An Introduction to Mathematical Reasoning
> Specifically, when learning a new field of math, the proofs sometimes don't feel rigorous to me right away, but once I get used to the the basics and the linguistic conventions, I'm more able to fill in the holes in my head the most complicated proofs i ever dabbled in were things like proofs of convergence for algorithms, but i encountered various types of proofs in several intro courses: computability and theory of computation, discrete math, introduction to higher math and then some upper div cs courses in ai/ml. what you say here rings absolutely true to me. as someone coming in with…
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…
"An Introduction to Mathematical Reasoning" by Peter J Eccels. Teaches the vocabulary of mathematics, just the basics you need to think like a mathematician, not a mathematics user like most science texts. "How To Solve It" George Polya. Heurists and problem solving skills, by a great mathematician. Do a google search, specially in the sci.math newsgroup. Again, read books by mathematicians for mathematicians; they're often far more enjoyable and actually far more straightforward (I was often confused by the examples in my school work; I didn't care for "vehicle moving at speed X" or…
I second the Eccles recommendation. It's a really great introduction to formal mathematics.
I've personally read An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Paperback by Peter Eccles and Chapter Zero by Carol Schumacher and would recommend them both. Sadly, this is not an easy thing to learn and requires a lot of work and most importantly discipline. You mustn't let yourself be at all complacent. It's very easy when doing exercises by yourself to believe that you've "got the idea" and you can "see how it works", but the key is to actually write _everything_ down, so that there is no room for handwaving at all. This is not easy to force yourself to do,…
The first upper-division course I took in college concerned itself solely with learning the art of mathematical proofs. I had an excellent professor, so I can't really say how much this book helps with the learning process when used by itself, but we were assigned An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, by Peter J. Eccles. Might be a good place to start!