Mathematical Logic

If, in 1901, a talented and sympathetic outsider had been called upon (say, by a granting-giving agency) to survey the sciences and name the branch which would be least fruitful in century ahead, his choice might well have settled upon mathematical logic, an exceedingly recondite field whose practitioners could all have fit into a small auditorium --- algebraists consumed by abstractive passion, or philosophers pursuing fantasies out of Leibniz and Ramon Llull, or (like Whitehead) both. It had no practical applications, and not even that much mathematics to show for itself: its crown was an exceedingly obscure definition of cardinal numbers. When, in 1910, it produced a work which the learned world was forced to notice --- the first volume of Whitehead and Russell's Principia Mathematica --- it was, so to speak, the academic Brief History of Time of its day, often mentioned, never used.

Our outsider would, of course, have been wrong. Mathematical logic was the inspiration for perhaps only half of twentieth-century philosohpy (that is, of honest philosophy; by volume, as Kolakowski says, Stalin was the century's most influential philosopher); many of our finest mathematicians, such as Norbert Wiener, John von Neumann and Andrei Kolmogorov cut their teeth on it, and notation (and notions) which began in the obscurities of Peirce and Peano are now to be found in every undergraduate math book. True, some early application --- one thinks particularly of Woodger's axiomatization of biology --- have, perhaps unfairly, gone nowhere, and McCulloch and Pitt's "A Logical Calculus of the Ideas Immanent in Nervous Activity" is more important for launching neural nets upon the world than for using Carnap's formalism. But in one extremely important field, however, it reigns supreme, and that is computation. Programming is, simply, mathematical logic in action; the melding of theory and practice is so complete that most practioners have no idea that their speech --- recursion, lexical scope, data abstraction, even those banes of C novices, pointers, referencing and dereferncing --- is prose. (Speaking of speech, Chomsky of course began as a logican, and his early work (air force and navy supported!) on formal languages is as much a part of logic as it is of linguistics or the theory of computation.) Of course, some of the computer's intellectual roots were more obviously useful --- but since these were the study of Brownian motion, and the physics of crystals and spectral lines, not much. (Its practical origins were military needs and vast quantities of government subsidies, which continue, but let's not disturb the myths about private enterprise any more than we must.)

I don't really know what the moral is, beyond the obvious one that useless knowledge isn't.

Things I want to understand better: Tarski's truth theory; the Russell-Whitehead "relation-arithmetic" and its descendants; model theory.

Boole, The Laws of Thought
Boolos and Jeffrey, Computability and Logic
Peter J. Cameron, Sets, Logic and Categories
Martin Gardner, Logic Machines and Logic Diagrams
Jaako Hintikka, The Principles of Mathematics Revisited ["Proposes a new basic first-order logic and uses it to explore the foundations of mathemaitcs. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity and truth in the same language. The famous impossibility results by Gödel and Tarski that have dominated the field for the past sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with all problems concerning the existence of sets and other higher-order entities." Homage to Russell's titles intended and fully appropriate.]
Warren McCulloch and Walter Pitts, "A Logical Calculus of the Ideas Immanent in Nervous Activity", in McCulloch's Embodiments of Mind
María Manzano, Model Theory
Peter Nidditch, The Development of Mathematical Logic [Very short, and, as it is written in Basic English (!), ungainly; but clear and adequate]
Willard Van Orman Quine
- Mathematical Logic [My review]
- Set Theory and Its Logic
Bertrand Russell
- Introduction to Mathematical Philosophy
- Logic and Knowledge
G. Spencer-Brown, The Laws of Form [Strictly for laughs]
John von Neumann, "The General and Logical Theory of Automata" (in Collected Works)
Alfred North Whitehead and Bertrand Russell, Principia Mathematica [I've read all of the abridgement, "To *56", plus a lot of the stuff on relation-numbers in vol. II...]
J. H. Woodger, Axiomatic Method in Biology [OK, so I haven't finished the last two chapters, but it really is good, even if it does use the theory of types]

Dirk Beyer and Andreas Noack, "CrocoPat 2.1 Introduction and Reference Manual", cs.PL/0409009 [" CrocoPat is an efficient, powerful and easy-to-use tool for manipulating relations of arbitrary arity, including directed graphs. This manual provides an introduction to and a reference for CrocoPat and its programming language RML."]
Rudolf Carnap
- Introduction to Semantics, and Formalization of Logic
- The Logical Syntax of Language
- Introduction to Symbolic Logic and Its Applications
- Meaning and Necessity
Alonzo Church, Introduction to Mathematical Logic
Thierry Coquand and Henri Lombardi, "A logical approach to abstract algebra", Mathematics Structures in Computer Science 16 (2006): 885--900
David Deutsch, Artur Ekert and Rossella Lupacchini, "Machines, Logic and Quantum Physics," math.LO/9911150
Anita Burdman Feferman and Solomon Feferman, Alfred Tarski: Life and Logic [Review in American Scientist]
Solomon Feferman, "Tarski's influence on computer science", cs.GL/0608062
Steven R. Givant, The Structure of Relation Algebras Generated by Relativizations
Robert Goldblatt, Logics of Time and Computation
Alessio Guglielmi, "A System of Order and Structure," cs.LO/9910023
Andrew Hodges, Alan Turing: The Engima
Richard W. Kaye, The Mathematics of Logic: A Guide to Completeness Theorems and their Applications [blurb]
H. Jerome Keisler and Sergio Fajardo, Model Theory of Stochastic Processes
Kleene, Introduction to Metamathematics
J. Lambek and P. J. Scott, Higher-Order Categorical Logic
Colin McLarty, Elementary Categories, Elementary Toposes
Emil Post, The Two-Valued Iterative Systems of Mathematical Logic
Hartley Rogers, Jr., Theory of Recursive Functions and Effective Computability
Erik Sandewall, Features and Fluents: The Representation of Knowledge about Dynamical Systems [Oxford Logic Guides, vol. 30]
Eric Schechter, Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions [Blurb, introductions]
Raymond Smullyan
- First-Order Logic
- Recursion Theory for Metamathematicians
- Theory of Formal Systems
Alfred Tarski
- Introduction to Logic and to the Methodology of the Deductive Sciences
- Logic, Semantics, Metamathematics
- Ordinal Algebras
- Cardinal Algebras
- "The Semantical Conception of Truth and the Foundations of Semantics"
R. F. Walters, Categories and Computer Science
William Weiss and Cherie D'Mello, Fundamentals of Model Theory [Free online]
Alexander S. Yessenin-Volpin, Christer Hennix, "Beware of the Gödel-Wette paradox!" math.LO/0110094

Notebooks

Mathematical Logic