* Read before - K

* Read before - K's look at the math QUEER academic thought. As far as I know, the subject of all disciplines are divided into two types. Studies on humans (humanities, social) and the study of nature (science). Of course, but not even a foot in both areas ranging as evolutionary biology that does not jump over the still human beings and nature that category incehesap itself (yet). But math is a little different. Average mathematics incehesap are often treated as a category of natural science. But really incehesap it? No. Mathematics is very natural and has nothing to do. Although there have helped with a lot of math to other sciences, but the study of mathematics itself is closer to the nature of our recognition system inside "something" is not. Think of the number. incehesap Among the most basic natural. Although the name of a natural number is a natural number, but the presence of God is not a substance. In fact, the person number 1? Of course not. Constant? Rational? incehesap A mistake? Complex? The same is true. Without understanding and accepting that we are starting from a primitive bunch of Shem are looking at a number of things to know, gained through speculation for its highly abstract mathematical patterns "concept". The same geometry as well as our points, lines, familiar concepts such as cotton and be aware of it, and take advantage of good living, but unconsciously points, lines and surfaces is things do not exist in the real world. The so-called "mathematical objects" is a reality that can be perceived through our senses, while none of that stuff just does not exist really. To be nevertheless able to describe the laws of nature so useful studies is very mysterious phenomenon. Many mathematicians "The world is just a simulation (模 寫) the idea" that the fact that the mathematical Plato who is sympathetic to understand the philosophy of Plato can be seen in this context. (Interestingly, the mathematical incehesap knowledge are the best gatjiman Is it really true immortality? That point is the subject of a long debate. Sooner or later we'll introduce a good article about it) Anyway birth in the human mind (or find) and mathematics, but Are the study of human beings, it is also not known if. Even if this study the function yunriron or political ideology is not derived. Of course, although the math says helped with a lot of humanities. If so, is the study of mathematics incehesap is what the hell? "Today is not the desire for self-control pattern." - Movie "A Beautiful Mind> John Nash's sleep at the end of the Ambassador Bill Devlin case mathematics is the" science of patterns. " Confusing to find a pattern in something to give an order, incehesap this is called mathematics. What went through high school age annoying everyone remember the beatings and problem solving? In fact, those things of nature and mathematics is the distance. Mathematics is also an important part of solving the problem, of course, but the core of mathematics to prove, namely mathematical objects after Ji Han of any pattern through proven processes to generalize put it on the rock of eternity. This is the essence of mathematics and mathematics rather than real people tasted the entrance is why you feel the charm of mathematics. We spread out the math book, I learned one thing it means that mathematical knowledge is to know the truth, one that never change muneojyeodo the world. So Does the study of mathematics as a discipline is defined as any form? These are the answers can be found especially in the field of mathematics called incehesap Foundations of mathematics called. Basic theory say the word. "What will you study?" As we say, when the math Means a theory about the fundamental question. Once the target from me Is not establish whether the study will or without the playing cards. But coming incehesap out of high school level math barely reflect my one thousand and one discussing the Foundations of mathematics euroseon incehesap knowledge that is difficult (the name is based on theory, but it's actually a high school mathematics) incehesap was brought fur one good article in Galloway, DC mathematics. I think people who are interested will be an enjoyable've incehesap intellectual stimulation. ------------------------------------------------------------------------------------------------------
9 1. What is the axiom of mathematical axioms, we would indeed incehesap seek the 'infinite rigor' in mathematics. So when we prove any proposition, I really do have to prove to D? Those who tried to prove how far from proof is omitted and will be pondering what to write down more since I've seen anywhere. But none of you from the beginning to the end you will not have seen that it is the perfect proof. incehesap For example, let's was to prove the following self-evident fact. Prove the problem) is 0 is present. People incehesap who saw it and pay the most to prove incorrect, 0 is the logic of wool or hereafter defined, because there is no need to prove. However, to prove the existence of a definition that is a separate issue. incehesap For example, I x + 1 = 0 for s hajamyeo Let's say that you have defined the s -1 of the year that it is not. If so, may I have s is defined, because there is an unconditional high? It will not. Therefore, the presence of a zero is to be verified.
Of course, the above proposition is very abstract sense yiraseo If you came back, you can not give it seemed like the example below. Cleanup) remaining Clean about two natural numbers incehesap a and b, a> b when there is a constant q and r satisfy the following. a = bq + r (0 r <b) proof) incehesap held for the largest q satisfying bq a. That is, bq a <b (q + 1) held to the q. Then, when r = a-bq in Let's get. The q and r is an integer and satisfies the condition. The proof of the above is copied almost exactly what came to downplay any middle school textbooks. Does the proof of the above is strictly? De is the question first, incehesap prove that q is the land question, as always present. Using the notion of Well-Order may be solved according as this thing is still a mountain. Is r are integers? Would always be able to say that an integer integer integer subtract? Moreover, how do bq is Jung Su-ra is not guaranteed? In terms neuleoman no one goes deeper questions. r = a-bq with a = bq + r why that's the equivalent expression? Any ideas why the equation is to be established, in addition to the same on both sides? incehesap In addition, why do commutative? These endless questions are being asked not to be different from that primordial byeolban '1 + 1 = 2, and the problem of proving please. So that embellish Euclid axioms are created. Means the axioms are "intuitively decided to admit the proposition to be true without proof." In other words, for that which is considered as the axiom "Do not argue whether a true or false." incehesap Is to say. Euclid has put forward five axioms in geometry to build your own, offered us the perfect one value is omitted no proof from him. Of course, incehesap this axiom must be set when "just say I see fit." For example, the first axiom of Euclid are as follows: A straight line connecting the first axiom) any two points of the Euclidean can draw. You know you see horses too obvious. incehesap But look at the five axioms. incehesap Euclid's fifth axiom) parallel axiom] There are two lines, incehesap when another line passing through the two, the sum of the diagonal between the two lines is less than 180 ever meet. Did you think you're looking at one of gilgin but we figured no different from learning natural horse and the words 'the sum of the two diagonal in parallel is a 180'. But the fifth axiom of Euclid, you get attacked by a famous mathematician David Hilbert. Hilbert would have pointed out the fact that Euclid's fifth axiom draw a straight line on a non-planar surface is immediately be destroyed. Therefore, we must keep in mind when you set up the axiom in the following incehesap respects. 1. not cause inconsistency in any situation. 2. will be simple and useful. Based on this modern mathematics has set up many kinds of axioms. However, a number of axioms incehesap are based on the nine axioms eventually be introduced in the future. ZFC axioms called nine axiom that sustains most of it is just math. 2. ZFC axioms ZFC axioms is named after the initials of the two mathematicians Zermelo, Fraenkel's initials and Choice (theorem). Of course, BG, MK, such as the number of axioms, but the most commonly used will be discussed based on the ZFC axioms. It also requires a logical axiom, however, mathematical logic, once we will be ignored. The basic axioms of ZFC axioms 9 are as follows: (.. strict mathematical expressions wrote loose translation of words, because it is difficult to understand that I was randomly) 1. The existence axiom: There is an empty set. 2. The equivalence axiom: The X Y, Y X if X = Y. 3. A pair of axioms: x, y when there is a set {x, y} is also present. 4. The agreement axiom: X, if Y is present, there is also a set XUY. 5. A set of axioms: when X is present, there are also meeting P (X) of the subset of X. 6. Replace axioms: y for the element of the set A, there is a set B of y to satisfy the proposition p. 7. Axiom of infinity: a collection N is the set of natural numbers. 8. The basic axiom: There is a b c ... a days. 9. theorem: Suppose A is any set consists of several split. In this case, remove the elements one by one in each division of A