http://hps.phil.pku.edu.cn/bbs/read.php?tid=1169
Although Professor Wang’s lecture this time was rather jumpy in content, the reading materials assigned this time are relatively readable. If you read the materials in advance, listening to Professor Wang’s lecture can serve to help the two illuminate each other. I hope everyone will be sure to read the relevant materials, and at the very least must read the second part of *Science and Hypothesis*. In tomorrow’s discussion class, the main speakers and the TAs should discuss more around the materials, and not wander too far afield.
Broadly speaking, the issues involved in this lecture and the reading materials are these: 1. What is space? 2. What is geometry?
Is “space” real? Is real space curved or flat? What do curved and flat mean? Can we experience non-Euclidean space? What relation do experience, geometry, and physics have to one another?
In the preface to the English translation of *Science and Hypothesis*, Poincaré says: “To analyse this concept [space] is not to sacrifice the reality to some I know not what phantom. The language of geometry is, after all, only a language. Space is but the name given to the aggregate of things that we have come to believe. What is the origin of this word, and what is the origin of the other words? What do they conceal? It is allowable to ask this question; on the contrary, the prohibition of asking it would be to allow oneself to be deceived by words; it is necessarily to adore metaphysical idols, as the savage prostrates himself before a wooden idol without daring to look inside what it contains.”
Newton’s “absolute space-time” once was precisely such a “metaphysical idol,” and Poincaré’s and Einstein’s relativity began from questioning this idol. The reason relativity is called relativity can already be glimpsed from Poincaré’s analysis of the concept of space.
But we need not steer the topic into the study of relativity, and even less do we need to understand more of the technical details of physics. We can simply understand things within the scope of the knowledge involved in Professor Wang’s lecture and the reading materials. Professor Wang himself actually left us a ready-made question: he claims he will explain why most scientists are willing to accept curved space but reject flat space? This is also one of the questions Poincaré answers. Poincaré’s answer is “convention.” Whether space is curved or flat is neither an a priori judgment nor an empirical fact; logically, one cannot determine which set of geometry is more correct, and empirically, one cannot distinguish what space actually is. So what, then, does the concept of “space” really mean? Poincaré says this concept is people’s “convention.” Through his analysis of space, Poincaré drew a boundary with Kant’s apriorism and articulated his own philosophical position, known as “conventionalism.”
What relation do geometry and human experience have? On the one hand, Poincaré says, “If there were no solids in nature, there would be no geometry”; but on the other hand, geometry does not conflict with any experience. What did Poincaré mean? Do you agree with Poincaré’s claim? Many similar questions can be thought about and discussed.
As for the question of handedness, the teacher did not say much in class, and there is no corresponding reading material, so it is not encouraged to involve it too much in the discussion class. I still recommend reading *Right and Left*, which is a very interesting book.
The question of handedness is also related to the question of space. “Left” and “right” are a unique pair of concepts: on the one hand, they seem to be some kind of absolute distinction. For example, you can turn a bottle that is “upward” over and make it “downward”; you can turn “front” around and make it “back.” But something that is “left-handed” can never, no matter what, be made “right-handed”; sometimes not even topological transformations will do. Obviously, in three-dimensional space, a left-handed trefoil knot and a right-handed trefoil knot are two different things. Yet on the other hand, left and right are also completely relative. Imagine, then, between that left-handed trefoil knot and the right-handed trefoil knot absolutely different from it, what exactly is the difference? In fact, only in relative terms can one even speak of difference. If not relative to right, “left” has no meaning at all. Of course, we can define the handedness of a trefoil knot by means such as the right-hand screw rule, but this presupposes that we already know in advance what “right” hand means.
Following the prompt in *Right and Left*, we can conduct a thought experiment: suppose we are communicating with a distant extraterrestrial civilization; we cannot meet face to face, cannot raise our left hand to indicate the direction of left, and can only communicate through written (digital) information. Perhaps first we can exchange knowledge of number theory through digital means; through successful conjectures and deciphering, perhaps we can exchange a great deal of information, and even find a way to talk about trefoil knots. Yet there is still no way to ensure that we mean the same trefoil knot. Even if we know that these two trefoil knots of different handedness are still two “different” things hundreds of light-years away, and that we will never confuse them, we still have no way at all to “distinguish” them.
Kant once used the properties of left and right to support the existence of absolute space, but later reduced this absolute space to the human a priori forms of cognition, whereas Poincaré reduced it to people’s conventions.
Translated from the Chinese original with AI assistance. The original text is authoritative.
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