Achilles can't catch up with the tortoise's philosophical answer.

Achilles can't catch up with the tortoise, relying on formal logic. Formal logic itself can't be used continuously without restriction, that is, it can't guarantee to deduce the correct conclusion from the correct premise. The more steps of deduction, the greater the distortion of logic, and even absurd conclusions can be drawn. To ensure that the correct premise does not lead to wrong inference, there should not be too many logical inference steps, and the fewer steps, the smaller the risk of distortion. The concrete form of self-proof in theory is relative in logic, objectivity and grammar.

[Keywords:] Achilles, formal logic, discontinuity

First, the dilemma of logic application.

"Achilles can't catch up with the tortoise" is a philosophical story in ancient Greece. Achilles was a good long-distance runner. Of course, he can catch up with the tortoise and solve it by equation method. Suppose Achilles' speed is A and the tortoise's speed is B. When Achilles starts chasing the tortoise, the tortoise is in front of Achilles. Assuming this distance is c, how long will it take Achilles to catch up with the tortoise? Let the required time be x, then ax=bx+c and x=c/(a-b). Since abc is unchanged, X can certainly find the solution. Of course, if the difference of ab is very small, then the solution can tend to infinity. But the philosophical story is that Achilles can't catch up with the tortoise no matter how fast he runs.

It can be seen that we can also solve problems through thinking, not just through practice or experiment. But when we introduced the problem of infinite partition, things changed. If you think so, Achilles must run the distance that the tortoise has exceeded before he can catch up with the tortoise. This is not a hypothesis, but something that should really be done. But this way of thinking is hypothetical. You can use this way of thinking or not. Once this way of thinking is adopted, the thinking process will not be completed, so that Achilles can't catch up with the tortoise. According to this train of thought, when Achilles passed the distance of the tortoise, the tortoise also advanced a distance during this time, although it was getting smaller and smaller. This kind of thinking can be repeated indefinitely. Every time I think about it, the result is the same. In this process, the logic did not go wrong. We can keep this kind of thinking indefinitely. The distance that the tortoise keeps moving forward will never be zero, although it tends to be infinitely small. Therefore, we can draw a conclusion with the method of formal logic: Achilles will never catch up with the tortoise.

The problem in this story was discovered by the ancient Greek philosopher Zhi Nuo. Achilles can't go home, he thought. Assuming the idea of infinite division, we can think like this: in the process of Achilles' going home, he must walk a distance first, and then he must walk a distance first, so that he can think infinitely, that is, this kind of thinking can be repeated indefinitely, so Achilles will never go home in human thinking. The essence of these two kinds of thinking is infinite division. You can be sure that the result of the first segmentation will not be zero, and the result of the second segmentation will not be zero. The following are all the same segmentation. If we are based on logic, there is no reason to think that the final division will be zero.

These philosophical puzzles also existed in ancient China, such as "a foot pestle, half of which is used every day, is inexhaustible", which means that you can never use up half a root every day. However, it should be noted that matter and space are different, and the infinite division of space is more complicated. According to the principles of contemporary physics, the infinite division of matter has two aspects. On the one hand, macroscopic matter cannot be divided indefinitely, and when it is divided into molecules or atoms, matter cannot maintain itself. But from the origin of matter, we still don't understand the boundary of infinite division of matter, which is a matter of physical structure.

Foreign philosophy also has many similar problems. For example, the story of "The Last Straw" tells that a donkey can bear a certain weight of straw. How can he bear an infinite number of straws, or how can he be sure which straw is the last straw to crush the donkey? You can add straw to the donkey's back one by one in this way. Set an "axiom", if there is a straw on the donkey's back, adding another straw will not crush the donkey, so continuous addition will not crush the donkey. In reality, many people have this kind of thinking, thinking that there will never be the last straw to crush the donkey, so they do whatever they want.

There is also a philosophical story about "baldness" in the west, which tells that a normal person's hair adds a hair or loses a hair, not the difference between baldness and baldness. If this statement can become an "axiom", pull out one hair at a time from a person with normal hair and ask, "Is he bald?" Of course, we can't find the basis for determining whether a hair is bald, so we can't answer whether he is bald. Then continue to think and practice like this, until finally all his hair is pulled out, and he won't be bald. Even if you want to define baldness and determine the number of baldness, it is difficult, mainly because it is difficult to find the physical or philosophical basis for this definition. Even through collective negotiation, it is difficult to define baldness. Of course, baldness can now be defined by biological or genetic knowledge, and perhaps baldness is qualitatively different from normal people in genes.

In real life, there are many such cases. People call it "cutting sausages". This is the first time to slice. Still sausage? Of course it is; Then cut another piece. Still sausage? I still do. Infinitely cut, ask again, but until the cut is finished, there is no essential difference between the previous one and the next one, but the whole has changed. Then, when did sausage change from "one" to "one"? There seems to be no boundaries.

The characteristics of the above problems are easy to solve on the whole, but things are infinitely divided, and there is no essential difference between the front and back divisions, so there is no connection point in the transition from "things to nothing". In other words, there is no boundary between quantitative change and qualitative change.

Second, the reflection on formal logic.

How to solve the above problems? Maybe we can use calculus. In the story that Achilles can't catch up with the tortoise, it involves the problem of "dividing the limited space infinitely in a limited time and at an infinite speed". This division is actually infinitesimal, and we can completely stipulate that this infinitesimal is equal to zero, so as long as there are infinitesimal phenomena or situations, we can think that zero will appear and the changes of things are certain.

The difference between us and the ancients is that we think infinitesimal is zero, while the ancients think infinitesimal can never be equal to zero. In fact, infinitesimal is a complete concept. Once finite, it is not zero. To find the boundary between zero and non-zero, we should actually think about an infinite object in a finite way, or turn a finite thing into infinity.

Perhaps it can be summarized by philosophy here. Logical discontinuity. Because infinitesimal and its concrete manifestations are discontinuous on the whole, the former is zero and the latter is not zero, that is, zero and non-zero are discontinuous. The problem of the ancients was that he wanted to deal with infinite division and infinitesimal in a continuous way, but now we deal with it in a discontinuous way.

To sum up with philosophy, we may come to the view that in the application of formal logic, even if the starting point is correct and the correct process is added, that is, the major premise and minor premise of syllogism are correct, the conclusion may not be correct, so we cannot be superstitious about syllogism and formal logic. Formal logic and syllogism are used in previous philosophical stories and analysis of these stories. In Achilles' story, the major premise is that division can be exhausted or the space can be zero after division, and the minor premise is that it can not be exhausted or zero. The conclusion is that Achilles can't catch up with the tortoise and can't go home.

It can be seen that for a complex problem, we can't try to deduce a correct conclusion from a correct conclusion, and the more steps of derivation, the greater the distortion of logic may be, and even absurd conclusions may be drawn. In other words, if you want to ensure that the correct premise does not lead to wrong inference, then you can't use too many steps in formal logic, you can only use a limited number of steps. The fewer steps you use, the smaller the danger of distortion. Therefore, the application of formal logic should be restricted. In the thinking of formal logic, the boundary between right and wrong can never be found.

The function of logic is only to help people make assumptions, and then on the basis of assumptions, guide people to engage in empirical and practical tests. Logic should never assume the function of testing knowledge. Einstein said: Imagination is more important than knowledge. (2) In the process of understanding, the role of imagination is the key, and the role of investigation is also very important. What is imagination? Of course, imagination has illogical imagination, but it is more logical to imagine. Logic can effectively promote people's cognitive activities, but it must be positioned correctly. In other words, logic is always a hypothesis. In Einstein's concept, logic can only stay in the stage of imagination. Any good logical reasoning is just a hypothesis, and the truth it contains can never be equated with truth and reality.

Therefore, in the process of understanding, we should use a variety of ways of thinking and look at things from a variety of perspectives, both from the whole and from the part; We should not only look at the whole, but also at the individual. Many methods are needed, but there is no absolute right. Of course, human beings can solve simple problems, and we should pay attention to the transformation between various methods for complex problems. The diversification of thinking mode provides an opportunity for human progress, especially there is no need to pursue singleness between rational and irrational, logical and perceptual thinking modes.

The vigilance against formal logic did not attract enough attention in ancient times. Ancient Euclidean geometry is a positive model of formal logic, but in the modern period of European ideological history, represented by rationalism, formal logic was pushed to the extreme. The essence of rationality advocated by rationalism is to fully explain the picture of the world with the formal logic exemplified by Euclid geometry. "appeal to reason, the essence is to resort to logic." (3) this kind of culture is summarized as modernism in contemporary times. Contemporary reflection on rationality and formal logic is deepening day by day. Foucault said, "Nothing is more useless than reason." Zheng Xueli, a Chinese-American scholar at the University of Hawaii, said: Logic is actually the most unrealistic. This kind of criticism is not necessarily correct everywhere, but it really needs a profound reflection on rational positioning. Professor Chen Xinxia believes: "In modern understanding and practice, we must go beyond traditional rationalism." To surpass traditional rationalism, we must rethink the rational ability of formal logic.

Third, the reflection on the logical form.

The essence of logic is language, and the reflection on it should go deep into language. Formal logic is actually thinking in language. The laws in logic are the laws and limitations that human beings have when using language to think. What is formal logic and what is the form of logic? Is language. There is a joke in ancient times that apes are very similar to people, bears are very similar to apes, and dogs are very similar to bears. Can dogs be said to be very similar to humans? Obviously not. It can be seen that language has the function of enlarging and narrowing, which can be used as the forerunner of cognition, but not as the whole of cognition. In the past, people called logic the law of thinking. This definition equates logic with thinking, which is incorrect. Logic is a phenomenon that thinking and language * * * work together, so thinking and language * * * should be used to define logic.

Therefore, logical discontinuity is actually the discontinuity of language. However, the consistency between language and objective reality deserves human reflection. We should pay attention to objective reality, not pure human language. Humans should directly reflect the objective world with words and consciousness, and should not be divorced from the objective world and indulge in the deduction of pure concepts. We can't independently deduce the picture of the objective world through words and thinking, and then we don't pay attention to the objective world, but stay within the scope of language and logic and try to test whether our thinking is correct. Once we think that our thinking is self-evident, we are optimistic that this thinking is foolproof, so we regard this thinking as the law and essence of the objective world itself and think that thinking and the world have reached complete identity. Under such thinking, it is inevitable to pursue ideological coercion and cultural indoctrination of others, which will lead to the rigidity of others' thoughts.

This also requires a reasonable definition of "self-justification". Self-proof is a grammatical concept, not an epistemological concept. Grammar should be well-founded, of course, grammar is not absolutely well-founded, there is no end. In logic and objectivity, we should understand that anything that is self-evident is relative, and it is equivalent to a product in economic activities, and its functional design is endless. Logically self-justifying theories and concepts are endless in logic and objectivity. In other words, the theory can never be absolutely justified in logic, objectivity and grammar.

Einstein said that the world is too simple for anyone to understand. The world is simple, there is not too much logical deduction, only simple and intuitive description, of course, intuitive rather than mysterious intuition. If we use logic to interpret the world, it will complicate the world and may distort it. Of course, logic is not worthless, but it must be properly positioned. It is a link in the cognitive process, which can help people put forward flexible and diverse assumptions, provide as many options as possible for the final understanding, and push the understanding forward through trial and error. Only in this way can we improve the creativity and flexibility of the subject, and also help poor Achilles catch up with the mysterious tortoise.