Perhaps, how a certain philosophical thought explains a certain mathematical phenomenon is a more objective expression.
Let's start with the definition. According to the strict ε -δ language, the definition of limit is as follows:
Let the function f(x) be defined in the eccentric neighborhood of point x0. If there is a constant a, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when X satisfies inequality 0,
| f(x)-A | & lt； ε.
Then the constant a is called the limit of the function f(x) when x→x0.
This definition defines what is infinite approximation-how small it can be, but it can't get itself.
It is explained that δ is determined by ε, and the independent variables tend to be as close as possible. And if the limit exists, the function can be as close as it wants, but not itself.
This definition is very strict, sweeping away the ambiguity caused by all specious basic concepts, and completely separating philosophy and metaphysics from calculus.
Philosophy has only some selective and far-fetched explanations for individual phenomena in higher mathematics.
Let's take the Soviet textbook Marxist Philosophy as an example to see how this system "explains" the mathematical limit:
Limit is the embodiment of the law of mutual variation of quality. The change from tangent to tangent at the intersection of straight line and curve is a process from quantitative accumulation to qualitative change.
When the quality changes with each other, the quantitative change can be the accumulation of quantity or the change of structure and order. When summing infinite series, if the order is changed or parentheses are added, the final convergence and divergence and limit value may change.
Of course, the infinite series of absolute convergence, no matter how to change the order and add brackets, the result is the same, which shows that internal cause is the basis of the development of things, external cause is the external condition of the development of things, and external cause must work through the content. External factors have a great influence on the development of things, and sometimes they can cause changes in the nature of things. However, no matter how big the external factors are, they should also be exerted through internal factors.
Limit is the embodiment of unity of opposites. For infinitesimal quantities, it is neither 0 nor 0. When we look at it alone, we can't judge whether it is 0 or not. Only by putting it into a specific limit can we know whether it is zero (high-order infinitesimal) or non-zero (same-order infinitesimal)-the existence of things depends on its opposite, as if there is only black in the world, there is no concept of color, and there is no black saying.
So there are a lot of ... basically, they shoot first and then draw the target.
It's not that I deny Marxist philosophy or any other philosophy. It is a rigorous modern mathematical axiom system and a strict and precise definition, which denies the role of philosophy in mathematics.
Whenever we talk about philosophy, we are dissolving philosophy-on the one hand, we think that consciousness is the reflection of matter in the mind, on the other hand, we think that philosophy can guide science and mathematics a priori, which is anti-Marxist in itself.
History is the history of mankind, and philosophy is the philosophy of mankind. People create history in production struggle and class struggle, and all those who observe and think construct different philosophical systems in their minds.
Natural science is a natural law, and mathematics is a logical law. In the final analysis, people can only follow them and use them if they find them and know them. They can't create natural laws out of thin air against facts, nor can they fabricate mathematical theorems out of thin air without logic.
I implore philosophy to put down natural science and mathematics.
Since the existence of a society with its own class, people's knowledge can be divided into production struggle and class struggle knowledge-the former is a weapon to fight against nature and the latter is a weapon to fight against people-and it is impossible to win by confusing the two kinds of struggle weapons.
Philosophy belongs to man, the essence of man, and in reality, it is the sum total of all social relations. So let philosophy shine in its proper field-social science, let the glory of capitalism eliminate all oppression and exploitation, liberate more free people, and constantly discover and deepen our understanding of natural science and mathematics.