The Word Problem

Abstract

The word problem is a decision problem in group theory. It turns out to be a problem that is formally undecidable. We will explain here the necessary background in group theory and what the word problem is, and we will provide some discussion of the undecidability issue.

Logical consequences are the scarecrows of fools and the beacons of wise men.—Thomas Henry Huxley

It is undesirable to believe a proposition when there is no ground whatever for supposing it to be true.—Bertrand Russell

Logic is a large drawer, containing some useful instruments, and many more that are superfluous.—C.C. Colton

How can one talk about ‘understanding’ a proposition and ‘not understanding’ a proposition? Surely it’s not a proposition until it’s understood?—Ludwig Wittgenstein

The philosophy of reasoning, to be complete, ought to comprise the theory of bad as well as of good reasoning.—J.S. Mill

Mathematical induction often arises as the finishing step ... of an inductive research.—George Polya

Multiplication is vexation, Division is as bad; The rule of three doth puzzle me, And practice drives me mad.—A Nursery Rhyme

There is something in the vanity of logic which addles a man’s brains.—E.A. Poe

Philosophy is the science which considers truth.—Aristotle

We have avoided both controversy and general philosophy, and made our statements dogmatic in form ... — Bertrand Russell and Alfred North Whitehead