Three mathematical crisis,Ancient Mathematics,Medieval Mathematics,Early Modern Mathematics,Modern Mathematics,Classification,Mathematics,The first mathematical crisis,In ancient Greece,Pythagoras put forward the famous proposition All is number.All the numbers can be marked as integer or integer ratio.,1,1,d=?,Pythagoras,Solution of the first mathematical crisis,In about 370 BC,Pythagoras s two students solved the first crisis of mathematics by introducing the concept of incommensurable quantities (不可约通量)in geometry or irrational number (无理数). But this crisis was completely solved in the Nineteenth Century, relying on the establishment of real number theory.,Influence of the first mathematical crisis,Influence of the first mathematical crisis is huge, which greatly promoted the development of mathematics and related sciences. First of all, the first mathematical crisis made people aware of the existence of the irrational number(无理数), after the irrational number was born, many mathematicians, formal study of irrational number gave strict definition of the irrational number and proposed a new classification of rational and irrational numbers,laying the foundation for the development of mathematical analysis.,The second mathematical crisis,The second mathematical crisiss germination（萌芽） is in about 450 BC, from the historical and logical point of view, this crisis is inevitable. It happened in seventeenth Century, when Newton founded the calculus. It was triggered by Bekeles revolt.,Bekele asked: infinitesimal(无穷小量) as a quantity, is it exactly 0?,The famous paradox- Bekele paradox,Bekele,Newton,Solution of the second mathematical crisis,Until the nineteenth Century, Cauchy detailedly and systematically developed of the limit theory. Weistrass founded the limit theory.In addition ,the theory of real numbers, set theory were established, thus freeing infinitesimal （无穷小量） from metaphysical(形而上学的) fetters（束缚）.,Cauchy,Weistrass,Influence of the second mathematical crisis,The second mathematical crisis began with peoples exploration of the infinite quantity, while Bekele paradox is the direct cause of this crisis. The process of solving this crisis led to the glamour (辉煌)of analytics in the eighteenth Century , so it is called the age of analysis(分析时代).,The third mathematical crisis,The second half of the nineteenth Century, Cantor founded the famous set theory.Mathematicians found that from the natural numbers(自然数) and the Cantors set theory could establish the whole mathematics building.,But, in 1903 ，it came a shock news to the mathematical community(数学界) : Set theory is flawed(有缺陷的)! This is the British mathematician Russell proposed the famous Russell paradox.,Cantor,Russell,The German mathematician Zermelo thought that appropriate axiom（公理）system can restrict the concept of the set , to pure the set logically.,Thank you！,