Mathematics made difficult
Раньше слышал об этой книге, но не искал. Прочитав предисловие, уже понял, что зря.
Цитаты (взяты с reddit):
[Dedication] "To Clement V. Durell, M.A., without whom this book would not have been necessary"
[p.10] "Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige. Mathematics is prestidigitation."
He manages to pose several confusing questions about even the most basic facts. Leave alone "Question 4. Whether 1 is a number?", who can ever answer ""Question 5. Whether one should count with the same numbers he adds with, up to isomorphism?" :-)
[p.23] "This section is about addition. The fact that the reader has been told this does not necessarily mean that he knows what the section is about, at all. He still has to know what addition is, and that he may not yet know. It is the author's fond hope that he may not even know it after he has read the whole section."
[p.28] "With a few brackets it is easy enough to see that 5+4 is 9. What is not easy to see is that 5+4 is not 6."
[p.40] He defines a cancellable number x as one for which x+p = x+q never holds unless p=q. He first proves that if x and y are cancellable so is x+y, then with great care proves that 1 is cancellable, and therefore all numbers are cancellable.
[p.44-48]. In just a few pages, he gives a category-theoretic construction of the group of integers. Surely, this has never been done before.
[p.25] (On mathematical "beliefs".) "Like the world of a science-fiction story, a system of beliefs need not be highly credible-it may be as wild as you like, so long as it is not self-contradictory-and it should lead to some interesting difficulties, some of which should, in the end, be resolved."
[p.37] "unfortunately, there is a flaw in the reasoning. [..] to say that each of two numbers cannot be bigger than the other is to repeat the statement that is to be proved. It is not correct in logic to prove something by saying it over again; that only works in politics, and even there it is usually considered desirable to repeat the proposition hundreds of times before considering it as definitely established."
[Starred exercise] "Show that 17 × 17 = 289. Generalise this result."
Ссылка на DJVU версию (1.7 MB)
Ссылка на PDF версию (24.3 MB)
Цитаты (взяты с reddit):
[Dedication] "To Clement V. Durell, M.A., without whom this book would not have been necessary"
[p.10] "Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige. Mathematics is prestidigitation."
He manages to pose several confusing questions about even the most basic facts. Leave alone "Question 4. Whether 1 is a number?", who can ever answer ""Question 5. Whether one should count with the same numbers he adds with, up to isomorphism?" :-)
[p.23] "This section is about addition. The fact that the reader has been told this does not necessarily mean that he knows what the section is about, at all. He still has to know what addition is, and that he may not yet know. It is the author's fond hope that he may not even know it after he has read the whole section."
[p.28] "With a few brackets it is easy enough to see that 5+4 is 9. What is not easy to see is that 5+4 is not 6."
[p.40] He defines a cancellable number x as one for which x+p = x+q never holds unless p=q. He first proves that if x and y are cancellable so is x+y, then with great care proves that 1 is cancellable, and therefore all numbers are cancellable.
[p.44-48]. In just a few pages, he gives a category-theoretic construction of the group of integers. Surely, this has never been done before.
[p.25] (On mathematical "beliefs".) "Like the world of a science-fiction story, a system of beliefs need not be highly credible-it may be as wild as you like, so long as it is not self-contradictory-and it should lead to some interesting difficulties, some of which should, in the end, be resolved."
[p.37] "unfortunately, there is a flaw in the reasoning. [..] to say that each of two numbers cannot be bigger than the other is to repeat the statement that is to be proved. It is not correct in logic to prove something by saying it over again; that only works in politics, and even there it is usually considered desirable to repeat the proposition hundreds of times before considering it as definitely established."
[Starred exercise] "Show that 17 × 17 = 289. Generalise this result."
Ссылка на DJVU версию (1.7 MB)
Ссылка на PDF версию (24.3 MB)