The Thorough Axiomatization of Algebra
The Thorough Axiomatization of Algebra
Late-nineteenth-century concerns about the meaning of number led, in the early twentieth century, to attempts to provide an axiomatic foundation for algebra and the theory of numbers. Building on the earlier work of Niels Abel and Evariste Galois on algebraic equations and using the framework provided by Cantor's theory of sets, a group of mathematicians led by Emmy Noether formalized definitions for a number of algebraic structures. Emphasis in mathematical research shifted from finding solutions to equations to the structures that such sets of solutions exhibit. The theory of groups, in particular, provided an important tool for theoretical physics.
As mathematics has developed over the centuries, the concept of number has been broadened and generalized. Originally only the counting numbers, 1,2,3,..., or positive integers were recognized. The introduction of the zero and the negative integers turned arithmetic into a far more powerful tool for commerce, while the introduction of rational numbers or fractions allowed for a notion of proportion that was essential to the development of geometry and architecture.
Geometry, and especially the Pythagorean theorem, brought to light the existence of irrational numbers that could not be expressed as the ratio of two integers. Greek mathematicians knew that the square root of two is irrational but were disturbed by the knowledge. The disciples of Pythagoras (582-497 b.c.) treated this fact as a secret to be kept within the group.
With the development of algebra during the Renaissance, there appeared a need for numbers to represent the square roots of the negative numbers. A fully satisfactory treatment of complex numbers was not attained until the early nineteenth century, by which time other interesting generalizations of the number concept—quaternions and matrices, for example—were appearing.
A further classification of numbers arose from studies of algebraic equations involving rational coefficients and whole number powers of one unknown. Exact solutions had been found for the quadratic, cubic, and quartic equations by the sixteenth century. These solutions were expressed in terms of rational numbers and radicals—the square root, cube root, or fourth root of a positive or negative rational number. It came then as somewhat of a surprise when Norwegian mathematician Niels Henrik Abel (1802-1829) published a proof that the general fifth order equation did not have a solution of this form. Studies of the cases in which higher order equations were so solvable by the young mathematical prodigy Evariste Galois (1811-1832) involved studying the behavior of the solutions under a group of interchange operations, and gave rise to the group concept.
The group concept would be formalized further, making use of the set theory introduced by German mathematician Georg Cantor (1845-1918). A group is a set on which an operation is defined that matches each pair of elements in the set with one element of the set. Using the notation ☉ to indicate the operation, so that "a☉b = c" means that element c is produced when a is allowed to operate on b, the set is said to form a group if three conditions are met:
(1) for all elements a, b, and c, in the set, a☉(b☉c) = (a☉b)☉c , where the operation in parenthesis is understood to be performed first
(2) there is an identity element i, such that a☉i = i☉a = a for every element a in the set
(3) for every element a in the set there is an inverse element a', such that a☉a′ = a′☉a = i
These properties may then be taken as the axioms defining a group. An example of a group is provided by the integers—positive, negative, and zero—with ordinary addition as the operation and zero as the identity. The first property is termed the associative property since it states that the result of two operations is independent of how the terms are grouped. Groups with an operation that obeys a commutative axiom are called Abelian groups after Abel. These are written thus:
(4) for all elements a, b in the group a☉b = b☉a
A ring is a set for which two operations have been defined, which we might denote as ⨁ and ⨂, with the set being an Abelian group under the ⨁ operation, while the operation ⨂ is associative, and there is an identity operation for it as well. Further, there is a distributive axiom
(5) a⨂(b⨁c) = (a⨂b)⨁(a⨂c)
The integers under ordinary addition and multiplication are a ring. The power of the ring concept was highlighted in 1907, when Joseph Wedderburn (1882-1948), a professor of mathematics at Princeton University, published a paper treating hypercomplex numbers, generalizations of the complex numbers and quaternions, from the field point of view.
A field is a ring for which the ⨂ operation is commutative and for which
(6) if c is not zero, and a⨂c = b⨂c, then a = c
(7) every element except zero has an inverse under ⨂
The real numbers, but not the integers alone, constitute a field.
An ideal is a subset of a ring that is itself a ring and has the property that when any of its elements is multiplied by any element of the parent ring, a member of the subset results.
As the real numbers are an example of a ring, so the subset of real numbers that can appear as solutions of a given type of algebraic equation generate an ideal. German mathematician Emmy Noether (1882-1935) adapted the theory of rings and ideals to the study of algebraic equations, in the process developing the abstract description of these structures. Austrian-born mathematician Emil Artin (1898-1962), who spent a year working with Noether at the University of Göttingen, continued to contribute to field theory throughout his long career. Dutch mathematician Bartel Leendert van der Waarden (1903- ) was also an important contributor to the axiomatic school that developed around Noether.
The career of Emmy Noether reflects many of the difficulties facing female scholars at the beginning of the twentieth century, and also the rapid change in the status of women at about the time of the First World War. The daughter of a university mathematics professor, she was allowed to attend classes at the University of Erlangen but not permitted to enroll as a student because of her sex. By 1907 conditions had changed somewhat so that she could receive the doctorate degree from the university. From then until 1915 she wrote mathematical papers, supervised the research of advanced students, and occasionally took her father's place in the classroom, but without any official position or salary. In 1916 she joined the great German mathematician David Hilbert (1862-1943) at the University of Göttingen. Despite Hilbert's strenuous efforts, he was unable to obtain a salaried position for her until 1923. For the next decade she was one of the most influential mathematicians in Europe, until, like so many other intellectuals of Jewish descent, she fled the Nazi regime in 1933 for the United States. In America she accepted a teaching position at Bryn Mawr College, a select college for women, and became affiliated with the Institute for Advanced Study in Princeton, New Jersey, but died following surgery two years later.
Noether made important contributions to theoretical physics as well as to pure mathematics. Group theory provides a natural description of the symmetries of objects and of space itself. The set of possible rotations of a three-dimensional coordinate system, for example, form non-commuting groups. Noether was able to show that the independence of the laws of physics under rotations in space leads directly to one of the fundamental conservation laws in physics, that of the conservation of angular momentum. Further, the independence of the laws of physics under displacements in space or time, leads, respectively, to the laws of conservation of linear momentum and energy. Modern particle physics relies heavily on the connection between conservation laws and symmetry in developing theories of the behavior of elementary particles. Several new conservation laws have since been discovered, each connected to a basic symmetry of the underlying equations of motion. The importance of the fundamental relationship between symmetry and conservation is generally recognized, and the basic idea is included in many texts for physics undergraduates. Ironically, the individual responsible for this important theoretical insight is usually not identified.
Enthusiasm among mathematicians for group theory was perhaps excessive. The eminent French mathematician Jules Henri Poincaré (1854-1912) had said, "The theory of groups is, as it were, the whole of mathematics stripped of its subject matter and reduced to pure form." By 1930 it was clear that group theory, though important , would not continue to be seen as the essence of mathematics. At about the same time, group theory was embraced by the new field of quantum physics as an essential tool in understanding the quantum states of atoms, molecules, and nuclei. The more complex structures of rings, fields (in the mathematical sense), and ideals, have had some impact on mathematical physics, but not as much.
The axiomatization of algebra and its emphasis on the associative, commutative, and distributive laws also had an impact on elementary and high school mathematics teaching in the United States In the 1950s and 1960s a movement often described as the "New Math" spread through many school districts. Designed by university professors of mathematics and research mathematicians, the new curriculum emphasized understanding of the basic concepts of mathematics and the properties of numbers rather than the memorization of addition and multiplication facts. After some criticism from within the mathematics community, including the publication of a popular book, Why Johnny Can't Add, by New York University Professor Morris Kline, it became generally accepted that grade school students were not yet ready to acquire such advanced concepts. The effort was gradually replaced with an emphasis on the discovery of mathematical ideas by solving practical problems.
DONALD R. FRANCESCHETTI
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