In the sixteenth and seventeenth centuries, European capitalist countries rose one after another, and in order to compete for hegemony, it was urgent to develop navigation and arms industry. In order to develop navigation, it is necessary to determine the position of ships in the sea and the latitude and longitude on the earth; In order to fight, we also need to know how to make the shell hit accurately, which urges people to study various "movements" and the quantitative relations in various movements, which provides the basis for the generation of the concept of function.
/kloc-In the mid-7th century, Descartes introduced the concept of variable (variable) and formulated analytic geometry, thus breaking the understanding of the unknown that is limited to equations. Later, Newton and Leibniz independently established the differential theory. During this period, with the enrichment of mathematics content, a large number of specific functions appeared, but functions were not given a universal definition. Newton has always used the word "fluency" to express the relationship between variables since he began to learn calculus in 1665. 1673, Leibniz first used the term "fluency" in a manuscript. He uses a function to represent any quantity that changes with the change of points on the curve. (Definition 1) This can be said to be the first "definition" of a function. For example, the length of tangent, chord and normal and the horizontal and vertical coordinates are equal. Later, this term was used to represent power, that is, X, X2, X3, ... Obviously, the original meaning of the word "function" is very vague and inaccurate.
People will not be satisfied with such an inaccurate concept, and mathematicians have further discussed the function.
Second, the development and perfection of the concept of function 1. Based on the function concept of "variable", Bernoulli, a Swiss scientist and Leibniz student, gave a clear definition in 17 18: the function of a variable is an analytical expression composed of these variables and constants. (Definition 2) and the symbol φx of the function is given here. This definition makes the function analytic for the first time.
/kloc-In the mid-8th century, when famous mathematicians D'Alembert and Euler studied string vibration, they felt it necessary to give a general definition of function. D'Alembert thought that a function refers to any analytical formula, and in 1748 Euler defined it as: a function is an arbitrarily drawn curve. (Definition 3) Before that, in 1734, Euler also gave the symbol f(x) of a function, which we still use today.
In fact, these two definitions (definition 1 and definition 2) are two commonly used expressions of functions: analytical method and mirror method. Later, due to the appearance of Fourier series, the relationship between analytical expressions and curves was communicated. However, it is obviously one-sided to define functions with analytic expressions, because many functions have no analytic expressions, such as Dirichlet function.
1775, Euler gave a broader definition in the preface to the book Principles of Differential Calculus: If some variables depend on other variables in such a way that when the latter variable changes, the former variable also changes, then the former variable is called a function of the latter variable. (Definition 4) This definition simply reflects the dialectical factors in the function and the vivid process from "self-change" to "dependent change", but it does not mention the corresponding relationship between the two variables, so it does not reflect the characteristics of the real scientific function concept, but is only the "embryonic form" of the scientific definition of the function concept.
Function is a concept derived from the study of object motion, so the definitions of the first few function concepts only admit the relationship between variables, such as the falling distance of free fall and the amplitude of simple pendulum motion, which can be regarded as functions of time. Obviously, the understanding of the concept of function is limited only from the perspective of the "change" of variables in motion. For example, for constant functions, there is no explanation.
/kloc-At the beginning of the 9th century, Lacroix formally proposed that as long as one variable depends on another, the former is a function of the latter. 1834, Russian mathematician Lobachevsky (лобачевский) further put forward the definition of function: the function of X is a number that has a certain value for each X (definition This definition points out the necessity of correspondence (condition) and the idea of function correspondence, and the concept of correspondence is the essence and core of function concept.
/kloc-Cauchy, a French mathematician in the 0 th and 9 th centuries, gave a clearer definition: there are two interrelated variables, and the value of one of them can be changed arbitrarily within a certain range. Such a variable is called an independent variable, and the value of another variable changes with the value of the independent variable. This variable is called dependent variable, and the function as independent variable is called dependent variable. (Definition 6)
In 1829, Dirichlet gave the so-called Dirichlet function: when x is a rational number, y =1; Y=0 when x is irrational. This function is not complicated, but it cannot be expressed analytically. This idea is the beginning of the transformation of mathematics from the past study of "calculation" to the later study of "concept, nature and structure". In 1837, he defined a function as: in a certain change process, there are two variables X and Y. If for each fixed value of X in a certain range, Y has a unique fixed value corresponding to it according to a certain corresponding relationship, then Y is called a function of X; X is called an independent variable. (Definition 7) The advantage of this definition is that it directly emphasizes and highlights the "correspondence" relationship and grasps the essential attributes of the concept. Only one rule is needed, so that every value in the domain of this function has a certain y value corresponding to it, no matter whether this rule is a formula, an image, a table or other forms; Its disadvantage is that it omits and simplifies the vivid idea of functional change.
Function concept based on "set"
The concept of function develops with the development of mathematics. The definition of function is constantly improved, abstracted and perfected in the development of mathematics. In 1970s, German mathematician G Cantor put forward set theory. After entering the 20th century, with the development of set theory, the concept of function has also made new progress. It finally got rid of the shackles of the number domain, expanded to a broader research field, and realized the modernization of the concept.
At the beginning of the 20th century, the American mathematician Weblan gave the following definition of function: If there is such a relationship between the set of variable Y and the set of another variable X, that is, there is a completely determined value of Y corresponding to each value of X, then Y is said to be a function of variable X. (Definition 8) Since this definition, the concept of function has always been based on sets, while the first seven definitions are based on variables (numbers).
With the passage of time, the function is clearly defined as the corresponding relationship between sets, defined as: A and B are two sets. If any element of A has a unique element in B according to a certain correspondence, such correspondence becomes a function from set A to set B (definition 9). According to the concept of mapping, this definition establishes the concept of function from the point of view of "mapping", which can be described as follows: the mapping f from set A to set B: A→ B is called the function from set A to set B, which is called function F for short. (Definition 10) The above three definitions break the shackles of the number field and make the elements in the set abstract, which may or may not be numbers, but all other tangible or intangible things. For example, X is the set of all triangles, Y is the set of all circles, and F can be the mapping of each triangle to its circumscribed circle.
The definition of new function can be understood as follows: a function is a correspondence (rule), and for elements in a certain range (set), another element is determined according to this correspondence (rule). In this way, the concept of function is transformed from "change" in a narrow sense to "correspondence" in a broad sense, and function is correspondence (rule).
Compared with the primary stage, understanding the concept of function with "correspondence" ("rules") reveals the essence of the concept of function, but at least it does not meet our intention of using undefined terms. Because what is "correspondence" and how to understand "rules" need to be defined. For example, if the rules are different, will the functions be different? For example, f(x)=x and f(x)=( 1+x)- 1 are of course different rules, but they define the same function.
In order to solve this contradiction, at the beginning of the 20th century, especially after the 1960s, the definition of function involving only the concept of "set" was widely adopted, but set was not defined as the original concept. This definition is: Let A and B be any two sets, and F be a subset of Cartesian set A× B, which satisfies the following conditions: ① For any a ∈ A, there is a B ∈. Then call F a function from A to B, and write F: A → B. (Definition 1 1) This definition uses the concept of "relationship", and gives a general definition of a function that only involves the original concept "set", that is, it does not need to use "correspondence" to avoid the interpretation of "rules". As long as the set theory is applicable to all fields of mathematics, the function definition given in this way is always applicable. It is the most modern definition.
So far, the most perfect definition of "function" has been given (definition 1 1). As one of the most basic concepts in mathematics, the foundation has been directly established on the set, that is, the function is regarded as the correspondence from one set to another, which is actually the same as "mapping".
Third, the comparison between the old and new definitions The new definition (the definition based on set is called the new definition, and the definition based on variable (number) is called the old definition. ) and the old definition, there are two important differences: (1) The old definition is based on the basic concept of "variable", while the new definition is based on the basic concept of "set". What is a variable? It is usually understood as something that can be measured after a unit is selected, such as length, quality, time, etc. On the one hand, this understanding is too vague and can only be illustrated by examples, which is difficult to be accurate; On the other hand, because it involves the relationship between size and size, it is too narrow to reflect the universality of application. Secondly, even if there is no question of what is "quantity", as a variable, it must take a value in a certain range (not necessarily a numerical value), which is actually a set A (the domain of the function) that has to be presupposed. The so-called "variable value" is essentially a circuitous statement that "A belongs to A" in disguise. The concept of visible variables already contains the idea of set.
⑵ In the old definition, "dependent variable" is a function, and in the new definition, "correspondence" is a function. The essence of the concept of function is not that the dependent variable "changes" with the convenient quantity, but that there is a certain correspondence between the two sets. Obviously, the new definition can reveal the essence of function more directly.