1For the congress on The School of Giuseppe Peano between Mathematics, Logic, and Interlingua to be held6–7 October 2008 in Turin2008-06-19 09:49Definitions and Nondefinability in Geometry:Pieri and the Tarski School1James T. SmithSan Francisco State UniversityIntroductionIn 1886 Mario Pieri became professor of projective and descriptive geometry at the RoyalMilitary Academy in Turin and, in 1888, assistant at the University of Turin. By theearly 1890s he and Giuseppe Peano, colleagues at both, were researching related ques-tions about the foundations of geometry. During the next two decades, Pieri used,refined, and publicized Peano’s logical methods in several major studies in this area. Thepresent paper focuses on the history of two of them, Pieri’s 1900 Point and Motion and1908 Point and Sphere memoirs, and on their influence as the root of later work of AlfredTarski and his followers. It emphasizes Pieri’s achievements in expressing Euclideangeometry with a minimal family of undefined notions, and in requiring set-theoreticconstructs only in his treatment of continuity. It is adapted from and expands onmaterial in the 2007 study of Pieri by Elena Anne Marchisotto and the present author.Although Pieri 1908 had received little explicit attention, during the 1920s Tarskinoticed its minimal set of undefined notions, its extreme logical precision, and its use ofonly a restricted variety of logical methods. Those features permitted Tarski to adaptand reformulate Pieri’s system in the context of first-order logic, which was only thenemerging as a coherent framework for logical studies. Tarski’s theory was much simpler,and encouraged deeper investigations into the metamathematics of geometry.In particular, Tarski and Adolf Lindenbaum pursued the study of definability, extend-ing earlier work by the Peano School. They settled some questions about systems relatedto Pieri’s, and during the 1930s showed that in the first-order context, Pieri’s selectionof equidistance as the sole undefined relation for Euclidean geometry was optimal. Nofamily of binary relations, however large, can serve as the sole undefined relations.Tarski’s work itself went mostly unpublished for decades, but began to attract researchactivity during the 1950s. Tarski’s followers have extended his methods to apply to othergeometric theories as well as the Euclidean. The present paper concludes with a descrip-tion of the 1990–1991 discovery by Victor Pambuccian that Euclidean geometry can bebased on a single binary relation if the underlying logic is strengthened.Page 2 SMITH: DEFINITIONS AND NONDEFINABILITY2See his reflections after his list of twenty-three axioms and theorems in section 1.3Already in the introduction (Einleitung), Pasch discussed points without definition. Sections 1 and 2 beginby introducing betweenness and coplanarity. Pasch distinguished the notions of coplanar set (ebene Fläche)and plane (Ebene). Not until section 13 did he introduce congruence.4“Dipendentemente dal significato attribuito ai segni non definiti ... potranno essere soddisfatti, oppure no,gli assiomi. Se un certo gruppo di assiomi è verificato, saranno pure vere tutte le proposizioni che si dedu-cono ...” (Peano 1889, 24).2008-06-19 09:49PaschMoritz Pasch began his career around 1870 as an algebraic geometer, but his emphasischanged to foundations of analysis and geometry. To correct logical gaps in classicalEuclidean geometry and in G. K. C. von Staudt’s 1847 presentation of projective geome-try, Pasch published in 1882 the first completely rigorous synthetic presentation of ageometric theory.Pasch clearly indicated that, in contrast to earlier practice, he would discuss certainnotions without definition.2 Determining which ones he actually left undefined requiresclose reading.3 They are• point,•coplanarity of a point set,•segment between two points, • congruence of point sets.As proclaimed, he defined all other geometric notions from those. For example, he calledthree points collinear if they are not distinct or one lies between the other two, anddefined the line determined by two distinct points to be the set of points collinear withthem. Pasch developed incidence and congruence geometry, extended it to projectivespace, then showed (section 20) how to select a polar system to develop Euclidean or non-Euclidean geometry.Peano and MotionsGiuseppe Peano began intense study of fundamental principles of mathematics duringthe 1880s. His 1889 booklet on foundations of geometry contained some technicalimprovements over Pasch 1882. But more importantly, it divorced that discipline fromthe study of the real world:Depending on the significance attributed to the undefined symbols... the axioms can be satisfiedor not. If a certain group of axioms is verified, then all the propositions that are deduced fromthem will be equally true ... .4This freedom to consider various interpretations of the undefined notions, and thedistinction between syntactic properties of symbols and their semantic relationships tothe objects they denote, was essential for all later studies of definability.In 1894 Peano introduced the use of direct motion to replace congruence as an unde-fined notion in Euclidean geometry. A geometric transformation, this sort of motion doesSMITH: DEFINITIONS AND NONDEFINABILITY Page 35With additional work, indirect motions can be defined, which relate anticongruent figures.6Pieri 1889 is an annotated translation of Staudt 1847.7See Pieri 1900, P9§1, for the definition of collinearity; P28§1 and P7§3 for equidistance; P7§2 for midpoint;and P1,2,6§4 for betweenness.2008-06-19 09:49not involve time. Figures can be defined as congruent if some direct motion maps oneto the other.5Pieri and MotionsAfter earning the doctorate in 1884 at Pisa, Mario Pieri began his research career inalgebraic and differential geometry. Soon after Pieri’s appointments in Turin, CorradoSegre suggested that Pieri translate Staudt 1847, the fundamental work on that subject.6Evidently Pieri, like Pasch, became intrigued with its logic: he returned to study it againand again. Pieri’s senior colleague Giuseppe Peano was already investigating deep ques-tions in foundations, and in 1890 himself repaired a lapse in Staudt’s work. During the1890s, Pieri became a preeminent member of the Peano School. A series of his papersculminated in Pieri 1898, the first complete axiomatization