Reddit reviews The Princeton Companion to Mathematics
We found 49 Reddit comments about The Princeton Companion to Mathematics. Here are the top ones, ranked by their Reddit score.
Princeton University Press
Start with a good algorithms book like Introduction to algorithms. You'll also want a good discrete math text. Concrete Mathematics is one that I like, but there are several great alternatives. If you are learning new math, pick up The Princeton Companion To Mathematics, which is a great reference to have around if you find yourself with a gap in your knowledge. Not a seminal text in theoretical CS, but certain to expand your mind, is Purely functional data structures.
On the practice side, pick up a copy of The C programming language. Not only is K&R a classic text, and a great read, it really set the tone for the way that programming has been taught and learned ever since. I also highly recommend Elements of Programming.
Also, since you mention Papadimitriou, take a look at Logicomix.
This might be more than you want, but check out http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
The Princeton Companion to Mathematics is a really good resource for understanding the broad landscape of what's currently big in math research. It goes into great detail about the history of different branches, biographies of famous mathematicians, small summaries of key concepts, and applications in the modern world. The list price is close to $80, but there might be other ways of finding a copy (wink).
http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
Not a textbook per se, but I'll go with the Princeton Companion to Mathematics. It's a coherent (and surprisingly accessible) overview of just about all of pure math.
You wouldn't be doing much mathematics if you stuck to writing about those topics.
You could take a look at The Princeton Companion to Mathematics (or some other source that contains a lot of mathematics), find a topic that piques your interest (even if initially you don't understand much about it), try to figure out as much as you can about it (you could ask questions on this subreddit), and then try to write an introduction to it at the level of your fellow students.
I heard good things about Princeton Companion, but I have never read it.
Someone requests a guide and the response is there is no one comprehensive compendium? (a) Learn to fucking read - OP's not asking for a comprehensive compendium. (b) There are comprehensive compendiums of many, much more general subjects than "people of North East India" (eg. mathematics, history of the fucking world).
TLDR: STFU
I like Szekeres's A Course in Modern Mathematical Physics for referencing intro-grad-level material. It covers abstract linear algebra, differential geometry, measure theory, functional analysis, and Lie algebras, and teaches you some physics along the way.
More generally, the best "breadth" book on advanced mathematics is Princeton Companion to Mathematics by Gowers et al. and its slightly underachieving younger brother of a companion text, Princeton Companion to Applied Mathematics by Higham et al.. You won't properly learn advanced mathematics this way, but you'll get the bird's-eye view of modern research programs and the math underlying them.
If you want a more algebraic take on Szekeres's program to teach physicists all the math they need to know, check out Evan Chen's Napkin project, which is intended to introduce advanced undergrads (it's perfectly fine for grad students too) to a wide variety of advanced mathematics on the algebra side of things.
Since you're doing probability and statistics, check out Wasserman's All of Statistics and Knill's Probability Theory and Stochastic Processes for good, concise references for intro-grad-level material.
I will second what /u/Ovationification said, though. I didn't really learn anything with the above books, I just use them occasionally for reference or to think about pedagogy.
You might enjoy skimming through the Princeton Companion to Mathematics (amazon, princeton, maa review, wikipedia), which gives an overview of the main areas of mathematics.
While it's not on the bleeding edge, the Princeton Companion gives a good overview of recent mathematics.
Hi there,
For all intents and purposes, for someone your level the following will be enough material to stick your teeth into for a while.
Mathematics: Its Content, Methods and Meaning https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
This is a monster book written by Kolmogorov, a famous probabilist and educator in maths. It will take you from very basic maths all the way to Topology, Analysis and Group Theory. It is however intended as an overview rather than an exhaustive textbook on all of the theorems, proofs and definitions you need to get to higher math.
For relearning foundations so that they're super strong I can only recommend:
Engineering Mathematics
https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp/1403942463
Engineering Mathematics is full of problems and each one is explained in detail. For getting your foundational, mechanical tools perfect, I'd recommend doing every problem in this book.
For low level problem solving I'd recommend going through the ENTIRE Art of Problem Solving curriculum (starting from Prealgebra).
https://www.artofproblemsolving.com/store/list/aops-curriculum
You might learn a thing or two about thinking about mathematical objects in new ways (as an example. When Prealgebra teaches you to think about inverses it forces you to consider 1/x as an object in its own right rather than 1 divided by x and to prove things. Same thing with -x. This was eye opening for me when I was making the transition from mechanical to more proof based maths.)
If you just want to know about what's going on in higher math then you can make do with:
The Princeton Companion to Mathematics
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
I've never read it but as far as I understand it's a wonderful book that cherry picks the coolest ideas from higher maths and presents them in a readable form. May require some base level of math to understand
EDIT: Further down the Napkin Project by Evan Chen was recommended by /u/banksyb00mb00m (http://www.mit.edu/~evanchen/napkin.html) which I think is awesome (it is an introduction to lots of areas of advanced maths for International Mathematics Olympiad competitors or just High School kids that are really interested in maths) but should really be approached post getting a strong foundation.
If you really want to get a feel for what (pure) mathematics is and how it all interrelates, you should read the Princeton Companion to Mathematics. This book is kind of like wikipedia for pure math, though extremely well written (and by prominent mathematicians). I think the best audience for the book are undergraduate students in math, but anyone really interested in learning what "real" mathematics is about should enjoy it.
While not a strictly historical book, The Princeton Companion to Mathematics is a good one to have handy for all sorts of light mathematical reading and has a pretty extensive section on mathematicians.
The Princeton Companion to Mathematics. Seriously. This book is amazing. It gives a high-level overview every major branch of mathematics.
It's the best tool I've found for rapidly expanding the breadth of one's mathematical knowledge. A good starting point to find what really interests you.
The Princeton Companion to Mathematics is essentially a really general dictionary for most topics in known math. Entries vary from one half-page to three or four pages long, covering definitions, theories, and sometimes deeper understanding of certain subjects. I believe each entry is also personally written by a professional in the field.
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
If I'm not mistaken this is the draft of his article for the PCM.
This encyclopedia is fun.
This might also be some use to you: The Princeton Companion to Mathematics. Incredibly comprehensive math encyclopedia.
Right now I'm making my way through The Princeton Companion to Mathematics and it's a very good overview of all the different mathematical fields, going through what they work on and examples of how it's done.
http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
5.4 lbs of math goodness.
The closest thing I know of to what you're asking for is the Princeton Companion to Mathematics. It is over a thousand pages long.
You will never be able to obtain complete mastery over mathematics as a whole, certainly not in just one year. People spend their entire careers on tiny sub-areas of a single topic within mathematics.
However, if your goal is to attend college and study physics, you actually don't need anything like this. Instead, what you need to do is look up the physics program at the college you want to attend, see what level of mathematics they expect you to know coming in, and then use something like Khan Academy to work up to that level.
> I'm not looking for a book to help me become a set theory pro, I'm literally just looking for a book that will give me some challenging, enjoyable bedtime reading.
Are you sure that you want to read a book on axiomatic set theory or are you happy with any math subject and it's just that set theory is the only one that comes to your mind?
In the latter case, I would recommend Mathematics and its history by John Stillwell for bedtime reading (and it does have a bit of set theory, too). Also, the The Princeton Companion to Mathematics is highly recommended.
And in any case, the mathematics section of your local library provides more low cost bedtime reading than I could ever note here. :-)
Working your way through a beginning discrete math class is kind of an overview of the history of math. But here are some stand-alone books on it. Writing quality varies.
The World of Mathematics
A History of Mathematical Notation. Warning: his style is painful.
Journey Through Genius
The Princeton Companion to Mathematics. A reference book, but useful.
I had a similar request to yours, except I wanted to go beyond Calculus to get a broad survey of mathematical topics, using a ground up approach. The Princeton Companion to Mathematics is exceptional, I can't recommend it enough! It covers all the topics you wish your mathematics teachers had instilled in you, all within a comprehensive & comprehensible form. It has been years since I studied math. I've long since forgotten a majority of what I was taught but, I can still easily progress in this book and I feel like I finally understand many of the ideas that were impenetrable before.
I'm not alone in my positive review. You'll note that people have been heaping praise onto this volume on Amazon and in more formal book reviews as well.
This is a far more advanced version of what you're looking for, as a survey of "real" research level math topics. You would need far more than just calculus to be able to understand it though. I would say some of the videos by Numberphile, 3blue1brown, etc. would maybe suit your interests?
Talk to professors in your university. Also, check this book out.
The Princeton Companion is an excellent source that introduces you to an absolute ton of topics in a readable way. It's not cheap, but IMHO it's the type of book everyone should have, whether you're a professor or an aspiring student.
I recommend the following book for getting a good overview of modern mathematics: The Princeton Companion to Mathematics although it is a bit pricey (though less expensive than the average textbook). It is extremely well written, even if it doesn't necessarily hit all the details. It focuses more on an intuitive understanding of many modern mathematical concepts so that more formal and detailed treatments. The authors wrote the book to help math students get up to speed about various different fields of math as well as help working mathematicians better communicate across different disciplines.
Martin Gardeners books are good too. I specifically like The Colossal Book of Mathematics and The Colossal Book of Short Puzzles and Problems. His books tend to be very problem oriented rather than theory building, whereas the Princeton Companion is more expository. While Gardeners Colossal books are quite a bit shorter than the companion, I read them more slowly since I often stop to work on the problems he presents.
I think it helps to realize that there isn't any particular order to learning different kinds of math. High School and elementary schools set math up like there's a clear hierarchy to all the material, but that's not necessarily the case. For example, you don't need Calculus to do basic Graph Theory or elementary Set Theory.
There are lists of textbook recommendations on /r/math but these are the books I would recommend without knowing much about your current skills or interests.
For math history I enjoyed Journey Through Genius. Also, check out this thread linked in the r/math FAQ for some others.
As for books that survey the whole of higher mathematics at a layman's level... you might have a hard time finding one. That stuff isn't exactly easy to talk about without an assload of prior knowledge. The closest thing I can think of is The Princeton Companion to Mathematics, but that certainly isn't written for a general audience.
Here's another thread from a few weeks ago with some other book recommendations. Actually, Keith Devlin's Mathematics: The New Golden Age (see review in that thread) might be more what you're looking for.
In the first half of the twentieth century a great deal of energy was expended trying to provide a foundation for mathematics by mathematicians and philosophers. These efforts precipitated the development of both logic (mathematical and philosophical) and also played an important role in us developing our understanding computation (i.e.the Church-Turing thesis) because many philosophers and mathematicians believed that a mathematical proof should be able to be verified computationally.
Some major figures:
Here is a section from the wonderful Princeton Companion to Mathematics that discusses this period but focuses more on the foundations of math than computation.
he might already have these but
The Princeton Companion to Mathematics contains exactly what you're asking for, but restricted to the world of math. It has hundreds of short articles on many of the most important pure and applied math topics, and they're written by top experts in a way that's accessible to someone with a little bit of college math knowledge.
Maybe you are thinking of The Princeton Companion to Mathematics?. I believe he wrote and reviewed a number of sections to it.
I used it for college calc and linear algebra, but if you are looking for past that then no, they probably don't have much.
One book I'm unwilling to part with is the princeton companion to mathematics. It's an incredible book with an entry by well known mathematicians in their field, writing on every major field of math, in a way that is accessible to people with a college level math background. It would be a really good starting point to find some sub-topics you were interested in.
Something to find from the public library maybe?
http://www.amazon.com/dp/0691118809/
Of course. More just generalizable/widely applicable fundamentals.
Example: https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
Princeton companion to mathematics https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809?
Yeah. Part of me feels like I've just been lucky in finding easy problems that the "real" scientists in my field hadn't bothered to try yet.
I still don't really understand linear algebra or vector calculus, for instance. I have Linear Algebra Done Right, Div, Grad, Curl, and all that, and the Princeton Companion to Mathematics on my wish list, which may help.
I've got this book in pdf format called "The Princeton Companion to Mathematics". It's got a very nice overview of all of Mathematics in it. It's extremely brief but very broad. I don't think you'll learn much mathematics but it gives you an idea of what everything's about.
I didn't get the PDF by "traditional" means and I'm sure you could acquire it the same way.
I suggest The Princeton companion to Mathematics (wiki).
this book ? http://www.amazon.com/dp/0691118809/ref=asc_df_06911188091498544?smid=ATVPDKIKX0DER&tag=hyprod-20&linkCode=asn&creative=395093&creativeASIN=0691118809
Best thing you can do is read your little heart out. Find a copy (electronic or library) of something like the Princeton Companion and browse it over the course of a few weeks/months and pick out a few fields that particularly interest you. Then hit up How To Become A Pure Mathematician and start on your reading.
Eventually - and by this I mean "in a year or two" - you want to be able to email the prof in your dept who shares your interest and say "Hey, I've read the foundational texts on xyz, what would you recommend next?" and from there develop a relationship that'll hopefully lead to some undergrad "research" and a glowing letter of recommendation in your final year.
The other, equally important thing is to be a likable, sociable person. Unless you're some kind of wunderkind, collaboration is the name of the game and it gives you a huge advantage over the smelly nerd that no-one really wants around.
e: also lol undergrad pure maths research hahahahaha. if you can read a contemporary research paper in most pure maths subfields by senior year, you're ahead of the game.
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
Expensive, but worth it.
Have you considered buying him a book on mathematics? I know they can be expensive but there are some interesting ones out there.
For example, the bible of the mathematics.
I myself received a classical (danish) gymnasium education in the liberal arts; 3 germanic languages and a romanic language, latin, philosophy, history, theology, mathematics, physics - almost the whole works, but I never seemed to pick up on too much of the math. It seemed otherworldly and irrelevant to me at the time.
I've 'rediscovered' the beauty of math partly through my interest in history (we where never presented with the historical context of mathematical inventions - or their technical implications) and through a personal curiosity that was never satisfied in the rigid classes.
I found the following book useful to my personal studies: http://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/
I think there could do some really cool analysis tracing lines of thought and how they developed or comparing what was in vogue in math to world developments at the time. This book might be a good overview for modern developments and this one has a overview of the development of math through history
Rather than give descriptions of each of the fields you named, I'll just mention that you might want to peruse The Princeton Companion to Mathematics.
Discrete mathematics with ducks!
Contemporary Abstract Algebra
An Illustrated Theory of Numbers
Visual Complex Analysis
Number-Crunching
The Magic of Math: Solving for x and Figuring Out Why
The Princeton Companion to Mathematics
The Number Devil: A Mathematical Adventure
Professor Stewart's Cabinet of Mathematical Curiosities
Gödel, Escher, Bach: An Eternal Golden Braid
One Two Three . . . Infinity: Facts and Speculations of Science
Recently made a move and these are a few books on my shelf that stand out more.
There are so many beautiful mathematical images that could be chosen though and I do wish more publishers would have more creative book covers also.
You want the Princeton Companion to Mathematics