Foundations of Geometry in the School of Peano
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
KeywordsElementary Geometry Projective Geometry Projective Transformation Projective Point Point Transformation
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- 1.The citations in this paper of page numbers from Pieri’s publications refer to Opere sui fondamenti della matematica, a cura dell’Unione Matematica Italiana, Roma, Cremonese, 1980. Wherever possible paragraph references (§) are also given. The reader is invited to consult the list of Pieri’s publications in the larger context of his entire opus as specified in E.A. Marchisotto, J.T. Smith (2007), 373–399.Google Scholar
- 6.F. Schur gave a simplified version of Peano’s system due to G. Ingrami in F. Schur (1902), §1, 267 ff. Schur referred to axioms of connection and order given by D. Hilbert as “projective axioms”, and to the simplified version of Peano’s axioms as “another grouping” of Hilbert’s projective axioms.Google Scholar
- 7.U. Cassina called Peano’s postulates “graphic postulates”, observing that in their presence, two hypotheses are possible: Euclidean and non-Euclidean geometry (Cassina 1961b, §12.4, 319–320).Google Scholar
- 8.A. Brigaglia, G. Masotto (1982), 137.Google Scholar
- 9.See M. Avellone, A. Brigaglia, C. Zappulla (2002), §§3, 7. Segre also invited Pieri to edit a foundational paper of R. De Paolis (E.A. Marchisotto, J.T. Smith 2007, §2, 123–124).Google Scholar
- 10.Letter of C. Segre to M. Pieri, Torino 11 October 1887, in Arrighi 1997, Nr. 114, 113. Cf. M. Pieri (1889).Google Scholar
- 11.M. Pieri (1889), XXIV–XXV. Pieri was one of many, including, for example, Peano in 1891, who attempted to make Staudt’s reasoning rigorous. See E.A. Marchisotto (2006), §§6, 7.Google Scholar
- 12.M. Pieri (1889), §106, 43–44, footnote (*). Pieri built on the results of F. Klein, G. Darboux and T. Reye to construct his proof. For the details, see E.A. Marchisotto (2006), §7, 295–298. For a discussion of the evolution of proofs, see J.D. Voelke (2008).Google Scholar
- 13.L. Carnot had written a pioneering volume on projective geometry entitled Géométrie de position (L. Carnot 1803). Reye produced a study of projective geometry emanating from Staudt’s ideas entitled Die Geometrie der Lage (Reye 1886–92). Pieri cited Reye’s book in his translation of Staudt as well as in his own axiomatizations of projective geometry (Pieri 1889, XXV). Reye 1886–92 appeared in 5 editions up to 1923.Google Scholar
- 14.Pieri noted that he used the terms projective point, projective line, etc. to distinguish these primitives from the common physical understanding of point and line (Pieri 1895a, §2, 5–8 — Opere 1980, 15–18).Google Scholar
- 15.G. Vailati has been cited for characterizing the fundamental properties of the quaternary relation of separation of points of a closed line on the basis of incidence. See Veblen, Young 1908, §4, 362, Borga, Palladino 1992, 32. In G. Vailati (1895a) (Scritti 1911, 26) Vailati indicated that his “repeated discussions” with Pieri led him to such considerations. In G. Vailati (1895b) (Scritti 1911, 30), Vailati again acknowledged M. Pieri (1895a), §7.Google Scholar
- 16.E.A. Marchisotto (2006), §3, 281–283.Google Scholar
- 18.J. Barrow-Green, J. Gray (2006), 275.Google Scholar
- 21.See U. Cassina (1940), §2.9, U. Cassina (1961a), 414–416 and U. Cassina (1961b), 320–325.Google Scholar
- 22.Cassina said that by appealing to Peano’s graphic postulates and postulates of motion, in the presence of the postulate of Archimedes, the fundamental theorem of Staudt could be proved (Cassina 1940, §3.11).Google Scholar
- 23.The language proposed by Peano had a goal of obtaining “a highly synthetic and rigorous exposition” of mathematical theory, achieved through the expression of mathematical propositions “without the prolixities and ambiguities found in ordinary language” (Borga, Palladino 1992, 28).Google Scholar
- 25.Peano followed M. Chasles who had used the word homography instead of collineation (Chasles 1837, II, 695). In general, usage of the terms homographies and collineation varied among geometers, and many, like Peano, used them interchangeably.Google Scholar
- 26.Formula a ∈ b indicates that “a è un b” or a is an individual of the class b (Peano 1889d, Notazioni, 6 — Opere scelte, vol. 2, 1958, 59).Google Scholar
- 29.Pieri only used nominal definitions in his system and Peano also attempted to do so. See, for example, M. Pieri (1900a), Prefazione, 173 — Opere 1980, 183–184, G. Peano (1889d), 25–28 — Opere scelte, vol. 2, 1958, 77–80. F. Rodriguez-Consuegra (1991), §3.4.2, 121, reports that Peano only resorted to other types when he could not offer nominal ones.Google Scholar
- 30.De Paolis, for example, adopted Staudt’s interpretation, indicating for example that a segment E1 E2 is decribed by fixing the direction of the motion if a point E leaves from an initial position E1 and arrives at a final position E2 (De Paolis 1880-81, Parte 1 §1, 489). Darboux had used such a characterization: “Supposons qu’un point M se meuve de P′ en Q′ …” (Darboux 1880, 58). But Peano (1894c, 76) indicated that such statements as “a moveable point describes a line” should be excluded from geometry books.Google Scholar
- 31.In introducing segmental transformations, Pieri referred the reader to (G. Peano 1894g, §26, 31–32) where Peano had discussed different categories of functions, and where he had distinguished Sim from sim, the term used for bijective. In (M. Pieri 1896b, §13, 457 — Opere 1980, 69 footnote), Pieri explicitly observed that his segmental transformations are not required to be bijective are therefore more general than similes. Google Scholar
- 33.Reye had used the term harmonic projectivity (harmonische Projectivitaeten) in lecture 12 of the third edition of the second volume of his Die Geometrie der Lage (Reye 1886-1892).Google Scholar
- 35.G. Darboux (1880, 58) had already proved that in the presence of continuity separation can be defined from harmonicity. Pieri went one step further, defining separation from harmonicity without appealing to continuity. See E.A. Marchisotto (2006), §7.Google Scholar
- 36.M. Pieri (1906f), 1–5.Google Scholar
- 41.Russell believed that Pieri was the one who demonstrated the true nature of the purely projective method (Gandon 2004, 189).Google Scholar
- 42.G. Peano (1915j), 171.Google Scholar