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Mathematicians believe that they provide the source of philosophers' ideas. But in the 18th century, philosophers denied the forerunners of the truth of the material world. In the Theory of Human Nature, Hume emphasizes that we know neither spirit nor matter, both of which are illusory. We only accept feelings, simple concepts such as impressions, memories and thoughts are only vague reflections of these feelings, and any complex concept is a collection of simple concepts. In fact, the spirit is only the concentration of our feelings and concepts, we can not assume the existence of anything other than what we can perceive through direct experience, but experience can only produce feelings.

Hume is equally skeptical about matter. Who can guarantee that there is a physical world that exists forever? all we can know is how we feel about such a world. Perceiving a chair repeatedly does not prove that it does exist. Time and space are just the way and order in which we generate concepts. Similarly, causality is just a habitual connection between concepts. No matter time or space, or causality, it is not objective reality. We are confused by our perceptual ability, so we believe that such reality ∶ has an external world with definite attributes. In fact, this is just an unfounded inference, and the generation of perception is incomprehensible. We don't know whether it comes from the outside, from the bottom of the heart, or from God.

David Hume himself is just a single aggregator of feelings and thoughts, and he can only be there in this way. "self" is the convergence of different perceptions. Any attempt to understand yourself can only lead to understanding. All other people and the supposed existence of the outside world are just the understanding of one person, and there is no guarantee that they do exist. Therefore, it is impossible to have any scientific laws about an eternal and objective material world. Such a rule is just an appropriate summary of a feeling. Furthermore, since the concept of causality is not based on scientific proof but a habit of thinking in the usual order of frequent "events", it is impossible to know whether events perceived in the past will happen again in the future. In this way Hume denied the inevitability, eternity and indestructibility of the laws of nature.

By denying the belief that the outside world follows fixed mathematical laws, Hume also denied the value of the logical reasoning structure that represents reality. But mathematics also contains theorems about numbers and geometry, which are undoubtedly derived from hypothetical truths that contain numbers and geometry. Hume does not deny axioms, but he derogates them and the results derived from them. Axioms come from the perception of the physical world that is supposed to exist. The theorem is indeed the inevitable result of axioms, but it is nothing more than an accurate restatement of axioms. They are inferences, but they are only the reasoning of assertions implied in axioms. Therefore, axioms and theorems are synonymous repetitions, not truths.

Therefore, Hume answered the basic question of "how to obtain the truth"-he denied the existence of the truth, and it was impossible to distinguish the truth. Hume's work not only belittled the efforts and results made in science and mathematics, but also questioned the value of reasoning itself. For most 18th-century thinkers, such a denial of man's highest intellectual ability is rebellious. Hume's philosophy was contradictory and abhorrent to the vast majority of scholars in the 18th century, and so incompatible with the amazing achievements in mathematics and other sciences that it was refuted.

Kant, perhaps the most respected and profound philosopher in history, launched this challenge. But after a careful examination of Kant's painstaking conclusion, it is found that it is not more convincing than others. In his book introduction to Future Metaphysics, Kant does seem to side with scientists and mathematicians, ∶.

We can say exactly that pure transcendental comprehensive knowledge, pure mathematics and pure physics are real and predetermined, both of which contain some widely accepted and absolutely certain propositions. And independent of experience.

In his book critique of Pure reason, Kant even begins with more confident words. He affirms that all mathematical axioms and theorems are truth, but why? Kant asked himself. Is he willing to accept such a truth? It is clear that experience itself is not sufficient to prove their effectiveness. If you can answer a bigger question: is mathematics really a science? And you can answer that question. Kant's answer is that the form of ∶ time and space is determined by our minds, and the so-called time and space is just a pattern of our perception. This pattern of perception (what Kant calls intuition) is determined by the way the mind treats experience. We perceive, organize and understand experience according to these forms of intelligence. The mind adds these ways to the senses and impressions to make the feelings coincide with the inner patterns.

Since the intuition of space comes from the mind, the mind automatically accepts some properties of space, such as a straight line is the shortest path between two points, three points determine a plane and Euclid's parallel axiom. Kant calls these truths a transcendental hypothetical truth, and they are part of our minds. The scientific nature of geometry lies in the logical inference that reveals these truths. The fact that the mind treats experience through "spatial structure" shows that experience is consistent with basic principles and theorems. The order and rationality of the outside world that we think we perceive are imposed on it by our spirit and the way we think.

Since Kant created space from the human brain, he saw no reason why it should not be Euclidean space. He can't conceive of other geometric spaces. This led him to believe that there was no other space, and that the Euclidean geometry theorem was neither inherent in the universe nor designed by God, but was the result of systematizing and rationalizing man's perceptual knowledge. As for God, Kant said that the nature of God is not within the scope of rational knowledge, but we should still believe in God. Kant is more imprudent in geometry than bold in philosophy. He had never been 40 miles from his home in the East Prussian city of Konigsberg, yet he assumed that he could determine the geometry of the world.

What are the mathematical laws of science? Since all experiences are made up of the spiritual framework of time and space, mathematics must be consistent with all experiences. In the metaphysical basis of Natural Science, Kant admits that Newton's laws and their inferences are self-evident. He claims that the proved Newton's laws of motion can be derived from pure reasoning, and that these laws are the only hypotheses that can make nature understood. He said that what Newton gave us such a clear understanding of the universe will never change.

More generally, Kant believes that the world of science is a world of sensory impressions organized and controlled by the spirit and consistent with internal categories, such as space, time, cause and effect, and matter. The spirit contains the structure that the object must conform to. Sensory impressions do come from the real world, but unfortunately, the world is unknowable, and the so-called reality is only known through subjective classification with the help of perception. So there is no other way to organize experience except Euclidean geometry and Newtonian mechanics. With the increase of experience and the formation of new science, the mind will not extract and form new principles from these new experiences. Instead, the sleeping part of the mind is awakened to explain these new experiences. The observation of the mind is inspired by experience, which explains why some truths, such as the laws of mechanics, were discovered quite late, while others were known centuries ago.

Kant's philosophy is almost undisguised to advocate reason, but he believes that the role of reason lies not in the exploration of nature, but in the development of the barren places of the human mind. Because the perception from the outside world provides the original material of the spiritual organization, experience is recognized as the inevitable factor of knowledge, and mathematics is the revealer of the inevitable law of spirit.

Mathematicians are accustomed to the assertion that mathematics is a system of transcendental truth, but most people do not pay enough attention to how Kant came to this conclusion. Otherwise, his doctrine (what mathematicians prove is not inherent in the material world, but from the human spirit) will make all mathematicians stop working. Is the structure inherent in our reality the same as what we feel? Does the perceptual structure of this space have to be Euclidean? How do we know that? Unlike Kant, mathematicians and physicists still believe that there is an external world governed by laws independent of the human spirit. People just reveal the laws of their design and use them to predict what will happen in the outside world.

Kant's theory not only emancipates the mind, but also binds the mind. because it emphasizes that the spirit can organize, we do not really understand the experience in the world, he laid the foundation for the creation of concepts contrary to the concepts that people believed in at that time, but because he insisted on organizing spatial perception according to the law of Euclidean geometry, he hindered the acceptance of other ideas. If Kant had paid more attention to the work of contemporary mathematicians, he might not have been so stubborn about this view.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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