Boole's ingenious ideas are revealed in particular in the good notation he employed. Usually few attention is paid to mathematical notation, but in general it is crucial in problem solving to adopt the right notation that fully captures the essentials of the problem. This article aims to give a survey on the development of the notation underlying Boole's Algebra. It puts together bits of pieces to narrate a brief history of the notation of Boole's Algebra. Starting from the development of algebraic notation from Arabian times, it describes how the notation and laws were nally freed from their arithmetic interpretation.
Furthermore, we sketch De Morgan's and Boole's lives and works and show how Boole connected algebra and logic. Euclid's theorem states that the sum of the coloured squares and the two remaining rectangles is equal to the whole square. In the case of al-Khwarizm 's problem, the whole square has units since the straight line has ten units. The two coloured squares on the segments have fty eight units and so al-Khwarizm concludes that each rectangle amounts to twenty one units see Figure 1.
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To complete the solution of the problem we quote from Rosen's translation of al-Khwarizm 's Algebra: What must be the ammount of a square, which, when twentyone dirhems are added to it, becomes equal to the equivalent of ten roots of that square? Solution: Halve the number of the roots; the moiety is ve. Multiply this by itself; the product is twenty- ve. Subtract from this the twenty-one which are connected with the square; the remainder is four.
Extract its root; it is two. Subtract this from the moiety of the root, which is ve; the remainder is three. This is the root of the square which you required, and the square is nine.
George Boole, The Boole-De Morgan Correspondence, - PhilPapers
Or you may add the root of the moiety of the roots; the sum is seven; this is the root of the square which you sought for, and the square itself is forty-nine4. Due to lack of any symbols and notation the problem and solution are di cult to understand.
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They are based on a gurative approach as shown on the right in Figure 1. There, the base line is divided into two parts of equal size so that the rectangle bt equals md. Since bg covers twenty 3 Katz , p. Furthermore we know that the square cg equals twenty ve units, so that cn equals four and cm two. Finally, we subtract two from ve, the length of ct, and obtain the solution three and subsequently seven.
Such a gurative solution is straight-forward, but di cult to understand, since it does not use any notation for problem solving. The Arabian knowledge, being itself based on its Babylonian, Egyptian, Greek, and Indian roots, slowly spread through Italy over Europe. Concerning notation, he de wrote for example our 14? He even allowed zero and negative exponents so that 9x0 and 9x?
But the work was not printed until , so that it was of little in uence. The two symbols may have originated from marks chalked on chests of merchandise in German warehouses; they indicated the variation of the standard weight. Here the French Francois Viete introduced a convention as simple as it was fruitful: He used a vowel for unknown quantities and a consonant for known ones.
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But unfortunately, he did not use the advanced notation developed by his predecessors. For the rst time all these notations and conventions were brought together by the French philosopher and mathematician Rene Descartes All in all, he is the rst author whose writings are readable for us. Leibniz carried out tremendous work in philosophy, mathematics, law, politics, diplomacy, and history.
One particular project, that was also motivated by Leibniz' passion for world- and religious peace,7 concerned a philosophical language free of ambiguities. Thus it would help to facilitate human communication and understanding and would nally end all controversies that depend on reasoning. And if the syntactical level reveals the semantics of the word, then the language must be understandable everywhere because its words are not arbitrary anymore.
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However, Leibniz' project was even more ambitious. Leibniz assigned characteristic numbers to concepts. Basic concepts are assigned prime numbers and complex concepts non-primes, so that the composition of the complex concept out of basic ones is exactly revealed by the multiplication of primes giving a non-prime characteristic. In a similar fashion, he argues that an ape having the characteristic 10 is not 7 Eco , p. But both have 2, i. However, he was the rst one who tried to tie philosophy and mathematics together and who claimed that reasoning can be reduced to arithmetic calculation and thus be mechanised.
Both had developed the in nitesimal calculus independently. Newton's method of uxions, denoted by a dot, was motivated by physical considerations and his notation was quite unhandy. In contrast, Leibniz, who was more interested in de ning an abstract and elegant calculus, developed the notation of di erentials dx as used today. At about a controversy started, ending with Leibniz and Newton accusing each other of plagiarism. Thereupon British mathematics stuck to Newton's method and fell back behind Continental mathematics where Leibniz' notation led to fast progress.
Since the industrial revolution changed British society dramatically.
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The new inventions such as steam engine and railway and the growing enterprises produced a great need of engineering skills and indirectly also of mathematical education. In the second half of the 19th century Great Britain, then a wealthy and relatively safe empire, reacted with the formation of many colleges and universities.
Cambridge played a key role for the change of paradigm that overcame Britain's neglected symbolic access to mathematics. The new interest in symbolic manipulation opened new avenues in symbolic algebra. An important step towards a separation of algebra and arithmetic was taken by Peacock. De Morgan believed that one can create an algebraic system with arbitrary symbols and a set of laws under which these symbols are manipulated. Only afterwards one would give an interpretation of the laws and manipulations.
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Here is a symbolic calculus: how can it be made a signi cant one? In the following ways, among others. His family moved soon to England, where they lived rst at Worcester and then at Taunton. His early education was in private schools where he enjoyed a classic education containing Latin, Greek, Hebrew, and mathematics. In , he graduated as a fourth Wrangler and turned his back to mathematics to study to be a lawyer at Lincoln's Inn in London. But already a year later he revised this decision and applied for a position as professor of mathematics at the newly established University College in London.
At the age of 22, having no publications, he was appointed. The work that De Morgan produced in the years to come spanned a wide variety of subjects with an emphasis on algebra and logic. But surprisingly he was not able to connect them. In contrast to Boole, De Morgan was a traditional logician, who knew the medieval theory of logic and semantics.
But his main contributions are found in the theory of syllogisms, where he was the rst one since medieval times who extensively discussed quanti ed relations. He recognized that relational inferences are the core of mathematical inference and scienti c reasoning. In these works, he uses letters X , Y , Z , etc. De Morgan seemed not to be aware of the importance of a good notation, because he chose nearly arbitrarily the symbols to represent the traditional categorical form AOEI , as shown in Figure 2, and the symbols for E and I coincide even with disjunction and conjunction.
It is obvious that this inappropriate notation limited De Morgan's work. He worked on the history of mathematics and wrote biographies on Newton and Halley and a dictionary of mathematicians of the 17th century. He was also engaged in many associations. His diversity also covered educational subjects: He wrote essays on mathematical education, the concept of an Ecole Polytechnique and, the education of the deaf and dumb. De Morgan's professional career was not smooth and he proved a strong and sometimes eccentric personality in academic life.
Edited with an introductory study by Howard Foster Lowry.
Oxford: Clarendon Press, Boole, George. The Boole-De Morgan Correspondence, Edited by G. Bradley, Henry. Collected papers, with a memoir by Robert Bridges. Bridges, Robert. Correspondence of Robert Bridges and Henry Bradley, With a selection of letters by family and friends, edited by Margaret Smith.
Browning, Robert and Elizabeth Barrett.
Related The Boole-De Morgan Correspondence, 1842-1864
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