Utility of a general math book introducing subjects not in historical order?

[Cantor080]

I have Abstract Algebra Course, and my teacher is using the book, "Abstract Algebra" of David S. Dummit, and Richard M.Foote. This book, and other books which my teachers are using for the courses, seem to give definitions, theorems, and problems, in no historical order or are not mentioning the utility or conformation which might have created them. I don't know the utility of this order of giving data. These textbooks seem to be not historical, and seem to be not mentioning on for what the entire particular math (example: Abstract algebra "has its roots in the issue of solvability of equations by radicals"*) was constructed for, i.e. they seem to be not mentioning the utilities/reasons for which they might have been constructed for, and on how each of its components (or entire theory) got created for attaining them.

Order of Dummit and Foote book seem to be giving discrete theorems and definitions, which seem to have come for particular utility of knowing the bigger utility of solvability of equations. It seems that these textbooks (?) are not allowing us to participate (?) in the search of conformations for the attainment of the same utility from our new data, or seem to be not allowing us to modify the theories (create math?) for our own utilities for our custom needs (?). I am in Science college and not Engineering college, I thought they could have constructed syllabus to make creators of math, and not just make us know what is there in the math.

One of the reason my teacher of Abstract Algebra gave for not using historical approach is the time. It seems to me that historical approach books, (here, if historical approach mean to give the reasons for which the certain topic was started, and on how each of its component got created for the attainment of it) could be of ~500 pages, as that of Saul Stahl's mentioned below, and seems to be of not too great length.

And history seems to allow knowing math as it really is, and seems to allow knowing the recurring conformations (laws or principles) of math creation. I don't what other unknown consequences occur on changing this natural (?) order, and I don't know on who decides the order of data given in the textbooks, and on whether they have thought on the consequences of order or not.
I don't know why these books (as Dummit and Foote, or any other textbook?) have evolved (?) and are standing now, as they are. What utility do they have? Can we not have a book with historical order of data in math?

*extracted from Introductory Modern Algebra, A Historical Approach, Saul Stahl.

 

[FactChecker]

The history of mathematics has a lot of confusion and wrong turns. It is interesting and people do read mathematical history books, but it is not very effective in helping one to understand modern mathematics. It would be like asking someone for driving directions to a town and getting the entire history of roads, from dirt roads to the development of the interstate road system. By the time they were done, you might be more confused than you were to start with.

 

[Stephen Tashi]

What utility do they have? Can we not have a book with historical order of data in math?

As @FactChecker indicates, actual History can be complicated. What you are imagining is an approach where History is streamlined and simplified. It would not be the History Of Mathematics, but rather something like the History of Timmy The Math Student, showing his intellectual development as he proceeds from topics like solving quadratic equations to more and more abstract topics. I, myself, would probably enjoy reading about Timmy more than most texts on abstract algebra.

Abstract algebra did not develop exclusively to solve polynomial equations, but, in so far as it did, you are correct that most abstract algebra texts fail to make a clear connection between their content and solving equations. To understand the connection, people resort to specialized materials like http://www.science4all.org/article/galois-theory/ and https://arxiv.org/abs/1108.4593 and https://www.amazon.com/dp/082840268X/?tag=pfamazon01-20

In defense of typical advanced texts, they are the "bootcamp" approach. They force students who are accustomed to thinking in concrete and Platonic ways to convert to thinking abstractly. For example, students are forced to accept that definitions mean what they say rather than simply being descriptions of things that already exist. If we measure student progress in terms of amount of material covered, typical texts get more done than could be accomplished by motivating each topic. The speed comes at the cost of mental discomfort.

 

[mathwonk]

to learn math more in the historical order in which it was developed you could read some books written in older times, and I recommend it. Beginning with Euclid's Elements, you will learn what was available long ago not only in geometry, but how geometry was then used to do elementary algebra (to expand the square (a+b)^2, one constructs a square of sides a+b,a+b and notes that it consists of squares of sides a,a and b,b, and also two rectangles each of sides a,b), and also the start of number theory and even some beginnings of analysis, in the sense of computing volumes by approximation. Then a later text with more modern algebra wouod be Euler's Elements of Algebra, which includes the solutions of the equatiomns of second third and fourth degree, prior to the discovery that fifth degree equations are not solvable. I have no such favorite historical source for calculus, but perhaps Euler's Analysis of the Infinite as a possible "precalculus" text would be of interest. In the 20th century one has the excellent text of Courant on calculus, with historical comments. As a complement to Dummitt and Foote, you might consult Vander Waerden's Algebra, or under its earlier title "Modern Algebra".

 

[FactChecker]

Certainly, any Abstract Algebra book would benefit by mentioning the significant applications that motivated the concepts. I think it could be done judiciously, without spending too much time trying to explain how things developed before complex numbers, matrices, etc. But it could not be a very large part of the book.

One of the problems with Abstract Algebra, specifically, is that a lot of it is motivated by people noticing common trends and proofs in a wide variety of subjects. Hence the motivation to abstract the fundamental concepts for general application.

 

[mathwonk]

Galois was analyzing the problem of when polynomial equations have solutions that can be expressed using arithmetic operations plus extraction of roots. This turned out to be related to symmetry properties of the systems of roots and of the number systems they live in. E.g. a quadratic equation irreducible over the rational field can be solved by extracting a square root. The resulting number field in which the roots live is a two dimensional extension of the rational field, hence a rational vector space of dimension 2, with elements {a+b.sqrt(c)}, where a,b are any rational numbers and c is fixed, associated to the given equation.. The two roots can be permuted by the group of symmetries of two objects, but more remarkably this permutation extends to an automorphism of the entire field extension generted by the two roots over Q. I.e. the two roots are permuted by means of the "conjugation" automorphism of the field extension, taking a + b.sqrt(c) to a-b.sqrt(c).

More generally, Galois was led to study those permutations of the roots of an equation that can also be obtained by restricting a field automorphism of the number field generated by the roots. This is now called the "Galois group" of the equation, but Galois had to invent the theory of groups out of whole cloth, building on ideas of Legendre, I believe. Assuming the root field was indeed generated by simply adding in roots of earlier existying elements Galois could identify a special property of the Galois group, namely the group had to be built up in stages analogous to the stages of adding in new roots. Thus he had to introduce the ideas of subgroups and normal subgroups and decomposition series to describe this somewhat complicated structure that distinguished solvable equations.

Polynomial equations are intimately associted to polynomials, which form a "ring", and their solutions belong to "fields", which have automorphism "groups". Nowadays abstract algebra consists of the study of these three classes of objects: groups, rings, and fields, all of which Galois had to pretty much create to solve his problem. Today we just begin by introducing all these objects which will be needed to describe the solution by Galois of the problem of which equation have solutions by radicals, and ask you to accept that they will be useful. In fact today the other uses of these theories are far more interesting than the original problem of solving equations, but that is where they arose.

In one of the last times I taught this course I was frustrated that I had never quite gotten to Galois' theory so I began with that topic, to be sure to cover it. Sure enough before covering it I had to treat groups, rings, fields, and vector dimension, since all were needed in the solution of Galois' problem. I think the class appreciated it very much, and it motivated quite well every topic we covered. My notes are available free on my website, at UGA math dept. math 843, 844, and the later subjects of linear algebra, less well motivated, math 845.

http://alpha.math.uga.edu/~roy/


By the way, although Dummitt and Foote may not be well motivated, it is very clear, and I still have my copy. The problem sets are also quite extensive and useful. I have taught from Dummitt and Foote, as well as Lang, and Herstein, but my favorite was the book Algebra, by Michael Artin, which I recommend. Van der Waerden is not the easiest to read but is very insightful, oh and Jacobson is excellent but also challenging. The other general algebra books on my shelf are by Hungerford, Chi Han Sah, A.A. Albert (extremely terse), and Shifrin.

 

[Cantor080]

Related:
. . . our students of mathematics would profit much more from a study
of Euler’s Introductio in Analysin Infinitorum, rather than of the available
modern textbooks.
(Andr´e Weil 1979, quoted by J.D.Blanton 1988, p. xii)

 

[Cantor080]

Related:
..maxim of Niels Henrik Abel, “I learned from the masters and not from the
pupils” ..
(Extracted from Alexander Ostermann • Gerhard Wanner,
Geometry by Its History )