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A Radical Approach to Real Analysis: Second Edition (Classroom Resource Materials)

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I am a topologist by training who was Shanghaied into being an analyst when I was hired as a teacher. As a consequence of this, the Advanced Calculus course I taught was rather heavy on topology.

Over the course of time--having been transformed into more of an analyst that I would have ever dreamed--I've come to the conclusion that analysis is best learned before topology.

This is a text that accomplishes that by using the historical approach.

One learns how Newton approached problems, how Euler did, how Cauchy did. Not only is it interesting, it is enlightening. I've taught this course for 15 years now, and of all of the approaches I've taken, this has been the most fruitful.

My students have come from calculational courses, and the historical approach of this book provides a bridge over which they may come into the land of proof. They also see the issues that caused the need for modern rigor face to face

I do supplement the course with material that is more modern (Hardy's book A Course of Pure Mathematics) and material on the Riemann integral, but I've been spoiled for any other approach.

If you had calculus in high school or college then you learned about Newton, Leibnitz, and Riemann but probably did not encounter Lebesgue (pronounced le-bek). At the University of Alabama Huntsville learning about Lebesgue integration is key to advancing into graduate studies in mathematics. The natural follow-on course after Calculus I and II, etc. is Real Analysis. This book, using Lebesgue integration methods, is a good sequel to Lebesgue calculus.
I purchased this book, after reading about it in the Mathematical Association of America (MAA). For autodidacts like myself, it is a good first introduction to the topic.

This is not a bad book, but it does everyone a huge disservice by pretending to be historically informed when in fact it is propagating harmful and stupid myths that have no basis in historical fact whatever. An example should make this clear.

"Daniel Bernoulli suggested in 1753 that the vibrating string might be capable of infinitely many harmonics. The most general initial position should be an infinite sum of the form
(2.71) y(x) = a_1 sin(pi x / l) + a_2 sin(2 pi x / l) + a_3 sin(3 pi x / l) + ...
Euler rejected this possibility. The reason for his rejection is illuminating. The function in equation (2.71) is necessarily periodic with period 2l. Bernoulli's solution cannot handle an initial position that is not a periodic function of x. Euler seems particularly obtuse to the modern mathematician. We only need to describe the initial position between x=0 and x=l. We do not care whether or not the function repeats itself outside this interval. But this misses the point of a basic misunderstanding that was widely shared in the eighteenth century. For Euler and his contemporaries, a function was an expression: a polynomial, a trigonometric function, perhaps one of the more esoteric series arising as a solution of a differential equation. As a function of x, it existed as an organic whole for all values of x. ... To Euler, the shape of a function between 0 and l determined that function everywhere." (p. 53-54)

There is not a single line anywhere in any pre-19th century mathematical work that comes anywhere near making this sort of claim. Self-righteous "mathematicians" have invented these myths to justify their dogmatic and authoritative mode of "teaching" and their passionate hatred of intuition. In falsely lending these propaganda fabrications a veneer of historical truth, Bressoud is perhaps the worst lier of them all. It is not Euler who is "obtuse," but Bressoud. There was no "basic misunderstanding widely shared in the eighteenth century"; rather, the "basic misunderstanding" lies with Bressoud and his fellow poseur historians of today.

All of this is easily established by simply reading Euler. The relevant paper is E213, which is readily available online. Let me summarise what you will find if you read that paper.

First of all, Bernoulli never claimed that (2.71) can express any initial position of the string. He merely argued for a general series solution of the vibrating string equation which *implies* that the initial position is of the form (2.71). Hence Euler's main objection, which is this: I can bring the string into any position whatever, let go, and it will move according to the vibrating string differential equation. Thus Bernoulli's solution is not completely general insofar as (2.71) does not express any possible initial position of the string. And since Bernoulli has provided no argument that (2.71) can in fact express any initial position, nor in fact any method for calculating the coefficients a_i, we have no reason to believe that his solution is completely general.

At this point Euler preempts a hypothetical counterargument: perhaps, says Euler, some might argue that "owing to the infinite number of undetermined coefficients," equation (2.71) "is so general as to include all possible curves." This, however, is plainly false, Euler points out by noting the periodicity properties of (2.71). Now, at this point it would be possible for a Bernoullian to retreat still further and say that (2.71) can express any function, not on the real line, but on the interval [0,l]. This is a perfectly valid argument, but it is an argument which Bernoulli never raised and which Euler never claimed to have refuted.

So much for the periodicity argument, which Bressoud has obviously distorted most unfairly. But worse still is Bressoud's generalisation from this case to the alleged "basic misunderstanding." This is sheer stupidity and fabrication, as is plain to anyone capable of reading at a fourth grade level. In fact, every last word of it is plainly and unambiguously rejected by Euler in the very article in question when he points out that the initial position of the string can be any curve, which "often cannot be expressed by any equation, be it algebraic or transcendental, and is not even included in any law of continuity."