Let's get back to this ``(_)''. ``(_)'' represents ``something'' here.
Marvin: Again, ``something''... I am tired.
You may ask ``what is something?'' I understand. But the answer is still ``something'' Or ``something of something''. For example, if I limit the subject to money, I could say this something is ``something of money.'' If you put ``some(thing) of money'', exact the same ``some(thing) of money'' will be out. This is ``λ(_).(_).'' See figure 1. The output amount never larger or smaller than the input. Therefore, if you put 100 Altair dollar, the output is 100 Altair dollar. If you put 200 Altair dollar, the output is 200 Altair dollar. If you put 'some' Altair dollar, the output is exact same 'some' Altair dollar. This is the meaning of ``something''. I can not say more exact since it is abstracted. Some schools teach this ``something'' as 'x', so some people feel easy to understand as if you input x, then the output is x.
Figure 1. a Lambda expression.
Then, it does not matter if we replace ``(_)'' with ``(^).'' The same function we can write as ``λ (^).(^),'' ``λ x.x,'' or ``λ y.y.'' Conventionally, people wrote this as ``λ x.x.'' This is the meaning of ``Something.'' If you ask mathematicians, ``What is something here?'' then they will answer, ``something is something.'' They are not fooling you, that is the best answer they have.
Let's back to the analogy of vending machine. A simple machine can only accept 100 Altair dollar and can issue 100 Altair dollar ticket for Sirius. This vending machine is simple because it fixes the destination and the price. There were such vending machine on earth is described by Heron of Alexandria.
However, if the machine can accept other destinations and other prices, that would be more versatile. For example, it can also sell 150 Altair dollar ticket for Orion. Not only Altair dollar, but if it accepts a galactic credit card and you can get the ticket for the restaurant at the end of the universe. Or a concert ticket in Kreuzberg. I think you would agree the ``something'' is more general, this is more useful. The first example of ``something'' was just 100 Altair dollar. Then it becomes any price of Altair dollar, then credit card. The output started with a ticket for Sirius, then Orion, restaurant and concert. We would like to think about all kind of ``something'' here, that's the idea of function.
Now I hope you know what the meaning of ``something'' here. Marvin seems have a comment.
Marvin: ``One of my designer developed a machine, called a general exchanger. the difference between an usual exchanger and a general exchanger is the general exchanger accepts anything, and outputs something which has the equal value to the input. When you put your software, I presume you took more than three months to develop it, the output was an old bread with a cup of cold tea.''
I: ``That was broken, wasn't it?''
M: ``Yes, it was. I can not accept there was a cup of tea.''
4 weeks ago