35 of 36 people found the following review helpful
By
Jos Domingo Moll Vidal
- Published on Amazon.com

Format: Paperback

When I was at Hi-School, here in Spain, in the United States of Europe, our teacher used to mention this book to us lots of times. Later on, when at University, I left no stone unturned at several University libraries till I found a copy of the book, a Spanish edition. I read the book just in one session, due to how interesting I found it. Professor Morris Kline (he taught at New York University, I think I remember)shows throughout the book the rare ability to absolutely master every mathematical concept he talks about and at the same time being able to see those concepts with fresh eyes, as if they were new for him, as if he himself were a teenager encountering those ideas for the first time and trying to come to grips with them. True that modern math is far more abstract and powerful than what was the knowledge body of mathematics say in the 17th century for instance. But at the same time, emphasis in detaching ideas from any connections with the physical world, abstracting as an end per se, and letting 'rigorousness' and formalism prevail over intuition, has led some areas of modern math to something which looks like an esoteric exercice consisting of sipping through the symbols in a book. Kline makes for instance the point that too much insistence on 'rigorousness' is equivalent to finding snakes under the jewels, and says that when a mathematician does not care any more or is not sensible to problems such as the movements of planets around a star, the behaviour of density waves (sound) in a cavity, the movement of a mass hanging from a spring, etc., then mathematics are close to over. He says, for instance and amongst many other things, that when mathematics -within a determined mathematical field- go too far away from the physical concepts that may have inspired them and grow out of themselves for too long, then only the fact of that task being in the hands of men with an extremely developed intuition may prevent that mathematical field from finally becoming barren. I found it very interesting how many hypocrisies Morries Kline pointed out in the modern style of teaching mathematics. Some teachers use intuitive, supposedly 'non-rigorous' methods to work out things IN their heads, but try to transmit it to the students in a modern, 'rigorous', and absolutely non-pedagogical way. A double language that usually will have coupled with it an equally twofold moral -hypocrisy-. They think clear in their heads but offer muddy explanations to the students, demoralizing them and making it look as if perfectly assembled and refined mathematical ideas were actually coming out of their heads instantly. It is all vane presumption that does not good to actual teaching. The final victims are the students. This may be one of the most important factors in accounting for the superiority of Japanese students over American ones in several math contests and comparisons held through the years. I too think that intuition is paramount in math and that 'rigorousness' is secondary when confronted to that. Think of Thales from Miletus, Aristarchus, Archimedes, etc. They lacked a good mathematical formalism, they did not even have a positional number system, and even so they worked wonders (Archimedes' laws of statics, mechanics, etc.) and realized by pure intuition things that were only rediscovered more than two thousand years later (say for instance that the planets all revolve round the sun, that stars are other, distant suns, etc.). Nowadays, on the contrary, we seem to have too much formalism, even loads of it, but not that great intuition of the ancient scientists. Sometimes I wonder what Newton or Archimedes would be able to do if they had a period of updating, a Pentium and some other of the resources we have today. Ha ha ha! Kurt Gödel also said once that whereas along the last three centuries abstract mathematics have experienced huge progress, the solution of complicated numerical problems that can be stated in a handful of symbols of elementary arithmetics (consider for example Goldbach's conjecture, or the now 'solved' Fermat's Last theorem)is very backward. This too comes to support the comments made above. I definitely think that Mr Kline's criticisms must be taken very seriously, at least for the sake of giving children a better, or at least decent, mathematical upbringing, instead of killing vocations even before they are born. He speaks along the book with astounding clarity and honesty. Somehow he looks like the character in that telltale, saying that the king is naked whilst almost everybody else -blind people obviously would not see it- too realizes it but does not dare claiming it out 'because no one else does', 'because it goes against tradition', out of cowardice or due to any other obscure reasons. For the good of future generations, for a better education, this book should be read by every physicist and specially by every mathematician, let alone those mathematicians who also have to 'teach' their discipline. I would have given this book six stars if it were possible. Pity that it is currently out of print. I went trough hell to find a copy, but all the pains were worthy by far. Definitely a must for any mathematics library.