I don't know of any other subject which is taught in such an anti-historical way as mathematics. Although mathematicians are often fairly scrupulous in giving credit to the original discoverers of theorems, they also are energetic in restating these theorems in terms of concepts which the original discoverers would have been completely unfamiliar with.

When Emil Artin taught Galois Theory, he did apparently discuss Galois's own approach. He tells an anecdote to the effect that he asked one of his classes how much of his book on the subject Galois himself would have recognized, and one of his students suggested that probably the title would have been the only recognizable thing in the whole book. And then another student said, "No, he probably would say, 'Okay, Galois, that's me, but who's this guy Theory?' "

Artin's teaching in this respect was exceptional. In general, the teaching of mathematics gives students little way of understanding where mathematical ideas have come from and what the original motivation for the development of various mathematical topics was.

Graduate students learn all sorts of high-powered concepts and theorems about Banach spaces, for instance, before they ever have any idea of why mathematicians ever got interested in such spaces or what the theory they are learning is good for. (Many students never do learn this.)

In my opinion this has a lot to do with the fact that today we see a splintering of mathematics into zillions of tiny little subspecialties, many of whose practitioners know almost nothing about any mathematics except their own little splinter.

I am not a historian of mathematics by any means. Here, I simply
present a brief sketch of the development of modern Algebra
(sometimes called Abstract Algebra) taken from the book by Bourbaki,
*Elements of the History of Mathematics* (French title,
*Elélements d'Histoire des Mathématiques*).

We generally don't teach students how revolutionary the axiomatic approach is. Typically, in an undergraduate course in Modern Algebra (what is often referred to as a Herstein level course), we simply take the axiomatic approach for granted from Day One. This approach is so familiar and so comfortable to a contemporary mathematician that we seldom give much thought about how bewildering it is to students whose previous experience of mathematics has been limited to courses like calculus. (However sometimes students have seen a little of the axiomatic approach in Linear Algebra, but with very little explanation about why it was ever decided to deal with a seemingly concrete topic like vector spaces in such an abstract way.)

The axiomatic approach is not simply a matter of using axioms in mathematics. The use of axioms, after all, goes back as far as Euclid.

But Euclid, before giving his axioms, starts out by defining the primitive notions of geometry. A point is defined as, roughly speaking, "That which has position but no size." And a line is defined as, "That which has length but no breadth." (I don't remember how Euclid goes about defining the concept of straightness. I don't think he defined a straight line as being the shortest distance between two points.)

The contemporary axiomatic approach, on the other hand, is basically the attitude that when we do mathematics, we don't need to know what the things we are working with are. We only need to know what the rules are. (So that in geometry, we don't need to know what a point or a line is. We only need to know the axioms.)

This is very different from the teaching of mathematics in grammar school and high school and college calculus courses. There, it is considered very important that students understand what numbers are (albeit in a way that to mathematicians is shockingly informal) and what addition, subtraction, and multiplication are, before learning the rules that enable one to actually do arithmetic. And it is very important to thoroughly master arithmetic before going on to represent it in symbolic form in high school algebra. And it is very important to be familiar with a number of specific examples of functions and to understand the concepts of differentiation and integration before going on to learn the rules which enable one to actually differentiate and integrate functions. (In fact, calculus teachers are often annoyed when students, inventing the axiomatic approach on their own, as it were, discover that it is not really necessary to understand the concepts in order to do the calculations.)

But a typical undergraduate course in Modern Algebra starts out saying something like, "A group consists of a set of elements which can be multiplied in such as a way that the following three axioms are satisfied." (Four axioms if one includes closure, which was in fact the key axiom and to some extent was the only one in the original development of group theory, associativity and the existence of an identity element and inverses being taken as pretty much self-evident.)

It is understandable that a student might ask in bewilderment, "But what are these elements? And how does this multiplication work?" And the answer given by the Axiomatic Approach is, "It doesn't matter. Only the rules matter."

This attitude, that it is possible to study things without knowing what one is talking about, is an incredible cognitive leap, and it is the real foundation (not the mathematical foundation, but the psychological foundation) of abstract mathematics.

Of course in order to provide students with the sense that there is some tangible reality to what we are talking about, we immediately provide them with some familiar examples of groups (or rings, or whatever). And one strategy students might use when they can't cope with the level of abstraction they are given is to say, "Okay, when the professor says 'group,' I'm going to think he's talking about the integers. And when he says, 'multiplication,' I'm going to think about addition." (I've used this strategy myself sometimes, when learning a new kind of mathematics.) But this strategy doesn't work very well. It's misleading, because any particular example will have a number of special properties that will not be true of groups in general. (Addition of integers is commutative, for example, and the integers form a cyclic group.) And so if one needs to have concrete things to think about (and I think that almost all of us do), one needs to think not just in terms of one example, but in terms of a number of very dissimilar examples.

And my own experience was that even after I got to be good at this sort of abstract thinking, I would still come to certain concepts (such as the concept of the free product of non-abelian groups, or that of the tensor product) that were so abstract and where it was so difficult to find any natural examples, that for quite a while I still found them very difficult to think about.

But here the axiomatic approach can rescue you on a higher level. You don't really need to think about what a free product or a tensor product actually is ("what it looks like," in my words). You simply need to find a set of axioms that describe the way it behaves. (This has become pretty much the standard way of thinking about tensor products, and I always felt a sort of contempt for mathematicians who proved theorems about tensor products by starting with the construction.)

This is certainly one advantage of the axiomatic approach: that one can work with quite complicated objects (and most mathematical constructions, even the natural numbers, are actually quite complicated) without needing to think about what they "look like." But the primary advantage of the approach is that usually a single set of axioms will describe a very large number of vastly dissimilar mathematical systems, and so by starting from axioms, one can prove theorems that apply to a huge number of different things. (Most of us don't make a big point of using the axiomatic characterization of the real numbers, for instance, because the field of real numbers is the only mathematical thing to which that complete set of axioms applies.)

But how did this revolutionary new way of mathematical thinking come about?

It came about, actually, in a very gradual and somewhat natural way. It came about because over the course of the 19th century, mathematicians started becoming more and more interested in a new kind of subject matter having to do with algebra, but not algebra in the sense of solving equations (although the interest in solving algebraic equations was certainly one of the roots of this new interest). But rather this was algebra in more or less the sense we use the word today (but without thinking of it in abstract terms), namely the study of structures in which one could work in very much the same way that traditional algebra operates in the realm of rational numbers, real numbers, or complex numbers. Some of these structures were: the complex numbers, the quaternions, various algebraic number rings (certain subrings of the complex numbers), in addition to the algebra of matrices developed by Sylvester and Cayley and the algebra of logic developed by Boole. In addition, there was the study of permutation groups, which was originally not thought of as being algebra at all, I believe, but where the basic concepts were developed by Legendre, Abel, and Galois as an approach to understanding the solution of algebraic equations.

All of these subjects were originally studied for very natural and practical reasons having to do with questions in geometry, analysis, number theory, and the theory of equations.

What was new about all these subjects was the interest primarily in the structure as a whole, rather than in doing calculations within that structure. This was perhaps especially clear in the work of Legendre, Abel, and Galois on permutation groups, where what was important was the set of subgroups rather than the individual permutations.

Bourbaki identifies three main streams leading to the development of modern Algebra: (1) The theory of algebraic numbers, developed by Gauss, Dedekind, Kronecker, and Hilbert. (2) The theory of groups of permutations (and, later, groups of geometric transformations), where the work of Galois and Abel was fundamental. (3) The development of linear algebra and hypercomplex systems.

But as group theory was further developed by other mathematicians (Galois himself, of course, was killed in a duel, apparently because of his political activism, immediately after finishing his treatise), gradually it started becoming clear that the study of permutation groups actually had very little to do with permutations themselves.

And Jordan in 1868 began the study of infinite groups, specifically groups consisting of transformations of geometric space. This study was continued by Felix Klein and Poicaré, and was especially encouraged by Felix Klein's Erlanger Program for geometry. (At this point, there were a number of different kinds of geometry, such as Euclidean geometry, non-Euclidean geometry, projective geometry, affine geometry, and differential geometry. Klein suggested that each particular form of geometry should be characterized as the study of those properties which are invariant under a particular group of transformations. For instance, Euclidean geometry consists of the study of those geometric properties which are not changed by rigid motions.) The concepts and theorems which had been developed for permutation groups applied just as well to these groups of transformations.

By the late 19th century, Cayley and Dedekind and many other mathematicians were becoming very aware that what was really relevant in group theory was the law of composition (multiplication) in a group and not the nature of the objects making up the group.

But the importance of groups at this point still had to do with their concrete applications. Groups were still seen as consisting of operators of some sort and Dedekind and Cayley stopped short of defining groups in an axiomatic way and seeing them as structures which were of interest for their own sake.

where it is easy to prove that none of four factors in these two decompositions can be further decomposed into products within the ring consisting of all numbers of the form a + b\sqrt{-5}, with a and b ordinary integers. (Here \sqrt{-5} is used to denote the square root of -5 , which is, of course, a complex number.)

The theory of algebraic numbers was further developed by Dirichlet, Hermite, Kummer, Kronecker, and Dedekind. Kronecker and Dedekind used two different methods (which although very dissimilar are ultimately equivalent) to introduce of certain "ideal numbers" into algebraic number rings to remedy the lack of unique factorization. Dedekind's method was the invention of what we today call, in an arbitrary ring, ideals. In his work, Dedekind basically established the foundations of modern commutative ring theory. However the methods of Dedekind and Kronecker fell short of providing a proof of Fermat's Last Theorem, although they did enable proofs in many special cases.

The other main thread leading to modern commutative ring theory came from algebraic geometry, and I won't really discuss that here except to mention that mathematicians were becoming very aware that the algebra of functions defined on an algebraic curve or surface had a great deal in common with algebraic number rings. Here we see that importance of the fact that mathematicians working in what originally seemed very different specialties were familiar with each others work and influenced by it. (There existed still a third major example of commutative rings, namely those consisting of functions defined by power series.)

Bourbaki identifies the 142 page article by Steinitz in 1910 titled The Algebraic Theory of Fields as having given birth to the modern concept of Algebra. (One can also note that much earlier, Peano, in 1888, gave the axiomatic definition of a real vector space and defined the concept of a linear transformation between vector spaces.) The word field had first been used by Dedekind, whose concern was with certain fields contained within the complex numbers (algebraic number fields). And it was Dedekind and Hilbert who had first seen Galois Theory as a correspondance between subfields and subgroups of the Galois group. (Dedekind was the first to think of the Galois group as consisting of the automorphisms of the field extension rather than permutations of the roots of the polynomial in question.)

Steinitz in his 1910 article developed the notions of prime field (by this time there had been a lot of work on the theory of finite fields), separable extension, and transcendence degree, and proved that every field has an algebraically closed extension. But what makes his article thoroughly modern is that instead of defining a field as a set of complex numbers or congruence classes or the like, Steinitz simply defined a field to be a structure consisting of a set of elements in which two operations are defined (to be referred to as addition and multiplication) satisfying a certain set of rules.

The concept of a ring was first used by Dedekind, who used the word "order" (or rather, of course, its German equivalent, "ordnung.") The word "ring" (which is actually the same in German and in English) was introduced by Hilbert. The point is that in an algebraic number ring (or any finite integral extension of a base ring), if one looks at the powers of an element then one finds a point where subsequent powers can be expressed as linear combinations of the preceding ones. Thus the multiplication in a certain sense turns back on itself in a way that is somewhat like a geometric ring.

But it was not until 1914 when the first paper where the general notion of a ring is defined axiomatically: "On Zero Divisors and Decomposition of Rings," by Fraenkel. Although this gave the general definition, the paper itself was concerned with commutative artinian local rings where the unique prime ideal is principal.

In the same year, 1914, Hausdorff in his *Grundzüge
der Mengenlehre*, gave an axiomatic definition of
general topology. Of course this was a time when algebraists,
analysts, and topologists talked to each other, were interested
in each other's work, were influenced by each other, and in many
cases were actually the same individuals.

In 1878, Frobenius proved that the quaternions were the only possible (finite-dimensional) associative extension of the complex numbers in which division was possible and the only (finite-dimensional) non-commutative extension of the real numbers in which division was possible. This was independently proved two years later by C. S. Pierce. (Gauss had been convinced that the field of complex numbers was the only finite dimensional commutative field extension of the real number system. This was subsequently proved by Weierstrass.)

Later Cayley noted that there exists a set of two by two matrices satisfyiing the multiplication table of the quaternions. (The concept of a matrix is due to Sylvester, who introduced matrices as a shorthand for substitutions of variables, i.e. what we now call linear transformations.) But not until about 1870 was it noted, by the Americans B. Pierce and C.S. Pierce, that the set of square matrices of a given size form an algebraic system which permits addition, subtraction, and multiplication (i.e., in modern terminology, a ring).

The term "an algebra" seems to have been used by the Americans and British in pretty much its modern sense, i.e. a ring which is a finite-dimensional vector space over the complex numbers (or real numbers). On the other hand, the Germans generally preferred the term "hypercomplex system." Aside from Hamilton's quaternions, the main example before 1850 was Grassman's "exterior algebras," but the analogy to the quaternions and other algebras was only much later seen.

Other examples of algebras over the complex numbers were seen during the period 1850 to 1860, but the general study of algebras (and thus the roots of non-commutative ring theory) begins only in 1870 in the work of B. Pierce and C.S. Pierce, who introduce the concepts of idempotent and nilpotent elements and the decomposition of an idempotent element into a sum of orthogonal primitive idempotents.

Cayley and Sylvester and other British and American mathematicians then started working on the problem of classifying algebras of small dimension over the complex numbers.

During this time, the development of Lie groups and algebras (which are non-associative) was proceeding and some of the fundamental concepts in the theory of associative algebras (the concept of the radical, for instance) were developed first for Lie algebras.

Another key source of ideas and examples was the concept of a group algebra, which had been essentially defined by Dedekind in 1896, in a letter to Frobenius. Dedekind was very clear on the relation of this to the general theory of algebras, although the theory of group representations as developed by Burnside and Schur (around 1905) did not at that time explicitly use ring-theoretic methods.

The concept of a simple algebra over the complex numbers had been defined in 1893 by the German mathematician T. Molien, who then proved the first version of the Wedderburn Theorem, i.e. that a simple algebra over the complex numbers is isomorphic to the ring of n by n matrices over the complex numbers. It was at this point that the concept of a two-sided ideal became current and a lot of theorems were proved about them. But it is not clear from the Bourbaki survey whether the word "ideal" was originally used, and it is possible that the analogy to Dedekind's work on commutative rings was not immediately seen.

The concept of a semi-simple algebra was introduced by Elie Cartan. (Unfortunately, I don't have a date.)

The development around 1900 of the theory of finite fields by the American mathematicians E. H. Moore and L. E. Dickson was what motivated the generalization of the theory of algebras to the case where the base field was unrestricted. Wedderburn, another American, in 1905 proved that every finite skew field (a.k.a. division algebra) is in fact commutative.

In 1903, in a memoir on the algebraic solution of differential equations, Poincaré had defined the concepts of left ideal and right ideal for an algebra. (As mentioned, two-sided ideals had been essentially known since Molien's paper in 1893.) In this memoir, Poincaré proves that the minimal left ideals in the ring of n by n matrices have dimension n. However this result was not noticed by the algebraists.

The notions of left and right ideals were rediscovered in 1907 by Wedderburn, who proved that the radical was the largest nilpotent left ideal and proved his well known "Wedderburn Theorem" (later generalized by Emil Artin) which states that every semi-simple algebra over an arbitrary base field is a direct product of matrix rings over skew fields.

In 1920, Emmy Nöther and W. Schmeidler used the concepts of left and right ideal in a paper devoted to rings of differential operators. But otherwise, these concepts were ignored after Wedderburn's paper until 1927, when Emmy Nöther, and Brauer (and later A.A. Albert and Hasse) resumed the study of them.

By 1934, the basic theory of semi-simple rings was essentially complete.

Bourbaki's summary statement is, "The axiomatization of algebra was begun by Dedekind and and Hilbert, and then vigorously pursued by Steinitz (1910). It was then completed in the years following 1920 by Artin, Nöther and their colleagues at Göttingen (Hasse, Krull, Schreier, van der Waerden). It was presented to the world in complete form by van der Waerden's book (1930)."

What we see from all this (at least in my view) is that the development of modern Algebra was never motivated by mathematicians seeking abstraction for its own sake. Instead, algebraists working on quite concrete problems were trying to invent tools that might help with their investigation of these problems, and slowly (very slowly, as we look at their work in retrospect) began to notice that the same logical patterns recurred over and over again in different examples.