De Morgan, Augustus

Augustus De Morgan, (born June 27, 1806, Madura, India—died March 18, 1871, London, England), English mathematician and logician whose major contributions to the study of logic include the formulation of De Morgan’s laws and work leading to the development of the theory of relations and the rise of modern symbolic, or mathematical, logic.

De Morgan was educated at Trinity College, Cambridge. In 1828 he became professor of mathematics at the newly established University College in London, where, except for a period of five years (1831–36), he taught until 1866, when he helped found and became the first president of the London Mathematical Society. One of his earliest works, Elements of Arithmetic (1830), was distinguished by a simple yet thorough philosophical treatment of the ideas of number and magnitude. In 1838 he introduced and defined the term mathematical induction to describe the process that until then had been used with little clarity in mathematical proofs.

De Morgan was among the Cambridge mathematicians who recognized the purely symbolic nature of algebra, and he was aware of the possibility of algebras that differ from ordinary algebra. In his Trigonometry and Double Algebra (1849) he gave a geometric interpretation of the properties of complex numbers (numbers involving a term with a factor of the square root of minus one) that suggested the idea of quaternions. He made a useful contribution to mathematical symbolism by proposing the use of the solidus (oblique stroke) for the printing of fractions.

The laws that bear De Morgan’s name are a pair of dually related theorems that make possible the transformation of statements and formulas into alternate, and often more convenient, forms. Known verbally by William of Ockham in the 14th century, the laws were investigated thoroughly and expressed mathematically by De Morgan. The laws are: (1) the negation (or contradictory) of a disjunction is equal to the conjunction of the negation of the alternates—that is, not (p or q) equals not p and not q, or symbolically ∼(p ∨ q) ≡ ∼p·∼q; and (2) the negation of a conjunction is equal to the disjunction of the negation of the original conjuncts—that is, not (p and q) equals not p or not q, or symbolically ∼(p·q) ≡ ∼p ∨ ∼q.

Asserting that logic as it had come down from Aristotle was unnecessarily restricted in scope, De Morgan made his greatest contributions as a reformer of logic. The renaissance of logic studies, which began in the first half of the 19th century, came about almost entirely because of the writings of De Morgan and another British mathematician, George Boole. Alternate forms and generalizations of De Morgan laws exist in various branches of mathematics.