Ramble, Part 51: Up the Ladder
May. 30th, 2008 02:29 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowRather than continue with examples of the increasing extension and abstraction of algebra in the 19th century, I'd like to conclude this segment with an overview.
Number theory - the study of the whole numbers - probably has the strongest claim to being the progenitor of algebra, from Pythagoras on. Thus, it is appropriate to examine the path that led from number theory to abstract algebra.
At the hands of Fermat and his successors, questions concerning the representation of numbers as, for example, sums of squares had flowered. Gauss showed that, to study such questions, it was useful to consider other numbers and number-like objects - the Gaussian integers, integers modulo n, and so on. Lamé's work on Fermat's last theorem, and the work of Kummer and others who were inspired by it, revealed many more analogous situations. They also made it clear that what was important was not the numbers per se, but rather the structures - the rings - into which they were organized. The focus of number theorists then shifted from numbers to rings of numbers: collections of numbers which could be added, subtracted, or multiplied, to produce other numbers in the same collection. The similarities among, and differences between, these rings became of paramount interest.
At roughly the same time, the emergence of linear algebra presented mathematicians with new kinds of collection, with different operations and different properties. In vector spaces, addition and subtraction were available, but multiplication per se was not (although various analogous constructions were possible). Matrices - originally, merely shorthand devices for describing transformations of vector spaces - were seen to allow other operations, analogues of addition, subtraction, and multiplication, but with peculiarly different properties. The analogies between these and the rings that had arisen in number theory were noticed and studied.
Meanwhile, Évariste Galois, studying the problem of solving algebraic equations, introduced the notion of a transformation group, a set of transformations with only one operation, composition - i.e., following one transformation with another to produce yet another transformation. (For example: flipping a sheet of paper over a horizontal axis is one transformation; flipping it over a vertical axis is another; performing the two flips in succession has the effect of rotating the sheet through a half-turn. That is, the composition of the two flips is the rotation.) Transformation groups behaved in very different ways from the rings and vector spaces mentioned above, but there was a family resemblance: in each case, what was being studied was a set equipped with operations, satisfying certain rules.
Other, similar objects began appearing in profusion later in the century. Lie algebras - like rings, except that the "multiplication" was not even associative - emerged from mathematical physics; the Cayley Octonions arose from attempts to push Hamilton's quaternions yet further; Hermann Grassmann showed how to construct whole families of "hypercomplex numbers"; students of projective geometry devised "ternary rings"; on and on. The great Emmy Noether declared the worth of "abstract algebra", the study of all such structures however peculiar, without concern for any inspiration from geometry, analysis, or even number theory, and the field mushroomed. Jordan algebras, semigroups, groupoids, loops, universal algebras, monoids, superalgebras - the list grew ever longer.
Noether's proclamation held a danger: the detaching of algebra from its (more or less) concrete roots led to the risk of the fantastic elaboration of the utterly useless. The day was to come when a major work on "algebraic K-theory" was derided as "three monkeys algebra: see no geometry, hear no topology, do no analysis", and mocking satires on "hemidemisemiflipoids" were penned. Abstract algebra as described by Noether was a hothouse growth; in the last half-century or so, the strange fruits it produced have been brought outside. Some have withered, but many have proven enormously useful.
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Number theory - the study of the whole numbers - probably has the strongest claim to being the progenitor of algebra, from Pythagoras on. Thus, it is appropriate to examine the path that led from number theory to abstract algebra.
At the hands of Fermat and his successors, questions concerning the representation of numbers as, for example, sums of squares had flowered. Gauss showed that, to study such questions, it was useful to consider other numbers and number-like objects - the Gaussian integers, integers modulo n, and so on. Lamé's work on Fermat's last theorem, and the work of Kummer and others who were inspired by it, revealed many more analogous situations. They also made it clear that what was important was not the numbers per se, but rather the structures - the rings - into which they were organized. The focus of number theorists then shifted from numbers to rings of numbers: collections of numbers which could be added, subtracted, or multiplied, to produce other numbers in the same collection. The similarities among, and differences between, these rings became of paramount interest.
At roughly the same time, the emergence of linear algebra presented mathematicians with new kinds of collection, with different operations and different properties. In vector spaces, addition and subtraction were available, but multiplication per se was not (although various analogous constructions were possible). Matrices - originally, merely shorthand devices for describing transformations of vector spaces - were seen to allow other operations, analogues of addition, subtraction, and multiplication, but with peculiarly different properties. The analogies between these and the rings that had arisen in number theory were noticed and studied.
Meanwhile, Évariste Galois, studying the problem of solving algebraic equations, introduced the notion of a transformation group, a set of transformations with only one operation, composition - i.e., following one transformation with another to produce yet another transformation. (For example: flipping a sheet of paper over a horizontal axis is one transformation; flipping it over a vertical axis is another; performing the two flips in succession has the effect of rotating the sheet through a half-turn. That is, the composition of the two flips is the rotation.) Transformation groups behaved in very different ways from the rings and vector spaces mentioned above, but there was a family resemblance: in each case, what was being studied was a set equipped with operations, satisfying certain rules.
Other, similar objects began appearing in profusion later in the century. Lie algebras - like rings, except that the "multiplication" was not even associative - emerged from mathematical physics; the Cayley Octonions arose from attempts to push Hamilton's quaternions yet further; Hermann Grassmann showed how to construct whole families of "hypercomplex numbers"; students of projective geometry devised "ternary rings"; on and on. The great Emmy Noether declared the worth of "abstract algebra", the study of all such structures however peculiar, without concern for any inspiration from geometry, analysis, or even number theory, and the field mushroomed. Jordan algebras, semigroups, groupoids, loops, universal algebras, monoids, superalgebras - the list grew ever longer.
Noether's proclamation held a danger: the detaching of algebra from its (more or less) concrete roots led to the risk of the fantastic elaboration of the utterly useless. The day was to come when a major work on "algebraic K-theory" was derided as "three monkeys algebra: see no geometry, hear no topology, do no analysis", and mocking satires on "hemidemisemiflipoids" were penned. Abstract algebra as described by Noether was a hothouse growth; in the last half-century or so, the strange fruits it produced have been brought outside. Some have withered, but many have proven enormously useful.
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