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The Proof is in the Pudding: The Changing Nature of Mathematical Proof
This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.
- GenresMathematicsNonfiction
281 pages, Kindle Edition
First published January 30, 2010
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Displaying 1 - 3 of 3 reviews
February 28, 2020
Books that aim to address non specialists on mathematics face a quandary. If they assume (or, worse, demonstrate) technical content they quickly lose readership. But if they try to talk around the technical content, they produce nothing but meaningless hand waving.
I've read many books on the history of mathematics and I've seen many approaches to this problem. In THE PROOF, SK starts off on the meaningless hand waving side of the divide, but manages to find his footing in the later chapters, which include thoughtful accounts of the impact of computer driven proof engines, the transition of mathematics from a solitary to a highly social activity, and the blurring of clear cut outlines for determining whether something counts as a proof when the length of a proof stretches to thousands of pages.
I've read many books on the history of mathematics and I've seen many approaches to this problem. In THE PROOF, SK starts off on the meaningless hand waving side of the divide, but manages to find his footing in the later chapters, which include thoughtful accounts of the impact of computer driven proof engines, the transition of mathematics from a solitary to a highly social activity, and the blurring of clear cut outlines for determining whether something counts as a proof when the length of a proof stretches to thousands of pages.
March 21, 2021
I tremendously enjoyed reading the book.
The book is well-written, largely well-explained (I only found very few minor blunders in the description of mathematical problems, but the parts on (theoretical) computer science contain rather serious blunders). It does not go too deep into proofs but deep enough to showcase a few strategies and discuss them. The treatment is deeper than one can find in typical mainstream nonfiction books on maths, but is much lighter than in a mathematical publication. I enjoyed it and felt it was adequate. Also, I was impressed by the way Kantz explains. It reads like butter and is fun.
It is well structured and the personal judgement and view of Krantz is very interesting and worthy to ponder. Especially the personal account on the profession of a mathematician and differences between Europe and US. (Even though the picture he draws of Europe is not 100% accurate, he is right about the fixed salary schemes for positions at most continental European Universities)
I don't agree with everything he writes and also have a different view on the difference between computer science and mathematics.
Some minor critique: I found the statement that ALL proofs that are derived by contradiction are not constructive a bit strong. This may be true with the examples in the book but I have seen proofs by contradiction where the contradiction was derived by a construction and even the mathematician authoring the paper claimed it to be constructive. But it indeed might have been the case that the claimed theorem could have been formulated differently.
I also found it a bit verbose on history of mathematics (and history of computers and automatic theorem provers). While certainly interesting and entertaining (and I really enjoyed it and don't complain at all), I would have wanted to read more about proof methods mathematicians employ themselves as well. Proof by induction, etc.
I also wonder why the reference for A.L. Mann - A complete proof of the Robinson conjecture is cited as a preprint. It was published in 2003.
However, in any case it was a great read. I couldn't put it down as it was very entertaining.
The book is well-written, largely well-explained (I only found very few minor blunders in the description of mathematical problems, but the parts on (theoretical) computer science contain rather serious blunders). It does not go too deep into proofs but deep enough to showcase a few strategies and discuss them. The treatment is deeper than one can find in typical mainstream nonfiction books on maths, but is much lighter than in a mathematical publication. I enjoyed it and felt it was adequate. Also, I was impressed by the way Kantz explains. It reads like butter and is fun.
It is well structured and the personal judgement and view of Krantz is very interesting and worthy to ponder. Especially the personal account on the profession of a mathematician and differences between Europe and US. (Even though the picture he draws of Europe is not 100% accurate, he is right about the fixed salary schemes for positions at most continental European Universities)
I don't agree with everything he writes and also have a different view on the difference between computer science and mathematics.
Some minor critique: I found the statement that ALL proofs that are derived by contradiction are not constructive a bit strong. This may be true with the examples in the book but I have seen proofs by contradiction where the contradiction was derived by a construction and even the mathematician authoring the paper claimed it to be constructive. But it indeed might have been the case that the claimed theorem could have been formulated differently.
I also found it a bit verbose on history of mathematics (and history of computers and automatic theorem provers). While certainly interesting and entertaining (and I really enjoyed it and don't complain at all), I would have wanted to read more about proof methods mathematicians employ themselves as well. Proof by induction, etc.
I also wonder why the reference for A.L. Mann - A complete proof of the Robinson conjecture is cited as a preprint. It was published in 2003.
However, in any case it was a great read. I couldn't put it down as it was very entertaining.
October 17, 2021
Книга читается нелегко. Я так и не понял, какую аудиторию видел перед собой автор. Если это читатель неподготовленный, "не в теме", то он не споткнется разве что о многочисленные байки из жизни математического сообщества. Для любителей занимательной математики, занимательной математики там нет. Может кого-то интересует история математики? Но она если и излагается автором, то крайне фрагментарно, да и предвзято. Скажем, может подробно разжевываться биография Дирихле, но вот жизнеописания того же Гаусса там нет совсем. Наверное, для математика-профессионала она будет слишком проста, а вот я со своим математическим образованием в виде пяти семестров технического вуза на многое смотрел, как баран на новые ворота. Ибо, доказательство основанное на том, что "функция что-то отображает на себя", требует хотя бы самого общего объяснения, что понимается под "отображает на", поскольку такие вещи изучаются только будущими математиками.
Вообще книга бессистемна, по сути это набор эссе, свободных рефлексий автора. Но все-таки там есть ряд действительно оригинальных мыслей, сподвигших и меня на некоторые рефлексии также.
"Доказательство в математике — психологический инструмент, предназначенный для убеждения некоего лица или аудитории в том, что некоторое математическое утверждение истинно. Структуру и язык для построения такого доказательства выбирает его автор, но оно должно быть скроено по меркам той аудитории, которая будет его воспринимать и оценивать." Каково! У меня рука после этого не решилась поставить ниже четырех звезд, несмотря на все недостатки книги.
Вообще книга бессистемна, по сути это набор эссе, свободных рефлексий автора. Но все-таки там есть ряд действительно оригинальных мыслей, сподвигших и меня на некоторые рефлексии также.
"Доказательство в математике — психологический инструмент, предназначенный для убеждения некоего лица или аудитории в том, что некоторое математическое утверждение истинно. Структуру и язык для построения такого доказательства выбирает его автор, но оно должно быть скроено по меркам той аудитории, которая будет его воспринимать и оценивать." Каково! У меня рука после этого не решилась поставить ниже четырех звезд, несмотря на все недостатки книги.
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