Editing George Boole (section)

== 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.<ref name="EB1911" /> His works are in about 50 articles and a few separate publications.<ref>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.&nbsp;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.</ref><ref name="auto" />

In 1841, Boole published an influential paper in early [[invariant theory]].<ref name=SED /> He received a medal from the [[Royal Society]] for his memoir of 1844, "On a General Method in Analysis".<ref name="EB1911" /> It was a contribution to the theory of [[linear differential equation]]s, moving from the case of constant coefficients on which he had already published, to variable coefficients.<ref>[[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 }}.</ref> The innovation in operational methods is to admit that operations may not [[commutative law|commute]].<ref>[[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 }}.</ref> In 1847, Boole published ''The Mathematical Analysis of Logic'', the first of his works on symbolic logic.<ref>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).</ref>

=== Differential equations ===
Boole completed two systematic treatises on mathematical subjects during his lifetime. The ''Treatise on [[Differential Equations]]''<ref>George Boole, ''A treatise on differential equations'' (1859), [https://archive.org/details/atreatiseondiff06boolgoog Internet Archive].</ref> appeared in 1859, and was followed, the next year, by a ''Treatise on the [[Calculus]] of Finite Differences'',<ref>George Boole, ''A treatise on the calculus of finite differences'' (1860), [https://archive.org/details/treatiseoncalcul00geor  Internet Archive].</ref> a sequel to the former work.<ref name="EB1911" />
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",<ref>{{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}}</ref> in which he studied the sum of [[residue (complex analysis)|residues]] of a [[rational function]]. Among other results, he proved what is now called Boole's identity:
:<math>\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} </math>
for any real numbers ''a''<sub>''k''</sub>&nbsp;>&nbsp;0, ''b''<sub>''k''</sub>, and ''t''&nbsp;>&nbsp;0.<ref name=cmr>{{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}}</ref> Generalisations of this identity play an important role in the theory of the [[Hilbert transform]].<ref name=cmr />

=== Binary logic ===
{{Main|Boolean algebra}}
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.<ref name="EB1911" /> 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.<ref>[[John Corcoran (logician)|John Corcoran]], Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol.  24 (2003), pp. 261–288.</ref> Boole's initial involvement in logic was prompted by a current debate on [[quantification (logic)|quantification]], between [[Sir William Hamilton, 9th Baronet|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.<ref name=ODNB>{{ODNBweb|id=2868|title=Boole, George|first=I.|last=Grattan-Guinness}}</ref> It founded what was first known as the "algebra of logic" tradition.<ref name=Marc>Witold Marciszewski (editor), ''Dictionary of Logic as Applied in the Study of Language'' (1981), pp. 194–195.</ref>

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 article<ref>[[John Corcoran (logician)|Corcoran, 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).</ref> 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 ====
<blockquote>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.<ref>George Boole. 1854/2003. ''The Laws of Thought'', facsimile of 1854 edition, with an introduction by [[John Corcoran (logician)|John Corcoran]]. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review.24 (2004) 167–169.</ref></blockquote>

==== 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.<ref name=KY>[[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 }}.</ref> 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 (set theory)|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.<ref name=Marc /><ref>{{cite SEP |url-id=algebra-logic-tradition |title=The Algebra of Logic Tradition |last=Burris |first=Stanley}}</ref>

In fact, there is the other possibility generalizing Boole's original partial operation, that + should be read as [[disjunction|non-exclusive or]].<ref name=KY /> Handling this ambiguity was an early problem of the theory, reflecting the modern use of both [[Boolean ring]]s and Boolean algebras (which are simply different aspects of one type of structure). Boole and [[William Stanley Jevons|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.<ref name=SED />

=== Probability theory ===
The second part of the ''Laws of Thought'' contained a corresponding attempt to discover a [[Probability theory|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.<ref>{{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}}</ref><ref name="EB1911" />