Formalization, Primitive Concepts, and purity

We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are algebraic and even metamathematical. In particular, the converse to Desargues cannot be read as: the Desargues proposition implies there are non-coplanar points. Rather, Hilbert showed that Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a 3-dimensional geometry in which the given plane can be embedded. We (with W. Howard) give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem and examine the issue of purity for this embedding theorem. Mathematical logic formalizes normal mathematics. We analyze the meaning of ‘formalize’ in that sentence and use this analysis to address several recent questions. In the first section we establish some precise definitions to formulate our discussion and we illustrate these notions with some examples of David Pierce. This enables us to describe a variant on Kennedy’s notion of formalism freeness and connect it with recent developments in model theory. In the second section we discuss the notion of purity in geometric reasoning based primarily on the papers of Hallet [17] and Arana-Mancosuo [3]. In an appendix written with William Howard we give a geometric proof (differing from Levi’s in [29]) of Hilbert’s theorem that a Desarguesian projective plane can be embedded in three-space. Our general context is that there is some area of mathematics that we want to clarify. There are five components of a formalization. The first four 1) specification of primitive notions, 2) specifications of formulas and 3) their truth, and 4) proof provide the setting for studying a particular topic. 5) is a set of axioms that pick out the actual subject area. We will group these five notions in various ways through the paper to make certain distinctions. Our general argument is: while formalization is the key tool for the general foundational analysis and has had significant impact as a mathematical tool 1 there are spe1Examples include the theory of computability, Hilbert’s 10th problem, the Ax-Kochen theorem, ominimality and Hardy fields, Hrushovski’s proof the geometric Mordell-Lang theorem and current work on motivic integration.