Mathematical Logic Questions And Answers Pdf
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Once I was asked by a student if he could use my formal system to prove his own theorems. I explained that he could, but it would be tedious, and not very interesting. He asked how difficult it was to prove something about natural numbers. I told him that the question was meaningless, since he could only answer it by referring to theorems in a more complicated theory. This is the essence of the circle that you mention.
A third aspect of mathematics, as pointed out by Tom, is its use in physics. This is done by formulating the laws of physics in a certain formal language, using some logical notions. This is a more recent development, but already it is producing interesting results. For example, the theoretical physicists have written down what they regard as a complete system of logic, and it is believed that everything that can occur will occur in the universe, when all the stars have died out. This may not be the ultimate theorems of logic, but it is a beautiful system, and there is a sense in which the theory of everything is about the axioms of this system. It is an interesting line of thought to ponder whether this system is self-consistent; but I am not certain that any one has thought about this question.
I do not believe that it is possible to do math without using formal logic. If someone tells you that he does not believe in formal logic, this is a sign of very poor taste. I was always amazed at the thought of people who believed that a proof was not a proof if it was not written down. The truth is that without the use of formal logic, there is no such thing as mathematical logic. The great authors of mathematical logic were mathematicians who realized that mathematics was not mathematics unless written down in symbols. In the course of time, mathematicians have devised formal systems which are equivalent to each other in many ways.
Logic is beautiful because its structure is so strict. The logic of mathematics is often messy, but logic is orderly, and it is a joy to work in. If you can set aside the feeling that you are using a dubious method and are only doing mathematics, then you may find that it is good fun. The nice thing about working with formal logic is that the rules of the game are well-established, and if you understand the rules, you can play the game at the highest level in the world.
The problem that mathematics provides us with a foundation for our knowledge of the world is, in my opinion, the most fundamental problem we can ask of our knowledge of the world. It is not a matter to be solved in a day, nor to be solved in a year or in a hundred years. It is a problem that is as old as mathematics itself. It is an ultimate question that we must answer for ourselves.
First, mathematics is a human activity, and one of the most important human activities. If we fail to provide a satisfactory foundation for mathematics, then we will not succeed in developing the most important human activity. 827ec27edc