Mathematics: A Concise History and Philosophy (Undergraduate Texts in Mathematics) by W.S. Anglin (PDF)

Description

This is a concise introductory textbook for a one-semester (40-class) course in the history and philosophy of mathematics. It is written for mathemat­ ics majors, philosophy students, history of science students, and (future) secondary school mathematics teachers. The only prerequisite is a solid command of precalculus mathematics. On the one hand, this book is designed to help mathematics majors ac­ quire a philosophical and cultural understanding of their subject by means of doing actual mathematical problems from different eras. On the other hand, it is designed to help philosophy, history, and education students come to a deeper understanding of the mathematical side of culture by means of writing short essays. The way I myself teach the material, stu­ dents are given a choice between mathematical assignments, and more his­ torical or philosophical assignments. (Some sample assignments and tests are found in an appendix to this book. ) This book differs from standard textbooks in several ways. First, it is shorter, and thus more accessible to students who have trouble coping with vast amounts of reading. Second, there are many detailed explanations of the important mathematical procedures actually used by famous mathe­ maticians, giving more mathematically talented students a greater oppor­ tunity to learn the history and philosophy by way of problem solving.

User’s Reviews

Editorial Reviews: Review “…The book is well written and will help those who look for a deeper understanding of mathematical culture.” — MATHEMATICAL REVIEWS

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

⭐This book is really a theological treatise by Anglin. I’m not sure how Axler, Gehring, and Ribet (the editorial board) gave this book a green light for publication. It essentially follows the outdated, ethnocentric approach of Morris Kline, except that Anglin throws in his religious viewpoints on nearly every page.Consider for instance the following question taken from the book: ‘Psalm 104 praises God “who laid the foundations of the earth, so that it should not be moved forever.” Write a short essay showing that this verse can be interpreted in a way that respects both the truth of God’s revelation and the truth of Science.’ (p 160)The second question on the same page reads as follows: ‘Comment on the following. Socrates and Jesus were willing to die for what they believed was right, but Galileo recanted because he was a coward.’ (p 160)Do these questions belong in a history of mathematics course or in a course on the role of theology in the sciences?I bought this book as a resource to include good mathematical questions in a history of mathematics course, but having read it now, I am disappointed. There’s a lot of theology in this book, but the mathematics is thin.Amazon does not give the option to give a negative number of starts, so I give a rating of 1 star.

⭐It’s a good book, but it took at least three weeks for it to be shipped to my address.

⭐I have been both confused and challenged by math texts in the past, but never before have I felt outright belittled and insulted on the basis of my choice of religion. The author seems to feel that the purpose of mathematics is to gain insight into the mind and workings of the Christian God, and this view dominates the entire volume. He grants a lot of leeway to the ancient Greeks before the “Christian revelation” reached their corner of the world. After that, he blames a centuries-long drought of mathematics advancements on a single generation of Greek mathematicians who did not jump on the Christian bandwagon, as opposed to the individuals at the top of the power structure determining what sorts of research were and were not important for development. He claims that Galileo did nothing of importance, and he is mentioned exclusively by anti-Christian historians. (Galileo may not have made contributions to math, but his contributions to the framework and methodologies of science are as significant as Euclid’s work was to the framework of math.) Each of the 40 lessons ends with exercises, many of which include religion-based essay answers. While these may appear to invite discussion, the questions are invariably phrased in a pro-Christian manner.Beyond that, the works of mathematicians who actively published religious papers advocating Christianity seem to gain additional importance. Newton, Leibniz and Pascal (among others) get their own chapters, while Euler (who is probably the single most prolific mathematician in human history) gets a page and a half. Galois founded group theory and launched the field of abstract algebra when he showed that there is no general methodology for finding the roots of polynomials of degree 5 or higher, and he is heavily glossed over by Anglin, who credits the creation of abstract algebra to other individuals (who made undeniable contributions) while nearly dismissing the result of Galois with a single clause in a sentence stating that the result was found in that century. Galois isn’t even mentioned by name.To give a clear idea of the importance of Christianity to the author, let me say this: his concluding sentence, designed to summarize the key point of the entire book, is the following Ramanujan quote: “An equation for me has no meaning unless it expresses a thought of God.”I’m not saying there isn’t good material buried in here, but none of it stands out in memory as clearly as the feeling that I was being preached at. For those who agree with Einstein’s philosophy that studying science and math is akin to studying the mind of God, there is much worthwhile content here. There is interesting material for those just looking for math too. I did find it interesting to learn that some major mathematical developments were motivated by Biblical numerology and the like. However, that material can be portrayed accurately through an impartial viewpoint. When I pick up a math text, I don’t expect to be able to deduce the author’s religious views, as they should not be relevant to the content in any way, shape or form.As it stands, I can’t recommend this volume. Those interested in the subject matter will find more complete, less opinionated, and cheaper resources quite readily. I personally prefer the now out-of-print two volume set by David E. Smith titled “History of Mathematics.” Both volumes combined are half the price of this one, and the page count builds to over 1300 pages, each of which has more content than a page of Anglin’s book.

⭐This book is utterly worthless in every conceivable way. It is a mystery to me why Springer have brought disgrace upon themselves by publishing this inept drivel. A complete account of Anglin’s incompetence would require a review as thick as the book itself, but hopefully a few deterring examples will suffice.First, there are many blatant factual errors, e.g.:”There were five planets (or so Kepler thought) and five regular polyhedra. This could not be an accident!” (p. 158)Since Kepler’s work on Mars, Jupiter and Saturn are mentioned on the same page, one wonders whether it is Mercury, Venus or the Earth that Anglin imagines Kepler to have been ignorant of.Other statements cannot even be called false since they are such ludicrous nonsense, e.g.:”Just as many people before Lobachevsky thought that Euclid’s parallel postulate was a kind of sacred truth, so many people before Hamilton thought that the law of commutativity for multiplication was ineluctable. For us it is a commonplace that this law need not hold, since we have a ready example of noncommutativity in matrix multiplication.” (p. 195)The notion that there is some sort of “law” out there about commutativity of multiplication that may or may not hold is a very childish misconception. Whether we call certain operations with matrices and quaternions “multiplication” or not is purely a matter of convention. Thus the alleged “law” that “many people” allegedly held for “ineluctable” has no meaning whatsoever other than as a thoroughly inconsequential claim about naming conventions. Anglin’s stupidity is particularly disturbing in light of his immodest description of his own book as offering “a deep penetration into the key mathematical and philosophical aspects of the history of mathematics”, “giving the student an opportunity to come to a full and consistent knowledge” (p. viii).Let us give another example of what Anglin considers to be “a deep penetration” into philosophical issues. We read that “there are various objections to formalism,” e.g. that “formalism offers no guarantee that the games of mathematics are consistent.” Now, presumably in order to “give the student an opportunity to come to a full and consistent knowledge,” Anglin professes to offer the other side of the coin: “the formalist can reply [that] although some of the games of mathematics are indeed inconsistent, and hence trivial, others are not” (pp. 218-219). It is not clear in what sense Anglin fancies the assertion that mathematics is consistent to be a “reply” to the challenge to prove as much.Finally, the book is full of unsubstantiated revisionist history motivated by unabashed Christian propaganda (“God” is the entry in the index with the most references by far; more, in fact, than Euclid, Archimedes, Newton and Riemann combined), e.g.:”Most of the mathematicians at the Academy and the Museum rejected the new truths [sic] of Christ’s revelation. This is unfortunate because … if the mathematicians had joined the Christians, the Dark Ages would have been brightened by a dialogue between reason and faith. As it was, this dialogue was postponed to the later Middle Ages, when thinkers like Thomas Aquinas (1225-1275) advanced philosophies that were influenced as much by the Elements as by the Bible.” (p. 111)What a loss for mathematics that we had to wait so many centuries for the great geometer Aquinas!

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