HN Books @HNBooksMonth

Book is a collaborative effort of many mathematicians from different soviet regions

https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Grad...

but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out.

The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that

Integral over the boundary = Integral over the interior of the divergence, or more poetically

Int_(dA)A = Int_A dA

Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total.

So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a).

The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b).

I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone.

I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

⬐ floxy

Thanks for the recommendation. I'd probably need the undergraduate version unfortunately. But this internet stranger wants to encourage you write the book for kids someday.

I studied physics in undergrad many years ago, and it's been a long time since I used higher level math on a regular basis. I just picked up Mathematics: Its Content, Methods, and Meaning, on the recommendation of someone here. It's over 1000 pages, so it's going to be a lifetime reading project for me, but it's been wonderful to start reading. The first part of the book traces the earliest origins of math, and everything was grounded in real-world physical problems.

I've been a high school math teacher for most of my life, and I have deep frustrations with how removed from meaning math is presented to most students. Just because the teacher knows and states the possible relevance doesn't mean students should be expected to take the relevance at face value.

I was mostly focused on teaching algebra 1 classes, which is why I didn't use higher math all that often. But my understanding of higher math grounded my teaching of lower level concepts all the time, and I often spoke of higher level concepts with my students to help demystify math. My 8yo son loves math for now, and the moment school makes math meaningless to him I am planning to find some way to intervene.

https://www.amazon.com/gp/product/0486409163/

"Mathematics : Its Content, Methods and Meaning" by A. D. Aleksandrov, A. N. Kolmogorov ,M. A. Lavrent'ev. (3 Volumes)

This is a classic and exactly what you are seeking for. I think it was originally published in 1962.

https://www.goodreads.com/book/show/405880.Mathematics

https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

⬐ kwoff

Free on archive.org: https://archive.org/details/MathematicsItsContentsMethodsAnd...

⬐ cosmic_ape

I'd add V. Arnold's books on Differential Equations and Geometry to that list.

Wikipedia summarizes it nicely: "His views on education were particularly anti-Bourbaki." :)

⬐ m31415

Some problem collections by Arnold:

* Problems for Children from 5 to 15: http://jnsilva.ludicum.org/HMR13_14/Arnold_en.pdf

* A mathematical trivium pt. 1: http://www.physics.montana.edu/avorontsov/teaching/problemof...

* A mathematical trivium pt. 2: https://www.latroika.com/mathoman/exos/arnold-exos-math-engl...

⬐ reactor4

Thank you, Is there an answer key to those?

⬐ m31415

Not that I know of, but many of the problems have been discussed online, e.g, calculating the average of the 100th power of sin in under 5 minutes [1], which according to Arnold, if you cannot solve, you don't understand mathematics.

[1]: https://math.stackexchange.com/questions/24533/find-the-aver...

⬐ SSLy

There is some quality in Russian math books that western authors seem to lack.

⬐ m31415

Yes, and quite a few Mir books are available freely [1]

[1] https://mirtitles.org

⬐ sifar

Thank you for the link. A treasure trove. Their archive page [1].

[1] https://archive.org/details/%40mirtitles?&sort=-publicdate&p...

⬐ rramadass

Somebody from Russia/Former Soviet Countries from Eastern Europe ABSOLUTELY NEEDS to setup a publishing company (eg. Dover Publications) to bring all of their Science/Engineering books back into print. They will do very well in today's education market where the emphasis seems to have shifted to huge tomes/useless multiple editions/pretty colouring etc. rather than succinctly presenting the knowledge itself. The Mir (and other) publishers books were huge in many countries of the world and are remembered fondly to this day.

⬐ mlazos

I remember when one of my professors told me how AVL trees were invented by mathematicians in the Soviet Union. It was in a paper titled “An algorithm for organizing information”. It was a side project for them, and it’s humbling to hear how simple the idea was to geniuses.

⬐ rramadass

Hey, that's a neat fact! I did know that AVL stood for "Adelson-Velskii and Landis" but not that they were Soviet-era Mathematicians.

⬐ weinzierl

I absolutely agree about the quality of Russian math books. "Bronshtein and Semendyayev" and "Abramowitz and Stegun" come to my mind.

I doubt that a publishing company bringing back those titles into print would be very successful. One thing is that many of these titles are mostly of historical value. Who needs a book with mathematical tables nowadays? Many of the books are still available as used books too.

The most compelling argument though is that they are easily available on libgen (like this very post proves). So in a sense the publisher you wish for already exists, just not in a the form you probably thought of.

EDIT: Oops, I just learned that Milton Abramowitz and Irene Stegun are actually Americans.

⬐ rramadass

I disagree with you on the publishing front. Mathematics is fundamental and timeless (what do you mean "many of these titles are mostly of historical value"?) They may need some trivial editing (though i would much prefer that they be published as they were with a note explaining the historical aspects) but otherwise they were information dense and succinct with an eye to Applications. They were all excellent across the board. They were directly responsible for educating a lot of poor people in many countries due to their very low cost and affordability. I would say this was one of the biggest successes of the Soviet ideology i.e. the education of the masses in Science & Technology fields. Current day Russians/Eastern Europeans/Central Asians can justifiably be very proud of this part of their History.

Much of "modern" textbooks are full of excessive verbiage obscuring the essentials, "pretty printing" disguised as "easy comprehension" and a racket for the publishers to make money. Why in the world do i need so many editions of books containing Mathematics which has not changed in centuries? Why do they cost an arm and a leg? Education is as fundamental as Health services and both should be affordable in service of the population.

So again, somebody setup a publishing company (eg. Dover Publications) and bring ALL the forgotten books from the Soviet era back into print :-)

⬐ weinzierl

> Mathematics is fundamental and timeless

Absolutely and you certainly have a point with what you wrote. May opinion is more along the lines of: "the content is still as valuable as it ever was but the presentation is not."

Take one of the examples I mentioned. Abramowitz and Stegun is a collection of mathematical tables. If you needed to calculate the sine of a value, would you rip out your chuffed copy of Abramowitz and Stegun or would you use your calculator? Even for the more obscure tables there is probably nothing in the book that isn't in Mathematica. If I really needed to look into the book for some reason I would be too lazy find my copy, given that online versions[1] as well as extended and improved versions[2] are just a few mouse clicks away.

Now, a book of mathematical tables is like an extreme example but I still feel the same sentiment for all my old math books. Why bother with a physical copy if I have a searchable online version right at my fingertips? When i comes to the books from the Soviet era I guess libgen has them all and I think most people would not buy a physical copy anyway.

[1] http://people.math.sfu.ca/~cbm/aands/intro.htm

[2] https://dlmf.nist.gov/

⬐ rramadass

> but the presentation is not

You are very wrong here. It is the very presentation in those books viz; succinct and concise, no frills approach, high information density and with an eye to applications which makes them so valuable today. It is the best way of Science teaching distilled from the brains of a whole lot of smart people.

I am not sure why you are fixated on one book of tables. It is irrelevant in the broader scheme of things. For example none of the Mir books that i have, have anything to do with pre-calculated tables other than a few appendices.

There are a huge swath of students across the world who do not have the same access to technology as we do. Printed books are still the norm amongst the majority of students in the world. Printed books will also outlast any Digital media presentation of books due to its simplicity and robustness i.e no problems like DRM, unreadable extinct formats, availability of good ereaders, health aspects, etc (there is a whole lot more i can elaborate here).

Finally, and most important, research is beginning to show that we retain/understand less when using ebooks/ereaders than when we read a printed book. This is very much true of technical books (borne out by my own experience) where you need concentrated attention with body and mind. For example, we intuitively jump back and forth across pages, use our fingers as book marks, subconsciously create spatial maps of what we are reading etc. all of which have no analogues with current day ereaders. Cognitive Science is still trying to figure out how best to use modern technology. So don't throw away your old Maths books just yet :-)

⬐ amagumori

Oh shit, it’s authored by the Kolmogorov!

⬐ weinzierl

I love this one, it's from 1962 but already had a chapter on

ELECTRONIC COMPUTING MACHINES

§1. Purposes and Basic Principles of the Operation of Electronic Computers

§2. Programming and Coding for High-Speed Electronic Machines

§3. Technical Principles of the Various Units of a High-Speed Computing Machine

§4. Prospects for the Development and Use of Electronic Computing Machines

⬐ rsj_hn

I can second this. An amazing volume. If you want more details, I'd stick with the Soviet math school, which was amazing in it's pedagogical soundness and user friendly texts. Anything from the EMS -- Encyclopedia of Mathematical Sciences -- is worth picking up.

⬐ mikorym

There are also the three MIT volumes called Fundamentals of Mathematics.

⬐ andrepd

Love at first sight. It's a Landau but for mathematics. Love it.

⬐ juskrey

Exactly this

⬐ sonabinu

The best!

⬐ andrepd

Also libgen (since the authors are all dead):

http://gen.lib.rus.ec/search.php?req=Mathematics+methods+mea...

This book is fantastic and pretty much takes you through an entire undergrad mathematics course: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

⬐ throwawaymath

The topics covered in that book are undergrad level, but that book is not suitable for learning the topics. It’s more like a high level discussion of the topics looping them together with historical background. It’s more appropriate for people who are already familiar with the material.

I’m going to go with a few assumptions here:

a) You don’t do this full time.

b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.

c) You are skilled with programming/software in general.

In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.

Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

I’d recommend getting great and cheap russian recap of mathematics up to 60s [1] and a modern coverage of the field in relatively light essay form [2].

Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.

[1] https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

[2] https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

⬐ aphextron

>Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

I'd strongly disagree with this. To the mathematically literate, concepts like "imaginary numbers", "prime numbers" and "logarithms" are just simply understood things which are familiar and have always been a part of your lexicon. These are actually wildly complex, abstract ideas which take years to fully grasp as an adult being first exposed to the material. Developing a mathematical intuition to the level of an advanced high schooler is no small feat for an adult with zero mathematical training. I'd strongly suggest anyone actually starting from zero mathematical knowledge to go back and spend time doing basic remedial math courses from the point of simple algebra and arithmetic with a good teacher to truly understand numbers first.

⬐ synthmeat

I actually think I agree with everything you've said, but here's why I think it's moot - majority of visitors here have finished high school and I wouldn't be surprised that majority have at least started on tertiary education. Numbers are pretty high worldwide too. [1]

So, terms like "mathematically [i]literate", "adult with zero mathematical training", taken at face value, don't apply to most of us in the world, and almost certainly not to the OP either.

[1] https://ourworldindata.org/primary-and-secondary-education#c...

⬐ sidcool

What's your opinion of Mathematics for computer science?

⬐ synthmeat

Have no opinion - haven't read it and Amazon has no preview for it.

⬐ ruraljuror

If we're talking about the same book, it is available for free: https://courses.csail.mit.edu/6.042/spring17/mcs.pdf

I bought the book for sale on Amazon. The printed version seems like a print-on-demand copy of the free PDF. The paper size is 8.5x11 and the layout is the same. I'm a little suspicious of the publisher.

I have only used the book as a reference for a few sections. The style is very approachable.

⬐ gradschool

The Princeton Companion to Mathematics is a good resource consisting of a huge collection of detailed articles on many mathematical subjects by knowledgeable contributors. It requires no specialized background and is curated by Fields Medalist Tim Gowers. Whoever reads it from cover to cover is my hero, but failing that there's always an interesting article to jump to.

Don't just be a consumer but write something as soon as you're inspired. I wish there were more emphasis on writing mathematics in school prior to the graduate level. Leslie Lamport says if you're thinking but not writing you're not really thinking; you only think you're thinking. For Feynman the act of discovery wasn't complete until he had explained it to someone. There's also the rule of thumb that if you can't explain a mathematical concept to a ten year old, you don't understand it yourself.

Edit: typo

⬐ playing_colours

Can you please elaborate more how to be a producer rather than just a consumer in maths? Advice like to do more maths rather than read it is clear and a regular undergrad can follow it. Producing in maths sounds like writing papers to me that I can hardly imaging as an undergrad student. It is actually a problem for me as for a software engineer: in programming I can produce rather early and it creates a motivational feedback plus helps learing things better. With maths I cannot get the feeling of creating anything, the only pleasure is in solving problems from books.

⬐ vram22

>Don't just be a consumer but write something as soon as you're inspired. I wish there were more emphasis on writing mathematics in school prior to the graduate level. Leslie Lamport says if you're thinking but not writing you're not really thinking; you only think you're thinking. For Feynman the act of discovery wasn't complete until he had explained it to someone. There's also the rule of thumb that if you can't explain a mathematical concept to a ten year old, you don't understand it yourself.

Fantastic quotes and points, thanks for sharing.

⬐ synthmeat

> The Princeton Companion to Mathematics is a good resource...

I think Princeton Companion to Physics curated by Frank Wilczek, a Nobelist, is due to be published this year.

> Whoever reads it from cover to cover is my hero...

Yeah, I'd die an accomplished man if I would grok just a few books I treasure, amongst which are TPCTM and MICMAM.

> Don't just be a consumer but write something as soon as you're inspired.

Absolutely. That's why I recommend just a small amount of comprehensive resources. It's hard to get motivated by a pile of books complemented with synthetic problems related to a particular chapter. The idea is to just go about your daily life and start to slowly see more and more math problems everywhere around you; it does wonders to motivation.

⬐ mtreis86

What is MICMAM?

Whitepapers, lectures, and speech transcriptions are also good motivation, and useful resources. Sometimes overwhelming, especially if reading mathematical text is as a foreign language. And sometimes it takes you down a rabbit hole.

My biggest block for learning math has really been all the unlearning. After a while ideas like negative numbers and zeros and processes like addition and subtraction stop making as much sense as I thought.

Here is my favorite rabbit hole:

http://www.turingarchive.org/viewer/?id=465&title=01

leads to:

http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002278499...

leads to:

http://www.jamesrmeyer.com/pdfs/godel-original-english.pdf

Where to go from there - philosophy or computation? Lambda calculus is only a couple clicks away. Lisp papers, perhaps?

⬐ synthmeat

> What is MICMAM?

Check my [1] at root.

> Where to go from there - philosophy or computation?

For me, there's plenty of fun in mathematics without venturing even near the edges of it. Maybe one day I'll grow bored of it, who knows - it's a lifelong process.

⬐ mtreis86

I didn't mean to insinuate that the process I describe is one to be taken out of boredom. Let me try to explain what I am thinking:

I have been studying lisp and wanted to understand more about the origins. So I went back to the beginning of the language and read the various McCarthy papers. But what he was thinking is not entirely clear to me. So I wonder, what papers was he studying himself when he wrote this? That is easy to answer as he put the references right there in the back of the paper for me to track down. So I start reading papers written by Church and Godel. I repeat this process recursively while looking for shared references. That network of interconnected papers is a treasure trove of useful information. Reading the same papers an author was reading during their writing process is a valuable way to expand your understanding of their work.

This is awesome, thanks.

Obligatory 'zoomout' recommendation: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V..., which I learned about from HN (http://hackernewsbooks.com/book/mathematics-its-content-meth...). Wish I had read/pondered this before grad math classes.

This book is fantastic and should get you pretty far: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

⬐ tunesmith

btw I bought this on your recommendation. It's... big! :) I haven't started reading it yet, I'm in the middle of Gödel Escher Bach right now.

This is always a good topic with numerous HN submissions and comments:

Just from a search, there are some great results:

1. Mathematics for Computer Science - HN Submissions: https://news.ycombinator.com/item?id=9311752, https://news.ycombinator.com/item?id=3694448

2. How to Read Mathematics - HN Submissions: https://news.ycombinator.com/item?id=4030812, https://news.ycombinator.com/item?id=1576969

__

Here are some other excellent mathematics books:

1. Mathematics: Its Content, Methods and Meaning

Containing the thoughts and direction of numerous mathematicians including Kolmogorov, this is a great survey of the field of mathematics. It touches upon Analysis, Analytic Geometry, Probability, Linear Algebra, Topology, and more. [1.]

2. Concrete Mathematics: A Foundation for Computer Science

Containing the thoughts and direction of mathematician and computer scientists such as Donald Knuth, this is a great reference for computer science related mathematical concepts focusing on continuous and discrete concepts. [2.]

__

[1.] http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

[2.] http://www.amazon.com/Concrete-Mathematics-Foundation-Comput...

⬐ throwahdhw

Any HN recommendations for a good thorough book on Lambda Calculus? Preferably creative commons if such a text exists.

⬐ dpflan

Have you seen this post on StackExchange/Mathematics? This may be a good start. I can't personally recommend any. Of course, good luck learning!

http://math.stackexchange.com/questions/967/learning-lambda-...

There are hundreds of good math textbooks to recommend, it really depends on your interests.

For a broad overview at an undergraduate level, with a great job explaining the context of various mathematics topics, these Russian books from the 50s, Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrentiev, are pretty fun. Amazon link to the one-volume Dover reprint (but I’d recommend finding a used three-volume hardback copy): http://amzn.com/0486409163

Or check out John Stillwell’s Mathematics and its History: http://amzn.com/144196052X

Buy those: http://gcpm.rutgers.edu/books.html

and this: http://www.amazon.com/dp/0486409163/?tag=stackoverfl08-20

⬐ rajeshmr

Unable to open the amazon link, could you please repost or mention the name of the book ?

⬐ thatcat

Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics)

⬐ Enzolangellotti

Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics)

Also, you might want to get Richard Courant's "What is mathematics?" and a book (I'm reading multiple as I'm in your same spot, didn't have much mathematics during high-school because I thought I didn't possess the acumen, then I realized I really liked the subject) on proofs.

Gelfand's books are very very very good, trust me on this one, they build on the fundamentals. The books are not short of flaws though, namely the writing is informal, the author assumes some preexisting knowledge (that's why they are often not used as class books but as supplementary notes) and do not offer many exercises. But if you get the whole bunch you'll have covered the high-school curriculum (and more). The AMA olympiad books are good reads, same with those "Art of Problem Solving" books. But personally, I'm not starting these until I've gained enough confidence, I still can't solve elaborate problems or mathematical olympiad kind of questions (but I'm getting smarter).

This [0] book is to mathematics what Feynman Lectures are to physics (debatable, but I think it's worth the read).

[0] http://www.amazon.com/dp/0486409163/ (Mathematics: Its Content, Methods and Meaning)

Usually good books travel from English speaking world to other languages as translations.

One of the masterpieces that has gone the opposite direction is:

Mathematics: Its Content, Methods and Meaning (three volumes bound as one) by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev (18 authors total) http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

This book is really good companion for autodidacts. It's basically overview of mathematics.

⬐ bkcooper

This is a good book, although I agree with others that truly learning from it would be difficult. If you like this, another book you might really enjoy is the Princeton Companion to Mathematics.

⬐ dsuth

This is an absolutely beautiful book, aesthetically, and does a good job of conveying some of the wonder of mathematics.

⬐ pflats

>Usually good books travel from English speaking world to other languages as translations.

How do you know that?

⬐ kryptiskt

There are some numbers for this in David Bellos "Is that a fish in your ear?"[1], chapter 19.

An UNESCO study of translations between Swedish, Chinese, Hindi, Arabic, French, German and English over a decade showed that 104,000 of the 132,000 translations made between all those languages were translations from English.

[1] http://www.amazon.com/That-Fish-Your-Ear-Translation/dp/0865...

⬐ coldtea

>Usually good books travel from English speaking world to other languages as translations

Yeah, no. Usually good books get translated, period.

From Homer and the Bible to Pascal, Leibiz, Dostoyevsky, Godel and Einstein, good books fly the other way around all the time.

⬐ edmccard

>Yeah, no. Usually good books get translated, period.

Great books certainly get translated in every direction. But for merely good books, I wouldn't be surprised if readers outside the US consumed more books translated from English than readers in the US consume books translated from other languages.

⬐ LBarret

Quite americano-centric, no ?

Considering that the average reader reads more in Europe than in the US and that there is a very dynamic domestic industry in many of these countries, I would say the opposite.

And I didn't even take India and China into account...

source: http://www.prnewswire.com/news-releases/nop-world-culture-sc...

⬐ jules

Why?

I don't think so for the simple reason that academic books in English usually aren't translated because people in countries outside of the US can read English. Even academic books that do not have any native English speaking authors are usually written in English. Books are more likely to be translated to English than from English, because translating to English multiplies the size of the audience many times, whereas the other way around does not.

⬐ Pyret

I own this book and I'll tell you it's not easy learning math from this book. Most of it is just very light overview. There are much better rigorous textbooks that are simpler and more complete.

There are some great textbooks translated from Russian. Analysis by Kolmogorov, (rigorous) Linear Algebra by Shilov, Complex Analysis by Markushevich to name a few.

⬐ nabla9

That's why I said it's good companion.

The book covers too much to be thorough. Each chapter gives good introduction to the subject matter and ends with list of suggested reading.

I always read the relevant parts from this book before going deeper. Not everyone is going to dwell into non-euclidean geometry, functional analysis and topology.

Furthermore, I don't think typical self studying engineer in Hacker News wants to learn math using rigorous introduction to analysis. You can get good working knowledge and intuition without knowing what delta epsilon is.

⬐ Pyret

Well, there is no shortage of very good and VERY SIMPLE and rigorous intros to analysis books out today:

How to Think About Analysis by Lara Alcock.

Understanding Analysis by Stephen Abbot.

Mathematical Analysis and Proof by David Stirling.

Numbers and Functions: Steps into Analysis by Burn.

Analysis: With an Introduction to Proof by Steven Lay.

A First Course in Mathematical Analysis by David Brannan.

⬐ jeffreyrogers

I agree with you, that book is great for giving an overview of the general areas of mathematics and for providing context before going deeper into an area. I've used it to get some background in the courses I'm taking classes on before the semester starts and have found that really helpful.

This isn't a course but this book has been consistently word-of-mouth recommended as a comprehensive survey of undergraduate mathematics.

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

⬐ hackerboos

I bought this on recommendation from HN.

I'd also recommend - Engineering Mathematics by KA Stroud. All three volumes are fantastic.

Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

Covers something like three years of an undergraduate degree in mathematics. Lots of words - but that text is used to develop an understanding of the concepts and images. Considered a masterpiece. An enjoyable read.

I've looked at a bunch of these math compendiums while researching what to include in my book, and this one seemed the best so far: http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do... The writing isn't very hand-holdy, but it covers a lot of important topics, and without too much fluff.

For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.

I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.

Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov et al.

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

Not the same type of book, but you could do a lot worse than reading through Mathematics: Its Content, Methods and Meaning, by M. A. Lavrent’ev, A. D. Aleksandrov, A. N. Kolmogorov. It's an amazing book which gives a mathematical (but not rigorous in the sense of proofs etc.) overview of most of mathematics.

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

⬐ alok-g

+1.

The best thing I like about this book is that it often first gives a real-life example of a problem, then gives some history of how the problem was solved, and what the solution was. Even the first chapter was highly illuminating.

⬐ hf

Absolutely astounding: I have been looking for this book ever since I pored over it in the Wolfson Reading Room in Manchester Central Library 5 years ago. I didn't take down the authors' names, though, referring to it as "that yellow mathematics book" then and ever since.

I credit that book with much if not all my mathematical insight.

Thank you.

(Just seeing that cover leaves me all tear-eyed, reminiscing over that wonderfully irresponsible time.)

⬐ oskarth

Glad to help. Amazing that an off-hand comment can have so much impact on a human being across the globe. I've certainly been on the other side of it several times before.

⬐ auvrw

> "that yellow mathematics book"

replace "that" with "those" ( http://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics ).

the dover books are also a good series, and pretty much anything by Artin is good. i also looked at the Halmos book a couple of people have mentioned.

perhaps one thing to be aware of, though, is that you're not always going to learn things in the linear way they're layed out on the page. moreover, it might be helpful to have more than one book for any given subject. Lang's Algebra, for example, is a really good reference, but a tome if you read it like a text. so you might pick up something small and subject-oriented with a lot of exercises like Artin's Galois Theory or Atiyah's Commutative Algebra, and supplement it with a reference like Dummit and Foote or Lang's texts on Algebra as a whole.

oh, right: whatever book you choose, do the exercises.

⬐ pervycreeper

But that's not a Springer book, it appears to be published by Dover, and the covers look totally different. The GTM series is also highly variable in quality.

⬐ auvrw

perhaps the GTM series has some bad titles; i haven't read them all. but it has some good titles, and from the description original poster's background, it's probably as far as the person wants to stretch right now. sure, the cambridge advanced math series might be more "quality," but those also tend to be both very focused (in terms of specificity of the topic) and very dense (in terms of delivering a lot of results with almost zero fluff).

Recently, I've been reading this mega book that covers a lot of advanced topics in math by Kolmogorov et al. It is nice because it covers a lot of topics w/o in a fairly short text --- it is a brick but it covers 2000 years of knowledge! http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

For first-year stuff, I would recommend No bullshit guide to math and physics[1] and the No bullshit guide to linear algebra[2] of which I am the author.

[1] http://minireference.com/ [2] http://gum.co/noBSLA

May I present one of the greatest math books for general audience: Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev.

http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin...

Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come even close to this. Reading this book together with the The Princeton Companion to Mathematics was real treat.

⬐ alok-g

I second the book as well as the above praise for the book as one of the greatest. I have read several chapters from the book and have loved them all.

Question: I haven't read The Princeton Companion to Mathematics. How would you compare these two?

⬐ nabla9

The Princeton Companion is more like encyclopedia. Main part of the book is articles describing 100 or so mathematical concepts in alphabetical orders. Then it has articles describing major mathematical problems.

This book looks awesome. Mathematics is so vast that as a student its hard to know what classes you should take, or even what the map of the territory looks like. That is one of the reasons I got frustrated with my pure math BS.

The Russian school put out some pretty massive volumes, like

http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin...

but I found those to be a little too caught up in the proposition/proof cycle to be useful as a guide to the uninitiated.