Jumat, 18 Desember 2009
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Mathematical Research
This is Description about the opinion how to do Mathematical Research. There are several ways or steps to do the Mathematical Research. All of the steps must be done continuously because each step based on the others. We have to do all of the steps without reduce one of them.
The steps of Mathematical Research are:
I. In deep Study References
II. Identify and Formulate The Problem
III. Develop Method of Research
IV. Process of Research
V. Finding
VI. Publish
VII. Journal
Before starting to do research of mathematics, we must also do the preparation. This preparation determine the mathematician what he is reasonable or not to do the research. The preparations are:
1. We need references to do Mathematical research
We can do many ways to get references, in the modern era like now, there are many tools and infrastructures supporting our works. The ways we can do such as:
• Browsing Internet
The Internet provides opportunities galore, and can be used for a variety of things. Some of the things that you can do via the Internet are:
E-mail: E-mail is an online correspondence system. With e-mail you can send and receive instant electronic messages, which works like writing letters. Your messages are delivered instantly to people anywhere in the world, unlike traditional mail that takes a lot of time. Access Information: The Internet is a virtual treasure trove of information. Any kind of information on any topic under the sun is available on the Internet. The ‘search engines’ on the Internet can help you to find data on any subject that you need.
So we can get anything we needed in internet that can support our preparation to start Mathematical Research.
• Reading the Books
Books are portable. You can take them almost anywhere. As such, you can learn almost anywhere too. Books for professionals contain arguments for or against the actions within. A book on cooking argues that Chili powder goes well with beef and goes poorly with ice-cream. A book on building a business argues that testing an idea for profitability before setting up is a smart strategy and argues against just barreling forward with the idea without testing. Books take you places and you learn from books too. Books can help you understand things.
• Reading the magazines, scientist paper, etc.
2. Knowing The Nature of Mathematics
There are a lot of definitions about the nature of mathematics, all of definitions have the same idea about mathematics. The natural mathematics is a science or system which is constructed by deductive method composing definition, axiom, and theorem in such a way that there is no contradiction inside.
So we can say that mathematics is the science and mathematics is the system. As the science, mathematics is knowledge that very large in mind and it can be applied in our life. There are many contents in mathematics that become our task to find out them.
As the system, mathematics is constructed by definition, axiom and theorem.
Definition is a formal passage describing the meaning of a term (a word or phrase). The term to be defined is the definiendum (plural definienda). A term may have many subtly different senses or meanings. For each such sense, a definiens (plural definientia) is a cluster of words that defines that specific sense of the term.
Axiom is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths. In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).
Theorem is statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proofs of theorems have two components, called the hypotheses and the conclusions.
The examples of mathematics system are:
• Group Theory
• Ring Theory
• Numbers Theory
• Eucledian Geometry
• Non-Eucledian geometry
• Numbers system
3. Knowing The Purpose to do Mathematical Research
Before doing everything we must know the aim inside. It is the same if we want to do research of mathematics. The Aim of Mathematical Research is examine and establish the new system of mathematics. We can find many aspects in mathematics that examined the new system. The aspects are definition, axiom, theorem, pattern, rule, procedure, low etc.
4. Finding the supporting factors to do Mathematical Research
The factors that support the research of mathematics such as:
• Knowledge of Mathematics
• Knowledge of History of Mathematics
• Knowledge of philosophy of Mathematics
• Knowledge of world famous mathematician
• Experiences of developing Mathematics
• Asking
The Example, we learn about history and philosophy of mathematics, we can learn about Godel Theorem and Hilbert theory.
Godel theorem issued to reject or pushed away the Hilbert theory. In his Theory, Hilbert come to make one and only one system of mathematics, he certain that the system of mathematics can be composed by one system. Then, this theory is rejected by Godel with the theorem called Godel theorem.
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory
The substance of godel theorem are:
- Complete system of mathematics is not consistent
- Consistent system of mathematics is not complete
The conclution of godel theorem is impossible that there is only one system of mathematics. The proof is theorem of incompleteness. Gödel's incompleteness theorems are two theorems of mathematical logic that state inherent limitations of all but the most trivial axiomatic systems for mathematics. They state that any consistent system of axioms whose theorems can be listed by a computer program is incapable of proving certain truths about arithmetic. This means that any consistent computable formal theory which can prove some arithmetic truths cannot prove all arithmetic truths. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. The result is widely interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem.
The last, we can not said a truth Mathematician if we never do the research of mathematics. If we want to start research, we have to do the preparation above and also do each steps of mathematical research.
This is Description about the opinion how to do Mathematical Research. There are several ways or steps to do the Mathematical Research. All of the steps must be done continuously because each step based on the others. We have to do all of the steps without reduce one of them.
The steps of Mathematical Research are:
I. In deep Study References
II. Identify and Formulate The Problem
III. Develop Method of Research
IV. Process of Research
V. Finding
VI. Publish
VII. Journal
Before starting to do research of mathematics, we must also do the preparation. This preparation determine the mathematician what he is reasonable or not to do the research. The preparations are:
1. We need references to do Mathematical research
We can do many ways to get references, in the modern era like now, there are many tools and infrastructures supporting our works. The ways we can do such as:
• Browsing Internet
The Internet provides opportunities galore, and can be used for a variety of things. Some of the things that you can do via the Internet are:
E-mail: E-mail is an online correspondence system. With e-mail you can send and receive instant electronic messages, which works like writing letters. Your messages are delivered instantly to people anywhere in the world, unlike traditional mail that takes a lot of time. Access Information: The Internet is a virtual treasure trove of information. Any kind of information on any topic under the sun is available on the Internet. The ‘search engines’ on the Internet can help you to find data on any subject that you need.
So we can get anything we needed in internet that can support our preparation to start Mathematical Research.
• Reading the Books
Books are portable. You can take them almost anywhere. As such, you can learn almost anywhere too. Books for professionals contain arguments for or against the actions within. A book on cooking argues that Chili powder goes well with beef and goes poorly with ice-cream. A book on building a business argues that testing an idea for profitability before setting up is a smart strategy and argues against just barreling forward with the idea without testing. Books take you places and you learn from books too. Books can help you understand things.
• Reading the magazines, scientist paper, etc.
2. Knowing The Nature of Mathematics
There are a lot of definitions about the nature of mathematics, all of definitions have the same idea about mathematics. The natural mathematics is a science or system which is constructed by deductive method composing definition, axiom, and theorem in such a way that there is no contradiction inside.
So we can say that mathematics is the science and mathematics is the system. As the science, mathematics is knowledge that very large in mind and it can be applied in our life. There are many contents in mathematics that become our task to find out them.
As the system, mathematics is constructed by definition, axiom and theorem.
Definition is a formal passage describing the meaning of a term (a word or phrase). The term to be defined is the definiendum (plural definienda). A term may have many subtly different senses or meanings. For each such sense, a definiens (plural definientia) is a cluster of words that defines that specific sense of the term.
Axiom is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths. In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).
Theorem is statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proofs of theorems have two components, called the hypotheses and the conclusions.
The examples of mathematics system are:
• Group Theory
• Ring Theory
• Numbers Theory
• Eucledian Geometry
• Non-Eucledian geometry
• Numbers system
3. Knowing The Purpose to do Mathematical Research
Before doing everything we must know the aim inside. It is the same if we want to do research of mathematics. The Aim of Mathematical Research is examine and establish the new system of mathematics. We can find many aspects in mathematics that examined the new system. The aspects are definition, axiom, theorem, pattern, rule, procedure, low etc.
4. Finding the supporting factors to do Mathematical Research
The factors that support the research of mathematics such as:
• Knowledge of Mathematics
• Knowledge of History of Mathematics
• Knowledge of philosophy of Mathematics
• Knowledge of world famous mathematician
• Experiences of developing Mathematics
• Asking
The Example, we learn about history and philosophy of mathematics, we can learn about Godel Theorem and Hilbert theory.
Godel theorem issued to reject or pushed away the Hilbert theory. In his Theory, Hilbert come to make one and only one system of mathematics, he certain that the system of mathematics can be composed by one system. Then, this theory is rejected by Godel with the theorem called Godel theorem.
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory
The substance of godel theorem are:
- Complete system of mathematics is not consistent
- Consistent system of mathematics is not complete
The conclution of godel theorem is impossible that there is only one system of mathematics. The proof is theorem of incompleteness. Gödel's incompleteness theorems are two theorems of mathematical logic that state inherent limitations of all but the most trivial axiomatic systems for mathematics. They state that any consistent system of axioms whose theorems can be listed by a computer program is incapable of proving certain truths about arithmetic. This means that any consistent computable formal theory which can prove some arithmetic truths cannot prove all arithmetic truths. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. The result is widely interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem.
The last, we can not said a truth Mathematician if we never do the research of mathematics. If we want to start research, we have to do the preparation above and also do each steps of mathematical research.
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