George Boole

George Boole (around 1860)

George Boole [ ˌdʒɔːdʒ ˈbuːl ] (born November 2, 1815 in Lincoln , England ; † December 8, 1864 in Ballintemple, County Cork , Ireland ) was an English mathematician ( self-taught ), logician and philosopher . He is best known for the fact that Boolean algebra, which is fundamental to computer technology , was named after him.

Life

George Boole was born in Lincolnshire. He had not attended any secondary schools other than primary education. He taught himself ancient Greek, French and German by himself. At the age of 16 he became an assistant teacher to support his family financially. At the age of 19, Boole started his own school. Due to his scientific work he became a mathematics professor at Queens College in Cork (Ireland) in 1848 , although he had not attended university himself. There he met Mary Everest , his future wife. She was interested in mathematics, worked as a librarian and dealt with the didactics of mathematics. Her uncle George Everest was the namesake of the highest mountain in the world. George and Mary had five daughters, including the author and musician Ethel Lilian Voynich (1864-1960) and Alicia Boole Stott (1860-1940), who, as a mathematician with no formal academic education, managed to classify the regular polyhedra in four dimensions. Boole was awarded the Royal Medal by the Royal Society in 1844 . In 1847 he published his epoch-making logic work The Mathematical Analysis of Logic and in 1854 his more detailed book An Investigation of the Laws of Thought . In 1857 he was elected a member ("Fellow") of the Royal Society.

Booles headstone in St Michael's Cemetery, Blackrock, Ireland

George Boole's House, Bachelor's Quay, Cork

Early death

George Boole died of a feverish cold on December 8, 1864 at the age of only 49. On his footpath he walked two miles in the pouring rain to the university, where he then gave his lecture in soaked clothes. He caught a cold, developed a high fever, and did not recover from it later. His wife was a supporter of the naturopathy of the time, which used to treat "like with like". She is said to have poured buckets of cold water over her husband, who was sick with the feverish cold, in bed. Pleural effusion was given as the cause of his death .

Main work

In his work The Mathematical Analysis of Logic from 1847, Boole created the first algebraic logic calculus and thus founded modern mathematical logic , which differs from the logic that was customary up to now through a consistent formalization. He formalized classical logic and propositional logic and developed a decision-making process for the true formulas using a disjunctive normal form . Boole anticipated the solution of the problems posed by David Hilbert well over 70 years before Hilbert's program for a central area of logic - since the decidability of classical logic implies its completeness and freedom from contradictions . As generalizations of Boole's logic calculus, the so-called Boolean algebra and the Boolean ring were later named after him.

In 1964 the lunar crater Boole was named after him, as was the asteroid (17734) Boole in 2001 .

Boolean original calculus

Boole used the common algebra, which today is specified as a power series ring over the field of real numbers. He embedded classical logic in it by calling for the conjunction AND as multiplication and negation as the difference to the defined and, for logical terms, idempotency , that is: ${\displaystyle 1}$

${\displaystyle x}$ AND ${\displaystyle y\ :=\ x\land y\ :=\ xy}$

NOT ${\displaystyle x\ :=\ \neg x\ :=\ 1-x}$

${\displaystyle xx=x}$ for all logical terms ${\displaystyle x}$

It is an embedding in which not all terms have a logical sense; for example, because the sum is logically meaningless, which is why Boole called it uninterpretable. The addition is only a partial operation in the logical domain, which is why he spoke of elective symbols, elective functions, and elective equations for the logical terms and operators. This fact was criticized by his successors. But his method is completely correct. Because the logical area is operationally closed: It is the structure created by the idempotent indeterminate, the 1, the multiplication and the negation, since is idempotent and with and also and are idempotent, as one can easily calculate. This means that all logical operators that can be derived by definition also have an effect in this area, in particular the inclusive and the exclusive disjunction: ${\displaystyle 2\cdot 2\neq 2}$ ${\displaystyle 1+1}$ ${\displaystyle x+y}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle xy}$ ${\displaystyle 1-x}$

${\displaystyle x}$ OR ${\displaystyle y\ :=\ x\lor y\ :=\ x+y-xy}$

EITHER OR ${\displaystyle x}$ ${\displaystyle y\ :=\ x\oplus y\ :=\ x-2xy+y}$

Both definitions belong to the logical area:

${\displaystyle x\lor y\ =\ x+y-xy\ =\ 1-(1-x)(1-y)\ =\ \neg (\neg x\land \neg y)}$

${\displaystyle x\oplus y\ =\ x-2xy+y\ =\ x(1-y)+y(1-x)-x(1-y)y(1-x)\ =\ (x\land \neg y)\lor (y\land \neg x)}$

Its OR definition evidently supplies all axioms of the later Boolean algebra and its EITHER-OR definition all axioms of the later Boolean ring , whereby the additions and must be strictly distinguished. ${\displaystyle \oplus }$ ${\displaystyle +}$

Boole designed his calculus primarily as conceptual or class logic , in which the universe (the universal class) is and the indefinite classes (concepts) represent. Within this calculus he then presented the scholastic syllogistics with systems of equations. He represented its basic predicates with equations: ${\displaystyle 1}$ ${\displaystyle x,y,z,\ldots }$

ALL ARE    with equivalent forming ${\displaystyle x}$ ${\displaystyle y\ :=\ (xy=x)}$ ${\displaystyle x(1-y)=0}$

NO ARE ${\displaystyle x}$ ${\displaystyle y\ :=\ (xy=0)}$

Secondly, Boole also used his calculus as propositional logic , in which the indefinite represent statements and and the truth values: ${\displaystyle x,y,z,\ldots }$ ${\displaystyle 1}$ ${\displaystyle 0}$

${\displaystyle x}$ IS TRUE ${\displaystyle \ :=\ (x=1)}$

${\displaystyle x}$ IS WRONG ${\displaystyle \ :=\ (x=0)}$

He supplemented his logical decision-making process using a normal form with an equivalent semantic decision-making process with truth value substitutions in Boolean functions that assign a truth value to every logical term used. This procedure corresponds to the decision procedure with truth tables , which is used to determine tautologies .

Modifications of Boole's calculus

Under the Boolean algebra today not Boole's algebra is understood original, but the Boolean Association , the Boole developed successor. In 1864 William Stanley Jevons removed the logically meaningless mathematical terms from Boole and gave the addition a logical sense as an inclusive OR with the rule . Boole, who corresponded with him, disagreed with this reinterpretation of addition because the rules of usual algebra are violated, because implied in it . Nevertheless, this modification of Boole's calculus prevailed, significantly influenced by Ernst Schröder , who formulated the first complete system of axioms in 1877, which Giuseppe Peano brought into the modern non-additive form in 1888. ${\displaystyle x+x=x}$ ${\displaystyle x+x=x}$ ${\displaystyle x=0}$

Boolean calculus can also be modified in such a way that logically meaningless terms no longer occur and the usual calculation rules for addition are preserved. For this, the addition must be completed in the logical area and fulfill the idempotency; then specifically what implies holds , so that also holds and self-inverse terms are present. This gives the addition the sense of the exclusive EITHER-OR. Ivan Iwanowitsch Schegalkin gave this calculus variant for the first time in 1927 together with a complete axiomatization. This creates a so-called Boolean ring , which Marshall Harvey Stone gave the name in 1936. Boolean rings are computationally elegant, because the math rules that are familiar from school apply here. The normal form necessary for the decidability of a formula is created here simply by distributive multiplication and deletion of double factors and summands with the idempotency and the additional rule . ${\displaystyle (x+1)(x+1)=(x+1)}$ ${\displaystyle x+x=0}$ ${\displaystyle -x=x}$ ${\displaystyle xx=x}$ ${\displaystyle x+x=0}$

Both calculus variants are implicitly contained in Boole's original calculus, since one can derive both axiom systems with its definitions.

Fonts

• George Boole: The mathematical analysis of logic: being an essay towards a calculus of deductive reasoning , 1847.
  • translated, commented and with an afterword by Tilman Bergt: The mathematical analysis of logic. Hallescher Verlag , Halle (Saale) 2001, pp 195, ISBN 3-929887-29-0 .
  • Abridged and translated from English, reprinted in Karel Berka , Lothar Kreiser: Logic texts. Annotated selection on the history of modern logic , 4th edition, Akademie Verlag, Berlin 1986 (first edition 1971), pages 25–28 DNB 850989647 .
• George Boole: An Investigation of The Laws of Thought , London 1854; Reprint: Dover, New York, NY 1958, ISBN 0-486-60028-9 .
• George Boole: Selected Manuscripts on Logic and its Philosophy edited by Ivor Grattan-Guinness and Gérard Bornet, Birkhäuser, Berlin / Basel / Bosten, MA 1997, ISBN 3-7643-5456-9 (Berlin) / ISBN 0-8176-5456- 9 (Boston) (= Science networks , Volume 20, (English)).

literature

• James Gasser (Ed.): A Boole Anthology. Recent and Classical Studies in the Logic of George Boole. Kluwer Academic Publishers Dordrecht 2000, ISBN 0-7923-6380-9 . Current state of research.
• Marshall Harvey Stone : The Theory of Representations for Boolean Algebras . In: Transactions of the American Mathematical Society . Volume 40, 1936, pp. 37-111.
• Desmond MacHale: George Boole: His Life and Work . Boole Press, Dublin 1985.
• PD Barry (Ed.): George Boole: a miscellany . Cork 1969.
• R. Harley: George Boole: an essay, biographical and expository . London 1866.
• GC Smith: The Boole-De Morgan correspondence, 1842-1864 . New York 1982.
• Ian Stewart : The greats of mathematics: 25 thinkers who wrote history , rororo, Reinbek bei Hamburg 2018, ISBN 978-3-499-63394-2 , pp. 229–246
• Isaac Asimov : Biographical Encyclopedia of Natural Sciences and Technology , Herder, Freiburg / Basel / Vienna 1974, ISBN 3-451-16718-2 , p. 280
• TAA Broadbent: Boole, George . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 2 : Hans Berger - Christoph Buys Ballot . Charles Scribner's Sons, New York 1970, p. 293-298 .

Web links

Commons : George Boole  - Collection of Images, Videos and Audio Files

• Literature by and about George Boole in the catalog of the German National Library
• Boole: The mathematical analysis of logic , 1847 digitized
• Boole: An Investigation of The Laws of Thought , London 1854, digitized
• Boole: An Investigation of The Laws of Thought , E-Book, 2005 at Gutenberg.org
• Stanley Burris:  George Boole. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
• Entry on Boole: George (1815–1864), Mathematician in the Archives of the Royal Society , London
• John J. O'Connor, Edmund F. Robertson :  George Boole. In: MacTutor History of Mathematics archive .
• Author profile in the database zbMATH
• Spektrum.de: George Boole (1815–1864): We owe the computer to his work . 1st September 2018.

See also

• Boolean variable
• Boolean operator
• Boolean retrieval
• Boolean function
• Boolean hierarchy

Individual evidence

1. ↑ George Boole's 200th birthday: The man who made it possible for us to search online. Spiegel online from November 2, 2015.
2. ↑ ^{a b} Boole: The Mathematical Analysis of Logic . S. 60ff, defined by MacLaurin series .
3. ↑ "[...] it is interesting to see that the methods Boole introduced can be applied in a mechanical fashion. In effect he has given what is now called a decision procedure "( William and Martha Kneale: The Development of Logic. Oxford: Clarendon Press 1962, paperback 1984, ISBN 0-19-824773-7 , page 240).
4. ^ Gazetteer of Planetary Nomenclature
5. ↑ Minor Planet Circ. 41942
6. ^ Boole: The Mathematical Analysis of Logic . P. 18: "Properties which they possess in common with symbols of quantity, and in virtue of which, all the processes of common algebra are applicable to the present system." These include in particular the division P. 73 and Taylor series developments P. 60ff. Quantity means sizes, the expression for real numbers at that time.
7. ^ Boole: The Mathematical Analysis of Logic . P. 15 conjunction , p. 17 idempotence, p. 20 negation . ${\displaystyle xy}$ ${\displaystyle 1-x}$
8. ^ Boole: An Investigation of the Laws of Thought . P. 66: “The expression seems indeed uninterpretable, unless it be assumed that the things represented by and the things represented by are entirely separate; that they embrace no individuals in common. " ${\displaystyle x+y}$ ${\displaystyle x}$ ${\displaystyle y}$
9. ^ Boole: The Mathematical Analysis of Logic . P. 16.
10. ^ William Stanley Jevons: Pure logic , London 1864, p. 3: The forms of my system may, in fact, be reached by divesting his system of a mathematical dress, which, to say the least, is not essential to it.
11. ↑ Schröder: “The operating circle of the logic calculus”, 1877, foreword p. III: “Ballast of algebraic numbers”, “symbols that cannot be interpreted such as 2, -1, 1/3, 1/0”.
12. ^ Boole: The Mathematical Analysis of Logic . P. 53 (30) inclusive OR, P. 53 (31) exclusive EITHER-OR.
13. ^ Boole: The Mathematical Analysis of Logic . Pp. 31-47.
14. ^ Boole: The Mathematical Analysis of Logic . P. 21 (4) (5).
15. ^ Boole: The Mathematical Analysis of Logic . P. 51 (25) (26).
16. ^ Boole: The Mathematical Analysis of Logic . Pp. 62-64, Prop. 1 with corollaries; he spoke here of “modules of a function”.
17. ^ William Stanley Jevons: Pure logic , London 1864, p. 26 (69) A + A as "A or A" with rule A + A = A.
18. ↑ On the correspondence between Boole and Jevons: Article George Boole , in: Stanford Encyclopedia of Philosophy, 5.1 Objections to Boole's Algebra of Logic.
19. Jump up ↑ Ernst Schröder: The Operations Group of the Logic Calculus , Leipzig 1877, pp. 8-17 (2) (3) (5) (6) (7).
20. ^ Giuseppe Peano: Calcolo geometrico , Torino 1888, pp. 3–5, in: Boolean Algebra # Definition
21. ↑ Ivan Ivanovich Shegalkin: О технике вычислений предложений в символической логике, in: Matematicheskij Sbornik 34 (1927), 9-28; there p. 11f the axiom system [1]