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## MATHEMATICS

### Pure Mathematics

*A Mathematical Analysis of Logic*

*An Investigation of the Laws of Thought*

### Claude Shannon and circuit theory

George Boole’s work shows us that mathematics is a most creative and forward-looking subject, a way of seeing new structures. An unorthodox genius who was largely self-taught, he laid down foundations which set mathematics on a new path.

Boole transformed mathematics as a discipline. His 1854 book *An Investigation of the Laws of Thought *was described by the philosopher and mathematician Bertrand Russell as ‘the work in which pure mathematics was discovered’.

As his thinking developed, Boole came to regard mathematics as abstract, the manipulation of symbols to which no particular meaning is attached. Freeing algebra from arithmetic, he used algebraic symbols to represent logical statements, and established an algebra of symbols which can be added or multiplied. This powerful abstract approach caused a paradigm shift, giving modern mathematics an enormous scope and potency.

While in his teens, Boole had become fascinated by the work of Sir Isaac Newton (a fellow-native of Lincolnshire). His first love was differential calculus, which inspired and motivated many of his later discoveries.

From 1840, he published papers in the *Cambridge Mathematical Journal* on a range of topics, stressing repeatedly the importance of manipulation of symbolic operators in various areas of mathematics. His 1841 paper *An Exposition of a General Theory of Linear Transformations*,introduced a new branch of mathematics — now called Invariant Theory — which later became part of the inspiration for Einstein’s Theory of Relativity.

In 1843, Boole began applying algebraic methods to the solution of differential equations. He wrote a lengthy paper *On a General Method in Analysis, *published in 1844 in *Philosophical Transactions, *the prestigious scientific journal of the Royal Society in London.

The paper investigates differentiation and differential equations from an operator point of view and introduces Boole’s new ‘algebra of classes’. This ‘boolean algebra’ contributed to freeing mathematics from number systems, and pushed mathematics a step further towards abstraction. For this paper, Boole received the Royal Society’s Gold Medal, the first ever awarded for mathematics. It was a turning point in Boole’s career and brought his name to the attention of leading mathematicians and scientists.

In the mid-1840s, Boole made a major conceptual leap by combining algebra with logic. This prompted his first book *A Mathematical Analysis of Logic*,published in 1847*.*

He intended this book as ”. . . a first step towards understanding the thought processes of the human mind, as expressed in speech . . .” This phrase signals his greatest discovery. A gifted linguist, Boole was intrigued by the way an idea is reasoned and expressed in different languages; he discerned mathematical structures in everyday speech that hitherto had gone unnoticed.

*A Mathematical Analysis of Logic* proposed that ”. . . classes of objects and logical operations could be represented by mathematical symbols, and that algebraic operations could be used to process those classes”. With this realisation, Boole originated symbolic logic.

After being appointed Professor of Mathematics at Queen’s College Cork, Boole re-wrote and expanded his 1847 book. In 1854 he published *An Investigation of the Laws of Thought, *regarded as his *magnum opus*.

In this work, Boole demonstrates that logical propositions can be expressed as mathematical equations, and that the algebraic manipulation of symbols in those equations offers a fail-safe method of logical deduction. *The Laws of Thought *extends his exploration of logic, and introduces another ground-breaking concept, mathematical probability.

In the 20th century, Boolean algebra was taken up most effectively by the American engineer Claude Shannon, whose 1938 paper *A Symbolic Analysis of Relay and Switching Circuits *builds on Boole’s *Analysis of Logic. *

Today, Boole’s legacy is associated principally with Shannon and circuit theory, leading to the construction of modern computers. Less well-recognised but equally significant is the impact of Boole’s mathematical logic on computer programming, and how computers handle data.

In George Boole, we find a remarkable combination of high creativity with firm rigour. An independent thinker who explored the diversity of mathematics, he marched intently into the unknown while more orthodox figures chose a slower path.

George Boole’s work as a mathematician influences almost every aspect of modern life. His revolutionary advances are today fundamental aspects of computer science and electronics.