Notebooks
http://bactra.org/notebooks
Cosma's NotebooksenMathematical Logic
http://bactra.org/notebooks/2004/09/08#mathematical-logic
<P>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 <a href="whitehead.html">Whitehead</a>) 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 <a
href="bertrand-russell.html">Russell</a>'s <cite>Principia
Mathematica</cite> --- it was, so to speak, the academic <cite>Brief History of
Time</cite> of its day, often mentioned, never used.
<P>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 <a href="kolakowski.html">Kolakowski</a> says,
Stalin was the century's most influential philosopher); many of our finest
mathematicians, such as <a href="wiener.html">Norbert Wiener</a>, <a
href="von-neumann.html">John von Neumann</a> 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 <a href="neural-nets.html">neural nets</a> upon the
world than for using Carnap's formalism. But in one extremely important field,
however, it reigns supreme, and that is computation. <a
href="programming.html">Programming</a> 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, <a href="chomsky.html">Chomsky</a> 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
<a href="military-industrial.html">military needs and vast quantities of
government subsidies</a>, which continue, but let's not disturb the myths about
private enterprise any more than we must.)
<P>I don't really know what the moral is, beyond the obvious one that useless
knowledge isn't.
<P><em>Things I want to understand better:</em> Tarski's truth theory; the
Russell-Whitehead "relation-arithmetic" and its descendants; model theory.
<P><em>See also:</em>
<a href="godels-theorem.html">Gödel's Theorem</a>;
<a href="computation.html">Computation</a>;
<a href="logical-positivism.html">Logical Positivism</a>;
<a href="math.html">Math I Ought to Learn</a>
<ul>Recommended:
<li>Boolos and Jeffrey, <cite>Computability and Logic</cite>
<li>Peter J. Cameron, <cite>Sets, Logic and Categories</cite>
<li>Martin Gardner, <cite>Logic Machines and Logic Diagrams</cite>
<li>Jaako Hintikka, <cite>The Principles of Mathematics
Revisited</cite> ["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.]
<li>Warren McCulloch and Walter Pitts, "A Logical Calculus of the
Ideas Immanent in Nervous Activity", in McCulloch's <cite>Embodiments of
Mind</cite>
<li>María Manzano, <cite>Model Theory</cite>
<li>Peter Nidditch, <cite>The Development of Mathematical Logic</cite>
[Very short, and, as it is written in Basic English (!), ungainly; but clear
and adequate]
<li>Willard Van Orman <a href="quine.html">Quine</a>
<ul>
<li><cite>Mathematical Logic</cite> [<a
href="../reviews/mathematical-logic/">My review</a>]
<li><cite>Set Theory and Its Logic</cite>
</ul>
<li><a href="bertrand-russell.html">Bertrand Russell</a>
<ul>
<li><cite>Introduction to Mathematical Philosophy</cite>
<li><cite>Logic and Knowledge</cite>
</ul>
<li>G. Spencer-Brown, <cite>The Laws of Form</cite> [Strictly for
laughs]
<li><a href="von-neumann.html">John von Neumann</a>, "The General and
Logical Theory of Automata" (in <cite>Collected Works</cite>)
<li>Alfred North <a href="whitehead.html">Whitehead</a> and Bertrand
Russell, <cite>Principia Mathematica</cite> [I've read the abridgment, "To
*56", plus some of the stuff on relation-numbers in vol. II...]
<li>J. H. Woodger, <cite>Axiomatic Method in Biology</cite> [OK, so I
haven't finished the last two chapters, but it really is good, even if it does
use the theory of types]
</ul>
<ul>To read:
<li>Steve Awodey, <cite>Category Theory</cite>
<li>Boole, <cite>The Laws of Thought</cite>
<li>Rudolf Carnap
<ul>
<li><cite>Introduction to Semantics, and Formalization of
Logic</cite>
<li><cite>The Logical Syntax of Language</cite>
<li><cite>Introduction to Symbolic Logic and Its
Applications</cite>
<li><cite>Meaning and Necessity</cite>
</ul>
<li>Alonzo Church, <cite>Introduction to Mathematical Logic</cite>
<li>Thierry Coquand and Henri Lombardi, "A logical approach to abstract
algebra", <a
href="http://dx.doi.org/10.1017/S0960129506005627"><cite>Mathematics Structures
in Computer Science</cite>
<strong>16</strong> (2006): 885--900</a>
<li>David Deutsch, Artur Ekert and Rossella Lupacchini, "Machines,
Logic and Quantum Physics," <a
href="http://arxiv.org/abs/math.LO/9911150">math.LO/9911150</a>
<li>Anita Burdman Feferman and Solomon Feferman, <cite>Alfred Tarski:
Life and Logic</cite> [<a
href="http://www.americanscientist.org/template/BookReviewTypeDetail/assetid/40743">Review
in <cite>American Scientist</cite></a>]
<li>Solomon Feferman, "Tarski's influence on computer science",
<a href="http://arxiv.org/abs/cs.GL/0608062">cs.GL/0608062</a>
<li>Steven R. Givant, <cite>The Structure of Relation Algebras
Generated by Relativizations</cite>
<li>Robert Goldblatt, <cite>Logics of Time and Computation</cite>
<li>Alessio Guglielmi, "A System of Order and Structure," <a
href="http://arxiv.org/abs/cs.LO/9910023">cs.LO/9910023</a>
<li>Robert A. Herrmann, "Logic for Everyone", <a href="http://arxiv.org/abs/math/0601709">arxiv:math/0601709</a>
<li>Andrew Hodges, <cite>Alan Turing: The Engima</cite>
<li>Richard W. Kaye, <cite><a href="http://cambridge.org/9780521708777">The Mathematics of Logic: A Guide to
Completeness Theorems and their Applications</a></cite>
<li>H. Jerome Keisler and Sergio Fajardo, <cite>Model Theory of
Stochastic Processes</cite>
<li>Kleene, <Cite>Introduction to Metamathematics</cite>
<li>J. Lambek and P. J. Scott, <cite>Higher-Order Categorical
Logic</cite>
<li>Colin McLarty, <cite>Elementary Categories, Elementary
Toposes</cite>
<li>Emil Post, <cite>The Two-Valued Iterative Systems of Mathematical
Logic</cite>
<li>Hartley Rogers, Jr., <cite>Theory of Recursive Functions and
Effective Computability</cite>
<li>Erik Sandewall, <cite>Features and Fluents: The Representation
of Knowledge about Dynamical Systems</citE> [Oxford Logic Guides, vol. 30]
<li>Eric Schechter, <cite>C<a href="http://pup.princeton.edu/titles/8119.html">lassical and Nonclassical Logics: An
Introduction to the Mathematics of Propositions</a></cite>
<li>Gunther Schmidt, <cite><a href="http://www.cambridge.org/9780521762687">Relational Mathematics</a></cite>
<li>Raymond Smullyan
<ul>
<li><cite>First-Order Logic</cite>
<li><cite>Recursion Theory for Metamathematicians</cite>
<li><cite>Theory of Formal Systems</cite>
</ul>
<li>Antonin Spacek, "Statistical Estimation of Semantic Provability",
<a href="http://projecteuclid.org/euclid.bsmsp/1200512187">pp. 655--668 in
vol. I of <cite>Proceedings of the Fourth Berkeley Symposium on Mathematical
Statistics and Probability</cite></a>
<li>Keith Stenning and Michiel van Lambalgen, <cite><a href="http://mitpress.mit.edu/9780262195836"Human
Reasoning and Cognitive Science</a></cite>
<li>Alfred Tarski
<ul>
<li><cite>Introduction to Logic and to the Methodology of the
Deductive Sciences</cite>
<li><cite>Logic, Semantics, Metamathematics</cite>
<li><cite>Ordinal Algebras</cite>
<li><cite>Cardinal Algebras</cite>
<li>"<a href="http://www.ditext.com/tarski/tarski.html">The
Semantical Conception of Truth and the Foundations of Semantics</a>"
</ul>
<li>Jouko Vaananen, <cite><a href="http://cambridge.org/0521876591">Dependence Logic: A New Approach to Independence Friendly Logic</a></cite>
<li>R. F. Walters, <cite>Categories and Computer Science</cite>
<li>William Weiss and Cherie D'Mello, <cite>Fundamentals of Model
Theory</cite> [<a href="http://at.yorku.ca/i/a/a/i/10.htm">Free online</a>]
<li>Alexander S. Yessenin-Volpin, Christer Hennix, "Beware of the
Gödel-Wette paradox!" <a
href="http://arxiv.org/abs/math.LO/0110094">math.LO/0110094</a>
</ul>