Mathematician's delight

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1. throwaway67114May 2021
   Mathematician's Delight by W. W. Sawyer seems a good book.
   thread
2. RobertKeransDec 2020
   Yep, I definitely do something similar. I respect Bret Victor greatly (his Magic Ink essay has had more of an effect on what I do as a career than almost any other single thing), but I've never really been completely on board with this idea -- though in fairness it's a response to an implementation of a very specific programming environment, it's not a general solution.

   Re mentally visualising, I'm reading a book called Mathematician's Delight[1] at the minute, and its first chapter has one of the neatest pieces of writing on abstract reasoning that I think I've ever read, it's reified a lot of instinctual ideas I had about how I program. I've clipped quite a lot of notes from it -- this in particular is relevant I think:

   "Beginners in geometry are sometimes puzzled by being told that straight lines have no thickness. We shall never, we are told, meet a straight line in real life, because every real object has a certain thickness. One of Euclid's lines, however, has no thickness. Two lines meet in a point, and a point has no size at all, only position. We shall never meet a point in real life, either, for all real objects have a certain size, as well as a certain position It is not surprising that pupils wonder how we know anything about objects which no one has ever seen or ever can see.

   This difficulty is a good example of the confusion which can come from misunderstanding the methods of abstract thought. We have seen that Euclid's geometry grew out of the methods used for building, surveying and other work in ancient Egypt, This work was done with actual ropes or strings, real linen threads with actual thickness. What does Euclid mean when he says that a line has no thickness, although he is using results suggested by the use of thick ropes? He means that in laying out a football field or in building a house you are not interested in the size or the shape of the knots made where one rope is joined to another. If you allowed for the fact that ropes possess a definite thickness, if you carefully described all the knots used by a bricklayer, you would make the subject extremely complicated, and no advantage would be gained. Euclid therefore says, if an actual rope has thickness, neglect this in order to keep the subject reasonably simple.

   The position is not that Euclid's straight lines represent a perfect ideal which ropes and strings strive in vain to copy. It is the other way round. Euclid's straight lines simplified account of the complicated way in which actual ropes present a rough, behave."

   [1] https://archive.org/details/mathematicians-delight-w.-w.-saw...

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3. rramadassOct 2019
   In addition to all the other good suggestions, the following are recommended (have not seen these mentioned so far);

   - Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

   - Methods of Mathematics Applied to Calculus, Probability, and Statistics - https://www.amazon.com/Methods-Mathematics-Calculus-Probabil... (all books by Richard Hamming are recommended)

   - Calculus: An Intuitive and Physical Approach - https://www.amazon.com/Calculus-Intuitive-Physical-Approach-...

   For a Textbook reference, the following are quite good;

   - Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences - https://www.amazon.com/Mathematical-Techniques-Introduction-... (easy to read and succinct)

   - Mathematics for Physicists: Introductory Concepts and Methods - https://www.amazon.com/Mathematics-Physicists-Introductory-C...

   For General reading (all these authors other books are also worth checking out);

   - Mathematics, Queen and Servant of Science - https://www.amazon.com/Mathematics-Queen-Servant-Science-Tem...

   - Mathematics and the Physical World - https://www.amazon.com/Mathematics-Physical-World-Dover-Book...

   - Mathematician's Delight - https://www.amazon.com/Mathematicians-Delight-Dover-Books-Ma...

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4. krnAug 2018
   > The OP is talking about teaching community college mathematics. That's not introductory university mathematics; if you are lucky it's Pythagoras' Theorem and quadratic equations.

   W. W. Sawyer's books, such as Mathematician's Delight[1] and Vision in Elementary Mathematics[2], seem to be exactly for this audience.

   [1] https://www.amazon.com/dp/0486462404/

   [2] https://www.amazon.com/dp/048642555X/

   thread
5. jorgetownAug 2009
   Some years ago I took the same approach to start again from ground zero. I found Polya to be good, but a Mathematician's Delight is better and more accessible: http://www.amazon.co.uk/gp/product/0486462404?ie=UTF8&ta...
   thread
6. tokenadultAug 2009
   That thread recommends many very few good books, but probably mostly books too hard at first for the participant who has posted this new thread.

   I'll recommend a couple of books from that thread:

   http://www.springer.com/physics/book/978-0-306-45036-5

   http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

   I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.

   Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.

   http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

   Get that right away.

   Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).

   http://www.amazon.co.uk/gp/product/0486462404/

   Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.

   http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

   (Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)

   http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038...

   (Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)

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