II.6 The Development of the Idea of Proof, Nineteenth-Century Mathematics and the Formal Conception of Proof: Pages 138 – 139.

Cauchy, Weierstrass, Cantor and Dedekind aimed at eliminating intuitive arguments and concepts instead using more elementary statements and definitions.  The idea of pursuing an axiomatic basis of mathematical theories was pursued in the nineteenth century by George Peacock, Charles Babbage, John Herschel and Hermann Grassmann.  Giuseppe Peano in 1889 “Postulates for the Natural Numbers”  used an artificial language that would allow a completely formal treatment of mathematical proofs and applied an axiomatic approach to arithmetic.  This did not lead Peano to a more formal conception of proof.  Mario Pieri developed a theory to handle abstract formal theories.   

At the end of the nineteenth century Hilbert published his “Grundlagen der Geometrie”.  In this he synthesized and completed the prior trends of geometric research.  He introduced a generalized analytic geometry.  In these the coordinates may be from a variety of different number fields rather than just the real numbers.  One of the purposes of the logical analysis was to avoid being misled by diagrams.  His axioms for geometry were of five types: axioms of incidence, of order, of congruence, of parallels and of continuity.  He also required the axioms be independent, consistent and simple.

Eliakim Moore in the beginning of the twentieth century turned the study of systems of postulates into a mathematical field in its own right.Felix Hausdorff in 1904 was among the first mathematicians to associate Hilbert’s work with  a new formalistic view of geometry.  Other scientists who reacted to Hilbert’s approach are: Frege, Boole, De Morgan, Grassmann, Charles S. Peirce and Ernst Schroder. 

The inclusion of the understanding of logical quantifiers universal, \forall, and existential, $, was very significant in a step toward a new formal conception of logic.  Cauchy, Bolzano and Weierstrass

Fregein 1879 proposed a system in his work “Begriffsschrift”.There were similar ones proposed by Peano and Russel.  Frege had a formal system with possibility of syntactic checking of any deduction.  “It is the truth of axioms, asserted Frege, that certifies their consistency, rather than the other way around, as Hilbert suggested.”

 The foundational research in geometry and analysis converged to create a new conception of mathematical proof.   A mathematical proof is seen as a purely logical construct validated in purely syntactic terms, independently of any visualization through diagrams.  This conception has dominated mathematics ever since.

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