George Boole

{{Short description|English mathematician and philosopher (1815–1864)}}

{{Infobox philosopher

| honorific_suffix = {{post-nominals|size=100%|FRS}}

| image = File:Portrait of George Boole.png

| caption = Portrait of Boole, from
The Illustrated London News, 21 January 1865

| birth_date = {{birth date|df=y|1815|11|2}}

| birth_place = Lincoln, England

| death_date = {{death date and age|df=y|1864|12|8|1815|11|2}}

| death_place = Ballintemple, Cork, Ireland

| education = Bainbridge's Commercial Academy

| known_for = {{ubl|Boolean algebra|Boole's expansion theorem|Boolean function|Boole's inequality|Boolean polynomials|Boolean ring|Boole's rule|Abstract algebraic logic|Invariant theory|Wholistic reference}}

| spouse = {{marriage|Mary Everest|1855}}

| awards = {{ubl|Royal Medal (1844)|Keith Medal (1855–1857)|FRS (1857)}}

| discipline = Mathematics

| work_institutions = {{ubl|Lincoln Mechanics' Institute|Free School Lane, Lincoln|Queen's University, Cork}}

| era = 19th-century philosophy

| region = Western philosophy

| school_tradition = British algebraic logicIvor Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, 2002, ch. 5.1.

| main_interests = Mathematics, logic, philosophy of mathematics

}}

George Boole ({{IPAc-en|b|uː|l}} {{respell|BOOL}}; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854), which contains Boolean algebra. Boolean logic, essential to computer programming, is credited with helping to lay the foundations for the Information Age.{{Cite book |last=Nahin |first=Paul J. |title=The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age |publisher=Princeton University Press |year=2012 |jstor=j.cttq957s |isbn=978-0-691-17600-0 |author-link=Paul J. Nahin}}{{Cite web |last=Mulcahy |first=Colm |author-link=Colm Mulcahy |date=2015-11-01 |title=The Bicentennial of George Boole, the Man Who Laid the Foundations of the Digital Age |url=https://blogs.scientificamerican.com/guest-blog/the-bicentennial-of-george-boole-the-man-who-laid-the-foundations-of-the-digital-age/ |access-date=2023-09-30 |website=Scientific American Blog Network}}

Boole was the son of a shoemaker. He received a primary school education and learned Latin and modern languages through various means. At 16, he began teaching to support his family. He established his own school at 19 and later ran a boarding school in Lincoln. Boole was an active member of local societies and collaborated with fellow mathematicians. In 1849, he was appointed the first professor of mathematics at Queen's College, Cork (now University College Cork) in Ireland, where he met his future wife, Mary Everest. He continued his involvement in social causes and maintained connections with Lincoln. In 1864, Boole died due to fever-induced pleural effusion after developing pneumonia.

Boole published around 50 articles and several separate publications in his lifetime. Some of his key works include a paper on early invariant theory and "The Mathematical Analysis of Logic", which introduced symbolic logic. Boole also wrote two systematic treatises: "Treatise on Differential Equations" and "Treatise on the Calculus of Finite Differences". He contributed to the theory of linear differential equations and the study of the sum of residues of a rational function. In 1847, Boole developed Boolean algebra, a fundamental concept in binary logic, which laid the groundwork for the algebra of logic tradition and forms the foundation of digital circuit design and modern computer science. Boole also attempted to discover a general method in probabilities, focusing on determining the consequent probability of events logically connected to given probabilities.

Boole's work was expanded upon by various scholars, such as Charles Sanders Peirce and William Stanley Jevons. Boole's ideas later gained practical applications when Claude Shannon and Victor Shestakov employed Boolean algebra to optimize the design of electromechanical relay systems, leading to the development of modern electronic digital computers. His contributions to mathematics earned him various honours, including the Royal Society's first gold prize for mathematics, the Keith Medal, and honorary degrees from the Universities of Dublin and Oxford. University College Cork celebrated the 200th anniversary of Boole's birth in 2015, highlighting his significant impact on the digital age.

Early life

{{multiple image

| header = Boole's House and School in Lincoln

| image1 = 3 Pottergate - geograph.org.uk - 657140.jpg

| caption1 = House and School at 3 Pottergate

| image2 = BoolePlacque.jpg

| caption2 = Plaque from the house

}}

Boole was born in 1815 in Lincoln, Lincolnshire, England, the son of John Boole Snr (1779–1848), a shoemaker{{cite web |url=http://www.lincolnboolefoundation.org/john-boole/ |title=John Boole |publisher=Lincoln Boole Foundation |access-date=6 November 2015 |archive-url=https://web.archive.org/web/20160308133149/http://www.lincolnboolefoundation.org/john-boole/ |archive-date=8 March 2016 |url-status=dead}} and Mary Ann Joyce.{{cite web |url=https://georgeboole.com/boole/life/family/ |title=George Boole's Family Tree |access-date=2021-04-12 |archive-date=24 February 2021 |archive-url=https://web.archive.org/web/20210224173738/https://georgeboole.com/boole/life/family/ |url-status=live}} He had a primary school education, and received lessons from his father, but due to a serious decline in business, he had little further formal and academic teaching.{{Cite book |title=Math and mathematicians: the history of math discoveries around the world |first=Leonard C. |last=Bruno |date=2003 |orig-year=1999 |publisher=UXL |editor-last=Baker |editor-first=Lawrence W. |isbn=0-7876-3813-7 |location=Detroit, Mich. |page=[https://archive.org/details/mathmathematicia00brun/page/49 49] |oclc=41497065 |url=https://archive.org/details/mathmathematicia00brun |url-access=registration}} William Brooke, a bookseller in Lincoln, may have helped him with Latin, which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern languages.Hill, p. 149; [https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA149 Google Books] {{Webarchive|url=https://web.archive.org/web/20160317034644/https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA149 |date=17 March 2016 }}. In fact, when a local newspaper printed his translation of a Latin poem, a scholar accused him of plagiarism under the pretence that he was not capable of such achievements.{{sfn|Bruno|2003|pages=[https://archive.org/details/mathmathematicia00brun/page/49 49]–50 }} At age 16, Boole became the breadwinner for his parents and three younger siblings, taking up a junior teaching position in Doncaster at Heigham's School.Rhees, Rush. (1954) "George Boole as Student and Teacher. By Some of His Friends and Pupils", Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. Vol. 57. Royal Irish Academy He taught briefly in Liverpool.{{MacTutor Biography|id=Boole}}

File:Greyfriars, Lincoln - geograph.org.uk - 106215.jpg

Boole participated in the Lincoln Mechanics' Institute, in the Greyfriars, Lincoln, which was founded in 1833.{{Cite web |title=Society for the History of Astronomy, Lincolnshire. |url=http://www.freewebs.com/sochistastro/lincolnshire.htm |url-status=dead |archive-url=https://web.archive.org/web/20170301151123/http://www.freewebs.com/sochistastro/lincolnshire.htm |archive-date=1 March 2017 |access-date=2 September 2019 |website=}} Edward Bromhead, who knew John Boole through the institution, helped George Boole with mathematics books{{ODNBweb|id=37224|title=Bromhead, Sir Edward Thomas French|first=A. W. F.|last=Edwards}} and he was given the calculus text of Sylvestre François Lacroix by the Rev. George Stevens Dickson of St Swithin's, Lincoln.{{cite SEP |url-id=boole |title=George Boole |last=Burris |first=Stanley}} Without a teacher, it took him many years to master calculus.

At age 19, Boole successfully established his own school in Lincoln: Free School Lane.[https://georgeboole200.ucc.ie/boole/life/lincoln/selfeducation/ George Boole: Self-Education & Early Career] {{Webarchive|url=https://web.archive.org/web/20171122142141/https://georgeboole200.ucc.ie/boole/life/lincoln/selfeducation/ |date=22 November 2017 }} University College Cork Four years later he took over Hall's Academy in Waddington, outside Lincoln, following the death of Robert Hall. In 1840, he moved back to Lincoln, where he ran a boarding school. Boole immediately became involved in the Lincoln Topographical Society, serving as a member of the committee, and presenting a paper entitled "On the origin, progress, and tendencies of polytheism, especially amongst the ancient Egyptians and Persians, and in modern India".A Selection of Papers relative to the County of Lincoln, read before the Lincolnshire Topographical Society, 1841–1842. Printed by W. and B. Brooke, High-Street, Lincoln, 1843.

Boole became a prominent local figure, an admirer of John Kaye, the bishop.Hill, p. 172 note 2; [https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA172 Google Books] {{Webarchive|url=https://web.archive.org/web/20160610152358/https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA172 |date=10 June 2016 }}. He took part in the local campaign for early closing. With Edmund Larken and others he set up a building society in 1847.Hill, p. 130 note 1; [https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA130 Google Books] {{Webarchive|url=https://web.archive.org/web/20160427154156/https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA130 |date=27 April 2016 }}. He associated also with the Chartist Thomas Cooper, whose wife was a relation.Hill, p. 148; [https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA148 Google Books] {{Webarchive|url=https://web.archive.org/web/20160504074433/https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA148 |date=4 May 2016 }}.

From 1838 onwards, Boole was making contacts with sympathetic British academic mathematicians and reading more widely. He studied algebra in the form of symbolic methods, as far as these were understood at the time, and began to publish research papers.

Professorship and life in Cork

File:Boole House Cork 2012.jpg, in which Boole lived between 1849 and 1855, and where he wrote The Laws of Thought (picture taken during renovation)]]

Boole's status as a mathematician was recognised by his appointment in 1849 as the first professor of mathematics at Queen's College, Cork (now University College Cork (UCC)) in Ireland. He met his future wife, Mary Everest, there in 1850 while she was visiting her uncle John Ryall who was professor of Greek. They married in 1855.{{EB1911|inline=y|wstitle=Boole, George |volume=4 |pages=235–236 |first=William Stanley |last=Jevons |author-link=William Stanley Jevons}}Ronald Calinger, Vita mathematica: historical research and integration with teaching (1996), p. 292; [https://books.google.com/books?id=D21wKHoYGg0C&pg=PA292 Google Books] {{Webarchive|url=https://web.archive.org/web/20160427073547/https://books.google.com/books?id=D21wKHoYGg0C&pg=PA292 |date=27 April 2016 }}. He maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution.Hill, p. 138 note 4; [https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA138 Google Books] {{Webarchive|url=https://web.archive.org/web/20160527141944/https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA138 |date=27 May 2016 }}.

In 1861, Boole was involved in a Judgement in the Court of Queen's Bench in Ireland against one John Hewitt Wheatley of Craig House, Sligo for the sum of £400, whereby Wheatley's estate and interest in lands of Maghan/Mahon, County Cork became vested in Boole.Registry of Deeds, Dublin. Memorial: 1863-007-257 (extract). A Memorial of an Indenture of Re-Conveyance bearing date the seventh day of February one thousand eight hundred and sixty three made between William Hutchinson Massey of Mountmassey in the County of Cork, Esquire, of the first part, George Boole of Blackrock in the County of Cork, Esquire, Professor of Mathematics of the second part, Wilhelmina Smithwick of Dunmanway in said County of Cork, Spinster, of the third part and John Hewitt Wheatley of Craig House in the County of Sligo, Esquire, of the fourth part reciting a certain Indenture of Mortgage bearing date the fifth day of January one thousand eight hundred and sixty one whereby the said John Hewitt Wheatley in consideration of the sum.... And reciting that George Boole by the name and description of George Boole of Blackrock in the County of Corke, Professor of Mathematics did in or as of Trinity Term one thousand eight hundred and sixty one obtain a Judgement in the Court of Queen's Bench in Ireland against said John Hewitt Wheatley for the sum of four hundred pounds debt besides costs And reciting that said Judgement was duly registered on the ninth of November one thousand eight hundred and sixty one whereby pursuant to the statute in such Case made and provided the Estate and interest of the said John Hewitt Wheatley in said lands and premises became vested in one George Boole but subject to Redemption...

In March 1863, Boole leased Litchfield Cottage, Cork, the house in which he would live with his wife Mary until his death in December of the following year.Registry of Deeds, Dublin. Memorial: 1863-011-164 (extract). Registered: 30/03/1863. Memorial of an Intended Deed ... made between Edwards Casey then of Waterloo Place in the City of Cork, Esquire ... and George Boole then of Blackrock in the County of Cork, Esquire, L.L.D. then Professor of Mathematics in the Queens College at Cork ... After reciting that by Indenture of Lease bearing date the Twenty seventh day of March one thousand eight hundred and fifty six, John Litchfield then of Ballymaloo in the County of Cork, Esquire, did for the considerations therein mentioned demise and set unto the said William Jackson Cummins All That and Those the dwelling house with the premises and the Garden and walled in field to the rere thereof hereinafter particularly mentioned and described To Hold the said demised premises ... To Hold the said dwelling house and premises with the appurtenances unto the said George Boole, his Executors, Administrators and Assigns, from thenceforth for the then residue of the said term of one hundred years then to come and unexpired vested in him the said Edwards Casey... The premises was described in the deeds as "all that and those the dwelling house called Litchfield Cottage with the premises and appurtenances thereunto belonging and the Garden and Walled in field to the rere thereof". Boole's will bequeathed all his 'estate term and interest' in the lease of Litchfield Cottage unto his wife.Registry of Deeds, Dublin. Memorial: 1865-030-121 (extract). Registered: 20/10/1865. Memorial of a certain Deed of Assignment bearing date the Twenty first day of August one thousand eight hundred and sixty five and made between Mary Boole of 68 Harley Street, London, Widow and Executrix of the Last Will and Testament of George Boole late of Litchfield Cottage Blackrock in the County of Cork Esquire L.L.D. deceased of the one part and Francis Heard of Ballintemple in the County of Cork, Esquire, Captain in her Majesty's eighty seventh Regiment of South Cork, Militia of the other part Whereby after reciting that by Indenture of Lease bearing date the twenty seventh day of March one thousand eight hundred and fifty six made between John Litchfield of Ballymaloo in the County of Cork, Esquire, of the one part and William Jackson Cummins of the City of Cork, Doctor in Medicine of the other part, the said John Litchfield demised unto the said William Jackson Cummins All that and those the dwelling house called Litchfield Cottage with the premises and appurtenances thereunto ... also reciting that the said George Boole ... having before his death duly made and published his last Will and Testament in writing and thereby bequeathed all his Estate term and interest in said hereinbefore recited Indenture of Lease and premises thereby demised unto the said Mary Boole party of said deed of which this is the Memorial and said Will was afterwards duly proved by the said Mary Boole in the Court of Probate District of Cork... witnesses as to the execution of said Deed and this Memorial by the said Mary Boole are witnessed by John Knights, Porter at Queens College, Harley Street, London and Jane White, Housekeeper 68 Harley Street, London. In August 1865, some 8 months after his death, Mary (by then living at 68 Harley Street, London) passed the house on to Francis Heard of Ballintemple, Cork, Esquire, a captain in her Majesty's 87th Regiment of South Cork.

Honours and awards

{{multiple image

| header = Stained glass window in Lincoln Cathedral dedicated to Boole

| image1 = BooleWindow(bottom third).jpg

| caption1 = Detail depicting his favourite Bible passage (content suggested by his widow), God's calling of the prophet Samuel ([https://www.biblegateway.com/passage/?search=1+sam+3%3A1-10&version=NLT 1 Samuel 3:1–10]), a child dedicated to God by his parents{{Cite web |url=https://www.visitlincoln.com/things-to-do/lincoln-cathedral#george-boole-trail |title=Lincoln Cathedral | Things to do |access-date=16 November 2019 |archive-date=9 November 2019 |archive-url=https://web.archive.org/web/20191109114811/https://www.visitlincoln.com/things-to-do/lincoln-cathedral#george-boole-trail |url-status=live}}

| image2 = BoolePlaque2.jpg

| width2 =

| caption2 = Plaque beneath window

}}

In 1844, Boole's paper "On a General Method in Analysis" won the first gold prize for mathematics awarded by the Royal Society.{{cite book |title=The Life and Work of George Boole: A Prelude to the Digital Age |last=MacHale |first=Desmond |page=97}} He was awarded the Keith Medal by the Royal Society of Edinburgh in 1855{{cite journal |url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8735503 |title=Keith Awards 1827–1890 |journal=Earth and Environmental Science Transactions of the Royal Society of Edinburgh |date=January 1892 |volume=36 |issue=3 |pages=767–770 |publisher=Cambridge Journals Online |doi=10.1017/S0080456800037984 |s2cid=251574207 |access-date=29 November 2014 |archive-date=20 August 2016 |archive-url=https://web.archive.org/web/20160820015246/http://journals.cambridge.org/action/displayAbstract?fromPage=online |url-status=live}} and was elected a Fellow of the Royal Society (FRS) in 1857. He received honorary degrees of LL.D. from the University of Dublin and the University of Oxford.Ivor Grattan-Guinness, Gérard Bornet, George Boole: Selected manuscripts on logic and its philosophy (1997), p. xiv; [https://books.google.com/books?id=pzg7UFsIVJIC&pg=PR14 Google Books] {{Webarchive|url=https://web.archive.org/web/20160522113300/https://books.google.com/books?id=pzg7UFsIVJIC&pg=PR14|date=22 May 2016}}.

Works

Boole's first published paper was "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 his friendship with Duncan Farquharson Gregory, the editor of the journal. His works are in about 50 articles and a few separate publications.A list of Boole's memoirs and papers is 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 22 articles in all. In the third and fourth series of the Philosophical Magazine are found 16 papers. The Royal Society printed six 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 Mechanics' Magazine in 1848.

In 1841, Boole published an influential paper in early invariant theory. He received a medal from the Royal Society for his memoir of 1844, "On a General Method in Analysis". It was a contribution to the theory of linear differential equations, moving from the case of constant coefficients on which he had already published, to variable coefficients.Andrei Nikolaevich Kolmogorov, Adolf Pavlovich Yushkevich (editors), Mathematics of the 19th Century: function theory according to Chebyshev, ordinary differential equations, calculus of variations, theory of finite differences (1998), pp. 130–132; [https://books.google.com/books?id=Mw6JMdZQO-wC&pg=PA130 Google Books] {{Webarchive|url=https://web.archive.org/web/20160510024525/https://books.google.com/books?id=Mw6JMdZQO-wC&pg=PA130 |date=10 May 2016 }}. The innovation in operational methods is to admit that operations may not commute.Jeremy Gray, Karen Hunger Parshall, Episodes in the History of Modern Algebra (1800–1950) (2007), p. 66; [https://books.google.com/books?id=zMSl6QLlJZsC&pg=PA66 Google Books] {{Webarchive|url=https://web.archive.org/web/20160516223338/https://books.google.com/books?id=zMSl6QLlJZsC&pg=PA66 |date=16 May 2016 }}. In 1847, Boole published The Mathematical Analysis of Logic, the first of his works on symbolic logic.George Boole, [https://books.google.com/books?id=zv4YAQAAIAAJ&pg=PP9 The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning] {{Webarchive|url=https://web.archive.org/web/20160511151325/https://books.google.com/books?id=zv4YAQAAIAAJ&pg=PP9 |date=11 May 2016 }} (London, England: Macmillan, Barclay, & Macmillan, 1847).

= Differential equations =

Boole completed two systematic treatises on mathematical subjects during his lifetime. The Treatise on Differential EquationsGeorge Boole, A treatise on differential equations (1859), [https://archive.org/details/atreatiseondiff06boolgoog Internet Archive]. appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences,George Boole, A treatise on the calculus of finite differences (1860), [https://archive.org/details/treatiseoncalcul00geor Internet Archive]. a sequel to the former work.

Shortly after his death, Todhunter republished Boole's treatise with some of Boole's revisions, along with a supplement that was originally intended to be merged in the making of the second edition.

= Analysis =

In 1857, Boole published the treatise "On the Comparison of Transcendent, with Certain Applications to the Theory of Definite Integrals",{{cite journal |title=On the Comparison of Transcendent, with Certain Applications to the Theory of Definite Integrals |first=George |last=Boole |journal=Philosophical Transactions of the Royal Society of London |volume=147 |year=1857 |pages=745–803 |jstor=108643 |doi=10.1098/rstl.1857.0037 |doi-access=free}} in which he studied the sum of residues of a rational function. Among other results, he proved what is now called Boole's identity:

: $\mathrm{mes} \left\{ x \in \mathbb{R} \, \mid \, \Re \frac{1}{\pi} \sum \frac{a_k}{x - b_k} \geq t \right\} = \frac{\sum a_k}{\pi t}$

for any real numbers a_k > 0, b_k, and t > 0.{{cite book |mr=2129737 |last1=Cima |first1=Joseph A. |last2=Matheson |first2=Alec |last3=Ross |first3=William T. |chapter=The Cauchy transform |title=Quad domains and their applications |pages=79–111 |series=Oper. Theory Adv. Appl. |volume=156 |publisher=Birkhäuser |location=Basel |year=2005}} Generalisations of this identity play an important role in the theory of the Hilbert transform.

= Binary logic =

In 1847, Boole published the pamphlet Mathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system and wanted An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities to be seen as the mature statement of his views. Contrary to widespread belief, Boole never intended to criticise or disagree with the main principles of Aristotle's logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability.John Corcoran, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288. Boole's initial involvement in logic was prompted by a current debate on quantification, between Sir William Hamilton who supported the theory of "quantification of the predicate", and Boole's supporter Augustus De Morgan who advanced a version of De Morgan duality, as it is now called. Boole's approach was ultimately much further reaching than either sides' in the controversy.{{ODNBweb|id=2868|title=Boole, George|first=I.|last=Grattan-Guinness}} It founded what was first known as the "algebra of logic" tradition.Witold Marciszewski (editor), Dictionary of Logic as Applied in the Study of Language (1981), pp. 194–195.

Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic. A 2003 articleCorcoran, John (2003). "Aristotle's Prior Analytics and Boole's Laws of Thought". History and Philosophy of Logic, 24: 261–288. Reviewed by Risto Vilkko. Bulletin of Symbolic Logic, 11(2005) 89–91. Also by Marcel Guillaume, Mathematical Reviews 2033867 (2004m:03006). provides a systematic comparison and critical evaluation of Aristotelian logic and Boolean logic; it also reveals the centrality of holistic reference in Boole's philosophy of logic.

== 1854 definition of the universe of discourse ==

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them, the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilised men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.George Boole. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by John Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review.24 (2004) 167–169.

== Treatment of addition in logic ==

Boole conceived of "elective symbols" of his kind as an algebraic structure. But this general concept was not available to him: he did not have the segregation standard in abstract algebra of postulated (axiomatic) properties of operations, and deduced properties.Andrei Nikolaevich Kolmogorov, Adolf Pavlovich Yushkevich, Mathematics of the 19th century: mathematical logic, algebra, number theory, probability theory (2001), pp. 15 (note 15)–16; [https://books.google.com/books?id=X3u5hJCkobYC&pg=PA15 Google Books] {{Webarchive|url=https://web.archive.org/web/20160517144440/https://books.google.com/books?id=X3u5hJCkobYC&pg=PA15 |date=17 May 2016 }}. His work was a beginning to the algebra of sets, again not a concept available to Boole as a familiar model. His pioneering efforts encountered specific difficulties, and the treatment of addition was an obvious difficulty in the early days.

Boole replaced the operation of multiplication by the word "and" and addition by the word "or". But in Boole's original system, + was a partial operation: in the language of set theory it would correspond only to the union of disjoint subsets. Later authors changed the interpretation, commonly reading it as exclusive or, or in set theory terms symmetric difference; this step means that addition is always defined.{{cite SEP |url-id=algebra-logic-tradition |title=The Algebra of Logic Tradition |last=Burris |first=Stanley}}

In fact, there is the other possibility generalizing Boole's original partial operation, that + should be read as non-exclusive or. Handling this ambiguity was an early problem of the theory, reflecting the modern use of both Boolean rings and Boolean algebras (which are simply different aspects of one type of structure). Boole and Jevons struggled over just this issue in 1863, in the form of the correct evaluation of x + x. Jevons argued for the result x, which is correct for + as disjunction. Boole kept the result as something undefined. He argued against the result 0, which is correct for exclusive or, because he saw the equation x + x = 0 as implying x = 0, a false analogy with ordinary algebra.

= Probability theory =

The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities. Here the goal was algorithmic: from the given probabilities of any system of events, to determine the consequent probability of any other event logically connected with those events.{{cite book |last=Boole |first=George |title=An Investigation of the Laws of Thought |publisher=Walton & Maberly |year=1854 |location=London |pages=265–275 |isbn=978-0-7905-9242-8 |url=https://archive.org/stream/investigationofl00boolrich#page/264/mode/2up}}

Death

File:Grave of George Boole in Ireland.jpg, Cork, Ireland]]

In late November 1864, Boole walked, in heavy rain, from his home at Lichfield Cottage in Ballintemple{{cite web |url=http://www.buildingsofireland.ie/niah/search.jsp?type=record&county=CC®no=20868114 |title=Dublin City Quick Search: Buildings of Ireland: National Inventory of Architectural Heritage |access-date=3 November 2016 |archive-date=4 November 2016 |archive-url=https://web.archive.org/web/20161104021511/http://www.buildingsofireland.ie/niah/search.jsp?type=record&county=CC®no=20868114 |url-status=live}} to the university, a distance of three miles, and lectured wearing his wet clothes.{{cite news |url=http://www.irishexaminer.com/property/features/have-a-look-inside-the-home-of-ucc-maths-professor-george-boole-336830.html |title=Have a look inside the home of UCC maths professor George Boole |last=Barker |first=Tommy |date=13 June 2015 |newspaper=Irish Examiner |access-date=6 November 2015 |archive-date=3 July 2019 |archive-url=https://web.archive.org/web/20190703203707/https://www.irishexaminer.com/property/features/have-a-look-inside-the-home-of-ucc-maths-professor-george-boole-336830.html |url-status=live}} He soon became ill, developing pneumonia. As his wife believed that remedies should resemble their cause, she wrapped him in wet blankets – the wet having brought on his illness.{{sfn|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/52 52]}}{{Cite book |url=https://plato.stanford.edu/archives/sum2018/entries/boole/ |title=The Stanford Encyclopedia of Philosophy |first=Stanley |last=Burris |editor-first=Edward N. |editor-last=Zalta |date=2 September 2018 |publisher=Metaphysics Research Lab, Stanford University |access-date=2 September 2019 |via=Stanford Encyclopedia of Philosophy |archive-date=2 September 2019 |archive-url=https://web.archive.org/web/20190902094837/https://plato.stanford.edu/archives/sum2018/entries/boole/ |url-status=live}} Boole's condition worsened and on 8 December 1864,{{cite encyclopedia |title=George Boole |encyclopedia=Encyclopædia Britannica |date=30 January 2017 |url=https://www.britannica.com/biography/George-Boole |access-date=7 December 2017 |archive-date=7 December 2017 |archive-url=https://web.archive.org/web/20171207192647/https://www.britannica.com/biography/George-Boole |url-status=live}} he died of fever-induced pleural effusion.

He was buried in the Church of Ireland cemetery of St Michael's, Church Road, Blackrock (a suburb of Cork). There is a commemorative plaque inside the adjoining church.{{cite web |url=http://georgeboole.com/boole/life/ucc/death/ |title=Death-His Life-- George Boole 200 |access-date=10 February 2015 |archive-date=7 February 2020 |archive-url=https://web.archive.org/web/20200207005751/https://georgeboole.com/boole/life/ucc/death/ |url-status=live}}

Legacy

File:Bust of George Boole at University College Cork - 133760 (38218465931) (2).jpg]]

Boole is the namesake of the branch of algebra known as Boolean algebra, as well as the namesake of the lunar crater Boole. The keyword Bool represents a Boolean data type in many programming languages, though Pascal and Java, among others, both use the full name Boolean.P. J. Brown, Pascal from Basic, Addison-Wesley, 1982. {{isbn|0-201-13789-5}}, page 72 The library, underground lecture theatre complex and the Boole Centre for Research in Informatics{{cite web |url=http://www.bcri.ucc.ie |title=Boole Centre for Research in Informatics |access-date=18 December 2021 |archive-date=16 August 2019 |archive-url=https://web.archive.org/web/20190816105620/http://www.bcri.ucc.ie/ |url-status=live}} at University College Cork are named in his honour. A road called Boole Heights in Bracknell, Berkshire is named after him.

= 19th-century development =

Boole's work was extended and refined by a number of writers, beginning with William Stanley Jevons, who also authored the article about Boole in the Encyclopædia Britannica. Augustus De Morgan had worked on the logic of relations, and Charles Sanders Peirce integrated his work with Boole's during the 1870s.Ivor Grattan-Guinness, Gérard Bornet, George Boole: Selected manuscripts on logic and its philosophy (1997), p. xlvi; [https://books.google.com/books?id=pzg7UFsIVJIC&pg=PR46 Google Books] {{Webarchive|url=https://web.archive.org/web/20160425113513/https://books.google.com/books?id=pzg7UFsIVJIC&pg=PR46|date=25 April 2016}}. Other significant figures were Platon Sergeevich Poretskii, and William Ernest Johnson. The conception of a Boolean algebra structure on equivalent statements of a propositional calculus is credited to Hugh MacColl (1877), in work surveyed 15 years later by Johnson. Surveys of these developments were published by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis.

= 20th-century development =

File:Hasse2Free.png on basic propositions p and q arranged in a Hasse diagram. The Boolean combinations make up 16 different propositions, and the lines show which are logically related.]]

In 1921, the economist John Maynard Keynes published a book on probability theory, A Treatise of Probability. Keynes believed that Boole had made a fundamental error in his definition of independence which vitiated much of his analysis.Chapter XVI, p. 167, section 6 of A treatise on probability, volume 4: "The central error in his system of probability arises out of his giving two inconsistent definitions of 'independence' (2) He first wins the reader's acquiescence by giving a perfectly correct definition: "Two events are said to be independent when the probability of either of them is unaffected by our expectation of the occurrence or failure of the other." (3) But a moment later he interprets the term in quite a different sense; for, according to Boole's second definition, we must regard the events as independent unless we are told either that they must concur or that they cannot concur. That is to say, they are independent unless we know for certain that there is, in fact, an invariable connection between them. "The simple events, x, y, z, will be said to be conditioned when they are not free to occur in every possible combination; in other words, when some compound event depending upon them is precluded from occurring. ... Simple unconditioned events are by definition independent." (1) In fact as long as xz is possible, x and z are independent. This is plainly inconsistent with Boole's first definition, with which he makes no attempt to reconcile it. The consequences of his employing the term independence in a double sense are far-reaching. For he uses a method of reduction which is only valid when the arguments to which it is applied are independent in the first sense and assumes that it is valid if they are independent in the second sense. While his theorems are true if all propositions or events involved are independent in the first sense, they are not true, as he supposes them to be, if the events are independent only in the second sense." In his book The Last Challenge Problem, David Miller provides a general method in accord with Boole's system and attempts to solve the problems recognised earlier by Keynes and others. Theodore Hailperin showed much earlier that Boole had used the correct mathematical definition of independence in his worked out problems.{{cite web |url=http://zeteticgleanings.com/boole.html |title=ZETETIC GLEANINGS |access-date=10 March 2009 |archive-date=18 July 2011 |archive-url=https://web.archive.org/web/20110718152811/http://zeteticgleanings.com/boole.html |url-status=live}}

Boole's work and that of later logicians initially appeared to have no engineering uses. Claude Shannon attended a philosophy class at the University of Michigan which 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 optimise 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 Sofya Yanovskaya, Gaaze-Rapoport, Roland Dobrushin, Lupanov, Medvedev and Uspensky. 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 Information Age."That dissertation has since been hailed as one of the most significant master's theses of the 20th century. To all intents and purposes, its use of binary code and Boolean algebra paved the way for the digital circuitry that is crucial to the operation of modern computers and telecommunications equipment."{{cite news |url=https://www.theguardian.com/science/2001/mar/08/obituaries.news |newspaper=The Guardian |date=8 March 2001 |title=Claude Shannon |first=Andrew |last=Emerson |access-date=14 December 2016 |archive-date=10 April 2019 |archive-url=https://web.archive.org/web/20190410141635/https://www.theguardian.com/science/2001/mar/08/obituaries.news |url-status=live}}{{Clear}}

= 21st-century celebration =

{{Quote box|width=30%|align=right|quote="Boole's legacy surrounds us everywhere, in the computers, information storage and retrieval, electronic circuits and controls that support life, learning and communications in the 21st century. His pivotal advances in mathematics, logic and probability provided the essential groundwork for modern mathematics, microelectronic engineering and computer science."|source=—University College Cork.{{cite news |title=Who is George Boole: the mathematician behind the Google doodle |url=https://www.smh.com.au/technology/technology-news/who-is-george-boole-the-mathematician-behind-the-google-doodle-20151102-gkofyg.html#ixzz3qIcv6ii2 |newspaper=Sydney Morning Herald |date=2 November 2015 |access-date=20 February 2020 |archive-date=4 September 2017 |archive-url=https://web.archive.org/web/20170904113449/http://www.smh.com.au/technology/technology-news/who-is-george-boole-the-mathematician-behind-the-google-doodle-20151102-gkofyg.html#ixzz3qIcv6ii2 |url-status=live}}}}

The year 2015 saw the 200th anniversary of Boole's birth. To mark the bicentenary year, University College Cork joined admirers of Boole around the world to celebrate his life and legacy.

UCC's George Boole 200{{cite web |url=http://georgeboole.com |archive-url=https://web.archive.org/web/20140921124050/http://georgeboole.com/ |url-status=dead |archive-date=21 September 2014 |title=George Boole 200 – George Boole Bicentenary Celebrations}} project, featured events, student outreach activities and academic conferences on Boole's legacy in the digital age, including a new edition of Desmond MacHale's 1985 biography The Life and Work of George Boole: A Prelude to the Digital Age,{{Cite web |url=http://www.corkuniversitypress.com/SearchResults.asp?Search=boole&Submit= |title=Cork University Press |access-date=6 November 2014 |archive-date=8 November 2015 |archive-url=https://web.archive.org/web/20151108081951/http://www.corkuniversitypress.com/SearchResults.asp?Search=boole&Submit= |url-status=live}} 2014.

The search engine Google marked the 200th anniversary of his birth on 2 November 2015 with an algebraic reimaging of its Google Doodle.

File:George Boole House 2017.jpg

File:Statue of George Boole at Lincoln Central Train Station.jpg. The design, by sculptor Antony Dufort, was funded in part by the Heslam Trust.]]

In September 2022, a statue of George Boole in his role as a teacher was unveiled at Lincoln Central Train Station, in Boole's home town of Lincoln.

Views

Boole's views were given in four published addresses: The Genius of Sir Isaac Newton; The Right Use of Leisure; The Claims of Science; and The Social Aspect of Intellectual Culture. The first of these was from 1835 when Charles Anderson-Pelham, 1st Earl of Yarborough gave a bust of Newton to the Mechanics' Institute in Lincoln.James Gasser, A Boole Anthology: recent and classical studies in the logic of George Boole (2000), p. 5; [https://books.google.com/books?id=A2Q5Yghl000C&pg=PA5 Google Books] {{Webarchive|url=https://web.archive.org/web/20160510201103/https://books.google.com/books?id=A2Q5Yghl000C&pg=PA5 |date=10 May 2016 }}. The second justified and celebrated in 1847 the outcome of the successful campaign for early closing in Lincoln, headed by Alexander Leslie-Melville, of Branston Hall.Gasser, p. 10; [https://books.google.com/books?id=A2Q5Yghl000C&pg=PA10 Google Books] {{Webarchive|url=https://web.archive.org/web/20160511030622/https://books.google.com/books?id=A2Q5Yghl000C&pg=PA10 |date=11 May 2016 }}. The Claims of Science was given in 1851 at Queen's College, Cork.{{cite book |last=Boole |first=George |title=The Claims of Science, especially as founded in its relations to human nature; a lecture |url=https://books.google.com/books?id=BAlcAAAAQAAJ |access-date=4 March 2012 |year=1851 |archive-date=1 February 2014 |archive-url=https://web.archive.org/web/20140201175251/http://books.google.com/books?id=BAlcAAAAQAAJ |url-status=live}} The Social Aspect of Intellectual Culture was also given in Cork, in 1855 to the Cuvierian Society.{{cite book |last=Boole |first=George |title=The Social Aspect of Intellectual Culture: an address delivered in the Cork Athenæum, May 29th, 1855: at the soirée of the Cuvierian Society |url=https://books.google.com/books?id=PFWkZwEACAAJ |access-date=4 March 2012 |year=1855 |publisher=George Purcell & Co. |archive-date=1 February 2014 |archive-url=https://web.archive.org/web/20140201175254/http://books.google.com/books?id=PFWkZwEACAAJ |url-status=live}}

Though his biographer Des MacHale describes Boole as an "agnostic deist",{{cite book |title=Semiotica, Volume 105 |year=1995 |publisher=Mouton |page=56 |chapter=A tale of two amateurs |quote=MacHale's biography calls George Boole 'an agnostic deist'. Both Booles' classification of 'religious philosophies' as monistic, dualistic, and trinitarian left little doubt about their preference for 'the unity religion', whether Judaic or Unitarian.}}{{cite book |title=Semiotica, Volume 105 |year=1996 |publisher=Mouton |page=17 |quote=MacHale does not repress this or other evidence of the Boole's nineteenth-century beliefs and practices in the paranormal and in religious mysticism. He even concedes that George Boole's many distinguished contributions to logic and mathematics may have been motivated by his distinctive religious beliefs as an "agnostic deist" and by an unusual personal sensitivity to the sufferings of other people.}} Boole 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 was said to have chosen Unitarianism.^[reference?] Boole came to speak against what he saw as "prideful" scepticism, and instead favoured the belief in a "Supreme Intelligent Cause".Boole, George. Studies in Logic and Probability. 2002. Courier Dover Publications. pp. 201-202 He also declared "I firmly believe, for the accomplishment of a purpose of the Divine Mind."Boole, George. Studies in Logic and Probability. 2002. Courier Dover Publications. p. 451Some-Side of a Scientific Mind (2013). pp. 112–193. The University Magazine, 1878. London: Forgotten Books. (Original work published 1878) In addition, he stated "To infer the existence of an intelligent cause from the teeming evidence of surrounding design, to rise to the conception of a moral Governor of the World, from the study of the constitution and the moral provisions of our own nature;--these, though but the feeble steps of an understanding limited in its faculties and its materials of knowledge, are of more avail than the ambitious attempt to arrive at a certainty unattainable on the ground of natural religion. And as these were the most ancient, so are they still the most solid foundations, Revelation being set apart, of the belief that the course of this world is not abandoned to chance and inexorable fate."Concluding remarks of his treatise of "Clarke and Spinoza", as found in Boole, George (2007). An Investigation of the Laws of Thought. Cosimo, Inc. Chap . XIII. p. 217-218. (Original work published 1854)Boole, George (1851). The claims of science, especially as founded in its relations to human nature; a lecture, Volume 15. p. 24

Two influences on Boole were later claimed by his wife, Mary Everest Boole: a universal mysticism tempered by Jewish thought, and Indian logic.Jonardon Ganeri (2001), Indian Logic: a reader, Routledge, p. 7, {{isbn|0-7007-1306-9}}; [https://books.google.com/books?id=t_nOiqFmxOIC&pg=PA7 Google Books] {{Webarchive|url=https://web.archive.org/web/20160419081712/https://books.google.com/books?id=t_nOiqFmxOIC&pg=PA7 |date=19 April 2016 }}. 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 ... 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."Boole, Mary Everest Indian Thought and Western Science in the Nineteenth Century, Boole, Mary Everest Collected Works eds. E. M. Cobham and E. S. Dummer, London, Daniel 1931 pp. 947–967

In Ch. 13 of Laws of Thought Boole used examples of propositions from Baruch Spinoza and Samuel Clarke. The work contains some remarks on the relationship of logic to religion, but they are slight and cryptic.Grattan-Guinness and Bornet, p. 16; [https://books.google.com/books?id=pzg7UFsIVJIC&pg=PR16 Google Books] {{Webarchive|url=https://web.archive.org/web/20160508164733/https://books.google.com/books?id=pzg7UFsIVJIC&pg=PR16 |date=8 May 2016 }}. Boole was apparently disconcerted at the book's reception just as a mathematical toolset:

George afterwards learned, to his great joy, that the same conception of the basis of Logic was held by Leibniz, 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 Robert 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 that there was profound influence – via her uncle George Everest – of Indian thought in general and Indian logic, in particular, on George Boole, as well as on Augustus De Morgan and Charles Babbage:Kak, S. (2018) George Boole's Laws of Thought and Indian logic. Current Science, vol. 114, 2570–2573

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 1830–65. What share had it in generating the Vector Analysis and the mathematics by which investigations in physical science are now conducted?

Boole maintained that:{{Blockquote|text= No general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.{{cite book |last=Boole |first=George |title=Studies in Logic and Probability |publisher=Dover |location=Mineola, New York |date=2012 |edition=Reprint |isbn=978-0-486-48826-4 |orig-year=1952 |url=https://books.google.com/books?id=pr35F8N9eaoC&pg=PA273 |page=273 |editor-last=Rhees |editor-first=Rush |editor-link=Rush Rhees |access-date=27 October 2015 |archive-date=5 May 2016 |archive-url=https://web.archive.org/web/20160505104821/https://books.google.com/books?id=pr35F8N9eaoC&lpg=PP1&pg=PA273 |url-status=live}}}}

Family

In 1855, Boole married Mary Everest (niece of George Everest), who later wrote several educational works on her husband's principles.

The Booles had five daughters:

Mary Ellen (1856–1908){{cite web |url=http://georgeboole.com/boole/life/family/ |title=Family and Genealogy – His Life George Boole 200 |publisher=Georgeboole.com |access-date=7 March 2016 |archive-date=28 August 2017 |archive-url=https://web.archive.org/web/20170828185508/http://georgeboole.com/boole/life/family/ |url-status=live}} who married the mathematician and author Charles Howard Hinton and had four children. After the sudden death of her husband, Mary Ellen committed suicide in Washington, D.C., in May 1908.`My Right To Die´, Woman Kills Self in The Washington Times v. 28 May 1908 ([http://chroniclingamerica.loc.gov/lccn/sn84026749/1908-05-28/ed-1/seq-1.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20120605010718/http://chroniclingamerica.loc.gov/lccn/sn84026749/1908-05-28/ed-1/seq-1.pdf |date=5 June 2012 }}); Mrs. Mary Hinton A Suicide in The New York Times v. 29 May 1908 ([https://timesmachine.nytimes.com/timesmachine/1908/05/29/104804272.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20210225012041/https://timesmachine.nytimes.com/timesmachine/1908/05/29/104804272.pdf |date=25 February 2021 }}).
George Hinton (1882–1943), mining engineer and botanist
H. E. Hinton (1912–1977), entomologist
Geoffrey Hinton (born 1947), cognitive psychologist and computer scientist, Nobel prize for physics 2024, noted for work on artificial neural networks.{{cite web |url=https://georgeboole.com/news/george-booles-great-great-grandson-prof-geoffrey-hinton-wins-nobel-prize-of-computing-.html |title=|George Boole 200|News}}
Eric Hinton (born 1884)
William Hinton (1886–1909)"Smothers In Orchard" in The Los Angeles Times v. 27 February 1909.
Sebastian Hinton (1887–1923), lawyer, inventor of the jungle gym
Jean Hinton (married name Rosner) (1917–2002), a peace activist.
William H. Hinton (1919–2004) visited China in the 1930s and 40s and wrote an influential account of the Communist land reform.
Joan Hinton (1921–2010) worked for the Manhattan Project and lived in China from 1948 until her death on 8 June 2010; she was married to Sid Engst.
Margaret (1858–1935), married Edward Ingram Taylor, an artist.
Their elder son Geoffrey Ingram Taylor became a mathematician and a Fellow of the Royal Society.
Their younger son Julian Taylor was a professor of surgery.
Alicia (1860–1940), who made important contributions to four-dimensional geometry.
Her son Leonard Stott, a medical doctor and tuberculosis pioneer, invented a portable X-ray machine, a pneumothorax apparatus, and system of navigation based on spherical coordinates.D. MacHale, The Life and Work of George Boole: A Prelude to the Digital Age, Cork University Press, 2014. cited in [https://www.ams.org/journals/notices/201911/rnoti-p1853.pdf The Extraordinary Case of the Boole Family] {{Webarchive|url=https://web.archive.org/web/20191116073129/https://www.ams.org/journals/notices/201911/rnoti-p1853.pdf |date=16 November 2019 }} by Moira Chas
Lucy Everest (1862–1904), who was the first female professor of chemistry in England.
Ethel Lilian (1864–1960), who married the Polish scientist and revolutionary Wilfrid Michael Voynich and was the author of the novel The Gadfly.

= Concepts =

Boolean algebra, a logical calculus of truth values or set membership
Boolean algebra (structure), a set with operations resembling logical ones
Boolean circuit, a mathematical model for digital logical circuits.
Boolean data type is a data type, having two values (usually denoted true and false)
Boolean expression, an expression in a programming language that produces a Boolean value when evaluated
Boolean function, a function that determines Boolean values or operators
Boolean model (probability theory), a model in stochastic geometry
Boolean network, a certain network consisting of a set of Boolean variables whose state is determined by other variables in the network
Boolean processor, a 1-bit variables computing unit
Boolean ring, a ring consisting of idempotent elements
Boolean satisfiability problem
Boole's syllogistic is a logic invented by 19th-century British mathematician George Boole, which attempts to incorporate the "empty set".
Laws of thought
Principle of wholistic reference

= Other =

Notes

References

Walker, A. (ed) (2019) George Boole's Lincoln, 1815–49. The Survey of Lincoln, Vol.16. {{ISBN|9780993126352}}
University College Cork, George Boole 200 Bicentenary Celebration, [https://web.archive.org/web/20140921124050/http://georgeboole.com/ GeorgeBoole.com].
Ivor Grattan-Guinness, The Search for Mathematical Roots 1870–1940. Princeton University Press. 2000.
Francis Hill (1974), Victorian Lincoln; [https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA149 Google Books] {{Webarchive|url=https://web.archive.org/web/20160317034644/https://books.google.com/books?id=-A89AAAAIAAJ&pg=PA149 |date=17 March 2016 }}.
Des MacHale, George Boole: His Life and Work. [https://web.archive.org/web/20110812192528/http://boolepress.com/ Boole Press]. 1985.
Des MacHale, The Life and Work of George Boole: A Prelude to the Digital Age (new edition). [http://www.corkuniversitypress.com/SearchResults.asp?Search=boole&Submit= Cork University Press] {{Webarchive|url=https://web.archive.org/web/20151108081951/http://www.corkuniversitypress.com/SearchResults.asp?Search=boole&Submit= |date=8 November 2015 }}. 2014
Stephen Hawking, God Created the Integers. Running Press, Philadelphia. 2007.

External links

{{Sister project links| wikt=no | commons=Category:George Boole | b=no | n=no | q=George Boole | s=Author:George Boole | v=no | voy=no | species=no | d=q134661}}

[http://www.rogerparsons.info/george/boole.html Roger Parsons' article on Boole]
{{Gutenberg author |id=5861| name=George Boole}}
{{Internet Archive author |sname=George Boole}}
[https://www.maths.tcd.ie/pub/HistMath/People/Boole/CalcLogic/CalcLogic.html The Calculus of Logic] by George Boole; a transcription of an article which originally appeared in Cambridge and Dublin Mathematical Journal, Vol. III (1848), pp. 183–198.
[http://undersci.ucc.ie/wp-content/uploads/sites/12/2014/11/George_Boole.pdf George Boole's work as first Professor of Mathematics in University College, Cork, Ireland] {{Webarchive|url=https://web.archive.org/web/20171119095217/http://undersci.ucc.ie/wp-content/uploads/sites/12/2014/11/George_Boole.pdf |date=19 November 2017 }}
[https://web.archive.org/web/20140921124050/http://georgeboole.com/ George Boole website]
[https://web.archive.org/web/20160303202535/https://zbmath.org/authors/?q=ai:boole.george Author profile] in the database zbMATH
{{YouTube|Hljir_TyTEw|The Genius of George Boole}}