Mathematics, Description, and the Limits of Understanding

A Fundamental Divergence from Roger Penrose

By Geoffrey Chen

Before addressing whether artificial intelligence can constitute a form of knowledge without a knowing subject, it is necessary to clarify a more fundamental divergence. This divergence does not arise at the level of technical detail, but at the level of how mathematics, physical theory, and their relation to the world are understood.

One influential line of thought, exemplified by Roger Penrose, treats mathematics as a structure with independent ontological status. On this view, mathematics is not a tool invented for calculation or convenience, but a fundamental component of reality itself. The remarkable effectiveness and internal unity of mathematical structures in modern physics are taken as evidence that the physical world is, at its deepest level, mathematical in nature. Human understanding—particularly mathematical intuition—is therefore seen as a non-computational access point to this underlying structure. From this premise follows Penrose’s well-known conclusion that genuine understanding cannot be reduced to, or replicated by, purely computational systems.

The position adopted in this book diverges precisely at this foundational assumption. Here, mathematics is understood first and foremost as a formal system built upon human-defined functional assumptions. Its core consists of logical rules and methods of derivation, concerned with internal coherence and deductive consequence rather than with direct truth-claims about the world. The success of mathematics in physical theory does not, on this view, establish a necessary correspondence between mathematics and reality itself. This is because modern physical theories are not identical with the world, but represent one historically contingent mode of describing it—one that has, from its inception, been shaped by the requirement of mathematical formalizability.

In this sense, modern physics and modern mathematics did not evolve independently and later converge by chance. They co-evolved within a shared methodological framework that privileged precision, abstraction, and operational tractability. Their compatibility therefore reflects an internal consistency within this framework, rather than a privileged window onto the intrinsic structure of reality. Once we accept that contemporary physical theories constitute only one among several possible ways of describing the world, the close fit between mathematics and physics no longer suffices to demonstrate that mathematics reveals the world’s ontological foundation.

It is precisely here that both the strength and the limitation of Penrose’s position become apparent. His theory appears compelling and self-consistent because it presupposes that the intelligibility of the world must ultimately rest on a form of structure that cannot be dissolved into computation. Yet this same presupposition constrains his ability to accommodate contemporary developments in artificial intelligence. When systems emerge that can reliably generate, preserve, and apply knowledge-like outputs without any corresponding understanding or conscious subject, such phenomena cannot, within his framework, count as genuine knowledge. They can only be interpreted as further evidence of what computation lacks.

The project pursued here takes a different route. Rather than asking whether machines possess understanding, it asks whether understanding remains a necessary condition for knowledge at all. If knowledge can already function in practice without a knowing subject—if its effectiveness no longer depends on comprehension or intuition—then the privileged status traditionally accorded to understanding must itself be reconsidered. The disagreement with Penrose is therefore not a dispute over correctness, but a divergence in starting point. He seeks to secure a final ontological grounding for understanding; this work explores the possibility that, under contemporary technological and epistemic conditions, knowledge has begun to operate independently of the subject altogether.

From this perspective, Penrose’s framework is not something to be corrected, but something to be situated. Its boundaries are clear, and within those boundaries it remains powerful. Yet those same boundaries prevent it from recognizing the emergence of non-subjective knowledge systems now manifested by artificial intelligence. Clarifying this divergence is not an exercise in rejection, but a necessary step in establishing the theoretical foundation for what it means to think after the knowing subject.

在讨论人工智能是否可能构成一种“无主体的知识系统”之前，有必要先澄清一个更基础的分歧。这个分歧并不发生在具体技术层面，而发生在对数学、物理理论以及它们与现实世界关系的理解方式上。

以 Roger Penrose 为代表的一条思想路径，往往将数学视为一种具有独立实在性的结构体系。在这一视角中，数学并非人类为了操作与计算而设定的工具，而是世界本身所内嵌的秩序。物理理论之所以能够如此深刻而稳定地依赖数学形式，正是因为现实世界在最根本的层面上是“数学化的”。人类的理解，尤其是数学直觉，被视为通向这一深层结构的非计算性入口。正是在这一前提下，彭罗斯坚持认为，真正的理解不可能被纯粹的计算系统所替代。

而我所采取的立场，恰恰在这一前提处发生了分岔。在我看来，数学首先是一套建立在人为设定的功能假设之上的形式系统。它依靠逻辑规则和演绎机制展开，其内部只涉及自洽与可推导，而并不天然承担关于现实世界的真值陈述。数学与物理之间的高度兼容，并不能直接被理解为数学与世界之间存在必然对应关系，因为现代物理学本身就并非“世界本身”，而是一种在历史进程中逐步形成的、以数学可形式化为前提的描述体系。

换言之，现代物理学与现代数学并非独立发展后偶然契合，而是在同一套方法论、同一套可操作性标准中同步生长的结果。它们之间的高度一致性，首先是一种内部一致性，而非对世界本体结构的直接揭示。一旦承认现代物理理论只是诸多潜在世界描述路径中的一种，那么数学对这一描述的成功适配，就不足以证明数学本身具有对现实世界的必然指涉。

正是在这里，彭罗斯理论的力量与局限同时显现出来。他的理论之所以显得自洽而有吸引力，是因为它建立在一个前提之上，即世界的可理解性最终必须落实为某种不可被计算消解的结构事实。然而，这一前提同时也限制了他对当代人工智能现象的接受能力。当一个系统能够在没有理解主体、没有直觉经验的情况下，稳定地产生、维持并应用“知识性结果”时，这种现象并不会被他视为知识形态的转变，而只会被视为理解缺席的证明。

而我试图推进的，正是对这一理解特权的系统性反思。如果知识在实践中已经可以脱离认知主体而运行，如果其有效性不再依赖于理解的存在，那么问题就不再是“机器是否具备理解”，而是理解是否仍然是知识成立的必要条件。在这一点上，我与彭罗斯的分歧并非对错之争，而是出发点的不同。他试图为理解找到最终的本体位置，而我则接受这样一种可能性，即在当下的技术与实践结构中，知识已经开始以一种不再等待主体的方式继续存在。

正是基于这一判断，我的工作并不试图修正彭罗斯的理论，而是将其视为一个清晰而有边界的参照系。理解这些边界，并不是为了否定他，而是为了为“认知主体之后”的讨论奠定一个更稳固、也更符合当代经验现实的理论基础。