FOM: British 2000 abstract

[Harvey Friedman]

THE MATHEMATICAL MEANING OF MATHEMATICAL LOGIC
abstract
year 2000 celebration of the British Mathematical Colloquium
19 April, 2000
http://www.amsta.leeds.ac.uk/bmc/programme.html

Mathematical logic, in the sense that we know it today, arose out of
fundamental work on the foundations of mathematics roughly from the
philosopher Gottlob Frege in the years 1879-1903  through the work of the
mathematician Kurt Godel in the years 1929-1940. There was much other
influential work before 1940. Through this work,
the then urgent issues in the foundations of mathematics of general
interest to the mathematical community came to be treated in a thoroughly
mathematical way, with unexpectedly powerful results.

The subsequent development of what is called mathematical logic has
primarily focused on intense investigations into the mathematical concepts
and structures that originated in this early period. These include
propositional and predicate calculus, interpretability, consistency and
relative consistency, first order arithmetic, ordinal notations, first
order algebra, recursive functions, intuitionistic formal systems, first
order set theory, constructible sets, and large cardinals. This subsequent
development formed the basis for additional advances in the foundations of
mathematics, which again spawned intense investigations into further
mathematical concepts and structures. Among these are forcing,
computational complexity, and reverse mathematics (1963,1965,1974).

These investigations into the mathematical concepts arising out of
advances in the foundations of mathematics constitute  the main bulk of the
subject of mathematical logic as we know it today, with its principal
subdivisions of model theory, proof theory, recursion theory, and set
theory.

The general mathematical community does not perceive any currently urgent
issues in the foundations of mathematics, and the development of
mathematical logic proceeds without being driven by the core mathematical
traditions of arithmetic, algebra, and geometry. [There is a tradition of
forging certain topics in mathematical logic into unexpected tools in
certain core mathematical contexts (including Macintyre, Wilkie, Zilber and
others  in the UK). Only more time will reveal if parts of mathematical
logic become essential tools for core mathematicians, or if these successes
are episodic and/or the methods are conveniently eliminable.]

This largely autonomous development of the bulk of mathematical logic -
being allied with the great foundational tradition instead of the great and
much older arithmetic/algebraic/geometric tradition - cannot be expected to
capture the imagination of the current general mathematical community
without special explications of the basic concepts and structures that
drive this development.

In this talk, we seek to give presentations of several principal
structures and concepts of mathematical logic in clear and attractive
terms specifically designed for the general mathematical community.