Series 3: Mathematics, King of the World: Casual Remarks on Foundational Issues in Modern Mathematics!
2006/3/6 17:49:05
The a priori foundations of human knowledge were already made quite clear by Kant. Although infinitely many mathematical systems can be constructed, the fact that different peoples can all spontaneously discover and accept natural numbers naturally invites many mystical associations. There are many ways to explain this. One is to regard natural numbers as innate, a priori, and necessary. In truth, there is nothing extraordinary about this. Natural numbers can be fully constructed. We can accept natural numbers simply because we happen to exist in a world that allows us to accept them. The "naturalness" of natural numbers is actually contingent, and this contingency, in a certain sense, constitutes the a priori nature of human cognition.
Of course, there is also this intellectual game: reducing a mathematical system to ever simpler ones, a reduction primarily concerned with consistency. For example, the consistency of an arithmetic system can logically guarantee the consistency of larger mathematical systems, such as certain axiomatized grand systems of mathematics. But the logic of this reduction, like the a priori nature of natural numbers, holds no great mystery. To be reducible is to be equivalent to the logic it is reduced to; otherwise, one could easily design a logical pathway that renders such reduction completely meaningless.
If we take "reality" to mean the world we inhabit, then the part of mathematics that relates to reality is only a very small portion. The a priori nature of this portion's logic is actually the logic of the a priori nature of our world. Therefore, mathematics reveals the most profound secrets of the world. Mathematics is the prerequisite for all climaxes of the world, and naturally also contains the climax of all climaxes. Mathematics is the Queen of the World. However, mathematics can also adopt a non-realistic perspective. Here, mathematics has the broadest imagination to construct space-time belonging only to itself, free to paint lavishly purely for the sake of beauty. This is the true art among all arts, transcending space-time, transcending all so-called laws of reality. On mathematics' turf, mathematics is King.
Obviously, real-world mathematics is not so carefree. The academic norms of real-world mathematics are tied to real-world interests. For instance, all scholarship that can be academized must unfold within a common axiomatic system, and what unfolds therein is really nothing more than fame, status, money, and similar things. Real-world mathematics is actually like reality itself—just a game of ideology. Of course, one day mathematics could also become a purely intellectual game, where people form various gaming alliances according to their preferred axiomatic systems. In different systems, the non-existence of straight lines could be permitted, or points on a line could be discrete, and so on—anyone could choose whichever system they like. As for the system relevant to reality, it could easily be left to those with relatively limited intellect and imagination—more likely to be male—to work on within a relatively simple branch of mathematics. This branch could, of course, be given a nicer-sounding name, one that suggests a certain IQ level, something rather manly, such as: Theoretical Physics.
(To be continued)
缠中说禅 2006/10/7 17:36:44
The humanization of nature—this was articulated very clearly by Marx. A purely external, mechanical natural world is a delusion of humanity. Contemporary physics has the so-called anthropic principle, which can be referenced for comparison.