An Episodic History of Mathematics: Mathematical Culture Through Problem Solving by Steven G Grantz (PDF)

Description

An Episodic History of Mathematics will acquaint students and readers with mathematical language, thought, and mathematical life by means of historically important mathematical vignettes. It will also serve to help prospective teachers become more familiar with important ideas of in the history of mathematicsboth classical and modern.Contained within are wonderful and engaging stories and anecdotes about Pythagoras and Galois and Cantor and Poincar, which let readers indulge themselves in whimsy, gossip, and learning. The mathematicians treated here were complex individuals who led colorful and fascinating lives, and did fascinating mathematics. They remain interesting to us as people and as scientists.This history of mathematics is also an opportunity to have some fun because the focus in this text is also on the practicalgetting involved with the mathematics and solving problems. This book is unabashedly mathematical. In the course of reading this book, the neophyte will become involved with mathematics by working on the same problems that, for instance, Zeno and Pythagoras and Descartes and Fermat and Riemann worked on.This is a book to be read, therefore, with pencil and paper in hand, and a calculator or computer close by. All will want to experiment; to try things; and become a part of the mathematical process.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐At most colleges, a history of mathematics course assumes multivariable Calculus. It attempts to give a necessarily episodic overview of 4,000 years of history, slighting modern topics that are advanced. It it taught out of a mathematics department, and so uses a problem-solving approach to make it hands-on. It is meant to give mathematics majors a culturally rich setting for mathematics. Thus you would think that a book entitled “An Episodic History of Mathematics: Mathematical Culture through Problem Solving” would be ideal. The present book is not suitable for such a course for two essential reasons.First, the audience for this book is not clear. What mathematics student needs to be told how a derivative is defined but then is asked to follow a proof that every principal ideal domain is a unique factorization domain? What mathematics student needs to be shown how to complete the square in a trinomial algebraically but is then introduced to complex numbers abstractly as Hamilton did as a formal system of ordered pairs? The resulting treatment is uneven.Second, the book gives no flavor of the methods that were actually used historically. An historical vignette is given, but then immediately the notation and approach for the related problems are modern. One should solve quadratic equations geometrically as the Babylonians did. One should give a nod to Bishop Berkeley’s objections to Newton’s “ghosts of departed quantities” when introducing limits of quotients so as to emphasize the cultural difficulties in accepting Newton’s approach. One should mention Archimedes’ geometric method of exhaustion when introducing limits of sequences, and not just introduce limits algebraically via Zeno’s paradoxes (32-33).Krantz misses opportunities to ground the mathematics historically. For example, when did Hadamard live (233)? Of what benefit to a book on history is mentioning three-dimensional coordinate systems (104-108) with no historical references whatsoever? I suppose–given that an internet search allows us to find sources for ourselves–Krantz can be forgiven for not giving Roger Cook’s book on Kovalevskaya as his source for quoting Mittag-Leffler (314), but I believe that a book on the history of mathematics should model the historian’s craft as well as modeling the mathematician’s craft. Historians do not cite without reference.A discerning teacher could weave a path through the book that picks the right level for her students, especially given Krantz’s fine choice of exercises and further reading. For example, one of the further reading sources does in fact show how to “complete the square” geometrically.Krantz introduces several modern topics to heighten student interest in further investigation. The works of Green and Tao (233), of Knuth and Graham (245-246), and of Ulam, Stone, and Tukey (301) are presented with invitations to “talk it up in class,” to cite a few examples.I have found four things I would like to change. First and most minor, I would like to alphabetize the Bibliography (see [COSM], [WIKIGN], and [WIKIPR]). Second, I would like to draw hyperbolas (5) showing that they have asymptotes. Third, I would change “truth” to “provability” in a proof by contradiction. I am not sure what “all the known true statements in mathematics” (226) means. Known to whom? It is not the “known body of mathematics” that one assumes consistent and correct, but more modestly provable statements in number theory. Fourth, I would add to the index the topics in the exercises, such as “amicable numbers” (71) and “quaternions” (286).This last change would at least make the book useful for a course in problem solving with a mere tip of the hat to history.

⭐Krantz is not a polished writer, and this book is stylistically uneven because of it. In his preface he talks about the book being used by undergraduate math majors “engaged in a capstone experience” (see how clunky a writer he is) and the book also being used in a course for non-majors in “mathematical culture” or for a math course for “teacher preparation”, as well as for “a course in problem-solving”. He doesn’t appear to have had a clear readership in mind, which makes the book uneven in content.There isn’t much history in the book. It is not only “episodic” as the title says, but what little historical narrative Krantz provides for each chapter functions less as an informative discussion of a period in the history and development of the subject and more as a prologue to introduce some aspect of modern mathematics.Most of the book should be accessible to anyone with an active understanding of pre-calculus mathematics (but see below). This makes it less valuable to a fourth-year math major, and relatively more useful to others less schooled who are interested in the history of mathematics (although, again, this isn’t actually a history). As I said, Krantz isn’t clear in his own mind, so sometimes he writes to one readership and at other times it seems to another. Usually he assumes less acquaintance with his topic than what an undergraduate focused on math or science would have.Main sections within chapters conclude with suggestions for reading (usually two articles, but sometimes one or three, and a few times four) from The Two-Year College Mathematics Journal, The College Mathematics Journal, or The American Mathematical Monthly.Within his discussion of Greek mathematics (chapter 1), Krantz has a good presentation of how Archimedes’ “method of exhaustion” can be used to derive ever closer approximations of the value of pi. Within his discussion of the algebra of Al-Khwarizmi (chapter 4), he has a good presentation of “completing the square” to derive the quadratic formula.In chapter 7, Krantz begins to discuss calculus. He discusses the tangent and slope, and the derivative of a function, and works out a few examples. The formal definition of limit is not given anywhere in the book. Readers unacquainted with calculus should be able to follow along. What Krantz fails to do in this chapter is explain where he’s headed and thereby to motivate the reader to care. Through calculus, we can infer significant properties of curves. The reader may not care about properties of curves, but at least the point of all the tedium of calculus would be clear at the start. This chapter includes showing how the maximum and minimum values of a function (the hilltops and valleys of a curve) can be inferred using calculus. Any astute teacher would have introduced these ideas as motivation before going into the details of calculus. Krantz waited until he had the technique of finding the derivative.In chapter 8, he introduces integration. His exposition is uneven, partly because he doesn’t always explain where he’s going or why he’s taking the route he does. Readers who haven’t seen integration before are not going to be enlightened. This isn’t a textbook on calculus, of course, but Krantz is writing here as if he’s assuming the reader hasn’t seen integration before, and he’s not being careful enough to help those readers along. Out of nowhere, with absolutely no clarification, he writes (140) of “a monotone increasing or monotone decreasing, continuously differentiable, function.” Most of these terms have not been mentioned, much less explained or defined, prior to this; and Krantz seems mindlessly unaware he’s left out important information. Continuity is one of the unexplained notions, and the term isn’t in the index, either. It is finally discussed, along with maxima and minima again, in chapter 12, but chapters 7 and 8 make no reference to chapter 12.Krantz writes in his preface as if he hopes this book will be used as a textbook in a variety of courses. It shouldn’t be. This book is muddled by Krantz’s inattention to clear, coherent writing. What makes the book frustrating is that he vacillates between writing with care and clarity and writing clunky, poorly thought-out sentences or leaping over conceptual details that are vital to understanding his presentation. It’s as if the book were written by two authors, one alert and conscientious, the other half asleep and indifferent. A small example is his use of the notation of uppercase sigma and pi for sums and products. He explains the meaning of sigma notation (38) but uses uppercase pi as if its meaning were already understood (180, 185). This is sloppy teaching. If one notation needs explaining, so does the other. Krantz’s mental unclarity about his readership, and his inattention to good writing, results in an uneven text.Additional failures to teach well include:Section 10.5 _ Quadratic reciprocity is discussed with no historical motivation, except to mention the names of Euler, Legendre, and Gauss. The mathematical context for their interest in it isn’t given. This book doesn’t show the structural relationships within mathematics. Krantz sometimes makes mathematics look like a collection of frivolous puzzles. Certainly, it is not that.Section 10.6 _ Gaussian integers are not compared to complex numbers, reminding the reader that in the complex number a+bi that a and b are real numbers, whereas in the Gaussian integer a+ib (Krantz’s notation) that a and b are integers. Gaussian integers are therefore a subset of the complex numbers. Krantz doesn’t provide this clarification.Also in this section, the notions of ring and generator are used without definition (rings are explicitly introduced later in section 20.3). A Principal Ideal Domain is then explained in terms of the unexplained notion of generator. The term ‘Unique Factorization Domain’ is used in the statement of a lemma and not explained until afterwards in the proof. This proof uses the unclarified notion of Principal Ideal Domain. A later proof in this section refers to “the multiplicative group for the field Z/p.” This proof also refers to the notions of cyclic group, order of a group, and equivalence class. Clearly, Krantz is now, without warning or acknowledgment, writing for advanced undergraduate math majors.Krantz attributes the notion of ideal (for which he gives a definition without explanation) to “Emmy Noether and others” rather than to her progenitors Kummer and Dedekind. It is well known that Noether said: “Es steht alles schon bei Dedekind.” It’s already in Dedekind. Ideals are mentioned (with better clarity) again in section 20.3.1 but, again, only Noether is named as a contributor to the theory. In section 14.1, Dirichlet is said to have done work that led “in the hands of Emmy Noether” to the theory of ideals. Dirichlet’s

⭐, edited by Dedekind are not mentioned. It was the extensive, far reaching supplements to these lectures, supplements written by Dedekind, to which Noether was mainly referring when she said “it’s already in Dedekind.” Krantz is egregiously misrepresenting the history of mathematics. Dedekind, in fact, is only referred to incidentally in the book: as a colleague of Gauss, Riemann and Cantor (172, 249, 262, 263) but not Dirichlet, and within the terms ‘Dedekind cuts’ (211, 282) and ‘Dedekind Completeness’ (283). Only the reference to page 283 is found in the index.Section 12.4 _ The term ‘ordered field’ is used without definition.Section 13.3 and 21.2 _ As is typical in math books (so Krantz is following convention), the principle of mathematical induction is stated in the logically less clear order. It is stated [for any integer n] that: If P(1) and If P(n) implies P(n+1), Then [for any positive n,] P(n). It is because of the well-ordering of the integers that mathematical induction works. Reverse the conjunction for clarity: If for any integer n, P(n) implies P(n+1), and If P(1), Then for any positive n, P(n).The logic of the principle is clearer in this form and therefore appears less mysterious and magical to beginners. P(1) being true initiates the recursion, starting at P(1), allowed by P(n) implying P(n+1). Since P(1), then having established that P(n) implies P(n+1), it follows that P(1+1), and from this inference of P(2) we can infer P(2+1), and so forth for every n.The principle of weak induction is stated and discussed more clearly in the excellent textbook, from 2009,

⭐, by William Johnston and Alex M. McAllister. “Let P(k) be a predicate defined on integers k[element of]Z and let a[element of]Z. If P(a) is true and, for all n[greater than or equal to]a, we have P(n) implies P(n+1), then P(n) is true for all integers n[greater than or equal to]a.” (Johnson and McAllister: 228) They also state the principle of strong induction.Section 13.4 _ The term ‘group’ is used without definition. It is not defined until section 20.2.Chapter 15_ Krantz proves he hasn’t a clue what level of mathematics this book presupposes. He now assumes understanding of calculus (which he wasn’t assuming through most of the main text) and motivates absolutely nothing of the ideas of Riemann or explains why any of it matters.Section 16.2 and 17.1 _ In 16.2 the natural numbers are said to be the set {1,2,3,4,…} and in 17.1 the natural numbers are said to arise from set construction beginning with the null set and to be enumerated as 0,1,2,…. In one case 0 is not a natural number, in the other it is. Krantz mentions the disparity and says that “standard mathematical discourse” excludes 0 from the natural numbers. Krantz continues in 17.1 by mentioning the Peano axioms, and there he has that the natural number 1 is not a successor of any natural number, which implies that 0 is not a natural number. Despite Krantz’s assertion about standard discourse, this uncertainty of where to first include 0 is common in math books. The terminology needs to be settled. Terminology should assist, not hinder, and should therefore make logical sense and not obscure possible symmetries of meaning.// My position is: The counting numbers begin with 1, and the natural numbers begin with 0. Thus the counting numbers (a kind of number usefully distinguished only in early stages of learning about numbers) are the positive integers, and the natural numbers are the non-negative integers. Following Krantz (275), the whole numbers are the same as the integers, and the fractions are the same as the rational numbers. When teaching children, it’s reasonable before they learn about negative numbers to use ‘whole number’ and ‘fraction’, just as it is to use ‘integer’ and ‘rational number’, to refer to positive numbers or 0; then with the introduction of negative numbers, one can speak of positive, non-negative, non-positive, and negative whole numbers, integers, fractions, and rational numbers. 0 is explained to be the single whole number, integer, fraction, and rational number that is neither negative nor positive. Prior to teaching of negative numbers, then, allow ‘whole number’ to mean natural number, which includes 0. Afterwards, it means integer.// In common usage, ‘fraction’ and ‘whole’ are antonyms. In elementary school, a fraction is a number of the type n/m and so even though a whole number n isn’t a fraction notationally (and semantically, one would suppose it isn’t one either) it can be “changed into a fraction” or “written as a fraction” by n/1. This notational and linguistic confusion doesn’t occur when speaking of rational numbers and integers. //Chapter 17 _ This entire chapter on the number systems is over-simplified and consequently somewhat muddled. Krantz gives two versions of natural numbers (as mentioned above), based upon two methods of construction or description: set theoretic and axiomatic via Peano. The integers are then constructed set theoretically via equivalence classes of natural numbers (which presumably arise via set theory also). The rational numbers are then constructed via equivalence classes of integers. He doesn’t mention how elaborate a set such a rational number would be, nor has he discussed the historical concerns and controversies that led to such bizarre definitions of numbers. The real numbers are then described via axioms and not as arising in some set theoretical way from the rationals. In chapter 12, the reals were constructed via Cauchy sequences of rational numbers, where the rationals were assumed as given, presumably intuitively. The complex numbers are defined in chapter 17 as ordered pairs of real numbers, and in chapter 9 they are also defined this way.In general: This is a poorly conceived book, written without care or attention to level of content, with nominal historical background that is not only simplified but at times misleading. Krantz has wildly uneven teaching skills, going from admirable clarity to simply dumping information without motivation or reasonable explanation. It is not all bad, but overall it is a disappointing and surprisingly erratic book.

⭐Not a good math book. Not a good history book. Not a good history of mathematics book. I don’t understand why the MAA would put their stamp on it. I will not use this book again to teach my History of Mathematics course.

⭐Very detailed book, with many example of solved historical problems. The presentation of the mathematics symbols are a little deficient, nonetheless they are readable.

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