The knowable world belongs to the rational world, and the sensible world is the world that our senses feel.
The initial difference between the world came: the difference between the two worlds. When we think about geometry, we enter a super-perceptual world, which is a knowable world.
Geometrists discuss triangles, for example, giving axioms, theorems and inferences about triangles. For example, there is a triangle theorem that the sum of three internal angles of a triangle is equal to 180 degrees, which is derived from axioms. Three straight lines enclose a closed space, and their quantitative relationship must be strictly determined to form a closed space, which we call a triangle.
This matter needs no experiment, it is inferred from the reasoning itself. Therefore, when discussing the properties, axioms and theorems of triangles, geometricians simply don't care whether there are triangles in the world. There are no triangles in the world, so geometricians are still discussing triangles. Once triangles appear in the universe, these things must conform to all the discussions about triangles in geometry.
Please pay attention to this, the proposition of geometry, its axioms, theorems and inferences are not summarized by a lot of perceptual experience. If so, geometry is an empirical science. Let's not think about things the other way around.
We should not think that geometry is based on a large number of observations and experiments on quantitative relations and spatial relations, which is not the case.
Geometry can discuss things we haven't seen, their positional relationship and quantitative relationship. Once we have such a thing, it must conform to all the discussions about it in geometry or mathematics, so we have to ask: in a known world and a sensible world, who is stipulating who? This shocked us. We China people should probably learn a little philosophy from junior high school. Philosophy in textbooks and philosophy in textbooks must tell us epistemology. What is epistemology? Human cognition is divided into two stages, perceptual cognition stage and rational cognition stage, which rises from perceptual cognition to rational cognition and then completes a leap. This gives us a strong impression that knowledge always starts with perceptual experience and then slowly rises to abstract theory. If we look at geometry from this point, it is not the case. Perceptual knowledge should be called knowledge, not a simple sensory state, and it must contain sensory materials stipulated by admissibility. If there is no rational norm, that perceptual substance cannot be called cognition. From this perspective, learning philosophy is a very interesting thing, that is, breaking our usual common sense understanding framework of things. For example, the concept of sphere comes from a large number of spheres we have observed, and we have summarized the spheres. On the basis of a lot of perception, geometry obtained the concept of sphere. We usually think so. If we think so, then we just want to turn it around.
When our senses perceive things with different shapes, our perception already contains a cognitive form, which is called "sphere". It is precisely because of the cognitive form of "sphere" that we classify such things as ball games, and football, oranges and apples as one class, which is called ball games. Because we have the concept of pure balls in our minds, we will regard them as a class.
It is not countless balls that give us the concept of pure balls, but the concept of pure balls in our hearts that classifies them as ball games. Now we have entered the basic cultivation of western philosophy, a very old school of philosophy-Pythagoras school, and the digital world outlook.
Now let's discuss these three judgments and look at their interrelationships:
The first judgment is this sentence: "This is a square table", and a square table is a square table.
The second judgment: "Everything in a square must have four right angles", and there must be four right angles.
The third judgment: "A square is a square because there are four right angles."
The first judgment is undoubtedly an individual judgment, a judgment of something-this is a square table.
The second judgment, "everything in the world must have four right angles", is called full-name judgment, and everything in the world is included.
Let's look at the third judgment, "a square is a square because there are four right angles", which is neither an individual judgment nor a universal full-name judgment.
Why? Because it doesn't point to anything, we only discuss the square itself. What is the difference between this judgment and the previous two judgments?
-the first two judgments point to actual things, and the first judgment points to individual things; The second judgment points to everything. In a word, it points to something that actually exists. The third judgment does not point to anything in this world, only discusses the square, or only discusses the square itself-this judgment belongs to geometric judgment. The first two judgments are practical experience judgments, and the third judgment has nothing to do with things that don't exist in the world. Geometric judgments are all such judgments.
Let's discuss now: in the relationship of these three judgments, who depends on whom? According to the philosophical epistemology in the philosophy textbooks we have studied before, it should be said that the second judgment depends on the first judgment. We must have a large number of individual things judged as square things in order to have a second judgment-everything in a square must have four right angles. That is to say, the second judgment depends on the first judgment. What about the third judgment? The third judgment depends on the second judgment-take away everything that is square, that is, if it is square, there must be four right angles. I'm afraid this is what we used to think about these three judgment relationships.
When we think like this, we are thinking upside down. If we don't have a pure square concept in mind, that is, the concept of the opposite sex itself, how can we distinguish between things in the world and so-called square things? If you can't even tell a square thing apart, how can you judge that this individual thing is a square table? What is the truth of the matter?
The truth is that the third judgment is the premise of the second judgment, and the second judgment is the premise of the first judgment. Here, I'm afraid we have encountered difficulties and great obstacles in understanding. This is a normal phenomenon in philosophy study, which requires breaking the framework of common sense. You must ask, the third judgment, that is, "a square must have four right angles", how do we understand it in our hearts? Where does this understanding of the opposite sex come from? This kind of questioning is normal.
There is no doubt that when we say that the square is square, it is not a summary of a large number of observations made by the other side. We can observe that square things are different from round things, but we already have the distinction between square and round things in our hearts.