We use abstract numbers, but we don't intend to associate them with concrete objects every time. What we learn in school is an abstract multiplication table-always a multiplication table of numbers, not the number of boys multiplied by the number of apples, or the number of apples multiplied by the price of apples, and so on.
Similarly, taking geometry as an example, it is a straight line rather than a taut rope, and in the concept of geometric line, all properties are abandoned, leaving only the elongation in a certain direction. In short, the concept of geometric figures is the result of abandoning all the attributes of real objects and leaving only their empty forms and sizes. All mathematics has this abstract feature.
The concept of integers, the concept of geometric figures-these are just some of the most primitive mathematical concepts. After that, many other concepts have reached the level of abstraction, such as complex number, function, integral, differential, functional, n-dimensional and even infinite-dimensional space.
These concepts seem to be more abstract than one, so that they seem to have lost all contact with life, so that "ordinary people" can't understand anything except "inexplicable". In fact, this is definitely not the case. Although the concept of n-dimensional space is really abstract, it has completely realistic content.
It's not that difficult to understand these contents. For example, it is necessary to study the influence of N factors such as light, moisture, fertilization and close planting degree on wheat crop yield. Maybe we need to set n variables here and put them in n-dimensional space to study.
Of course, we don't want to discuss the abstraction of mathematical concepts too much here.
In early ancient humans, although they could judge the number of a project in practice in their own way, there were often only one and many concepts. For example, after hunting, people can often record the number of prey captured by the number of knots on the rope or the number of notches on the cave wall or bone.
The concept of arithmetic reflects the relationship between the quantities of a group of objects. These concepts are abstracted on the basis of analyzing and summarizing a lot of practical experience, and they are gradually produced. At first, they are numbers associated with a specific object, then abstract numbers, and finally the concepts of general numbers and any possible numbers.
Each stage is prepared by applying the accumulated experience of previous concepts. For example, in the abstraction of integer concept, the concept of single number (such as number 1, 2,5, etc. ) was abandoned at first, and then the concept of arbitrary integer was introduced after further abstraction.
Like the development of arithmetic, geometry is abstracted on the basis of analyzing and summarizing a lot of practical experience. People extract geometric forms from nature itself. Such as the shape of the moon and sickle, the water level of the lake, the straightness of light or trees, people improve it into their own handicrafts.
The concepts of geometric quantities-length, area and volume-are also produced from production practice. Engaged in agricultural production, people need to measure the area of land. It is also necessary to calculate the capacity of warehouses or containers when putting grain into granaries or engaging in business.
Geometry came into being from practice, spread from Egypt to Greece geographically, got greater development from the ancient Greeks, and developed in the direction of accumulating new facts and clarifying the relationship between them.
Yes, geometry is engaged in the study of "geometric objects" and figures, and studies their quantitative relations and mutual positions. But the geometric object is nothing else, it is a realistic object that abandons other attributes such as density, color and weight, and only examines it from the perspective of its spatial form.
In this way, geometry takes the spatial form and relationship of real objects that have abandoned all other attributes, in other words, it takes "pure form" as its own object. It is this degree of abstraction that distinguishes geometry from other sciences that also study the spatial forms and relationships of objects.
For example, astronomy studies the mutual position of objects, but only the mutual position of celestial bodies, geodesy studies the shape of the earth, and crystallography studies the shape of crystals. In all these cases, the study of the form and position of specific objects is related to or interdependent with their other attributes.
This is one of the basic laws of the formation of mathematical concepts: mathematical concepts are produced through a series of abstraction and generalization processes on the basis of the experience accumulated by predecessors in abstract concepts.
Of course, we should also remember that in mathematics, we study not only the quantitative relations and spatial forms directly abstracted from the real world, but also the relations and forms defined on the basis of established mathematical concepts and theories within mathematics. This is the second basic law of the formation of mathematical concepts.