When I was studying physics, I found Feynman’s books in the library, read them all, and had the feeling I understand everything! But then I tried to solve some final exams from previous years, and realized the feeling is false. These books gave me great intuition - but they made all the math look deceivingly simple, and as a result it is hard to develop the actual problem solving skills and intuition. I know my exper…
People want intuitive explanations because it seems the easiest. It is, that's the problem. Learning is supposed to be painful. You have to get your hands dirty.
It's easier to feel that you know something than to actually know it.
Spivak's Calculus. It's "just" calculus... but it's also everything else leading up to it. It's a wonderful book, written in a very engaging style, and it shows you how mathematicians think and how they play. It shows you why we have proofs, why things go wrong, and all that had to happen before we came up with a definition of derivatives and integrals that we're happy with (and of course, all of the things we can do…
One of the biggest conceptual moments in my life was taking an intro class to complex analysis in high school (not through my school itself, this was at a nearby university that runs a weekend program for interested applicants). It's probably not a novel intellectual framework for anyone who's spent time learning math from a theoretical point of view, but the guy who taught it opened with an (admittedly ahistorical, but that wasn't the point) tour of the development of numbers, starting with the intuitive case of counting, moving on to algebra and the question of what type of number could satisfy an equation like x + 5 = 2, and so on. He wasn't taking any philosophical position re the question of discovery vs. invention, but merely inviting us to consider the particular case of operations over a set not being algebraically closed and what it looks like to extend that set to support algebraic closure in a rigorously defensible way. Reading Spivak's Calculus, with the way that it starts off its path of inquiry by showing that, beginning with the basis of a totally-ordered field, the axioms at hand don't suffice to demonstrate the existence of e.g. a number that satisfies x^2 = 2, made me feel right at home again. It's like a detective story.
When I was studying physics, I found Feynman’s books in the library, read them all, and had the feeling I understand everything! But then I tried to solve some final exams from previous years, and realized the feeling is false. These books gave me great intuition - but they made all the math look deceivingly simple, and as a result it is hard to develop the actual problem solving skills and intuition. I know my exper…
People want intuitive explanations because it seems the easiest. It is, that's the problem. Learning is supposed to be painful. You have to get your hands dirty. It's easier to feel that you know something than to actually know it.
Learning requires a lots of false starts, traps, etc. Once one has mastered, he/she can provide an intuitive explanation (a path through the wild forest that learning is).
My recommendation below is not the equivalent of a Feynman's series for math, but one that is pegged much lower, for someone interested in basic remedial math. It is called "Who is Fourier: A Mathematical Adventure". I was tremendously surprised by this unusual gem of a book. It covers the range from basic arithmetic to logarithms, trigonometry, calculus to fourier series. https://www.amazon.com/Who-Fourier-Mathemati…
And it's only $975!
Oh No! That is an unfortunate link from Amazon where somebody is trying to hustle money. You don't need the 2nd edition since there is no change from the 1st which you can get for $10+ from many sites.
The book is quite good. It is written like a "Manga" book and hence has tons of drawings to help develop intuition for the concepts. It is written by a group of ordinary people with help from Scientists (a quirky club named Transnational College of Lex from Japan - https://en.wikipedia.org/wiki/Hippo_Family_Club ) and thus is very accessible. Highly recommended for High school students and above.
Note that the same group has also published two other books in the same vein; a) What is Quantum Mechanics b) What is DNA; both of which are also highly recommended.
People want intuitive explanations because it seems the easiest. It is, that's the problem. Learning is supposed to be painful. You have to get your hands dirty. It's easier to feel that you know something than to actually know it.
Learning requires a lots of false starts, traps, etc. Once one has mastered, he/she can provide an intuitive explanation (a path through the wild forest that learning is).
After doing the hard learning, you can lecture your intuitive mental model you have. But it's difficult to install that mental model into a beginner's mind. Often the intuitions are illusory mnemonics for the deeper understanding, which if you never learned in the first place would just point to nothing. You have to do the hard learning to arrive at the "intuitive" mental model.
The closest I'm aware of is What Is Mathematics? by Courant, Robbins, and Stewart -- starts off developing the natural numbers, goes onto number theory, analysis, complex numbers, set theory, projective geometry, non-euclidean geometry, topology, calculus, optimization, and some chapters on recent developments (as of its republishing in 1996, book was originally published in 1941). A lesser known one that isn't quite…
I am also been interested for the longest time - even with a degree in Mechanical Engineering - to understand mathematic like Calculus or the equations you see in neural networks papers.
My longest problem has been I have no idea what is going in the formula or fundamental questions like, "why is there a square root there". It is hard to describe my issue, but I've been very horrible with math anyways. Can't do gas station math anyways.
For mathematics, I would recommend: 1. "What Is Mathematics? An Elementary Approach to Ideas and Methods" by Courant and Robbins -- a general book on mathematics in the spirit of Feynman lectures. 2. Strogatz's "Nonlinear Dynamics and Chaos" -- it's a bit narrow in scope (mostly dynamical systems with a little bit of chaos/fractals thrown in) but very good nonetheless. 3. Tristan Needham, "Visual Complex Analysis", b…
Just a note, "Visual Complex Analysis" is a terrible book to learn complex analysis. Proofs are very iffy, akin to sketches of proofs. With that said, it is a excellent supplement to another std. complex analysis textbook.
Is it just the normal difference between e.g. an engineer's approach to real analysis and a mathematicians (but complex analysis swapped in), or something else?
I can think of a lot of fields where a decent grasp of complex analysis concepts would be very helpful even without being able to do rigorous proofs.
I think that those who are down-voting people suggesting Rudin are missing the forest for the trees. The level is not as low as Feynman's lectures, but if you want a text that has the hallmarks of a true master at work, Rudin fits the bill. I personally always felt that what made Feynman's lectures what they are were, was that the man actually understood the subjects on such a deep level that his mind was able to hone in on those little simple thought experiments and ways of looking at things that gave you huge insight into what he was lecturing on. Rudin comes at analysis with the same level of understanding and that's why Real & Complex and Baby Rudin are still the gold standard for analysis texts. It is impossible to convey the sense of elegance that mathematicians speak about until you've seen it and been blown away by it. Rudin will do that and leave you speechless, he's the master. I can tell you that my first encounter with Real & Complex leaving me thinking that Rudin was on a fundamentally higher plane than any author I had read before, it was like watching a magician. If you're looking for a real Feynman in mathematics, IMHO Rudin is your man.