George Boole
Encyclopedia
George Boole was an English
mathematician
and philosopher.
As the inventor of Boolean logic
—the basis of modern digital computer
logic—Boole is regarded in hindsight as a founder of the field of computer science
. Boole said,
, the father gave his son his first lessons; but the extraordinary mathematical talents of George Boole did not manifest themselves in early life. At first, his favorite subject was classics
. By his teens, he had learned Latin, Greek, German, Italian, and French.
With these languages, he was able to read a wide variety of Christian theology. Combining his interests in mathematics and theology, he compared the Christian trinity of Father, Son, and Holy Ghost with the three dimensions of space, and was attracted to the Hebrew conception of God as an absolute unity. Boole considered converting to Judaism
but in the end chose Unitarianism
.
It was not until his successful establishment of a school at Lincoln
, its removal to Waddington, and later his appointment in 1849 as the first professor of mathematics of then Queen's College, Cork in Ireland
(now University College Cork
, where the library, underground lecture theatre complex and the Boole Centre for Research in Informatics are named in his honour) that his mathematical skills were fully realized. In 1855 he married Mary Everest
(niece of George Everest
), who later, as Mrs. Boole, wrote several useful educational works on her husband's principles.
Though Boole published little except his mathematical and logical works, his acquaintance with general literature was wide and deep. Dante
was his favourite poet, and he preferred the Paradiso to the Inferno. The metaphysics
of Aristotle
, the ethics
of Spinoza
, the philosophical works of Cicero
, and many kindred works, were also frequent subjects of study. His reflections upon scientific, philosophical and religious questions are contained in four addresses upon The Genius of Sir Isaac Newton
, The Right Use of Leisure, The Claims of Science and The Social Aspect of Intellectual Culture, which he delivered and printed at different times.
The personal character of Boole inspired all his friends with the deepest esteem. He was marked by true modesty, and his life was given to the single-minded pursuit of truth
. Though he received a medal from the Royal Society for his memoir of 1844, and the honorary degree
of LL.D. from the University of Dublin
, he neither sought nor received the ordinary rewards to which his discoveries would entitle him. On 8 December 1864, in the full vigour of his intellectual powers, he died of an attack of fever, ending in pleural effusion
, an accumulation of fluid around the lungs. He is buried in the Church of Ireland cemetery of St Michael's, Church Road, Blackrock (a suburb of Cork City). There is a commemorative plaque inside the adjoining church.
, the editor of the journal, which lasted until the premature death of the latter in 1844. A long list of Boole's memoirs and detached papers, both on logical and mathematical topics, are found in the Catalogue of Scientific Memoirs published by the Royal Society
, and in the supplementary volume on Differential Equations, edited by Isaac Todhunter
. To the Cambridge Mathematical Journal and its successor, the Cambridge
and Dublin Mathematical Journal, Boole contributed twenty-two articles in all. In the third and fourth series of the Philosophical Magazine are found sixteen papers. The Royal Society printed six important memoirs in the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh
and of the Royal Irish Academy
, in the Bulletin de l'Académie de St-Pétersbourg for 1862 (under the name G Boldt, vol. iv. pp. 198–215), and in Crelle's Journal
. Also included is a paper on the mathematical basis of logic, published in the Mechanic's Magazine in 1848. The works of Boole are thus contained in about fifty scattered articles and a few separate publications.In 1848 Boole published The Mathematical Analysis of Logic the first of his contribution to symbolic logic.
Only two systematic treatise
s on mathematical subjects were completed by Boole during his lifetime. The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus
of Finite Differences, designed to serve as a sequel to the former work. These treatises are valuable contributions to the important branches of mathematics in question. To a certain extent these works embody the more important discoveries of their author. In the sixteenth and seventeenth chapters of the Differential Equations we find, for instance, an account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation. His principal characteristic was perfect confidence in any result obtained by the treatment of symbols in accordance with their primary laws and conditions, and an almost unrivaled skill and power in tracing out these results.
During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential Equations much more complete than the first edition, and part of his last vacation was spent in the libraries of the Royal Society and the British Museum
; but this new edition was never completed. Even the manuscripts left at his death were so incomplete that Todhunter, into whose hands they were put, found it impossible to use them in the publication of a second edition of the original treatise, and printed them, in 1865, in a supplementary volume.
With the exception of Augustus De Morgan
, Boole was probably the first English mathematician since the time of John Wallis who had also written upon logic
. His novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. Speculations concerning a calculus
of reasoning had at different times occupied Boole's thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterward regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought (1854), on Which are Founded the Mathematical Theories of Logic and Probabilities
, should alone be considered as containing a mature statement of his views. This ushered in a new focus on the nature of evidence, argument, and proof. Nevertheless, there is a charm of originality about his earlier logical work that is easy to appreciate.
He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy
between the symbols of algebra
and those that can be made, in his opinion, to represent logical form
s and syllogism
s, that we can hardly help saying that (especially his) formal logic is mathematics restricted to the two quantities, 0 and 1. By unity Boole denoted the universe of thinkable objects; literal
symbols, such as x, y, z, v, u, etc., were used with the elective meaning attaching to common adjectives and substantives. Thus, if x = horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination
as algebraic symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. Thus, (1 – x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 – x) (1 – y) would give us all things neither horned nor sheep. By the use of such symbols proposition
s could be reduced to the form of equation
s, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.
Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, formed a general symbolic method of logical inference
. Given any propositions involving any number of terms, Boole showed how, by the purely symbolic treatment of the premises, to draw any conclusion logically contained in those premises. The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events.
In 1921 the economist John Maynard Keynes
published a book that has become a classic on probability theory, "A Treatise of Probability." Keynes's comments about Boole's theory of probability were generally taken to be the definitive statement on the subject. Keynes believed that Boole had made a fundamental error that vitiated much of his analysis. In a recent book, "The Last Challenge Problem," David Miller provides a general method in accord with Boole's system, and attempts to solve the problems recognized earlier by Keynes and others.
Boole proposed that logical propositions should be expressed as algebraic equations. The algebraic manipulation of the symbols in the equations provides a fail-safe method of logical deduction, i.e. logic is reduced to algebra. Boole replaced the operation of multiplication by the word 'and' and addition by the word 'or'. The symbols in the equations can stand for collections of objects (sets) or statements in logic. For example, if x is the set of all brown cows and y is the set of all fat cows, then x+y is the set of all cows that are brown or fat, and xy is the set of all cows that are brown and fat.
Let z = the set of all Irish cows. Then z(x+y) = zx+zy; in other words 'the set of Irish cows that are either brown or fat is the same as the collection of cows that are Irish and brown or Irish and fat'.
the niece of George Everest
: a universal mysticism tempered by Judaic thought, and by Indian logic. In addition to Mary Boole, the intellectual accomplishments of Boole (and to a lesser extent de Morgan
and Babbage) have also been claimed as being influenced by Indian thought, in particular Indian logic
.
In an open letter in defence of Indian thought, Mary Boole stated that an adolescent mystical experience provided for his life's work:
Despite the acceptance of the Laws, Boole was apparently disconcerted at the book's reception as only a mathematical toolset:
Mary Boole claimed profound influence (via her uncle George Everest) of Indian thought on Boole (as well as de Morgan
and Babbage):
is named after him.
Boole's work was extended and refined by William Stanley Jevons
, Augustus De Morgan
, Charles Sanders Peirce, and William Ernest Johnson
. This work was summarized by Ernst Schröder
, Louis Couturat
, and Clarence Irving Lewis
.
Boole's work (as well as that of his intellectual progeny) was relatively obscure, except among logicians. At the time, it appeared to have no practical uses. However, approximately seventy years after Boole's death, Claude Shannon attended a philosophy class at the University of Michigan
that introduced him to Boole's studies. Shannon recognised that Boole's work could form the basis of mechanisms and processes in the real world and that it was therefore highly relevant. In 1937 Shannon went on to write a master's thesis at the Massachusetts Institute of Technology
, in which he showed how Boolean algebra could optimize the design of systems of electromechanical relays, then used in telephone routing switches. He also proved that circuits with relays could solve Boolean algebra problems. Employing the properties of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers. Victor Shestakov
at Moscow State University (1907–1987) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians S.A. Yanovskaya, Gaaze-Rapoport, Dobrushin
, Lupanov, Medvedev, and Uspensky, though they presented their academic theses in the same year, 1938 . But the first publication of Shestakov's result took place only in 1941 (in Russian). Hence Boolean algebra became the foundation of practical digital circuit
design; and Boole, via Shannon and Shestakov, provided the theoretical grounding for the Digital Age.
The crater Boole
on the Moon is named in his honour.
The keyword Bool represents a Boolean datatype
in many programming languages, though Pascal
uses the full name Boolean.
English people
The English are a nation and ethnic group native to England, who speak English. The English identity is of early mediaeval origin, when they were known in Old English as the Anglecynn. England is now a country of the United Kingdom, and the majority of English people in England are British Citizens...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and philosopher.
As the inventor of Boolean logic
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
—the basis of modern digital computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
logic—Boole is regarded in hindsight as a founder of the field of computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. Boole said,
... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning ...
Biography
George Boole's father, John Boole (1779–1848), was a tradesman of limited means, but of "studious character and active mind". Being especially interested in mathematical science and logicLogic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the father gave his son his first lessons; but the extraordinary mathematical talents of George Boole did not manifest themselves in early life. At first, his favorite subject was classics
Classics
Classics is the branch of the Humanities comprising the languages, literature, philosophy, history, art, archaeology and other culture of the ancient Mediterranean world ; especially Ancient Greece and Ancient Rome during Classical Antiquity Classics (sometimes encompassing Classical Studies or...
. By his teens, he had learned Latin, Greek, German, Italian, and French.
With these languages, he was able to read a wide variety of Christian theology. Combining his interests in mathematics and theology, he compared the Christian trinity of Father, Son, and Holy Ghost with the three dimensions of space, and was attracted to the Hebrew conception of God as an absolute unity. Boole considered converting to Judaism
Judaism
Judaism ) is the "religion, philosophy, and way of life" of the Jewish people...
but in the end chose Unitarianism
Unitarianism
Unitarianism is a Christian theological movement, named for its understanding of God as one person, in direct contrast to Trinitarianism which defines God as three persons coexisting consubstantially as one in being....
.
It was not until his successful establishment of a school at Lincoln
Lincoln, Lincolnshire
Lincoln is a cathedral city and county town of Lincolnshire, England.The non-metropolitan district of Lincoln has a population of 85,595; the 2001 census gave the entire area of Lincoln a population of 120,779....
, its removal to Waddington, and later his appointment in 1849 as the first professor of mathematics of then Queen's College, Cork in Ireland
Ireland
Ireland is an island to the northwest of continental Europe. It is the third-largest island in Europe and the twentieth-largest island on Earth...
(now University College Cork
University College Cork - National University of Ireland, Cork
University College Cork is a constituent university of the National University of Ireland. The university is located in Cork....
, where the library, underground lecture theatre complex and the Boole Centre for Research in Informatics are named in his honour) that his mathematical skills were fully realized. In 1855 he married Mary Everest
Mary Everest Boole
Mary Everest Boole was a self-taught mathematician who is best known as an author of didactic works on mathematics, such as Philosophy and Fun of Algebra, and as the wife of fellow mathematician George Boole...
(niece of George Everest
George Everest
Colonel Sir George Everest was a Welsh surveyor, geographer and Surveyor-General of India from 1830 to 1843.Sir George was largely responsible for completing the section of the Great Trigonometric Survey of India along the meridian arc from the south of India extending north to Nepal, a distance...
), who later, as Mrs. Boole, wrote several useful educational works on her husband's principles.
Though Boole published little except his mathematical and logical works, his acquaintance with general literature was wide and deep. Dante
DANTE
Delivery of Advanced Network Technology to Europe is a not-for-profit organisation that plans, builds and operates the international networks that interconnect the various national research and education networks in Europe and surrounding regions...
was his favourite poet, and he preferred the Paradiso to the Inferno. The metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
of Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, the ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
of Spinoza
Baruch Spinoza
Baruch de Spinoza and later Benedict de Spinoza was a Dutch Jewish philosopher. Revealing considerable scientific aptitude, the breadth and importance of Spinoza's work was not fully realized until years after his death...
, the philosophical works of Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...
, and many kindred works, were also frequent subjects of study. His reflections upon scientific, philosophical and religious questions are contained in four addresses upon The Genius of Sir Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, The Right Use of Leisure, The Claims of Science and The Social Aspect of Intellectual Culture, which he delivered and printed at different times.
The personal character of Boole inspired all his friends with the deepest esteem. He was marked by true modesty, and his life was given to the single-minded pursuit of truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
. Though he received a medal from the Royal Society for his memoir of 1844, and the honorary degree
Honorary degree
An honorary degree or a degree honoris causa is an academic degree for which a university has waived the usual requirements, such as matriculation, residence, study, and the passing of examinations...
of LL.D. from the University of Dublin
University of Dublin
The University of Dublin , corporately designated the Chancellor, Doctors and Masters of the University of Dublin , located in Dublin, Ireland, was effectively founded when in 1592 Queen Elizabeth I issued a charter for Trinity College, Dublin, as "the mother of a university" – this date making it...
, he neither sought nor received the ordinary rewards to which his discoveries would entitle him. On 8 December 1864, in the full vigour of his intellectual powers, he died of an attack of fever, ending in pleural effusion
Pleural effusion
Pleural effusion is excess fluid that accumulates between the two pleural layers, the fluid-filled space that surrounds the lungs. Excessive amounts of such fluid can impair breathing by limiting the expansion of the lungs during ventilation.-Pathophysiology:...
, an accumulation of fluid around the lungs. He is buried in the Church of Ireland cemetery of St Michael's, Church Road, Blackrock (a suburb of Cork City). There is a commemorative plaque inside the adjoining church.
Work
To the broader public Boole was known only as the author of numerous abstruse papers on mathematical topics, and of three or four distinct publications that have become standard works. His earliest published paper was the "Researches in the theory of analytical transformations, with a special application to the reduction of the general equation of the second order." printed in the Cambridge Mathematical Journal in February 1840 (Volume 2, no. 8, pp. 64–73), and it led to a friendship between Boole and D.F. GregoryDuncan Farquharson Gregory
Duncan Farquharson Gregory , a Scottish mathematician, was the youngest son of James Gregory and Isabella Macleod .-Education:...
, the editor of the journal, which lasted until the premature death of the latter in 1844. A long list of Boole's memoirs and detached papers, both on logical and mathematical topics, are found in the Catalogue of Scientific Memoirs published by the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
, and in the supplementary volume on Differential Equations, edited by Isaac Todhunter
Isaac Todhunter
Isaac Todhunter FRS , was an English mathematician who is best known today for the books he wrote on mathematics and its history.- Life and work :...
. To the Cambridge Mathematical Journal and its successor, the Cambridge
Cambridge
The city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...
and Dublin Mathematical Journal, Boole contributed twenty-two articles in all. In the third and fourth series of the Philosophical Magazine are found sixteen papers. The Royal Society printed six important memoirs in the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh
Royal Society of Edinburgh
The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity, operating on a wholly independent and non-party-political basis and providing public benefit throughout Scotland...
and of the Royal Irish Academy
Royal Irish Academy
The Royal Irish Academy , based in Dublin, is an all-Ireland, independent, academic body that promotes study and excellence in the sciences, humanities and social sciences. It is one of Ireland's premier learned societies and cultural institutions and currently has around 420 Members, elected in...
, in the Bulletin de l'Académie de St-Pétersbourg for 1862 (under the name G Boldt, vol. iv. pp. 198–215), and in Crelle's Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
. Also included is a paper on the mathematical basis of logic, published in the Mechanic's Magazine in 1848. The works of Boole are thus contained in about fifty scattered articles and a few separate publications.In 1848 Boole published The Mathematical Analysis of Logic the first of his contribution to symbolic logic.
Only two systematic treatise
Treatise
A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject.-Noteworthy treatises:...
s on mathematical subjects were completed by Boole during his lifetime. The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
of Finite Differences, designed to serve as a sequel to the former work. These treatises are valuable contributions to the important branches of mathematics in question. To a certain extent these works embody the more important discoveries of their author. In the sixteenth and seventeenth chapters of the Differential Equations we find, for instance, an account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation. His principal characteristic was perfect confidence in any result obtained by the treatment of symbols in accordance with their primary laws and conditions, and an almost unrivaled skill and power in tracing out these results.
During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential Equations much more complete than the first edition, and part of his last vacation was spent in the libraries of the Royal Society and the British Museum
British Museum
The British Museum is a museum of human history and culture in London. Its collections, which number more than seven million objects, are amongst the largest and most comprehensive in the world and originate from all continents, illustrating and documenting the story of human culture from its...
; but this new edition was never completed. Even the manuscripts left at his death were so incomplete that Todhunter, into whose hands they were put, found it impossible to use them in the publication of a second edition of the original treatise, and printed them, in 1865, in a supplementary volume.
With the exception of Augustus De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
, Boole was probably the first English mathematician since the time of John Wallis who had also written upon logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
. His novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. Speculations concerning a calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
of reasoning had at different times occupied Boole's thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterward regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought (1854), on Which are Founded the Mathematical Theories of Logic and Probabilities
The Laws of Thought
The Laws of Thought, more precisely, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, is a very influential 19th century book on logic by George Boole, the second of his two monographs on algebraic logic...
, should alone be considered as containing a mature statement of his views. This ushered in a new focus on the nature of evidence, argument, and proof. Nevertheless, there is a charm of originality about his earlier logical work that is easy to appreciate.
He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
between the symbols of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
and those that can be made, in his opinion, to represent logical form
Logical form
In logic, the logical form of a sentence or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form...
s and syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
s, that we can hardly help saying that (especially his) formal logic is mathematics restricted to the two quantities, 0 and 1. By unity Boole denoted the universe of thinkable objects; literal
Literal
Literal may refer to:*Literal and figurative language, taken in a non-figurative sense*Literal translation, the close adherence to the forms of a source language text...
symbols, such as x, y, z, v, u, etc., were used with the elective meaning attaching to common adjectives and substantives. Thus, if x = horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination
Combination
In mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations...
as algebraic symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. Thus, (1 – x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 – x) (1 – y) would give us all things neither horned nor sheep. By the use of such symbols proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s could be reduced to the form of equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.
Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, formed a general symbolic method of logical inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
. Given any propositions involving any number of terms, Boole showed how, by the purely symbolic treatment of the premises, to draw any conclusion logically contained in those premises. The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events.
In 1921 the economist John Maynard Keynes
John Maynard Keynes
John Maynard Keynes, Baron Keynes of Tilton, CB FBA , was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments...
published a book that has become a classic on probability theory, "A Treatise of Probability." Keynes's comments about Boole's theory of probability were generally taken to be the definitive statement on the subject. Keynes believed that Boole had made a fundamental error that vitiated much of his analysis. In a recent book, "The Last Challenge Problem," David Miller provides a general method in accord with Boole's system, and attempts to solve the problems recognized earlier by Keynes and others.
Boole proposed that logical propositions should be expressed as algebraic equations. The algebraic manipulation of the symbols in the equations provides a fail-safe method of logical deduction, i.e. logic is reduced to algebra. Boole replaced the operation of multiplication by the word 'and' and addition by the word 'or'. The symbols in the equations can stand for collections of objects (sets) or statements in logic. For example, if x is the set of all brown cows and y is the set of all fat cows, then x+y is the set of all cows that are brown or fat, and xy is the set of all cows that are brown and fat.
Let z = the set of all Irish cows. Then z(x+y) = zx+zy; in other words 'the set of Irish cows that are either brown or fat is the same as the collection of cows that are Irish and brown or Irish and fat'.
Alleged philosophical background Laws of Thought
Two influences on Boole have been claimed by his wife, Mary Everest BooleMary Everest Boole
Mary Everest Boole was a self-taught mathematician who is best known as an author of didactic works on mathematics, such as Philosophy and Fun of Algebra, and as the wife of fellow mathematician George Boole...
the niece of George Everest
George Everest
Colonel Sir George Everest was a Welsh surveyor, geographer and Surveyor-General of India from 1830 to 1843.Sir George was largely responsible for completing the section of the Great Trigonometric Survey of India along the meridian arc from the south of India extending north to Nepal, a distance...
: a universal mysticism tempered by Judaic thought, and by Indian logic. In addition to Mary Boole, the intellectual accomplishments of Boole (and to a lesser extent de Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
and Babbage) have also been claimed as being influenced by Indian thought, in particular Indian logic
Indian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...
.
In an open letter in defence of Indian thought, Mary Boole stated that an adolescent mystical experience provided for his life's work:
My husband told me that when he was a lad of seventeen a thought struck him suddenly, which became the foundation of all his future discoveries. It was a flash of psychological insight into the conditions under which a mind most readily accumulates knowledge [...] From the first he connected his scrap of psychologic knowledge with sacred literature. For a few years he supposed himself to be convinced of the truth of "the Bible" as a whole, and even intended to take orders as a clergyman of the English Church. But by the help of a learned Jew in Lincoln he found out the true nature of the discovery which had dawned on him. This was that man's mind works by means of some mechanism which "functions normally towards Monism." Besides the information which reaches it from the external world, it receives knowledge direct from The Unseen every time it returns to the thought of Unity between any given elements (of fact or thought), after a period of tension on the contrast or antagonism between those same elements.[...] At this point all possibility of becoming a priest came to an end. George set to work to write a book (The Laws of Thought), in order to give to the world his great discovery. If he had stated it in words, he would have been entangled in an unseemly theological skirmish. He presented the truth to the learned, clothed in a veil so transparent that it is difficult to conceive how any human being could have been blinded by it.
Despite the acceptance of the Laws, Boole was apparently disconcerted at the book's reception as only a mathematical toolset:
George Boole said to me that neither Aristotle's Logic nor the Creed of Moses could have been enunciated unless the formula to which the Universities had now given the name of "Boole's Equation" had been, in some form or other, perfectly well known. George afterwards learned, to his great joy, that the same conception of the basis of Logic was held by Leibnitz, the contemporary of Newton. De Morgan, of course, understood the formula in its true sense; he was Boole's collaborator all along. Herbert Spencer, Jowett, and Leslie Ellis understood, I feel sure; and a few others, but nearly all the logicians and mathematicians ignored [953] the statement that the book was meant to throw light on the nature of the human mind; and treated the formula entirely as a wonderful new method of reducing to logical order masses of evidence about external fact.
Mary Boole claimed profound influence (via her uncle George Everest) of Indian thought on Boole (as well as de Morgan
De Morgan
De Morgan is a surname, and may refer to:*Augustus De Morgan, mathematician and logician.** De Morgan's laws , a set of rules from propositional logic....
and Babbage):
Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830-1865. What share had it in generating the Vector Analysis and the mathematics by which investigations in physical science are now conducted?
Family
The Booles had five daughters:- Mary Ellen, (1856–1908) who married the mathematician and author Charles Howard HintonCharles Howard HintonCharles Howard Hinton was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualising the geometry of...
and had four children: George (1882–1943), Eric (*1884), William (1886–1909) and Sebastian (1887–1923) inventor of the Jungle gymJungle gymThe jungle gym, monkey bars, or climbing frame, is a piece of playground equipment made of many pieces of material, such as metal pipe or rope, on which children can climb, hang, or sit. The monkey bar designation refers to the rambunctious, climbing play of monkeys.-History:The first jungle gym...
. Sebastian had three children:- William H. HintonWilliam H. HintonWilliam Howard Hinton was an American farmer and prolific writer. A Marxist, he is best known for his book Fanshen, published in 1966, a "documentary of revolution" which chronicled the land reform conducted by the Chinese Communist Party in the 1940s in Zhangzhuangcun , sometimes translated as...
visited China in the 1930s and 40s, and wrote an influential account of the communist land reform. - Joan HintonJoan HintonJoan Hinton was a nuclear physicist and one of the few women who worked for the Manhattan Project in Los Alamos. She lived in the People's Republic of China after 1949, where she and her husband Erwin Engst participated in China’s efforts at developing a socialist economy, working extensively in...
(1921–2010) worked for the Manhattan ProjectManhattan ProjectThe Manhattan Project was a research and development program, led by the United States with participation from the United Kingdom and Canada, that produced the first atomic bomb during World War II. From 1942 to 1946, the project was under the direction of Major General Leslie Groves of the US Army...
and lived in China from 1948 until her death on 8 June 2010; she was married to Sid Engst. - Jean Hinton (married name Rosner) (1917–2002) peace activist.
- William H. Hinton
- Margaret, (1858 – ?) married Edward Ingram Taylor an artist.
- Their elder son Geoffrey Ingram TaylorGeoffrey Ingram TaylorSir Geoffrey Ingram Taylor OM was a British physicist, mathematician and expert on fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as "one of the most notable scientists of this century".-Biography:Taylor was born in St. John's Wood, London...
became a mathematician and a Fellow of the Royal SocietyRoyal SocietyThe Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
. - Their younger son JulianJulian Taylor (surgeon)Professor Julian Taylor, C.B.E., M.S., F.R.C.S., Hon.F.R.A.C.S. was a specialist in neurological surgery, Senior Surgeon at University College Hospital, a former Vice-President of the Royal College of Surgeons and later Professor of Surgery at the University of Khartoum.Born in St...
was a Professor of Surgery.
- Their elder son Geoffrey Ingram Taylor
- AliciaAlicia Boole StottAlicia Boole Stott was the third daughter of George Boole and Mary Everest Boole, born in Cork, Ireland. Before marrying Walter Stott, an actuary, in 1890, she was known as Alicia Boole...
(1860–1940), who made important contributions to four-dimensional geometry - Lucy Everest (1862–1905), who was first female Professor of Chemistry in England
- Ethel LilianEthel Lilian VoynichEthel Lilian Voynich, née Boole was a British novelist and musician, and a supporter of several revolutionary causes. She was born in Cork. Her father was the mathematician George Boole. Her mother was feminist philosopher Mary Everest, niece of George Everest and an author for the...
(1864–1960), who married the Polish scientist and revolutionary Wilfrid Michael VoynichWilfrid Michael VoynichWilfrid Michael Voynich , born Michał Habdank-Wojnicz, was a Polish revolutionary, British and American antiquarian and bibliophile, and the eponym of the Voynich manuscript.- Biography :...
and was the author of the novel The GadflyThe GadflyThe Gadfly is a novel by Ethel Lilian Voynich, published in 1897 , set in 1840s Italy under the dominance of Austria, a time of tumultuous revolt and uprisings. The story centers on the life of the protagonist, Arthur Burton, as a member of the Youth movement, and his antagonist, Padre Montanelli...
.
Legacy
Boolean algebraBoolean algebra
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets...
is named after him.
Boole's work was extended and refined by William Stanley Jevons
William Stanley Jevons
William Stanley Jevons was a British economist and logician.Irving Fisher described his book The Theory of Political Economy as beginning the mathematical method in economics. It made the case that economics as a science concerned with quantities is necessarily mathematical...
, Augustus De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
, Charles Sanders Peirce, and William Ernest Johnson
William Ernest Johnson
William Ernest Johnson was a British logician mainly remembered for his Logic , in 3 volumes....
. This work was summarized by Ernst Schröder
Ernst Schröder
Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
, Louis Couturat
Louis Couturat
Louis Couturat was a French logician, mathematician, philosopher, and linguist.-Life:Born in Ris-Orangis, Essonne, France, he was educated in philosophy and mathematics at the École Normale Supérieure...
, and Clarence Irving Lewis
Clarence Irving Lewis
Clarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in...
.
Boole's work (as well as that of his intellectual progeny) was relatively obscure, except among logicians. At the time, it appeared to have no practical uses. However, approximately seventy years after Boole's death, Claude Shannon attended a philosophy class at the University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
that introduced him to Boole's studies. Shannon recognised that Boole's work could form the basis of mechanisms and processes in the real world and that it was therefore highly relevant. In 1937 Shannon went on to write a master's thesis at the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
, in which he showed how Boolean algebra could optimize the design of systems of electromechanical relays, then used in telephone routing switches. He also proved that circuits with relays could solve Boolean algebra problems. Employing the properties of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers. Victor Shestakov
Victor Shestakov
Victor Ivanovich Shestakov was a Russian/Soviet logician and theoretician of electrical engineering. In 1935 he discovered the possible interpretation of Boolean algebra of logic in electro-mechanical relay circuits...
at Moscow State University (1907–1987) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians S.A. Yanovskaya, Gaaze-Rapoport, Dobrushin
Dobrushin
Dobrushin is surname of:* Roland Lvovich Dobrushin , a Soviet* Dobrushin-Lanford-Ruelle equations...
, Lupanov, Medvedev, and Uspensky, though they presented their academic theses in the same year, 1938 . But the first publication of Shestakov's result took place only in 1941 (in Russian). Hence Boolean algebra became the foundation of practical digital circuit
Digital circuit
Digital electronics represent signals by discrete bands of analog levels, rather than by a continuous range. All levels within a band represent the same signal state...
design; and Boole, via Shannon and Shestakov, provided the theoretical grounding for the Digital Age.
The crater Boole
Boole (crater)
Boole is a lunar crater that lies along the northwestern limb of the Moon, to the northwest of the crater Gerard. At this location it is viewed nearly from the side, and is very oblong in shape due to foreshortening. The crater formation is nearly circular, however, with a wide inner wall that has...
on the Moon is named in his honour.
The keyword Bool represents a Boolean datatype
Boolean datatype
In computer science, the Boolean or logical data type is a data type, having two values , intended to represent the truth values of logic and Boolean algebra...
in many programming languages, though Pascal
Pascal (programming language)
Pascal is an influential imperative and procedural programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and efficient language intended to encourage good programming practices using structured programming and data structuring.A derivative known as Object Pascal...
uses the full name Boolean.
External links
- Roger Parsons' article on Boole -- THIS LINK 404s
- George Boole's work as first Professor of Mathematics in University College, Cork, Ireland
- Boole, G. (1854) An investigation of the laws of thought. Macmillan, London, at the Internet ArchiveInternet ArchiveThe Internet Archive is a non-profit digital library with the stated mission of "universal access to all knowledge". It offers permanent storage and access to collections of digitized materials, including websites, music, moving images, and nearly 3 million public domain books. The Internet Archive...
.