Between mathematics and physics: the quest for understanding in the context of LLMs

This paper explores the possibility of a common cognitive mechanism underlying three phenomena: understanding in physics, understanding in mathematics, and the transmission of the experience of beauty. The working hypothesis is that understanding is not a localized act of grasping a single piece of content – a theorem, a description of a phenomenon, a feature of a work of art – but rather a function of integrating that content into a broader, internally coherent picture of reality, one that the knowing subject builds up over time through their own intellectual labour. Understanding "at a deeper level" would accordingly be a matter of degree: the extent to which a new element is integrated into this picture, rather than a property of the information itself. In all three domains, it seems, there is an important distinction between a purely formal or informational mastery of a subject and a deeper, substantive, integrative grasp of it – a distinction that becomes especially salient in the context of artificial intelligence systems, which are capable of generating accurate, self-contained descriptions of phenomena without, it appears, constructing any lasting, coherent picture of reality.

With regard to physics, the paper draws on M. Heller's account, according to which the mathematical structures regarded as "beautiful" are those that we have come to accept as accurate descriptions of reality, that have been confirmed through experiment, and that have, over time, been absorbed – so far as possible – into the broader, coherent edifice of physics. This raises the further question of whether Einstein's category of inner perfection (innere Vollkommenheit) – understood as a postulate of logical simplicity, economy of assumptions, and internal consistency, and thus, in effect, a criterion of coherence — already constitutes a form of understanding of physical reality, or whether it in some sense precedes such understanding. Episodes from the history of science (Heisenberg's discovery of the matrix representation of observables, and, analogously, Einstein's own work on general relativity) suggest that a "sense" of the aptness of a formalism — a sense of how well it fits the emerging picture of reality – may precede full, discursive understanding. This recalls the question, first posed by Poincaré, of whether and how a sense of the beauty of a mathematical structure can play a heuristic role, leading to a true description of phenomena even before that description is fully understood and incorporated into the body of accepted knowledge.

R. Scruton's theory of the transmission of the experience of beauty proves useful here. Scruton treats the aesthetic experience of a work of art as a function of a particular resonance between the sensibility of the artist and that of the viewer. This raises the question of whether such resonance is purely local – a relation to a single work – or whether, analogously to the mechanisms described above for physics and mathematics, it requires the work first to be situated within a broader, relatively coherent horizon of aesthetic sensibility that the viewer has built up over time. The central thesis of this paper is that understanding "at a deeper level" – of a physical phenomenon, a mathematical proof, or a work of art alike – requires an analogous, integrative resonance: the active, subjective incorporation of new content into an already existing picture of reality, rather than the merely passive reception of correct information about it.

The implications of this thesis are particularly salient for systems based on large language models. Such systems generate responses on a point-by-point basis, in reaction to successive queries, without possessing any lasting, evolving picture of reality comparable to the one built up by a human knowing subject. Even a single response that is formally correct and internally consistent is not thereby part of a cumulative process of integration into a broader, temporally sustained network of beliefs – and it is precisely this kind of integration that, on the view defended here, constitutes the essence of understanding. A further question remains open: to what extent can a system that lacks a stable, subjective point of view – the first-person perspective that serves as the starting point for phenomenological method, for instance in the work of Edith Stein – serve as a vehicle for such integration at all, rather than merely as an efficient channel for the transmission of discrete pieces of information?