An interesting idea: http://networkologies.wordpress.com/2011...-language/

“I’ve basically taught myself to ‘read’ math as a ‘foreign language,’ from calculus on forward. That’s to say, I never took a calculus course (though I did tutor HS math for years, so I had a pretty solid foundation), but now I can read almost the most advanced math books and understand what they’re saying. That’s not to say I could solve problems and get the right answers, but I can follow when others do it (and in a pinch, I could find the way to get the right answers).

This has opened up whole new fields of math and science texts that would otherwise be off-limits as ‘for specialists only.’ And it all came from the books you see below. I’m convinced ANYONE can teach themselves to read math as a foreign language. But math is so often taught poorly. These books do it right.”

I flunked several calculus courses in college, though I have always found math somewhat interesting, I just couldn't get it.
But many of these books mentioned here look very interesting, so I might pick a few of them up.

So, what should do you differently to also be able use your knowledge to actively solve problems?

I don't really know if I agree with that list. None of them are really classic texts and an awful lot of them just seem to be history/reference books or expositions aimed at the layman. You'd be far better off reading a small handful of real textbooks.

If anyone's interested I could offer a reading list (assuming HS level math). Maybe magamo knows some good ones.

Quote:So, what should do you differently to also be able use your knowledge to actively solve problems?

Practice solving problems, of course!

Mathematics is not a foreign language and there is very little merit in just reading texts passively. There is no skill involved in reading maths other than concentration.

I've always wanted to learn set theory, logic, that sort of thing. Right now I'm reading a book called "The Logic of Knowledge Bases" which is interesting, but a lot of the vocabulary is indeed hard as a non-expert. Do you have any books that you'd recommend?

So what does it mean to read math 'as a foreign language'? Is this different from just learning math?

@Jaunty
I would suppose the books are just about theory and how different branches of maths can be used. Eg you could learn what calculus can be used for but not actually be able to integrate or differentiate.

Jaunty Wrote:So what does it mean to read math 'as a foreign language'?

It means being able to "read almost the most advanced math books and understand what they’re saying without taking a calculus course". I guess taking courses on Real Analysis, Set Theory, Logic etc is still fine. O_o

Amset Wrote:I've always wanted to learn set theory, logic, that sort of thing. Right now I'm reading a book called "The Logic of Knowledge Bases" which is interesting, but a lot of the vocabulary is indeed hard as a non-expert. Do you have any books that you'd recommend?

Read this one → Naive Set Theory, by Paul R. Halmos.

iSoron Wrote:Read this one → Naive Set Theory, by Paul R. Halmos.

I'll look for it, thanks!

Amset Wrote:I've always wanted to learn set theory, logic, that sort of thing. Right now I'm reading a book called "The Logic of Knowledge Bases" which is interesting, but a lot of the vocabulary is indeed hard as a non-expert. Do you have any books that you'd recommend?

Hmm, that's something I mean to study myself someday so I haven't read anything on the subject, but if it were me buying for myself I'd probably go for: http://www.amazon.com/Set-Theory-Logic-R...724&sr=1-1

I think I'll write down what I recommend for various topics just in case someone asks me something I don't know about...

I tried to list these in a rough pedagogical order but I may have made some mistakes! I starred the optional books. I've listed 8 non-starred texts, which will get you far further than the 19 in the OP's link in terms of real ability.

//////////////////////////////////////

The first text you should read (read it all, do the exercises at your discretion): http://www.amazon.co.uk/Calculus-Michael...376&sr=1-1

Introductory text to algebra and geometry: http://www.amazon.com/Algebra-Geometry-A...338&sr=1-2

For probability there are a lot of texts but this is what I learnt from: http://www.amazon.co.uk/Probability-Intr...179&sr=1-1

**What I learnt mechanics from: http://www.amazon.co.uk/Course-Mechanics...399&sr=1-1

Big tome on mathematical methods (dip in where you please): http://www.amazon.co.uk/Mathematical-Met...893&sr=8-1

For elementary topology this is a relatively easy read, has some overlap with the second calculus book: http://www.amazon.co.uk/Introduction-Met...20&sr=8-22

The second text you should read about calculus (perhaps skip the last chapter, but the penultimate chapter which formalises vector calculus is great): http://www.amazon.co.uk/Principles-Mathe...570&sr=1-4

The first text you should read about complex analysis: http://www.amazon.co.uk/Introduction-Com...252&sr=8-1

**A fun, graphical text on complex analysis which could serve as a second or a companion text (lots of nice pictures, explains everything using geometry): http://www.amazon.co.uk/Visual-Complex-A...315&sr=1-1

For algebra this will cover all you need to know including group theory and linear algebra, but you may find something that suits you better (if you go with this try and find one of the cheap international editions or go with a used copy): http://www.amazon.com/Topics-Algebra-I-N...881&sr=1-1

**Geometry (Hyperbolic, etc.) http://www.amazon.com/Curved-Spaces-Geom...261&sr=1-1

**Markov chains: http://www.amazon.com/Markov-Cambridge-S...245&sr=1-1

**Modern probability: These online notes are concise, easy to follow and cover most of what you need to know about the subject - http://www.statslab.cam.ac.uk/~beresty/t...mnotes.pdf For a more in depth treatment, see: http://www.amazon.co.uk/Probability-Meas...658&sr=8-1

**Principles of statistics: http://www.amazon.co.uk/Essentials-Stati...16&sr=1-51 For statistical models themselves, this seems alright, probably lots of others to choose from http://www.amazon.com/Statistical-Models...689&sr=1-2

Books that have been recommended to me/Books on my to read list:
(I haven't read these, so beware)

**Further analysis, measure theory, complex analysis, functional analysis http://www.amazon.co.uk/REAL-COMPLEX-ANA..._pr_sims_i

**Cheap ODE/PDE texts http://www.amazon.com/Differential-Equat...460&sr=1-1 / http://www.amazon.com/Introduction-Ordin...500&sr=1-1

**Algebraic topology: http://www.math.cornell.edu/~hatcher/AT/AT.pdf / http://www.amazon.co.uk/Algebraic-Topolo...176&sr=1-1

**Graph theory http://www.amazon.co.uk/Modern-Graph-The...851&sr=8-1

**Number theory http://www.amazon.com/Fundamentals-Numbe...794&sr=8-1

**Logic and set theory http://www.amazon.com/Set-Theory-Logic-R...724&sr=1-1

**Numerical analysis http://www.amazon.co.uk/Numerical-Differ...849&sr=1-1

**Mathematical finance http://www.amazon.co.uk/Concepts-Practic...884&sr=1-1

**Fluid mechanics http://www.amazon.co.uk/Elementary-Fluid...995&sr=1-2

**Statistical physics http://www.amazon.co.uk/Concepts-Thermal...071&sr=8-1

**Astrophysics http://www.amazon.co.uk/Introduction-Mod...456&sr=1-2

If anyone has suggestions for the following I'd like to hear:

Quantum mechanics
Electromagnetism
Special and general relativity

If anyone else wants to chime in with their own suggestions or criticise my choices feel free. I'm not particularly interested in abstract algebra so I may have neglected that area.

There's so much I want to learn and so little time. I imagine I'll be spending my life chasing that state of being where you feel "I know enough".

No. Just no. Mathematics is definitions and proofs. One does not simply learn to read. It requires active participation. Take the following statement, which contains no mathematical symbols: "A square matrix is invertible if and only if its determinant is non-zero." First, should we just believe this is true? No, let's try to prove it. Oh wait a second... what is a matrix? what is a square matrix? What does invertible mean? What is a determinant? Second, assuming the statement is true, what does it mean? This is an elementary yet extremely significant statement that would not be appreciated by someone just skimming through a math book trying to follow along.

shadysaint Wrote:No. Just no. Mathematics is definitions and proofs. One does not simply learn to read. It requires active participation. Take the following statement, which contains no mathematical symbols: "A square matrix is invertible if and only if its determinant is non-zero." First, should we just believe this is true? No, let's try to prove it. Oh wait a second... what is a matrix? what is a square matrix? What does invertible mean? What is a determinant? Second, assuming the statement is true, what does it mean? This is an elementary yet extremely significant statement that would not be appreciated by someone just skimming through a math book trying to follow along.

Hmm, not sure if I agree with you. I understand what your statement means, despite 1) having no idea how to prove it, and 2) being unable to recall how to determine the determinant. In fact I use matrices all the time in 3D games programming, but I no longer have a great understanding of why it works, but I know how to use them for 3D stuff, and using built in functions to the do the actual matrix maths.

Isn't that the point of the OP's list - as long as it's explained well (in terms designed for the "layman"), it is perfectly possible to understand mathematical concepts, language and symbols without totally being able to understand why they actually work or even how to apply them?

My point was that instead of going through OP's list, you could go through half as many books, actually understand the theory and be able to apply it. Surely that's far more stimulating (and useful) than reading some wishy-washy overview.

It's the difference between learning Japanese and reading about Japanese. The latter doesn't count as learning Japanese.

But what if I can't understand the theory? What if I find it too hard, or worse too tedious? I did maths A-level, and quite a bit at uni, and it's fine in theory, but for me (and many others) it is a pretty hard and boring subject.

The OP's list is of books that explain maths extremely well, in very easy to understand terms. Also importantly he points out that all the books are very interesting and enjoyable to read, describing one as a "page turner". Can you say the same about the regular maths textbooks on your list?

The OP's list just gives the wrong impression. Reading these sorts of "page turner" books won't teach you anything of substance, and certainly won't let you understand advanced texts. To learn anything you have to sit, stare and scratch your head.

I can't say that I've enjoyed reading every textbook I've ever read, but when there's a subject you really want to learn I wouldn't trade learning the actual theory for a non-technical overview for the world. Every hobby requires that bit of tedious effort to get good at it.

I think anyone can learn it if they're determined and really interested. I don't know why you'd want to read OP's list in the first place if you find it to be a boring subject.

dizmox Wrote:If anyone has suggestions for the following I'd like to hear:

Quantum mechanics
Electromagnetism
Special and general relativity

For Quantum Mechanics I'd recommend

http://www.amazon.co.uk/Quantum-Mechanic...=8-1-spell

or

http://www.amazon.co.uk/Quantum-Mechanic...=8-2-fkmr0

What exactly do you want to know about electromagnetism? Just a general introduction or something more specific?

Thanks! I actually bought Rae but never got round to reading it. For electromagnetism, I'd just like a general introduction, yeah. It just feels strange to learn quantum mechanics before electromagnetism (which is what happened at my university).

I plan to learn physics in depth after I've graduated and actually have time to devote to extra-curricular things.

Quote:I understand what your statement means

Ok, but would someone understand the significance? Would they be able to able to use the statement in a useful way? I mean, this isn't like something you need in order to buy groceries (although, you would probably find it useful if you were in charge of pricing the groceries), but if you pick up a book on differential equations some time down the road, it'll just be assumed that you know it.

Quote:being unable to recall how to determine the determinant.

Determinants are rarely calculated in real-world applications. Generally it is better to algebraically find the cases when a particular matrix has a zero or non-zero determinant. Why? Because a square matrix is invertible if and only if its determinant is non-zero. What does that mean?

Quote:I no longer have a great understanding of why it works...using built in functions to the do the maths.

And that is what separates engineers from mathematicians. Oh, and mathematicians don't have to worry as much if their work has any practical applications (yet).

dizmox Wrote:I think anyone can learn it if they're determined and really interested. I don't know why you'd want to read OP's list in the first place if you find it to be a boring subject.

Good point! I do sort of question what the OP is trying to achieve..

But at the same, I can see how maths can be interesting without knowing the calculations. The history, application, and general concepts can all be pretty fascinating. A 'Brief History of Time' is a good example - a great read on Cosmology and the physics and maths behind it, but only one equation in the entire book (E = mc2).

shadysaint Wrote:Determinants are rarely calculated in real-world applications.

Bad example; determinants are used all the time in computer graphics.

dizmox Wrote:Thanks! I actually bought Rae but never got round to reading it. For electromagnetism, I'd just like a general introduction, yeah. It just feels strange to learn quantum mechanics before electromagnetism (which is what happened at my university).

I plan to learn physics in depth after I've graduated and actually have time to devote to extra-curricular things.

It's a bit tricky to recommend an electromagnetism textbook as I never had a specific EM course. We did the basics using a general textbook (Physics by Tipler) in first year. After that we covered more advanced bits in other courses.

This textbook seems to have everything in it that I did though so maybe have a flick through it in a bookshop if you can. http://www.amazon.co.uk/Electromagnetism...237&sr=1-9

Thanks.

This recent progress in quantum computing has sparked my interest in physics and computer science...

Nice effort dizmox

and edit: For electromagnetism
http://www.amazon.com/Introduction-Elect...013805326X

Is by far the most widely used book across every university I know. He also has a book on quantum mech though I've never actually purchased it.

But the thing I love about him is that he writes it like a big ass blog, the maths is extremely simple to follow, no pretentious notations nor does he bog the reader down with Einstein (4 vector) notation either which is at i've seen many undergrad EM books do. I've read bits and pieces of the entire thing, but yeah it's a decent book, and I think any serious student of EM that reads this book to its entirety will really grasp EM.

And also on the topic of the actual thread, it's quite interesting and I'll try have a look at it. 'Reading' Physics to me was stuff I've acquired, surprisingly, through observation of how lecturers do it. I observe how *they* read (out aloud)

So you see them looking at a model of some sort
"ohh ok so this particle is waving here"
"the particle waves andddd it's subject by that force, and that force and that force, and its energy is deteriorating at this rate given by that random walk"

For me, it was just these simple things which really gave me the fluency to read physics.

good point..i do wonder what the OP thinks ppl will achieve by readingthe OP's list of books.

although on the other hand based on dizmox's suggestions, i further realize how different everybody is. Don't want to generalize so i'll give my perspective and backstory but Spivak's book is hard, so i wonder how hard the other books are on your list. I used spivak's calculus textbook in my 1st semester in undergrad and wow, what a shock to my system. Just so you know, I'm not a noob at math, I'd always been able to do it, at least the exercises and kinda get it in high school, having also been good enuf to represent my county at the new york state math team annual meeting, and i majored in math in undergrad, but it just felt totally different in undergrad.

I had never seen so many definitions before; math became a lot more like deep reading. Every week i'd spend at least 7 hours on a short homework assignment most of the time sitting blankly thinking of what to do. Sometimes I'd just give up on some problems. Still i got used to it kinda but never felt like i really accomplished or learned that much. It's so different that ppl who hated math in grade school may really end up liking it; I've seen it with some few ppl. then again i see it more likely that if you hated math in grade school, Dizmox's recommendations will do less to help you out than the OP's recs unless you have some serious interest and determination to just sit and think for long whiles.
I ended up acing my class and pretty much all my classes in undergrad but found out it wasn't what i liked compared to doing math team problems in high school.

If i have time i think i'll go through the OP's books as i think they can serve a great gateway for ppl to become interested in math and maybe it'll renew mine

Hmm, the thread took a tangent from the original post towards learning maths and physics and such rather than learning to read it, but some of the books seem interesting, so it's not a total loss. ;p