Von Plato J. The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age

Princeton: Princeton University Press, 2017. — 391 p.The information age owes its existence to a little-known but crucial development, the theoretical study of logic and the foundations of mathematics. The Great Formal Machinery Works draws on original sources and rare archival materials to trace the history of the theories of deduction and computation that laid the logical foundations for the digital revolution.
Jan von Plato examines the contributions of figures such as Aristotle; the nineteenth-century German polymath Hermann Grassmann; George Boole, whose Boolean logic would prove essential to programming languages and computing; Ernst Schröder, best known for his work on algebraic logic; and Giuseppe Peano, cofounder of mathematical logic. Von Plato shows how the idea of a formal proof in mathematics emerged gradually in the second half of the nineteenth century, hand in hand with the notion of a formal process of computation. A turning point was reached by 1930, when Kurt Gödel conceived his celebrated incompleteness theorems. They were an enormous boost to the study of formal languages and computability, which were brought to perfection by the end of the 1930s with precise theories of formal languages and formal deduction and parallel theories of algorithmic computability. Von Plato describes how the first theoretical ideas of a computer soon emerged in the work of Alan Turing in 1936 and John von Neumann some years later.
Shedding new light on this crucial chapter in the history of science, The Great Formal Machinery Works is essential reading for students and researchers in logic, mathematics, and computer science.An Ancient Tradition
Reduction to the Evident
Aristotle’s Deductive Logic
Infinity and Incommensurability
Deductive and Marginal Notions of Truth
The Emergence of Foundational Study
In Search of the Roots of Formal Computation
Grassmann’s Formalization of Calculation
Peano: The Logic of Grassmann’s Formal Proofs
Axiomatic Geometry
Real Numbers
The Algebraic Tradition of Logic
Boole’s Logical Algebra
Schröder’s Algebraic Logic
Skolem’s Combinatorics of Deduction
Frege’s Discovery of Formal Reasoning
A Formula Language of Pure Thinking
Inference to Generality
Equality and Extensionality
Frege’s Successes and Failures
Russell: Adding Quantifiers to Peano’s Logic
Axiomatic Logic
The Rediscovery of Frege’s Generality
Russell’s Failures
The Point of Constructivity
Skolem’s Finitism
Stricter Than Skolem: Wittgenstein and His Students
The Point of Intuitionistic Geometry
Intuitionistic Logic in the s
The Göttingers
Hilbert’s Program and Its Programmers
Logic in Göttingen
The Situation in Foundational Research around
Gödel’s Theorem: An End and a Beginning
How Gödel Found His Theorem
Consequences of Gödel’s Theorem
Two “Berliners”
The Perfection of Pure Logic
Natural Deduction
Sequent Calculus
Logical Calculi and Their Applications
The Problem of Consistency
What Does a Consistency Proof Prove?
Gentzen’s Original Proof of Consistency
Bar Induction: A Hidden Element in the
Consistency Proof