Formalization, Primitive Concepts, and Purity

Abstract

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. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem.Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity.

Completing Segre's proof of Wedderburn's little theorem

“…A purely geometric proof by Baldwin and Howard of the embedding of a Desarguesian projective plane in 3-space is given in [3,Appendix]. This proof does not rely upon the representation theorem of Desarguesian planes by division rings.…”

Why Prove it Again?

“…Of particular interest is the contention defended inBaldwin (2013) -consonant with the viewpoint taken here, outlined in chapter 1 -that "purely formal methods" are themselves "impotent" for the purpose of characterizing the notion of purity of method.…”

“…However, an alternative proof of Hilbert's embedding theorem that uses only geometric notions was given by F.W. Levy in 1939, and another, quite recently, by John Baldwin andWilliam Howard. (See Levy 1939 andBaldwin 2013.…”

“…Levy in 1939, and another, quite recently, by John Baldwin andWilliam Howard. (See Levy 1939 andBaldwin 2013. ) Hilbert gave his own assessment of the significance of his embedding theorem in the notes for his lectures from 1898/99 (published in Hallett and Majer 2004): Having shown that a geometric proof of the planar Desargues's Theorem must invoke spatial considerations, he believed he had "for the first time : : : put into practice a critique of means of proof " that gave substance to the notion of purity of method; for he had formally demonstrated that Desargues's Theorem in the Plane is a result that cannot be proved using just means "suggested by its content.…”

On formalism freeness: Implementing Gödel's 1946 Princeton bicentennial lecture

“…This was the period inaugurated, roughly, by Frege and continuing through the first part of the twentieth century, 2 during which worries essentially about consistency-to simplify matters only a little-motivated the development of various foundational formal systems; a development which, if it did not exactly set those worries to rest, at least increased confidence in the unlikelihood of their ever being realized. The foundationalist objective which eventually emerged was stated in a preliminary but exact form 3 by Hilbert and his school. In its full form what we are calling the formalism-oriented foundationalist program, was simply this: embed mathematics in a formal language with an exact proof concept and an exact semantics, such that the proof concept is sound and complete with respect to the associated semantics as well as syntactically complete in the sense that all propositions that can be written in the formalism are also decided.…”