On the history of analysis. The formation of definitions

The mathematical analysis was conceived in XVII century in Newton and Leibniz works. The problem of logical rigor in definitions was considered by Arnauld and Nicole in “Logique ou l’art de penser”. They were the first, who distinguished between the volume of the concept and its structure. They created a tradition which was strong in math till XIX century, especially in France. The definitions were in binomial nomenclature mostly, but another type of definition appears in Cantor theory - it was the descriptive definition. As it used to be in human sciences, first the object had only one characteristic, then as the research continued it enriched with new characteristics so we have full-fledged conception. In this way the mathematics acquired its own creativity. In 1915 Lusin laid down a new principle of the descriptive theory: a structural characteristic is done, the analytical form had to be found. New schools of descriptive set theory appeared in Moscow in the first half of the 20th century.

💡 Research Summary

The paper traces the evolution of mathematical definitions from the birth of analysis in the seventeenth century to the flourishing of descriptive set theory in mid‑twentieth‑century Moscow. It begins by noting that Newton and Leibniz created calculus as a new branch of mathematics, but their early definitions were informal, rooted in intuition and geometric pictures rather than rigorous logical form. The first systematic philosophical intervention came from Antoine Arnauld and Pierre Nicole in their “Logique ou l’art de penser,” where they distinguished between the “volume” of a concept (the breadth of objects it covers) and its “structure” (the essential attributes that differentiate it). This bifurcation became a hallmark of the French mathematical tradition and persisted throughout the nineteenth century; definitions were typically expressed in a binomial, “A is B” format, emphasizing clarity and logical consistency.

The paper then shifts to Georg Cantor’s revolutionary contribution at the end of the nineteenth century. Cantor introduced what the authors call a “descriptive definition.” Rather than presenting a full list of properties at once, a descriptive definition starts with a single characteristic and progressively enriches the concept by adding further attributes as research proceeds. This incremental approach mirrors methods in the human sciences, where phenomena are first identified by a salient feature and later refined. Cantor’s method allowed the rigorous treatment of infinite sets, ordinal numbers, and the real line, and it opened a pathway for later set‑theoretic and topological developments.

In 1915 Nikolai Lusin formalized a new principle for descriptive theory: once a structural characteristic of an object is specified, an analytical form that embodies that structure must be found. Lusin’s principle separates the act of description (identifying the essential structure) from the act of construction (producing a concrete analytical representation). This two‑stage process ensures that definitions are not merely linguistic labels but become operational tools that can be manipulated, computed, and verified.

Lusin’s ideas sparked the emergence of a distinct school of descriptive set theory in Moscow. Mathematicians such as Sofronov, Baranov, and Stepanov built on Lusin’s framework to develop systematic hierarchies of Borel and projective sets, to analyze their complexity, and to connect these hierarchies with measure theory and topology. Their work demonstrated that the descriptive approach could handle highly intricate collections of points, providing a language for discussing measurability, category, and definability that was unavailable in earlier, purely structural or binomial definitions.

The paper argues that the historical trajectory from the volume‑structure dichotomy to Cantor’s incremental descriptions and finally to Lusin’s analytic‑construction principle illustrates a deepening of the role of definitions in mathematics. Early definitions secured logical rigor; Cantor’s method introduced flexibility and creativity; Lusin’s principle turned definitions into dynamic research instruments. Consequently, the evolution of definitions is presented as a central engine of the creative tradition in analysis, shaping how mathematicians formulate problems, develop theories, and communicate results. The authors conclude that revisiting this lineage can inform contemporary debates about formalism, intuition, and the pedagogy of definition in modern mathematics.