본 논문에서는 문헌에 나타난 기하학의 발달과정을 중심으로, 오늘날 수학의 학적 체계를 갖추게 한 연역법에 대하여 다음과 같이 고찰하고자 한다. 먼저, 실용적인 것을 목적으로 한 그리이스 이전의 수학을 살펴보고, 그리이스 시대의 수학을 고찰하였다. 이 시대에 와서 자명한 공리에서 출발하여 연역적으로 추론하여 점차 자명한 것과는 먼 새로운 정리들을 얻어내는 체계가 완성되었는데, 이는 유클리드「기하학 원본」에 의하여 집대성 되었다. 또한 본 연구에서는 평행선 공리에 대해 고찰하였다. 유클리드의 제5공리인 평행선 공리는 자명해 보이지 않았으므로 다른 자명한 공리로 대체하거나 또는 증명해 보려는 노력으로 ...
본 논문에서는 문헌에 나타난 기하학의 발달과정을 중심으로, 오늘날 수학의 학적 체계를 갖추게 한 연역법에 대하여 다음과 같이 고찰하고자 한다. 먼저, 실용적인 것을 목적으로 한 그리이스 이전의 수학을 살펴보고, 그리이스 시대의 수학을 고찰하였다. 이 시대에 와서 자명한 공리에서 출발하여 연역적으로 추론하여 점차 자명한 것과는 먼 새로운 정리들을 얻어내는 체계가 완성되었는데, 이는 유클리드「기하학 원본」에 의하여 집대성 되었다. 또한 본 연구에서는 평행선 공리에 대해 고찰하였다. 유클리드의 제5공리인 평행선 공리는 자명해 보이지 않았으므로 다른 자명한 공리로 대체하거나 또는 증명해 보려는 노력으로 비유클리드 기하학이 탄생하였다. 비유클리드 기하학에서는 유클리드의 평행선 공리를 부정하여 모순이 없는 논리를 전개하였다. 이로 인해 수학에서의 연역법에 공리에 대한 개념이 바뀌었는데, 유클리드의 기하학에서의 공리는 절대적인 진리였던 것이 논리의 전개를 위한 가설로 되었다. 결국, 비유클리드 기하학이 탄생하게 된 계기로 수학의 공리체계에는 무정의 요소의 도입에 대한 필요성을 느꼈고, 모리츠 파쉬, 페아노, 힐베르트 등에 의해서 무정의 요소가 사용되었다. 또한 수학의 전개과정에서 무모순성에 대한 문제가 대두되기 시작했는데, 힐베르트는 완전 무모순성을 증명하려 하였고, 괴델은 어떤 수학체계에 있어서, 거기서 허용되는 논리만을 써서는 그 체계의 무모순성을 증명할 수 없다는 것을 그의 제 2 정리에서 증명하였다.
본 논문에서는 문헌에 나타난 기하학의 발달과정을 중심으로, 오늘날 수학의 학적 체계를 갖추게 한 연역법에 대하여 다음과 같이 고찰하고자 한다. 먼저, 실용적인 것을 목적으로 한 그리이스 이전의 수학을 살펴보고, 그리이스 시대의 수학을 고찰하였다. 이 시대에 와서 자명한 공리에서 출발하여 연역적으로 추론하여 점차 자명한 것과는 먼 새로운 정리들을 얻어내는 체계가 완성되었는데, 이는 유클리드「기하학 원본」에 의하여 집대성 되었다. 또한 본 연구에서는 평행선 공리에 대해 고찰하였다. 유클리드의 제5공리인 평행선 공리는 자명해 보이지 않았으므로 다른 자명한 공리로 대체하거나 또는 증명해 보려는 노력으로 비유클리드 기하학이 탄생하였다. 비유클리드 기하학에서는 유클리드의 평행선 공리를 부정하여 모순이 없는 논리를 전개하였다. 이로 인해 수학에서의 연역법에 공리에 대한 개념이 바뀌었는데, 유클리드의 기하학에서의 공리는 절대적인 진리였던 것이 논리의 전개를 위한 가설로 되었다. 결국, 비유클리드 기하학이 탄생하게 된 계기로 수학의 공리체계에는 무정의 요소의 도입에 대한 필요성을 느꼈고, 모리츠 파쉬, 페아노, 힐베르트 등에 의해서 무정의 요소가 사용되었다. 또한 수학의 전개과정에서 무모순성에 대한 문제가 대두되기 시작했는데, 힐베르트는 완전 무모순성을 증명하려 하였고, 괴델은 어떤 수학체계에 있어서, 거기서 허용되는 논리만을 써서는 그 체계의 무모순성을 증명할 수 없다는 것을 그의 제 2 정리에서 증명하였다.
Ⅰ. Introduction In this paper 1. deduction in mathematics will be discussed, 2. discovery of non-Euclidean geometry will be discussed, 3. introduction of undefined notions in axiomatic system of mathematics and problem of consistency will be discussed. Ⅱ. The establish of deduction Modern mathematic...
Ⅰ. Introduction In this paper 1. deduction in mathematics will be discussed, 2. discovery of non-Euclidean geometry will be discussed, 3. introduction of undefined notions in axiomatic system of mathematics and problem of consistency will be discussed. Ⅱ. The establish of deduction Modern mathematics is based on 'deduction', which is a method of deriving conclusions by strict reasoning from axioms. Early mathematics required practical experiences for its development. During pre-Greek era, people aquired truth of mathematics by means of their experiences. But Greek employed some 'proofs' in mathematics (first performed by Thales), and that is the main difference between above two eras. After Thales, deductive method was developed by many contributors, namely Pythagoras, and so on. Pythagoras proved many theorems including the famous 'pythagoras theorem'. The above mentioned Greek achievements in mathematics were collected and systematically arranged in Euclid's 'Elements'. The 'Elements' has been highly praised because of skillful selection of the propositions and their arrangement into a logic sequence, which followed deductive reasoning starting from a small handful of initial assumptions. This work is composed of thirteen books with a total of 465 propositions. In book I, Euclid commences with the necessary preliminary 23 definitions, 5 postulates, and 5 common notions. In his 'Elements', Euclid regarded these postulates and common notions as absolute truth but there is evidence that logical foundations of parallel postulate (the 5th postulate) gave the early Greek considerable troubles. This postulate lacks the terseness and simple comprehensibility of the others and in no sense possesses the characteristic of being 'self-evident'. Therefore it is natural to think that perhaps it could be derived as a theorem from the remaining 4 postulates and 5 common pptions, or that it could be replaced by a more acceptable equivalent. Of the many substitutes that have been devised to replace Euclid's parallel postulate, the one most commonly used is that made by John Playfair. It is: Through a given point not on a given line can be drawn only one line parallel to the given line. The attempts to derive the parallel postulate as a theorem from the remaining nine postulates and common notions occupied geometers for over two thousand years and culminated in some of the most fat leaching developments of modern mathematics. Gerolamo Saccheri did very important work on this problem and obtained many theorems of new mathematics which is now called non-Euclidean geometry. But Saccheri lamely forces into his development an unconvincing contradiction involving haze notions about infinite elements. If he had not been so eager to exhibit a contradiction and rather he had admitted his inability to find one, Saccheri would today unquestionably be credited with the discovery of non-Euclidean geometry. A Russian mathematician named Lobatchevsky also worked on this parallel postulate and he became thoroughly convinced that Euclid's fifth postulate cannot be proved on the basis of the other nine common notions and postulates. He made an assumption in direct conflict with the parallel postulate: Through a point A lying outside a line a there can be drawn on the plane more than one line which do not meet a. With this new postulate, Lobatchevsky deduced a harmonious geometrical structure having no inherent logical contradictions. With his work, Lobatchevsky aquired the honor of discovering this particular non-Euclidean geometry. With the study of the non-Euclidean geometry, it turn out to be true that 'undefined notions' were needed in axiomatic systems of mathematics. The Euclid's 'Elements' did have a deductive structure, but it was replete with concealed assumptions, meaningless definitions, and logical inadequacies. Hilbert understood that not all term in mathematics can be defined add therefore began his treatment of geometry with three undefined objects-point, line, and plane-and six undefined relations-being on, being in, being between, being congruent, being parallel, and being continuous in bin celebrated volume entitled 'Foundation of Geometry'. Ⅲ. Conclusion As we have see, 'deduction' in mathematics is the method of development of systems of mathematics which starts from axioms, which includes undefined notions, and through logical reasoning, obtains proved propositions, that is, theorems.
Ⅰ. Introduction In this paper 1. deduction in mathematics will be discussed, 2. discovery of non-Euclidean geometry will be discussed, 3. introduction of undefined notions in axiomatic system of mathematics and problem of consistency will be discussed. Ⅱ. The establish of deduction Modern mathematics is based on 'deduction', which is a method of deriving conclusions by strict reasoning from axioms. Early mathematics required practical experiences for its development. During pre-Greek era, people aquired truth of mathematics by means of their experiences. But Greek employed some 'proofs' in mathematics (first performed by Thales), and that is the main difference between above two eras. After Thales, deductive method was developed by many contributors, namely Pythagoras, and so on. Pythagoras proved many theorems including the famous 'pythagoras theorem'. The above mentioned Greek achievements in mathematics were collected and systematically arranged in Euclid's 'Elements'. The 'Elements' has been highly praised because of skillful selection of the propositions and their arrangement into a logic sequence, which followed deductive reasoning starting from a small handful of initial assumptions. This work is composed of thirteen books with a total of 465 propositions. In book I, Euclid commences with the necessary preliminary 23 definitions, 5 postulates, and 5 common notions. In his 'Elements', Euclid regarded these postulates and common notions as absolute truth but there is evidence that logical foundations of parallel postulate (the 5th postulate) gave the early Greek considerable troubles. This postulate lacks the terseness and simple comprehensibility of the others and in no sense possesses the characteristic of being 'self-evident'. Therefore it is natural to think that perhaps it could be derived as a theorem from the remaining 4 postulates and 5 common pptions, or that it could be replaced by a more acceptable equivalent. Of the many substitutes that have been devised to replace Euclid's parallel postulate, the one most commonly used is that made by John Playfair. It is: Through a given point not on a given line can be drawn only one line parallel to the given line. The attempts to derive the parallel postulate as a theorem from the remaining nine postulates and common notions occupied geometers for over two thousand years and culminated in some of the most fat leaching developments of modern mathematics. Gerolamo Saccheri did very important work on this problem and obtained many theorems of new mathematics which is now called non-Euclidean geometry. But Saccheri lamely forces into his development an unconvincing contradiction involving haze notions about infinite elements. If he had not been so eager to exhibit a contradiction and rather he had admitted his inability to find one, Saccheri would today unquestionably be credited with the discovery of non-Euclidean geometry. A Russian mathematician named Lobatchevsky also worked on this parallel postulate and he became thoroughly convinced that Euclid's fifth postulate cannot be proved on the basis of the other nine common notions and postulates. He made an assumption in direct conflict with the parallel postulate: Through a point A lying outside a line a there can be drawn on the plane more than one line which do not meet a. With this new postulate, Lobatchevsky deduced a harmonious geometrical structure having no inherent logical contradictions. With his work, Lobatchevsky aquired the honor of discovering this particular non-Euclidean geometry. With the study of the non-Euclidean geometry, it turn out to be true that 'undefined notions' were needed in axiomatic systems of mathematics. The Euclid's 'Elements' did have a deductive structure, but it was replete with concealed assumptions, meaningless definitions, and logical inadequacies. Hilbert understood that not all term in mathematics can be defined add therefore began his treatment of geometry with three undefined objects-point, line, and plane-and six undefined relations-being on, being in, being between, being congruent, being parallel, and being continuous in bin celebrated volume entitled 'Foundation of Geometry'. Ⅲ. Conclusion As we have see, 'deduction' in mathematics is the method of development of systems of mathematics which starts from axioms, which includes undefined notions, and through logical reasoning, obtains proved propositions, that is, theorems.
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