Victorian Maths

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GB Teddy Has His ‘Religious Experience” in 1832 in Doncaster

 

George Boole’s 1847 “Mathematical Analysis of Logic

 

NINETEENTH CENTURY MATHS

The early 19th century in Britain was a time when maths recovered from a prolonged period of stagnation. English maths had tended to be too insular and prolonged its adherence to Newtonian notation and approaches to calculus. European mathematics made use of Leibniz’s notation and had progressed much further as a result.1

In 1812, some of the best Cambridge mathematicians came together to form a group dedicated to promoting the continental notation for calculus. They called themselves “The Analytical Society”.2 Members of the Society included Herschel, Peacock and Babbage; among later followers were DeMorgan and Gregory both key in their different ways to the development of George Boole as a nineteenth century mathematician.3 One of the four key founding members was E.F. Bromhead* who, through illness, had to leave Cambridge before graduating.4 Luckily for Lincoln and mathematics, he ‘retired’ to his family seat in Lincolnshire; Thurlby Hall. Here he engaged in various social and intellectual projects. One was to encourage George Green, the Nottinghamshire miller’s son, to pursue his genius by going to Cambridge University.5 Another was to encourage George Boole to develop his mathematical genius and publish in the right journals.

So, by the 1820s, it was accepted in the highest of British mathematical circles (firstly in Ireland, then the rest of Britain6) that ‘continental’ texts on advanced maths needed to be translated for English-speaking students and that english publications using Leibniz’s notation were also required.7 Of course, highly educated men, such as those in the Analytical Society read the European texts in their original languages. Boole, with his early interest in languages could do this too.8

Cambridge was at the forefront of the development of advanced mathematics in Britain, this was partly due to the changes wrought by the Analytical Society, and partly to the peculiarities of the assessment procedure for undergraduates. The “Tripos”, as it was called, put mathematics at the heart of university education9 and this resulted in a mathematical ‘hot-house’ process which not only pushed talented young men to extreme mental feats, it also attracted really mathematically gifted youngsters in a spiralling of achievement.10 The best of the graduates could become teachers in the university the next year and some went straight into the top scientific posts in the land.11

The Victorian period, from the middle of the 19th century, was a time of rapid growth in science and technology. This created a huge interest in the mathematical underpinnings of science and both pure and applied maths as a result.12 In the early part of the century, most mathematically literate people were clerics, or had been trained as clerics, in the second half of the century men of God became less involved and professional mathematicians and scientists became the main audience for new mathematical ideas.13

 

GEORGE BOOLE’S MATHS

George Boole was famously moved to consider logic by an argument between one of his best friends, Augustus De Morgan (of De Morgan’s law fame) and Sir William Hamilton the Scottish logician.14 However, in addressing the issue of a coherent set of logical relations and the terminology required, Boole drew upon an insight which he claimed to have had when only 16.15

Boole spent most of his childhood being more concerned with languages than maths. Although precocious in all things intellectual and having completed the maths required of minimum university entry by the age of 10, young George spent the years prior to his first job in Doncaster studying languages.16 Latin was accompanied by Greek, Hebrew, French German and Italian. At 14, he was translating classical greek poetry.17 All this personal study of various languages had led him to wonder -as would Chomsky a century later,18 “could there be an underlying grammar, a shared ‘deep’ structure to all God-given language?” (the God part was quite important to Boole19). In Doncaster, having been bitten by the bug of calculus20 (which he was reading in French from a borrowed copy of Lacroix21) he had the understandable insight “could there be an underlying set of rules and relations which underpin all God-given mathematical truths?”. Einstein was to have a similar question a century later too22.

Fifteen years of mathematical endeavour later, Boole drew upon his earlier insight when in 1847 he published his “Mathematical Analysis of Logic”, hoping to smooth the argument between DeMorgan, Hamilton and their respective followers. (Remarkably, Boole and DeMorgan’s texts were published at the same time. DeMorgan had scrupulously kept both Boole’s and his own texts secret from one another to protect each other’s claims to originality. As the Professor at The University of London, he was well placed to eclipse Boole, it’s a measure of the man that he made sure he didn’t.23) Boole’s approach had better notation (developed and improved in his second book on logic “The Laws of Thought”). It also covered more ground and formed a more coherent whole. As is so often the case, almost as soon as he was published, Boole could see necessary and possible improvements to his text. He promised them in a discussion of his book published in “The Mathematical Journal” a few months after the book’s publication.24 The Journal, with its circulation to serious mathematicians worldwide was the key publication for the wider dissemination of Boole’s ideas and it primed that audience -already used to articles by Boole- to expect a more thorough work on logic from him.25

Of George’s forty-plus journal articles on mathematical topics, this ‘post “Analysis” article is the only substantial one he published on his logic. Most of his articles are on aspects of advanced calculus, the area for which he gained his Royal Society medal and his initial recognition as a gifted mathematician.26 Ironically, his efforts in this area have long-since been superceded or incorporated into the unremarkable practice of calculus.27 Despite his continued efforts in calculus it was his true/false logic that drew most plaudits from the wider academic world, leading to two honorary postdoctoral awards and the Fellowship of the Royal Society.28 The impact of his ideas on logic was largely felt after his second long work on the subject, promised in the 1848 Journal article, but not delivered until 1854 “An Investigation of The Laws of Thought”. Due to his untimely death in 1864 he never publicly returned to his Logic but we know he tried to write more several times. After his death, DeMorgan considered editing and publishing his work on the topic but considered that the notes and half-finished papers Boole had left behind would not add substantially to his existing work on logic.29

However, it was over threequarters of a century before Boolean Logic found its eventual home as the bedrock of electronic computing.

*Link needs academic library membership -or personal membership of ODNB, Wikipedia is an alternative starting point.

 

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