[History of] models of mathematics

Gottfried Leibniz's notion in the late 1600s of a "universal language" in which arguments in mathematics and elsewhere could be checked with logic can be viewed as an early idealization of mathematics. Starting in 1879 with his "formula language" (*Begriffsschrift*) Gottlob Frege followed a somewhat similar direction, suggesting that arithmetic and from there all of mathematics could be built up from predicate logic, and later an analog of set theory. In the 1890s Giuseppe Peano in his *Formulario* project organized a large body of mathematics into an axiomatic framework involving logic and set theory. Then starting in 1910 Alfred Whitehead and Bertrand Russell in their *Principia Mathematica* attempted to derive many areas of mathematics from foundations of logic and set theory. And although its methods were flawed and its notation obscure this work did much to establish the idea that mathematics could be built up in a uniform way.

Starting in the late 1800s, particularly with the work of Gottlob Frege and David Hilbert, there was increasing interest in so-called metamathematics, and in trying to treat mathematical proofs like other objects in mathematics. This led in the 1920s and 1930s to the introduction of various idealizations for mathematics—notably recursive functions, combinators, lambda calculus, string rewriting systems and Turing machines. All of these were ultimately shown to be universal (see page 784) and thus in a sense capable of reproducing any mathematical system. String rewriting systems—as studied particularly by Emil Post—are close to the multiway systems that I use in this section (see page 938).

Largely independent of mathematical logic the success of abstract algebra led by the end of the 1800s to the notion that any mathematical system could be represented in algebraic terms—much as in the operator systems of this section. Alfred Whitehead to some extent captured this in his 1898 *Universal Algebra*, but it was not until the 1930s that the theory of structures emphasized commonality in the axioms for different fields of mathematics—an idea taken further in the 1940s by category theory (and later by topos theory). And following the work of the Bourbaki group beginning at the end of the 1930s it has become almost universally accepted that structures together with set theory are the appropriate framework for all of pure mathematics.

But in fact the Mathematica language released in 1988 is now finally a serious alternative. For while it emphasizes calculation rather than proof its symbolic expressions and transformation rules provide an extremely general way to represent mathematical objects and operations—as for example the notes to this book illustrate.

(See also page 1176.)