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## BOOLE, G.(1815-1864) and DE MORGAN, A(1806-1871)

 George Boole was born in lincoln, England, in 1815.  His father was a struggling lower-class tradesman, so Boole had only a common school education, but he managed to teach himeself Greek and Latin.  Later, while working as an elementary-school teacher, he learned mathematics by reading the works of Laplace and Lagrange, studied foreign languages, and, through his friend De morgan, became interested in      formal logic. In 1847, Boole published a pamphlet entitled The Mathematical Analysis of Logic, which De Morgan praised as epoch making. In his work, Boole maintained that the essential character of mathematics lies in its form rather that in its content; mathematics is not (as some dictionaries today still assert) merely "the science of measurement and number," but, more broadly, any study consisting of symbols along with precise rules of operation upon those symbols, the rules being subject only to the requirement of inner consistency.  Two years later, Boole was appointed professor of mathematics at the newly founded Queen's College in Cork, Ireland.   In 1854, Boole expanded and clarified his earlier work of 1847 into a book entitled Inberstigation of the Laws of Thought, in which he established both formal logic and a new algebra-the algebra of sets known today as Boolean algebra.   In more recent times, Boolean algebra has found a number of applications, such as to the theory of electric switching ciruits.  In 1859 Boole published his Treatise on Differential Equations, and then, in 1860, his Treatise on the Calculus of Finite Differences.   The latter book has remained a standard work in its subject right into present times.  Boole died in Cork in 1864. Augustus De Morgan, was born(blind in one eye)in 1806 in Madras, where his father was associated with the East India Company.  He was educated at Trinity College, Cambridge, graduating as fourth wrangler, and in 1828 became a professor at the then newly established Unibersity of London (later renamed University College), where.   throgh his works and his students, he exercised a wide influence in English mathematics.  He was well read in the phliosophy and the history of mathematics, and wrote works on the foundations of algebra, differential caculus logic, and the theory of probability.  He was a highly lucid expositor. His witty and amusing book.  A budget of Paradoxes, still makes entertaining reading.        He continued Boole's work on the algebra of sets, enunciating the principle of duality of set theory, of which the so-called De Morgan laws are an illustration: If Aand B are subsets of a universal set.   then the complement of the union of Aand B is the intersection of the complements ofAand B, and the complement of the intersection of Aand B is the union of complement s of Aand B (in symbols:(A ¡ú B)' = A ¡û B' and (A ¡û B)' = A ¡ú B' , where prime denotes complement).  Like Boole, De Morgan regarded mathematics as an abstract study of symbols subjected to sets of symboicoperations.  De Morgan was an outspoken champion of scademic freedom and of relighious tolerance.  He pelformed beautifully on the flute and was always jovial company, and he was a confirmed lover of big-city life.  He had a fondness for puzzles and conundrums, and when asked either his age or his year of birth would reply.  "I was x years old in the year $x2$." He died in London in 1871.