Did Lobachevsky Have A Model Of His 'imaginary Geometry'?

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky’s geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky’s writings its role in Lobachevsky’s geometrical theory turns to be very unusual. Lobachevsky doesn’t consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky’s construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.

💡 Research Summary

The paper revisits the long‑standing question “Did Lobachevsky have a model of his ‘imaginary geometry’?” and argues that the answer depends on how we understand the notion of a model in the nineteenth‑century context. Modern model theory, shaped by Hilbert’s formal axiomatic method, treats a model as an independent mathematical structure that satisfies a given set of axioms. Lobachevsky, however, did not work within this later framework. Instead of searching for external realizations of hyperbolic geometry, he constructed a non‑standard “model” of the Euclidean plane inside hyperbolic three‑space.

The author shows that Lobachevsky selected a particular surface—essentially a complete, smooth two‑dimensional submanifold of hyperbolic 3‑space, such as a pseudosphere or a surface isometric to the Poincaré disk—and then defined Euclidean distance and angle on that surface. Because the surface lives inside hyperbolic space, the induced metric automatically satisfies the hyperbolic axioms. In other words, Lobachevsky “embedded” Euclidean geometry as a special case of his hyperbolic theory, using the surface as a bridge. This construction is not a model in the Hilbertian sense (a separate structure that validates the axioms) but rather a reinterpretation of an existing geometric object to serve as a concrete instance of the new theory.

Two major implications follow. First, Lobachevsky’s notion of a model is essentially “a part of another axiomatic system.” He treats the chosen surface as a piece of hyperbolic space that, when viewed from a Euclidean perspective, reproduces Euclidean relations. Thus the model is a translation between two geometries rather than an external witness. Second, his approach is analytic rather than constructive: he does not build a fresh structure from scratch but re‑examines an already known object, endowing it with new definitions that make it satisfy the hyperbolic axioms. This anticipates later categorical and structural ideas where objects are understood through their relationships rather than through an absolute set‑theoretic foundation.

The paper contrasts this with Hilbert’s program, which insists on a clear separation between axioms and models, and on the existence of models that are logically independent of the axioms they satisfy. Lobachevsky’s method blurs that boundary; his “model” is simultaneously a piece of the ambient hyperbolic space and a representation of Euclidean geometry. Consequently, his work fits more naturally into a non‑Hilbertian foundational framework, one that emphasizes internal reinterpretation and the fluid movement between different geometric languages.

Further, the author connects Lobachevsky’s surface construction to modern developments in Riemannian geometry and topology. A smooth surface in hyperbolic 3‑space carries a Riemannian metric; comparing this metric with the Euclidean metric on the same surface yields an implicit isometry between the two distance structures. Such an isometry is precisely the kind of object studied today in algebraic topology, geometric group theory, and modern model theory (e.g., categorical models of geometric theories).

The central thesis can be summarized in three points: (1) Lobachevsky possessed a proto‑model‑theoretic insight, but he did not cast it into Hilbert’s later formalism; (2) his non‑standard model treats an existing geometric object as a conduit between Euclidean and hyperbolic geometries, highlighting a bidirectional transfer of concepts; (3) this perspective aligns with contemporary structural and categorical approaches, suggesting that the historical development of non‑Euclidean geometry can be read through the lens of modern foundational trends.

In conclusion, the paper argues that Lobachevsky’s “model” is not a mere historical curiosity but a substantive example of a different way to relate axioms and structures. By embedding Euclidean geometry as a surface inside hyperbolic space, he demonstrated a deep interplay between the two systems that prefigures later ideas about internal models, functorial translations, and the fluidity of mathematical meaning. This reinterpretation enriches our understanding of 19th‑century geometry and bridges it to current research in geometry, topology, and the philosophy of mathematics.