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This chapter applies the account of explanatory proofs in mathematics from previous chapters to proofs of Desargues’s theorem. Mathematicians regard a nonmetrical proof exiting to the third dimension as explaining why Desargues’s theorem holds, even though this proof lacks purity by invoking a third spatial dimension absent from the theorem. Other proofs of Desargues’s theorem incorrectly depict it as coincidental. Furthermore, the theorem’s natural home is projective geometry rather than Euclidean geometry. Desargues’s theorem in projective geometry unifies what in Euclidean geometry are special cases requiring separate treatment. But this unification requires that projective concepts be natural properties in mathematics. Certain proofs qualify as explanatory only because certain properties qualify as natural, and vice versa. A proposal for resolving this circularity is offered. Mathematical explanations that involve subsumption under a theorem, rather than a proof, are investigated and shown to depend on explanatory proofs.

This chapter provides an overview of the basic questions associated with matroid representability and indicates how one actually goes about constructing representations. The key ideas are presented in Sections 6.1 and 6.3–6.6, which cover projective geometries, different matroid representations, constructing representations for matroids, representability over finite fields, and regular matroids, respectively. Section 6.2 looks at affine geometries, a class of highly symmetric structures that are closely linked to the projective geometries of Section 6.1. Section 6.7 discusses algebraic matroids, a class of matroids that properly contains the class of representable matroids and arises from algebraic dependence over a field. Section 6.8 focuses on characteristic sets, its main idea being concerned with how one can capture geometrically certain algebraic properties of a field. Section 6.9 examines modularity, a special property of flats that is important in several contexts including matroid constructions. Finally, Section 6.10 discusses an important class of matroids introduced by Dowling.

This chapter serves as a brief lesson on projective geometry. It starts by discussing two basic theorems of projective geometry, which are concerned with perspectives from a point and perspectives from a line. The discussion then looks at projectivities and perspectivities, which result from several perspectivities and relate a range, respectively. This is followed by the study of analytic projective geometry and the connection between projective and projection. The chapter concludes with a discussion on the complex projective line, the projective three-space, and Felix Klein's thoughts on the presence of non-Euclidean geometries in projective geometry, specifically the topology of the projective plane.

The connection between perspective and the indexical shows up in the sciences, for example, when talk of frames of reference is conducted in terms of observers (whose frames they are, so to speak). This metaphor can be eliminated to produce an objective theoretical representation. But in the use of such representations as basis for prediction and manipulation, the indexical element returns. Examining the character of perspective and the role of indexical judgments (such as self-attributions and self-locations) brings to fore a fundamental connection between perspective, measurement, and theoretical representation.

This chapter reviews the key concepts related to projective geometry, which is the field that has helped develop thinking on physics and mathematics. It first looks at the fact that Felix Klein admitted that projective geometry (e.g. the dual nature of objects) is important in developing thought. From here the discussion shifts to a study of the classical definition of time, before turning to the idea of space. The spatial model, light rays, the state space of quantum particles, and quantum foam are the other concepts that are reviewed in this chapter. The discussion ends with the main purpose of the book, which is as a proposal to remove the slicing model of four-dimensional figures and spacetime from one's thinking and replace it with the projection model to avoid blocks.