stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2007-12-22 06:39 pm
stoutfellow) wrote2007-12-22 06:39 pm
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Ramble, Part 40: Meanwhile, Off the Coast
While continental mathematicians were wrestling with the idea of the complex numbers, mathematics in Great Britain was undergoing a curious retrograde period. Where Gauss was leading the way to the acceptance of complex numbers, a movement led by one Francis Maseres was rejecting even the negative numbers. British mathematics had been in the doldrums since the Leibniz-Newton quarrel, but this was not merely stagnation; it was regression. Fortunately for Britain and for the mathematical world, it resulted in a powerful and conceptually explosive reaction. More under the cut.
Maseres' position is epitomized by the following quote:
Peacock had been pushing for reform in British mathematics for some time. Along with Charles Babbage, who conceived of the Difference Engine, a forerunner of the modern computer, and John Herschel, son of the famous astronomer William Herschel and a redoubtable astronomer in his own right, Peacock founded the Analytical Society, aimed at introducing the more flexible Leibnizian notation to British mathematics. Peacock himself, having attained an examiner's position at Cambridge, was able to force the change to begin there, and other universities soon followed suit.
But it is Peacock's work in algebra that deserves more attention. His efforts began as a rebuttal of Maseres, but ended in establishing, for the first time, the axiomatic method in algebra. He began with something of an end run on Maseres' position. Conceding that it made sense to subtract a number a from a number b only when b was greater than a, Peacock made a distinction between arithmetical algebra, in which the symbols are taken as names of unspecified numbers, and symbolical algebra, in which they are abstracted away from any such assignment. He then sought for laws which hold in arithmetical algebra, so long as the expressions make sense - e.g., that a-(b-c)=(a-b)+c, so long as b>c and a>b - and declared that in symbolical algebra, such laws must be taken as holding in general. This he called the Principle of the Permanence of Equivalent Forms; in essence, it is the kind of rule-of-thumb for generalization I have spoken of before.
On the basis of this principle, Peacock went on to establish a set of rules for algebra, including the familiar commutative, associative, and distributive laws. Not that these were previously unknown; they can be found, in geometric form, in Euclid. There, though, they were presented as theorems, derivable from the axioms of geometry; Peacock proposed treating them as axioms in their own right. He envisioned algebra acquiring the grand structure that geometry had long had; and so it did. But mathematics was entering a time of great and rapid change, and the end result was not something Peacock could have foreseen.
Peacock is largely forgotten now, but he intervened crucially at a key moment in mathematical history, and he deserves wider attention.
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Maseres' position is epitomized by the following quote:
If any single quantity is marked either with the sign + or the sign - without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of -5, or the product of -5 into -5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon.To a modern mathematician, this seems an extremely bizarre stance, but it dominated British mathematics for several decades in the late 18th and early 19th centuries. It was finally broken by a reformer by the name of George Peacock.
Peacock had been pushing for reform in British mathematics for some time. Along with Charles Babbage, who conceived of the Difference Engine, a forerunner of the modern computer, and John Herschel, son of the famous astronomer William Herschel and a redoubtable astronomer in his own right, Peacock founded the Analytical Society, aimed at introducing the more flexible Leibnizian notation to British mathematics. Peacock himself, having attained an examiner's position at Cambridge, was able to force the change to begin there, and other universities soon followed suit.
But it is Peacock's work in algebra that deserves more attention. His efforts began as a rebuttal of Maseres, but ended in establishing, for the first time, the axiomatic method in algebra. He began with something of an end run on Maseres' position. Conceding that it made sense to subtract a number a from a number b only when b was greater than a, Peacock made a distinction between arithmetical algebra, in which the symbols are taken as names of unspecified numbers, and symbolical algebra, in which they are abstracted away from any such assignment. He then sought for laws which hold in arithmetical algebra, so long as the expressions make sense - e.g., that a-(b-c)=(a-b)+c, so long as b>c and a>b - and declared that in symbolical algebra, such laws must be taken as holding in general. This he called the Principle of the Permanence of Equivalent Forms; in essence, it is the kind of rule-of-thumb for generalization I have spoken of before.
On the basis of this principle, Peacock went on to establish a set of rules for algebra, including the familiar commutative, associative, and distributive laws. Not that these were previously unknown; they can be found, in geometric form, in Euclid. There, though, they were presented as theorems, derivable from the axioms of geometry; Peacock proposed treating them as axioms in their own right. He envisioned algebra acquiring the grand structure that geometry had long had; and so it did. But mathematics was entering a time of great and rapid change, and the end result was not something Peacock could have foreseen.
Peacock is largely forgotten now, but he intervened crucially at a key moment in mathematical history, and he deserves wider attention.
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