Abstract: This paper expounds that mathematics is full of dialectics from the aspects of the unity of opposites in mathematical operation, the interrelation between different mathematical knowledge, the quantitative change to qualitative change in the development of mathematical theory, and the negative law of negation in mathematics.
[Keywords:] mathematical dialectics, unity of opposites, contradictions are interrelated
The world is an objective material world, which follows the laws of movement, change and development. Materialist dialectics means that the world is objective and the material world is universal and eternal. Mathematics is full of dialectics, and mathematicians at all times and all over the world regard the thought of dialectics of nature as the guiding ideology for studying mathematics, thus achieving one achievement after another. It is a meaningful work to study mathematics according to dialectical materialism.
First, the unity of opposites in mathematical operations.
In mathematics, addition, subtraction, multiplication and division, multiplication and division, exponential and logarithmic operations, trigonometric and inverse trigonometric operations, differential and integral operations, etc. , are reciprocal operations. Reciprocal operation is two opposite sides, which is a concrete reflection of positive and negative contradictions in the real world in mathematics. They are interdependent and inseparable. Transform each other under certain conditions. The existence and unity of the positive and negative of mathematical operations is a powerful lever to solve mathematical problems, so whether there is an inverse operation for a given operation and how it is formed has always been the central topic of mathematical research.
There are high and low levels in mathematical operations. Generally speaking, we call addition and subtraction, multiplication and division, multiplication and square root as primary, secondary and tertiary operations respectively. There is a certain relationship between superior operation and subordinate operation, which can be transformed into each other. For example, multiplication is addition with the same addend, and power is multiplication with the same factor. Derivation of multivariate function comes down to derivation of univariate function, integration of multivariate function comes down to differentiation of function, and differentiation and integration of univariate function are linked by Newton-Leibniz formula.
Second, mathematics is full of contradictions.
Constant and variable are two very important concepts in mathematics. Constant is a quantity that reflects the relative static state of things, and variable is a quantity that reflects the changing state of things. They are different. But they are relative and dependent, and can be transformed into each other under certain conditions, so they are unified.
The finiteness and infinity in the real world are reflected in mathematics and become the finiteness and infinity of quantity. In mathematics, people often know infinity through finiteness. On the one hand, infinity can exist as a finite sum, and it can also exist as the opposite of all finite; On the other hand, it can be used to describe the process of quantity change. There is a qualitative difference between finite and infinite. For example, there is no one-to-one correspondence between a finite set and any of its proper subset. But at infinite concentration, this is not the case. For example, natural number set can establish a one-to-one correspondence with proper subset. A finite number set must have a maximum number and a minimum number, but an infinite number set is not necessarily the case. Another example is that the finite sum of numbers satisfies the commutative law and associative law, but these laws cannot be applied arbitrarily in the infinite sum formula, otherwise it will lead to fallacious results.
Straight and curved are two different images. Geometrically, the former curvature is zero and the latter curvature is non-zero. Algebraically speaking, the former is a linear equation and the latter is a nonlinear equation, so the difference between straight and curved is extremely obvious. Engels said: "Geometry begins with the following development. Straight lines and curves are absolutely opposite. A straight line can't be represented by a curve at all, nor can a curve be represented by a straight line. Both are incommensurable, but the calculation of a circle is only possible if it is represented by a straight line. When the curve has an asymptote, the straight line becomes a curve completely, and the curve becomes a straight line completely. The concept of parallelism also tends to disappear. These two lines are not parallel, they are constantly changing. This is the dialectical thought that under certain conditions, straight and curved can be transformed into each other.
Third, the quantitative change to qualitative change in the development of mathematical theory.
The law of quantitative change and qualitative change points out that quantitative change and qualitative change are the two most basic States of the movement and change of things, and the development and change of things show the repeated process of quantitative change to qualitative change, and then qualitative change causes new quantitative change. Mathematical theory embodies the laws of quantitative change and qualitative change. On the one hand, the existence of every concept in mathematics has a specific quantitative boundary. If the quantitative change exceeds this limit, qualitative change will occur and another concept will be formed. This new concept has its own unique new quantitative change. For example, the range of the number of sides of a regular polygon is "a finite number greater than or equal to 3". If the number of sides changes beyond the above range, it is no longer a regular polygon, but a line segment or a circle. (When the number of sides is less than 3, it is a line segment; When the number of sides exceeds the range of finite number, that is, it is a circle when it tends to infinity. ) Line segments and circles have their own new quantitative changes. On the other hand, the formation process of mathematical theory is from quantitative change to qualitative change, from approximation to accuracy. For example, in order to find the area of the curved trapezoid, the curved trapezoid is divided into several small curved trapezoid. If the division is dense enough, these small curved trapezoids can be approximately regarded as small rectangles. Then the approximate area of each small curved trapezoid is obtained by finding the rectangular area, and the sum is the approximate area of the original curved trapezoid. Because it is only an approximation, the above process is a process of quantitative change, and there is no qualitative leap. If division is infinite encryption, that is, the maximum width of each small curved trapezoid tends to zero, the exact area of the original curved trapezoid can be obtained, which is the basic idea of definite integral theory.
Fourthly, the development of mathematical theory embodies the law of negation of negation.
The law of negation of negation reveals that the complete process of the development of things itself is: it goes through two negations and three stages, that is, from affirmation to self-denial, and then from negation to new affirmation-negation of negation. The development of every mathematical theory conforms to the law of negation of negation. When the theory was first formed, it was affirmative; With the need of practice and the deepening of research, the imperfections and inaccuracies of this theory are gradually exposed and denied; Then mathematicians began to study how to make the theory more perfect and accurate, and finally reached new conclusions and new affirmations. In addition, the results of mathematical operations also reflect the law of negation of negation. For example, a positive number is still positive after being negated twice. In propositional logic, two negatives of a proposition are still the original proposition.
In a word, mathematics contains philosophical thoughts everywhere, mathematicians are constantly maturing in the vicissitudes of philosophy, and philosophical views are constantly improving under the impetus of mathematical achievements.
References:
[1] Zhang shizao. Mathematics pedagogy in middle school. Jiangsu Education Press, 199 1.
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