I have a Japanese version of this page. There was a good question on my last article. ``What happen if the vending machine (a function) is broken?''
First of all, we need to define what broken means. Some of you would say, ``Broken is broken, what else?'' But, this answer does not make any sense for mathematicians. Are there no common sense in the head of mathematician? Maybe, yes. But, there is a reason.
I am a Sunday mathematician/programmer. Mainly I program a code to solve some of my problems. To tell my computer to solve my problem, I need to interpret my problem to a code which my computer can execute/understand. Many of mathematics formulation is really formulated, which means you do not need to understand what it is, they are just a procedure. Then my computer can execute to solve my problem. I formulate some problem since after that is done, I do not need to think about that. Rest of the problem is solved automatically. This is fun for me. ``Computer, search some information. Computer, find the shortest path from my home to the station. Computer, find the cheapest ticket under this condition...'' All these things must be translated to a code which my computer can understand. ``Broken is broken'' is not enough for the computer. Since computers are so stupid so far.
For example, Broken could mean 1. when you input something, but nothing comes out, 2. when you input anything, the output is always the same, or 3. when you input something, the output seems totally random. Which is the broken means for you? None of them? Current usual computer can not guess, we need to tell that.
Also, ``broken'' has subjective meaning. There are chips which have build-in self destruction mechanism. For example, some kind of decryption chips. These chips contain a secret encryption key, so some malicious people want to read it. When the chip detects such activity, it breaks itself. Some of the credit card chips and DVD copy detection chips have this function.
When you can not read the information, usually it means ``broken.'' But the designer of these chips designed to do that. If someone can still read the secret information, it is broken for the designer. Therefore, ``broken'' is subjective meaning. When someone's credit card is stolen, the owner usually does not want to his/her card available anymore. If the chip does not self destructed by the malicious one's attack, then the owner may sue the designer, ``The chip was not broken, because it was broken. If it is not broken, it should have broken itself.'' A human can understand this means, but, it is difficult for current machines.
Lambda calculus thinks about all kind of functions. Therefore, such ``broken'' function (whatever it means) must be described. If lambda calculus can not describe some of the functions, that is the limit of this calculus. But, a person must define what is the broken means if he/she want to describe it in lambda calculus. Mathematician is a such a lazy people, but they are also perfectionist. The art of mathematics is how to reach the ``perfect laziness.'' Therfore, all kinds of functions are of course considered. In the a few thousand years of mathematics history, one era is ended when the people found out the perfectness of mathematics system or limit of mathematics system. But this story is too large here and I would like to concentrate only at the some aspect of the lambda calculus. Although, I would like to mention about that if I could.
19 hours ago