Since antiquity the intuitive notions of continuous change, growth,and motion, have challenged scientific minds. Yet, the way to the understanding of continuous variation was opened only in the seventeenth century when modern science emerged and rapidly developed in close conjunction with integral and differential calculus, briefly called calculus, and mathematical analysis.
The basic notions of Calculus are derivative and integral:
derivative is a measure for the rate of change, the integral a measure for the total effect of a process of continuous change. A precise under-standing of these concepts and their overwhelming fruitfulness rests upon the concepts of limit and of function which in turn depend upon an understanding of the continuum of numbers. Only gradually, by penetrating more and more into the substance of Calculus, can one appreciate its power and beauty. In this introductory chapter we shall explain the basic concepts of number, function, and limit, at first simply and intuitively, and then with careful argument.
1.1 The Continuum of Numbers
The positive integers or natural numbers 1,2, 3,... are abstract symbols for indicating how many objects there are in a collection or set of discrete elements.
These symbols are stripped of all reference to the concrete qualities of the objects counted, whether they are persons, atoms, houses, or any objects whatever.
The natural numbers are the adequate instrument for counting elements of a collection or set. However, they do not suffice for another equally important objective: to measure quantities such as the length of a curve and the volume or weight of a body. The question, how much? cannot be answered immediately in terms of the natural numbers. The profound need for expressing measures of quantities in terms of what we would like to call numbers forces us to extend the number concept so that we may describe a continuous gradation of measures. This extension is called the number continuum or the system of real numbers (a non descriptive but generally accepted name).The extension of the number concept to that of the continuum is so convincingly natural that it was used by all the great mathematician sand scientists of earlier times without probing questions. Not until the nineteenth century did mathematicians feel compelled to seek a firmer
logical foundation for the real number system. The ensuing precise formulation of the concepts, in turn, led to further progress in mathematics. We shall begin with an unencumbered intuitive approach, and later on we shall give a deeper analysis of the system of real numbers.
Supplement
One of the great achievements of Greek mathematics was the
reduction of mathematical statement sand theorems in a logically
coherent way to a small number of very simple postulates or axioms,the well-known axioms of geometry or the rules of arithmetic governing relations among a few basic objects, such as integers or geometrical points. The basic objects originate as abstractions or idealizations from physical reality. The axioms, whether considered as evident from a philosophical point of view or merely as overwhelmingly plausible, are accepted without proof; on them the crystalized structure of mathematics rests. For many centuries the axiomatic Euclidean mathematics was accepted as a model for mathematical style and even imitated for other intellectual endeavors. (For example, philosophers,such as Descartes and Spinoza, tried to make their speculations more convincing by presenting them axiomatically or, as they said, more geometrico.)
The axiomatic method was discarded when after the stagnation during the Middle Ages mathematics in union with natural science started an explosively vigorous development based on the new calculus. Ingenious pioneers vastly extending the scope of mathematics could not be hampered by having to subject the new discoveries to consistent logical analysis and thus in the seventeenth century an invocation of intuitive evidence became a widely used substitute for deductive proof.Mathematicians of first rank operated with the new concepts guided by an unerring feeling for the correctness of the results, sometimes even with mystical associations as in references to infinitesimals or infinitely small quantities. Faith in the sweeping power of the new manipulations of calculus carried the investigators far along paths impossible to travel if subjected to the limitations of complete rigor.Only the sure instinct of great masters could guard against gross errors.
The uncritical but enormously fruitful enthusiasm of the early period gradually met with counter currents which rose to full strength in the nineteenth century but did not impede the development of constructive analysis initiated earlier. Many of the great mathematicians of the nineteenth century, in particular Cauchy and Weierstrass, played a role in the effort toward critical reappraisal. The result was not only anew and firm foundation of analysis, but also increased lucidity and simplicity as a basis for further remarkable progress.
An important goal was to replace indiscriminate reliance on imprecise'intuition by precise reasoning based on operations with numbers; for naive geometric thinking leaves an undesirable margin of vagueness as we shall see time and again in the following chapters. For example,the general concept of a continuous curve eludes geometrical intuition.A continuous curve, representing a continuous function, as defined earlier, need not have a definite direction at every point; we can even construct continuous functions whose graphs nowhere have a direction,or to which no length can be assigned.
Yet one must never forget that abstract deductive reasoning is merely one aspect of mathematics while the driving motivation and the great universal scope of analysis stem from physi
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Since antiquity the intuitive notions of continuous change, growth,and motion, have challenged scientific minds. Yet, the way to the understanding of continuous variation was opened only in the seventeenth century when modern science emerged and rapidly developed in close conjunction with integral and differential calculus, briefly called calculus, and mathematical analysis.
The basic notions of Calculus are derivative and integral:
derivative is a measure for the rate of change, the integral a measure for the total effect of a process of continuous change. A precise under-standing of these concepts and their overwhelming fruitfulness rests upon the concepts of limit and of function which in turn depend upon an understanding of the continuum of numbers. Only gradually, by penetrating more and more into the substance of Calculus, can one appreciate its power and beauty. In this introductory chapter we shall explain the basic concepts of number, function, and limit, at first simply and intuitively, and then with careful argument.
引 言
自古以来,关于连续地变化、生长和运动的直观概念，一直在向科学的见解挑战。但是，直到十七世纪，当现代科学同微分学和积分学(简称为微积分)以及数学分析密切相关地产生并迅速发展起来的时候,才开辟了理解连续变化的道路.
微积分的基本概念是导数和积分:导数是对于变化速率的一种度量，积分是对于连续李化过程总效果的度量.正确理解这些概念以及由此产生的大量丰富成果，有赖于对极限概念和函数概念的认识，而极限和函数的概念又基于对数的连续统的了解。只-.有越来越深刻地洞察微积分的实质，我们才能逐渐地赏识其威力和价值，在引言这一章里,我们将阐明数、函数和极限的概念。首先作一简单而直观的介绍,然后再仔细论证.
1.1 The Continuum of Numbers
The positive integers or natural numbers 1,2, 3,... are abstract symbols for indicating how many objects there are in a collection or set of discrete elements.
These symbols are stripped of all reference to the concrete qualities of the objects counted, whether they are persons, atoms, houses, or any objects whatever.
The natural numbers are the adequate instrument for counting elements of a collection or set. However, they do not suffice for another equally important objective: to measure quantities such as the length of a curve and the volume or weight of a body. The question, how much? cannot be answered immediately in terms of the natural numbers. The profound need for expressing measures of quantities in terms of what we would like to call numbers forces us to extend the number concept so that we may describe a continuous gradation of measures. This extension is called the number continuum or the system of real numbers (a non descriptive but generally accepted name).The extension of the number concept to that of the continuum is so convincingly natural that it was used by all the great mathematician sand scientists of earlier times without probing questions. Not until the nineteenth century did mathematicians feel compelled to seek a firmer
logical foundation for the real number system. The ensuing precise formulation of the concepts, in turn, led to further progress in mathematics. We shall begin with an unencumbered intuitive approach, and later on we shall give a deeper analysis of the system of real numbers.
实数连续统
正整数或自然数1,2,.....这些抽象的符号，是用来表示在离散元素的总体或集合中具有多少个”对象。
这些符号完全不涉及所计数的对象的具体性质，不管它们是人,是原子,是房子,还是别的什么。
自然数是计算一个总体或“集合”中元素的一种合适工具但是,为了达到一个同等重要的目的，如度量曲线的长度、物体的体积或重量等这样一些量，自然数便不够用rsquo;了. 我们不能直接用自然数来回答“是多少?”这一类的问题。由于极其需要用我们称之为数的事物来表示各种量的度量,我们就不得不将数的概念加以扩充,以便能够描述度量的连续变化。这种扩充了的数系称为数的连续统或“实数”系..(这是一个末加说明但一般都认可的名称.)数的概念向连续统概念的扩充是如此自然而令人信服，以致所有早期的大数学家和科学家都毫无疑议地予以采用.直到十九世纪，数学家们才感到必须为实数系寻求一个比较可靠的逻辑基础。随后产生的对_上述概念的正确表述，反过来又导致数学的进步。我们将首先从不难理解的直观描述入手，然后给出实数系的比较深入的分析'.
Supplement
One of the great achievements of Greek mathematics was the
reduction of mathematical statement sand theorems in a logically
coherent way to a small number of very simple postulates or axioms,the well-known axioms of geometry or the rules of arithmetic governing relations among a few basic objects, such as integers or geometrical points. The basic objects originate as abstractions or idealizations from physical reality. The axioms, whether considered as evident from a philosophical point of view or merely as overwhelmingly plausible, are accepted without proof; on them the crystalized structure of mathematics rests. For many centuries the axiomatic Euclidean mathematics was accepted as a model for mathematical style and even imitated for other intellectual endeavors. (For example, philosophers,such as Descartes and Spinoza, tried to make their speculations more convincing by presenting them axiomatically or, as they said, more geometrico.)
The axiomatic method was discarded when after the stagnation during the Middle Ages mathematics in union with natural science started an explosively vigorous development based on the new calculus. Ingenious pioneers vastly extending the scope of mathematics could not be hampered by having to subject the new discoveries to consistent logical analysis and thus in the seventeenth century an invocation of intuitive evidence became a widely used substitute for deductive proof.Mathematicians of first rank operated with the new concepts guided by an unerring feeling for the correctness of the results, sometimes even with mystical associations as in references to infinitesimals or infinitely small quantities. Faith in the sweeping power of the new manipulations of calculus carried the investigators far along paths impossible to travel if subjected to the limitations of complete rigor.Only the sure instinct of great masters could guard against gross errors.
The uncritical but enormously fruitful enthusiasm of the early period gradually met with counter currents which rose to full strength in the nineteenth century but did not impede the development of constructive analysis initiated earlier. Many of the great mathematicians of the nineteenth century, in particular Cauchy and Weierstrass, played a role in the effort toward critical reappraisal. The result was not only anew and firm foundation of analysis, but also increased lucidity and simplicity as a basis for further remarkabl
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