Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Real algebra [and axioms]

A notion of real numbers as measures of space or quantity has existed since antiquity. The development of basic algebra gave a formal way to represent operations on such numbers. In the late 1800s there were efforts—notably by Richard Dedekind and Georg Cantor—to set up a general theory of real numbers relying only on basic concepts about integers—and these efforts led to set theory. For purely algebraic questions of the kind that might arise in high-school algebra, however, one can use just the axioms given here. These add to field theory several axioms for ordering, as well as the axiom at the bottom expressing a basic form of continuity (specifically that any polynomial which changes sign must have a zero). With these axioms one can prove results about real polynomials, but not about arbitrary mathematical functions, or integers. The axioms were shown to be complete by Alfred Tarski in the 1930s. The proof was based on setting up a procedure that could in principle resolve any set of real polynomial equations or inequalities. This is now in practice done by Simplify and other functions in Mathematica using methods of cylindrical algebraic decomposition invented in the 1970s—which work roughly by finding a succession of points of change using Resultant. (Note that with n variables the number of steps needed can increase like 2^2^n.) (See the note about real analysis below.)

From Stephen Wolfram: A New Kind of Science [citation]