Similarly, lambda calculus does not treat each concrete function. It concerns all kind of functions. We can never predicted what kind of problem we need solve in the future. However, if we have a theory for any kind of functions and if this concept ``function'' is enough abstracted, we can use this theory to solve the problem arisen in the future. Mathematics is hard to be obsolete. For example, I doubt the idea of ``addition'' would be obsolete later in thousand years. The reason we think about this lambda calculus or abstraction is we expect the theory last long. I like a knowledge which last long since I do not need to learn all the time. Many study/science aimed this way, but it is not so easy.
Mathematics sticks at infinity, definition, or something mathematics stuffs since we want to expect that what correct today is also correct tomorrow. It is interesting for me that mathematics is the basics of almost all sciences and technology. A study, which does not change once it is established, drives all the world technology change. But if you think a bit more, this is natural. The advancement of technology is based on yesterday's advancement. If this base is changed every day, we could have never advance. The technology itself can change every day. But the base of technology can not be unstable. The basis of the technology is so stable, therefore, we can put some change based on that.
19 hours ago