Plato's Ghost: The Modernist Transformation of Mathematics

4 Modernism Avowed

4 MODERNISM AVOWED 4.1 Geometry 4.1.1 Abstract Italian Geometry We have already seen that Italian mathematicians were energetic students of projective geometry, which some, such as Corrado Segre and Federigo Enriques, extended to n dimensions and treated abstractly, the better to allow it to be interpreted in a variety of ways. Another of Segre’s students, who followed him in this work but was more sympathetic to the axiomatic approach, was Gino Fano. When Fano wrote about n-dimensional geometry, he tried systematically to show that each new postulate is independent of the previous ones, and in this way he made some interesting discoveries almost without noticing. For example, Fano’s fourth postulate asserts that each line contains more than two points. Its necessity is demonstrated by proposing a model where each line contains exactly two points (he took the three vertices of a triangle as points, and the three edges of the triangle as lines). This model satisfies the first three postulates but not the fourth, so the fourth postulate is independent. With these four postulates, Fano could construct the fourth harmonic point D, of three collinear points A, B, C. But is the fourth harmonic point D distinct from A, B, and C or not? Fano showed that it need not be by means of a model consisting of seven points and seven lines for which each line has exactly three points. (Take the three vertices of an equilateral triangle, the three midpoints of the edges, and the center of the triangle as points; take the edges of the triangle, the lines through the center to the opposite midpoint, and the three midpoints themselves as lines.)1 Accordingly, another postulate is needed to ensure that there are harmonic series that do not fold back on themselves in this fashion. That said, Fano missed the lasting significance of his new geometries, and saw them merely as counterexamples, whose sole purpose was to demonstrate the independence of the axioms. They were not presented as starting points for the de1 This counterexample, presented as a projective plane over the field of two elements, is nowadays called a Fano plane. velopment of new geometrical research, as was to be the view of the American geometers in the early 1900s. The most forceful and innovative axiomatizer of geometry was the still somewhat neglected figure of Mario Pieri (1860–1913), who graduated from Pisa in 1884, and also went to Turin to study with Segre and Peano.2 From Segre he learned that an axiomatic structure can be valuable even when completely independent of experience , which Peano had denied. From Peano, he learned rigor, and a formulation of geometry that blended the algebraic methods of Grassmann with the formalities of Pasch and further refined them with Peano’s way of writing mathematics as a hybrid of logic and set theory.3 Pieri’s work is marked by his complete abandonment of any intention to formalize what is given in experience. This was the first time this was attempted in geometry. Instead, he wrote that he treated projective geometry ‘‘in a purely deductive and abstract manner, . . . , independent of any physical interpretation of the premises.’’4 Primitive terms, such as line segments, ‘‘can be given any significance whatever, provided they are in harmony with the postulates which will be successively introduced .’’ Pieri presented nineteen axioms as the foundations for projective geometry, based on taking point, line, and motion (by which he meant congruence) as the primitive notions.5 The premises and the methods were entirely independent of intuition . He made extensive use of Peano’s notions of class and membership as a logical tool—he also wrote in the arid fashion of Peano—and in this way proved theorems without any recourse to external intuition or implicit perceptual, linguistic, or cultural experience. As he put it: A good ideographic algorithm is generally acknowledged as a useful tool to discipline thought, to eliminate ambiguities, mental limitations, unexpressed assumptions and other faults which are integral part of both spoken and written language, and that are so harmful to speculative investigation. Therefore it is vitally important to make use of the method of algebraic logic. Neglecting it, especially in this type of studies, seems to me a deliberate rejection of the most valid tool for the analysis of ideas that we have at our disposal today.6 Pieri went so far beyond Peano in allowing the mathematician to create a geometry that one can wonder what...