# Notes

## Section 3: The Phenomenon of Universality

History of universality

In Greek times it was noted as a philosophical matter that any single human language can be used to describe the same basic range of facts and processes. And with logic introduced as a way to formalize arguments (see page 1099), Gottfried Leibniz in the 1600s considered the idea of setting up a universal language based on logic that would provide a precise description analogous to a mathematical proof of any fact or process. But while Leibniz considered the possibility of checking his descriptions by machine, he apparently did not imagine setting up the analog of a computation in which something is explicitly generated from input that has been given.

The idea of having an abstract procedure that can be fed a range of different inputs had precursors in antiquity in the use of letters to denote objects in geometrical constructions, and in the 1500s in the introduction of symbolic formulas and algebraic variables. But the notion of abstract functions in mathematics reached its modern form only near the end of the 1800s.

At the beginning of the 1800s practical devices such as the player pianos and the Jacquard loom were invented that could in effect be fed different inputs using analogs of punched cards. And in the 1830s Charles Babbage and Ada Lovelace noted that a similar approach could be used to specify the mathematical procedure to be followed by a mechanical calculating machine (see page 1107). But it was somehow assumed that the specification of the procedure must be done quite separately from the specification of the data to which the procedure was to be applied.

Starting in the 1880s attempts to build up both numbers and the operations of arithmetic from logic and set theory began to suggest that both data and procedures could potentially be described in common terms. And in the 1920s work by Moses Schönfinkel on combinators and by Emil Post on string rewriting systems provided fairly concrete examples of this.

In 1930 Kurt Gödel used the same basic idea to set up Gödel numbers to encode logical and other procedures as numbers. (Leibniz had in fact already done this for basic logic expressions in 1679.) But Gödel then took the crucial step of showing that the process of finding outputs from all such procedures could in effect be viewed as equivalent to following relations of logic and arithmetic—thus establishing that these relations are in a certain sense universal (see page 784). This fact, however, was embedded inside the rather technical proof of Gödel's Theorem, and it was at first not at all clear how specific it might be to the particular mathematical systems considered.

But in 1935 Alonzo Church constructed a system in lambda calculus that he showed could be made to emulate any other system in lambda calculus if given appropriate input, and in 1936 Alan Turing did the same thing for Turing machines. As discussed on page 1125, both Church and Turing argued that the systems they set up would be able to perform any reasonable computation. In both cases, their original motivation was to use this fact to construct an argument that the so-called decision problem (Entscheidungsproblem) of mathematical logic was undecidable (see page 1136). But Turing in particular gradually realized that his notion of universality could be applied to practical computers.

Turing's results were used in the 1940s—notably in the work of Warren McCulloch and Walter Pitts—as a basis for the assertion that electric circuit analogs of neural networks could achieve the sophistication of brains, and this appears to have influenced John von Neumann's thinking about the general programmability of electronic computers. Nevertheless, by the late 1940s, practical computer engineering had also been led to the idea of storing programs—like data—electronically, and in the 1950s it became widely understood that general-purpose practical computers could be viewed as universal systems.

Many theoretical investigations of universality were made in the 1950s and 1960s, but gradually the emphasis shifted more towards issues of languages and algorithms.

From Stephen Wolfram: A New Kind of Science [citation]