Comprehending Meaning Through Number: The Transformation of Ideas from Ancient Doctrines to Artificial Intelligence Technologies

Abstract

The article explores the evolution of the idea of correlating numbers and meanings, from ancient numerological systems to modern models of natural language processing based on vector representations and neural networks. The authors demonstrate that the aspiration to uncover hidden properties of objects by associating them with numbers and performing operations on these numbers has been a common thread across various cultures for millennia. The article traces the stages in the formation of the concept of mathesis universalis (universal mathematics), starting from Aristotle and Proclus’s reflections on general mathematics and culminating in R. Descartes’s ideas about the existence of a universally applicable science of order and measure. Particular attention is devoted to G.W. Leibniz’s ambitious project of creating a universal characteristic – a formalized language in which all knowledge could be precisely expressed and logical inference could be carried out through calculations. It is noted that, despite inherent limitations, some of Leibniz’s intuitions have found embodiment in natural language processing technologies based on vector representations of words. However, the simple correlation of concepts with vectors is insufficient for the full realization of the idea of a calculus of reasoning, as it lacks a mechanism for deriving true statements from others. The development of the transformer architecture, which employs the mechanism of self-attention to construct context-dependent vector representations, has been a significant stride towards the automation of reasoning. Nevertheless, modern language models are based on the principle of maximizing likelihood rather than rigorous logical rules. The authors analyze conceptual solutions proposed for creating a comprehensive calculus of knowledge. In conclusion, it is emphasized that, while the idea of reducing reasoning to mathematical operations is undeniably attractive, one should be cautious about its maximalist interpretations. Cognition is constructive in nature, and even the most perfect model of formalization of reasoning cannot substitute for the practical wisdom and freedom of human thought.