Today in discussion section, I listened to Brother Donglin’s report, and it felt wonderfully intriguing; some of my earlier scattered thoughts have become much clearer.
What Brother Donglin discussed was the concept of “diagramming” in ancient Greek geometry. He pointed out that the geometrical constructions of the ancient Greeks cannot be understood as “construction” in the Kantian sense: the meaning of diagramming is not to provide some kind of “ontological proof.” Diagramming is undertaken for pedagogical and instructional purposes; the original meaning of “proof” is also a kind of showing, a process by which the teacher displays something to the student. “Geometrical diagramming, geometrical knowledge, and phronesis (practical knowledge, that is, knowledge of how to appropriately choose means according to ends) are associated with phronesis rather than with poiesis (making).” Of course, Brother Donglin’s study of classical mathematics as a whole will serve as a prelude to his research on Descartes.
At once I thought of the hidden connection between mathematics and mechanism in Descartes. Of course, in the eyes of modern people, mathematics is practically mechanism itself: the mathematization of nature is the mechanization of the world, and mathematics and mechanics have become one and the same stream, so that it is hard anymore to raise the question of “the relation between mathematics and mechanism.” But if one starts from ancient mathematics and looks back, one discovers that the combination of mathematics and mechanism was by no means so self-evident.
What status did geometry hold in the minds of the ancient Greeks? First of all, it was merely the closest thing to the world of Ideas, not the world of Ideas itself. In particular, the figures manipulated in the process of diagramming and solving problems did not belong to the existence of the world of Ideas; they were nothing more than means used in the teaching process for demonstration—whether demonstrated concretely with ink or in a sand tray, or worked through imaginatively, the “making” in diagramming does not possess the highest ontological significance. At the same time, the “making” of geometry does not belong to the realm of “production” either: the purpose of geometry is not to study how to “make,” but to reveal, through making appropriately, those forms of knowledge that people had already known potentially, but had forgotten. Through this sort of “making”—through something deduced or proven by mechanical, tightly interlocking steps—what is finally displayed awakens people’s latent understanding of it. This revealed thing—something that can be intuited but not spoken—is the aim of geometrical instruction. As for the tightly linked steps used in teaching, ontologically, they have always lacked standing.
“Practical knowledge” happens to occupy the middle position between theoretical knowledge and productive knowledge—if I were to distort it a little, theoretical knowledge is the capacity to be adept at intuiting things of the Idea, while productive knowledge is the capacity to be adept at using things of the tool, and practical knowledge is the capacity concerning how to appropriately select tools or methods in order to reveal or present the Idea. The rupture of this intermediate link has caused a muddling of the boundaries between Ideas and tools. If one can no longer weigh the use of tools according to ends, then knowledge of tool use can only submit to the logic of the tools themselves, namely ceaseless operation and the pursuit of ever greater efficiency. And if the Idea is no longer something finally presented through the revealing activity of tools, then the possible situations are either that the Idea is everywhere, with every step in the operation of tools being the existence of the Idea; or that the Idea exists nowhere, with the products of any mechanical motion no longer being the Idea itself. This is why modern people are both efficiency-ists (adept at using tools) and nihilists (having lost their ends); both rationalists (the mathematization of nature) and skeptics (truth no longer presents itself to human beings).
So how did this intermediary, “practical knowledge,” recede? In fact, the way to eliminate the mediating character of a certain medium is not to forget it, but to stare at it. Put it at the center and make it an object. The hallmark of modern thought is the self-consciousness of “method,” that is, to place intermediary things such as “steps,” “process,” and “tools” at center stage (this situation is indeed also related to printing). The Elements of Geometry, which had initially been used as a textbook, became a self-sufficient system. The self-sufficiency of intermediaries of course broke the original connection.
Among these, “education” is precisely a focal concept. In fact, the original meaning of the word “mathematics” is learning or what can be learned; and the word “education” derives from “educe,” whose basic meaning is to lead out. That is to say, the original meaning of education lies in presenting knowledge and awakening knowledge, not in constructing knowledge. In modern times, however, the process of education has become something self-sufficient in its own right: education is no longer about leading students toward knowledge, but about constructing knowledge. And the process of construction is self-sufficient; every step in building with blocks can be paused, and at that point the structure as it stands can already count as the final work. Every block is part of the final result. Every step in a modern mathematical proof is part of the mathematical system; every link in modern education is part of “knowledge”; every link we face in our own learning is an “object” of our learning. The purpose of learning is dissolved into the process of learning itself. “Examination-oriented education” is an extreme example—class is for exams, and exams are for testing the effectiveness of class, while class and exams are nothing more than two links within the educational process; means and ends circle around within education, and after a few rounds are finally abstracted into a “score,” a mere measure of the “efficiency” of learning. This pale and monotonous thing becomes the ultimate meaning of teaching. This situation not only resembles, in form, the modern situation in ethics—namely, the abstraction of ends into “happiness value”—but is in fact deeply connected to it; both are attributable to the loss of the entire link of “practical knowledge.”
Translated from the Chinese original with AI assistance. The original text is authoritative.
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