Exploration Of The Limit Idea In Philosophy And Mathematical History

As a kind of philosophical and mathematical idea, the development of the limit idea has undergone three stages—embryo, development and improvement. There is so much effort which philosopher and mathematicians had been making for in the long and tortuous evolutional process, with the shine of humans' wisdom. The formation of the limit theory provides not only a theory basis for calculus but also a strong tool for humans to recognize the infinite. It is an important idea of modern maths, which reveals the different features between higher maths and elementary maths from the methodology. It will have certain theoretical significance for our understanding of the history of mathematics and some problems of the mathematical philosophy, if we clarify the development skeleton of the limit idea and reveal the core of the limit idea and its inner relationship with philosophy.Zeno, an ancient Greek philosopher, proposed four famous paradoxes for defending his belief in the middle of the fifth century BC. These paradoxes were philosophical propositions, but they evoked people to research the limit idea of mathematics, and they were the key dynamics which produced by variable mathematics. The Cyclotomic of Liuhui in China and "Method of exhaustion", which proposed by Antiphon in ancient Greece and improved by Vieques in EU and developed by Archimedes, both implicated the prototype of modern limit concept. They have promoted for the further research of accurate limit concept.With the recognition and research of the infinite idea, the limit idea has developed constantly. In philosophy, Plato's actual infinite concept and Aristotle's potential infinite concept become the main evident for two parties of philosophers respectively. Both concepts are vital to the formation of the limit idea which improves more perfectly day by day.The establishment of calculus greatly promoted the development of the limit idea. According to the infinite idea, Newton and Leibnitz created calculus from different angles by using limit idea and method. The limit idea they created on the mathematical field took on great vitality even though their jobs are lack of the logical basis and mainly dependent on direct viewing. However, their efforts and achievements settle a solid basis for the further improvement of the limit ideaWith the widely using of calculus, the suspicions and critics for the unclear explanation of its basis, infinitesimal concepts, came from some sides, especially from philosopher Berkeley who was British Archbishop of Cloyne and critic. It caused a severization campaign of calculus even the whole analysis. The tightness of analysis begined from the work of Maclaurin, Cauchy, Abel, Connaught Boer and Dilikelei, and it developed further and finished by German mathematician Weill Shitelashi. He proposed the static definition of limit which not just won the legitimate status for the limit idea but provided the strict theoretical foundation for calculus by characterizing the trend of variable and getting rid of the intuitive marks of previous limit concept.The improvement of the limit idea is a need of social practice. It adds new dynamic for the growth of mathematics, and becomes the basis and starting point of modern mathematical idea. The rich dialectic idea contained by the limit idea is an excellent application of the unity and opposite sides of materialist dialectics in mathematical field. With the help of the limit idea, people can know from limit to infinite, from changing to unchanging, from straight figures to curved figures, from quantitative change to qualitative change, from approximation to exactitude. The establishment of the limit theorem makes maths get rid of many troubles of paradoxes about infinite. It is good for not only cultivating ways and qualities of thinking, but enhancing the ability of analyzing and solving problems.