![\"\"](\"https:\/\/i0.wp.com\/geoffreychen.com\/wp-content\/uploads\/2026\/02\/image-2.png?ssl=1\")
<\/a><\/figure>\n<\/div>\n\n\nBefore 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.<\/p>\n\n\n\n
One influential line of thought, exemplified by Roger Penrose<\/strong>, 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\u2014particularly mathematical intuition\u2014is therefore seen as a non-computational access point to this underlying structure. From this premise follows Penrose\u2019s well-known conclusion that genuine understanding cannot be reduced to, or replicated by, purely computational systems.<\/p>\n\n\n\n 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\u2014one that has, from its inception, been shaped by the requirement of mathematical formalizability.<\/p>\n\n\n\n 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\u2019s ontological foundation.<\/p>\n\n\n\n It is precisely here that both the strength and the limitation of Penrose\u2019s 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.<\/p>\n\n\n\n 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\u2014if its effectiveness no longer depends on comprehension or intuition\u2014then 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.<\/p>\n\n\n\n From this perspective, Penrose\u2019s 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<\/em>.<\/p>\n\n\n\n \u5728\u8ba8\u8bba\u4eba\u5de5\u667a\u80fd\u662f\u5426\u53ef\u80fd\u6784\u6210\u4e00\u79cd\u201c\u65e0\u4e3b\u4f53\u7684\u77e5\u8bc6\u7cfb\u7edf\u201d\u4e4b\u524d\uff0c\u6709\u5fc5\u8981\u5148\u6f84\u6e05\u4e00\u4e2a\u66f4\u57fa\u7840\u7684\u5206\u6b67\u3002\u8fd9\u4e2a\u5206\u6b67\u5e76\u4e0d\u53d1\u751f\u5728\u5177\u4f53\u6280\u672f\u5c42\u9762\uff0c\u800c\u53d1\u751f\u5728\u5bf9\u6570\u5b66\u3001\u7269\u7406\u7406\u8bba\u4ee5\u53ca\u5b83\u4eec\u4e0e\u73b0\u5b9e\u4e16\u754c\u5173\u7cfb\u7684\u7406\u89e3\u65b9\u5f0f<\/strong>\u4e0a\u3002<\/p>\n\n\n\n \u4ee5 Roger Penrose<\/strong> \u4e3a\u4ee3\u8868\u7684\u4e00\u6761\u601d\u60f3\u8def\u5f84\uff0c\u5f80\u5f80\u5c06\u6570\u5b66\u89c6\u4e3a\u4e00\u79cd\u5177\u6709\u72ec\u7acb\u5b9e\u5728\u6027\u7684\u7ed3\u6784\u4f53\u7cfb\u3002\u5728\u8fd9\u4e00\u89c6\u89d2\u4e2d\uff0c\u6570\u5b66\u5e76\u975e\u4eba\u7c7b\u4e3a\u4e86\u64cd\u4f5c\u4e0e\u8ba1\u7b97\u800c\u8bbe\u5b9a\u7684\u5de5\u5177\uff0c\u800c\u662f\u4e16\u754c\u672c\u8eab\u6240\u5185\u5d4c\u7684\u79e9\u5e8f\u3002\u7269\u7406\u7406\u8bba\u4e4b\u6240\u4ee5\u80fd\u591f\u5982\u6b64\u6df1\u523b\u800c\u7a33\u5b9a\u5730\u4f9d\u8d56\u6570\u5b66\u5f62\u5f0f\uff0c\u6b63\u662f\u56e0\u4e3a\u73b0\u5b9e\u4e16\u754c\u5728\u6700\u6839\u672c\u7684\u5c42\u9762\u4e0a\u662f\u201c\u6570\u5b66\u5316\u7684\u201d\u3002\u4eba\u7c7b\u7684\u7406\u89e3\uff0c\u5c24\u5176\u662f\u6570\u5b66\u76f4\u89c9\uff0c\u88ab\u89c6\u4e3a\u901a\u5411\u8fd9\u4e00\u6df1\u5c42\u7ed3\u6784\u7684\u975e\u8ba1\u7b97\u6027\u5165\u53e3\u3002\u6b63\u662f\u5728\u8fd9\u4e00\u524d\u63d0\u4e0b\uff0c\u5f6d\u7f57\u65af\u575a\u6301\u8ba4\u4e3a\uff0c\u771f\u6b63\u7684\u7406\u89e3\u4e0d\u53ef\u80fd\u88ab\u7eaf\u7cb9\u7684\u8ba1\u7b97\u7cfb\u7edf\u6240\u66ff\u4ee3\u3002<\/p>\n\n\n\n \u800c\u6211\u6240\u91c7\u53d6\u7684\u7acb\u573a\uff0c\u6070\u6070\u5728\u8fd9\u4e00\u524d\u63d0\u5904\u53d1\u751f\u4e86\u5206\u5c94\u3002\u5728\u6211\u770b\u6765\uff0c\u6570\u5b66\u9996\u5148\u662f\u4e00\u5957\u5efa\u7acb\u5728\u4eba\u4e3a\u8bbe\u5b9a\u7684\u529f\u80fd\u5047\u8bbe\u4e4b\u4e0a<\/strong>\u7684\u5f62\u5f0f\u7cfb\u7edf\u3002\u5b83\u4f9d\u9760\u903b\u8f91\u89c4\u5219\u548c\u6f14\u7ece\u673a\u5236\u5c55\u5f00\uff0c\u5176\u5185\u90e8\u53ea\u6d89\u53ca\u81ea\u6d3d\u4e0e\u53ef\u63a8\u5bfc\uff0c\u800c\u5e76\u4e0d\u5929\u7136\u627f\u62c5\u5173\u4e8e\u73b0\u5b9e\u4e16\u754c\u7684\u771f\u503c\u9648\u8ff0\u3002\u6570\u5b66\u4e0e\u7269\u7406\u4e4b\u95f4\u7684\u9ad8\u5ea6\u517c\u5bb9\uff0c\u5e76\u4e0d\u80fd\u76f4\u63a5\u88ab\u7406\u89e3\u4e3a\u6570\u5b66\u4e0e\u4e16\u754c\u4e4b\u95f4\u5b58\u5728\u5fc5\u7136\u5bf9\u5e94\u5173\u7cfb\uff0c\u56e0\u4e3a\u73b0\u4ee3\u7269\u7406\u5b66\u672c\u8eab\u5c31\u5e76\u975e\u201c\u4e16\u754c\u672c\u8eab\u201d\uff0c\u800c\u662f\u4e00\u79cd\u5728\u5386\u53f2\u8fdb\u7a0b\u4e2d\u9010\u6b65\u5f62\u6210\u7684\u3001\u4ee5\u6570\u5b66\u53ef\u5f62\u5f0f\u5316\u4e3a\u524d\u63d0\u7684\u63cf\u8ff0\u4f53\u7cfb\u3002<\/p>\n\n\n\n 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