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Foundations of Constructive Analysis, by Errett Bishop
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This book, Foundations of Constructive Analysis, founded the field of constructive analysis because it proved most of the important theorems in real analysis by constructive methods. The author, Errett Albert Bishop, born July 10, 1928, was an American mathematician known for his work on analysis. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. From 1965 until his death, he was professor at the University of California at San Diego.
- Sales Rank: #1030872 in Books
- Brand: Brand: Ishi Press
- Published on: 2012-07-31
- Original language: English
- Number of items: 1
- Dimensions: 9.00" h x .91" w x 6.00" l, 1.30 pounds
- Binding: Paperback
- 402 pages
- Used Book in Good Condition
About the Author
Errett Albert Bishop, born July 10, 1928, was an American mathematician known for his work on analysis. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. From 1965 until his death, he was professor at the University of California at San Diego. He died on April 14, 1983.
Most helpful customer reviews
9 of 9 people found the following review helpful.
A Brilliant Book
By Frank Cannonito
Errett Bishop was my friend and colleague and we had many discussions about this book and its subject matter. It is a difficult book because the way of thinking about the subject is unfamiliar to classically trained mathematicians, and this was a disappointment for Bishop. But in it Bishop found how to give, for example, a constructive proof of the Riemann Mapping Theorem - something which Goedel told Hilbert would not be possible (despite Ostrowskii's contemporary proof which was constructive except for the last step which was hanging by a hair). There is much more in this remarkable book and we are fortunate that Ishi Press International has reprinted it (with a New Forward by Michael Beeson). Highly recommended but difficult.
8 of 9 people found the following review helpful.
Few are Ready
By S.Z.
Because I learned to program computers before I learned calculus, the Riemann integral was "obviously" a computer program in disguise, except that the shrinking step-size "obviously" suggests the limit if the integrand is rational valued. On the other hand, I started to think that calling exp(9) an "exact" value was strange, since it's not exact in the same way that a rational number is exact. Along the same lines, xx -2 does not = 0 for any rational x, but only for a real number. But what is a real number? A Cauchy sequence of rational numbers, none of which satisfy x^2 - 2 = 0. In terms of rational numbers, you can find values of x such that xx - 2 < epsilon.
The point is that, if you want, you can base analysis on the natural numbers rather than on sets. You can reserve the term "exists" for actually computable numbers. To me this book, so far, presents analysis as a vast machine made of rational numbers. This makes certain proofs more complicated. At the same time their content is intuitively clearer, since (it seems to me) it all boils down to rational numbers and "high school algebra" (Zeilberger). Also important (and actually fundamental) is a rejection of the LEM. This appeals to me, since a math based on LEM is more theological/metaphysical perhaps than is necessary. I can't help but think that applied mathematics is constructive, since quantity and proportion seem to be where the rubber meets the road. To base analysis on natural numbers and concrete/particular constructions also gels with the math one learns on the way up to analysis. I personally think its important to learn mainstream analysis as well as constructive analysis. This particular constructive analysis book is written for readers who already know some classical analysis. It's dense. It's "motivated" in the specific sense that Bishop contrasts his version of analysis with the classical version. There's not a single picture in the book, nor are there many examples. Instead he covers lots of bases, including complex analysis and measure theory.
UPDATE
I still think highly of the constructive approach. I would have probably benefited more from a more elementary approach. Bishop speeds through basic analysis in order to get to the constructivized graduate level stuff, probably because constructivizing basic analysis was easy for him and even largely done already by Cauchy, Cantor, and others. It's not hard to get the basics of constructive analysis free, I now see. Anyway, I should also mention my own shift into anti-foundationalism in regards to math. I will be learning classical analysis in pursuit of my degree with less resentment now, even if constructive analysis seems truer and purer in some ways. The reason? Influenced by philosophers like Richard Rorty and Rueben Hersh, I'm coming around to see mathematics, as far as I am concerned with it, as a language for describing reality. I think it's picked up like a language. If you are looking into constructivism because you're skeptical about Cantor's infinity of infinities, for instance, you might want to read Hersh's book instead of Bishops. Zeilberger's blogs are also worth looking into. Chaitin's Metamath is also great.
To sum up, I still think this is great book, to the degree that I am worthy of it, but I've been benefited just as much by more philosophically explicit and less mathematically difficult writers. I read the philosophers in order to figure out where I should apply my effort. At the moment, I'm thinking that humans belong at the strategic rather than the tactical level. I don't think an applied mathematician will need foundations as much as he or she will need a harder-to-define "intuitive" know-how. I'll let you know.
4 of 4 people found the following review helpful.
Interesting for some
By Michael Nahas
I ran into constructive mathematics while looking at Homotopy Type Theory. Constructive mathematics takes a hard line on statements of existence: if you say a certain value exists, you have to show how to calculate it. Constructivists don't accept commonly held ideas like the Law of the Excluded Middle and the hierarchy of infinities. Without those tools, you can understand why most mathematicians thought that almost nothing could be done with constructive logic.
This book is probably the best known book in the constructive mathematics. Bishop showed that you can use constructive logic and still do real mathematics, like calculus.
The value of the book is seeing how the low-level definitions change and seeing how constructive logic is used in practice.
Some of the definition changes are expected. Real numbers are defined as sequences of rational numbers that get closer and closer to the true value. Other definitions are unexpected. For real numbers, =, >, and >= are all defined separately and not in terms of each other. You would think that >=, where we say "greater than or equal to", would be defined in terms of "greater than" and equality, but it isn't. This feels odd, but it doesn't affect many things.
It was good to see constructive logic used in practice. When I first heard about it, I thought it would be awful. Yes, it is occasionally requires turning things around. Yes, it sometimes lessens what can be proven. But most of the time, you just say to yourself that there's two or three things to keep in mind and it works fine. In many proofs, the logic was the same as a non-constructive proof; the only difference was the definition used for real numbers.
The book doesn't explain calculus or analysis or metric spaces. It expects you know those areas and that you've seen the non-constructive definitions and theorems. This book just rewrites the terms and theorems in a constructive fashion and proves the constructive theorems.
My background is limited, so I was only able to read 4 chapters in detail. The introductions weren't especially enlightening or inspiring, which was a disappointment. Bishop's constructive set theory is not very interesting to me, since I'm unlikely to use it and I think Martin-Lof's Dependent Type Theory does a much better job. Most of my enjoyment came from satisfying my curiosity on how the definitions and logic are used to build more complex pieces in mathematics and what limits they impose on it.
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