Foundations of Geometry in the School of Peano

Abstract

The question that motivated this paper — why Pieri made an analogy to Peano’s affinities when he introduced segmental transformations — revealed several plausible answers. But perhaps more importantly, seeking to answer the question provided an opportunity to explore the commonalities and differences about the scholars’ views and treatments of projective geometry and its transformations. In this regard, there is one more avenue to explore. What is not evident from his axiomatizations, but is clear from his lectures37 to students, is the evolution of Pieri’s thoughts about projective geometry. In his (1891) notes for a course in projective geometry at the Military Academy — prior to writing his first axiomatization, but after he had translated Staudt (1847) — Pieri took the same approach to projective geometry as had Peano³⁸. But in his notes for the University of Parma (1909–10), after he had written all his foundational papers in projective geometry, Pieri alerted students to the more “desirable” direction of Staudt as opposed to that pursued by J. Poncelet, Möbius, J. Steiner and Chasles, who studied projective geometry as an extension of elementary geometry.

Pieri had learned well from Peano, but was not reluctant to forge his own path. For example, Pieri would adopt Peano’s ideas on point transformations, but took their use to new levels. In (1898b), he demonstrated the possibility of constructing real projective geometry entirely on the basis of point and a projective point transformation that preserves lines. In (1900), he constructed absolute geometry, the theory common to Euclidean and Bolyai-Lobatchevskian geometry, solely on the undefined notions of point and motion³⁹. In that paper, Pieri observed that although the distinction between the synthetic concept of a congruence transformation (motions) from points to points rather than from figure to figure is not, from a logical perspective, significant, the first idea is more “manageable” to the deductive process. Pieri acknowledged Peano (1889d) and Peano (1894c), noting that Peano’s primitives and postulates could be derived from his Pieri (1900a, Prefazione, 174 — Opere 1980, 184).

And the path would come full circle. Peano would be inspired by Pieri’s fertile ideas. For example in 1903 Peano proposed a construction of geometry based on the ideas of point and distance⁴⁰. His proposal combined Pieri’s plan (announced in Pieri 1901b) to establish elementary geometry on the basis of point and two points equidistant from a third (that would be realized in Pieri 1908), with his own Peano (1898c) construction on point and vector. Using Pieri’s idea of equidistance, Peano was able to define the equivalence of vectors instead of taking it as primitive, as he had previously, and reformulate definitions (include that of vector) on solely on the basis of it. He produced a systemization of geometry founded on three primitives (point, the relation of equidifference between pairs of points, and inner product of two vectors) and nineteen postulates (reducible to seventeen).

It is impossible to exclude the influence of Peano on Pieri’s immersion into the world of foundations. After his (1895–1896) Notes, Pieri would continue to refine his ideas on projective geometry and ultimately produce what Russell (1903)⁴¹ called “the best work” on the subject. As I have observed, Peano was involved in a substantial way in propelling Pieri on the path to that achievement. And he shared Russell’s evaluation of it. Peano wrote: “The results reached by Pieri constitute an epoch in the study of foundations of geometry, and all those who later treated the foundations of geometry have made ample use of Pieri’s work and have echoed Russell’s evaluation”. Pieri had made his mentor proud!⁴²