Lx=d: SundayResearch

Japanese version Blaise Pascal made a calculator for helping his father's job (tax calculation). Usually it is painful to calculate a huge amount of accounting. I am not good at calculation. So, I thought if I have a computer, I do not need to compute anything myself. That's one of the motivation I took a computer science course in my University. Some people totally misunderstand that a computer scientist is good at arithmetic. No. If someone is good at arithmetic, why does she/he need to learn that? If a human can fly faster than sound, maybe we do not need to use a plane. If people can communicate without speaking between thousand kilometers away, why we need a telephone? I can not do arithmetic, therefore I learned computer science. So, let's make a computer. Figure 4 is a computer SUCC mark 1 by Sirius Cybernetics corp. This computer gets one Church number as an input, and outputs another Church number. This computer does not understand what the number is, but just exec...

Japanese version Last time, a circle was a symbol to represent a number. But, there is no such thing (a symbol to represent a number) in the Peano's axiom. Peano's axiom only define a Zero and a successor. we employed a square to represent Zero. But when we tell two numbers to a machine, we can not distinguish two numbers if we have only Zeros. See Figure 2. Therefore, we use a circle as a delimiter. One could say, we can use a space, but we also need to tell a space to a machine, otherwise any machine can not know a space exist. We need something like a number 0. 0 means ``there is nothing.'' If we write down nothing, how we could know something is missing. If we put 0, then we know nothing actively exists. 0 can represent ``existence of nothing.'' This is an excellent invention of human being. By the way, speaking about space, there is no space character in Japanese. I think also Korean and Chinese do not have space character. Therefore, a processing of Asia...

Last Saturday, I walked around the city with my friend. DDR buildings are a bit chilly... Or it is already winter here.

Japanese version Last time we were talking about how Peano defined the natural number. Because Lambda calculus defines the numbers based on that. Mathematical formulation makes the discussion (proof) more exact, this ``exact'' is important for mathematician. But, the formulation causes increasing the exactness, which means, there are no such thing, like ``You know about the numbers, just do something like calculation in appropriate way.'' Because even every single obvious issue should be defined in formulation. As the side effect, we could execute these rules on a machine --- we can make a computer! That's the interesting point for me. There are many ways to how to implement a computer as a machine. Pascaline and Charles Babbage's differential engine are gear based. Here our base is Peano's axiom. Before Marvin points out, this explanation is Masahiko Sato and Takafumi Sakurai's ``Basic theory of programming.'' (By the way, It is not so conven...

Japanese version This is continued from the last article. Please refer the Peano's axiom in the last article. The second definition means that we can create a next natural number from the current one. This function creates a successor number from the current number, therefore it called ``successor'' function. This defines ``plus one'' function. We have already the first natural number, zero, then we can make a successor number from zero. This successor number is ``something'' of zero. It is usually called one, but not necessary. This definition just said, it is something different from zero. We have now: 1. there is the first number, 2. we can make a successor number from a number. Out of these two definitions/rules, it seems we can make the whole natural numbers, but this is not enough for that. The third definition said that there is no loop of successor function, i.e., if you repeat the successor function staring with x, the result of them will never retu...

Japanese version A few weeks ago, I visited Dresden and saw a bridge called Blauses Wunder. The formal name of the bridge is Loschwitzer Bruecke, but even this name is on a map. The literal meaning of this word is ``Blue Wonder,'' but in German, there is another meaning -- very surprising in a bad way. This is bit strange... The originally this bridge was blue. Dresden is a beautiful city. I have friends there, so I visited there several times. I first know the word, Blauses Wunder in the book from Preussler, ``Der Raueber Hotzenploz.'' (Am Ende der Spur sollte jeder von beiden sein blaues Wunder erleben, dafuer hatte Hotzenploz vorgesorgt. in Chapter 6) Größere Kartenansicht

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. Zero is a number. If a is a number, the successor of a is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves equal. (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.'...

One of my friends is an artist. Recently, he made an article, ``Stereo Shadow.'' He used two light sources, one is blue, the other is red. Audiences see their shadow with wearing a red-blue filter glasses. Then they can observe their own ``3D'' shadow in the wall. There were several exhibitions of this in Japan. I heard he has prepared a video of this, and I am looking forward to seeing that. The black dots in the picture is for audience's safety. He found some children try to touch their shadow and hit the wall without noticing. So the dot shows ``Here, there is a wall'' sign.

Japanese version According to the Wikipedia's page , Lambda calculus was introduced by Church and Kleene in the 1930s as part of an investigation into the foundations of mathematics. By the way, I am an amateur Sunday mathematician, therefore, please do not believe this blog without check by yourself. This is just I think I understand these stuffs. So I think there must be many errors. I try to avoid errors, but, this is not my profession. I am also not confident about my English. Welcome the comments. The origin of this blog is Wikipedia . I could say in cool way, I was inspired by the Wikipedia's page. If you understand the Wikipedia's entry, I don't recommend to waste of time by reading this blog. There are bunch of interesting text around the world. When you read ``Lambda calculus was introduced by Church and Kleene in the 1930s as part of an investigation into the foundations of mathematics.'', and if you think ``I see, that's the reason of why lambda ...

Japanese version Standard mathematics books explain mathematical stuffs as definition, theorem, proof, definition, theorem, proof, .... This is quite simple and enough abstracted. Therefore, we can also explain lambda calculus in the standard way. But Marvin will sure complain that is so depressed. More abstracted theory could be more applicable to many things. It becomes less unnecessary stuffs, then, it becomes simpler and also more beautiful in a sense. The theory is to the point when more abstract. Japanese sword seeks for the beauty in the sword itself, it never decorates with some kind of jewels. Because a sword maker/master thinks the beauty comes from the sword itself. They shamed if they need to cover a sword with non-sword component. We can find many swords, staffs, ... are decorated with gold or some jewels. I can also see some kind of gorgeousness in that, however, I prefer beauty in these kind of simpleness. A French pilot said ``A designer knows he has achieved perfectio...