Language and the Rise of the Algorithm by Jeffrey M. Binder

Author:Jeffrey M. Binder [Binder, Jeffrey M.]
Language: eng
Format: epub, pdf
Tags: COM000000 COMPUTERS / General, COM042000 COMPUTERS / Artificial Intelligence / Natural Language Processing, COM051300 COMPUTERS / Programming / Algorithms, TEC056000 TECHNOLOGY & ENGINEERING / History
Publisher: University of Chicago Press
Published: 2022-11-25T00:00:00+00:00

Paradox and the Individual Will

The standard account of what happened to logical formalism goes something like this. Bertrand Russell and Alfred North Whitehead’s monumental effort in Principia struck readers as unsatisfactory, and David Hilbert and Wilhelm Ackerman tried to do better.16 Their project was supposed to produce a complete and consistent set of axioms along with a clearly defined procedure (one can now say with only slight anachronism, an algorithm) for deciding whether given mathematical statements were true. An early critique of Hilbert’s project appeared in the work of L. E. J. Brouwer, who advocated for an antiformalist approach he called intuitionism, and whose work would be an influence on Turing; this school of thought rejects the law of excluded middle and thus does not accept the existence of any mathematical entity that cannot be constructed through some realizable procedure.17 Gödel, Church, and Turing, within the span of just a few years, proved key elements of the Hilbertian program impossible. In papers published in 1936, Church and Turing proved independently that Hilbert’s decision algorithm cannot exist: any formal system must leave some problems unsolvable. The advent of paradox produced, as a side effect, the first precise definitions of computation, which would later form a central part of theoretical computer science.

Caught up in this intellectual tumult was yet another utopian attempt to replace words with symbols. One of Gödel’s mentors at the University of Vienna was Rudolf Carnap, whose 1928 book The Logical Structure of the World, usually referred to as the Aufbau (after its German title, Der logische Aufbau der Welt), outlines a “constructional system” in which complex concepts are reduced to simpler ones.18 This project, which Carnap likens to Leibniz’s universal characteristic, is supposed to unify the sciences by placing them all on the same conceptual basis.19 While this project is not as significant to the technical theory of algorithms as the work of Church and Turing, it is worth considering first because it shows how the Leibnizian program was faring in the early twentieth century. Following the practice of Russell and Whitehead, Carnap defines symbols using equations:

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