HN Books @HNBooksMonth

this is his article in the princeton companion to mathematics

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

a great (even if expensive) math book

⬐ apricot

Thank you! I was getting really strong "you've read this before" vibes from the article but couldn't place the source.

I suggest the following approach;

Start with some school textbooks for grades 8-12 i.e. Secondary Education. This is more for a refresher course in the absolute basics.

The above can be supplemented with the following books to develop intuition;

1) Who is Fourier - https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2n...

2) Functions and Graphs - https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...

After this is when you enter undergraduate studies and you have to fight the dragon of "Modern Maths" which is more abstract and conceptual. In addition to standard textbooks; i suggest the following;

1) Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

2) Mathematics: Its Content, Methods and Meaning - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

3) Mathematical Techniques (i am linking this so you can see the reviews but get the latest edition) - https://www.amazon.com/Mathematical-Techniques-Dominic-Jorda...

Finally, if you would like to learn about all the new-fangled mathematics your best bets are;

a) The Princeton Companion to Mathematics - https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

b) The Princeton Companion to Applied Mathematics - https://www.amazon.com/Princeton-Companion-Applied-Mathemati...

One important piece of advice that i have is to become comfortable with the Symbols, Notation and Formalism used in Mathematics. Most students are intimidated by the Formalism (which is nothing more than a precise form of shorthand to express abstract concepts) and give up on studying Mathematics altogether. This is a shame since it is merely the Form and not the Function of Mathematics.

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.

Nice! This book is modeled after the The Princeton Companion to Mathematics [1] which is simply awesome in every sense of the word, and extremely recommended for any person who is interested in mathematics.

I'm very glad to see more books being modeled after it, and I hope this trend will continue with things beyond math. My only hope is that those new books will match the quality of the original.

[1] - http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

⬐ rndn

Here is the Princeton Press link: http://press.princeton.edu/titles/8350.html

I have a hard copy of the Princeton Companion to Mathematics and find it incredibly useful -- highly recommended.

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

⬐ wuschel

I prefer the Handbook of Mathematics by Bronstein, which is a math reference bible for most science students in Germany.

http://www.amazon.com/Handbook-Mathematics-I-N-Bronshtein/dp...

⬐ weinzierl

The Bronstein is awesome and it has a fantastic index but I wouldn't call it a dictionary. Moreover it's far from comprehensive regarding to mathematics because it contains mostly topics relevant for engineering, physics and economics.

Vieweg Mathematik Lexikon is good, but in German. There must be something similar in English.

Please please please have a copy of these books in your house:

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

http://www.amazon.com/How-Solve-Mathematical-Princeton-Scien...

Its certainly too advanced for a 6 year old (or even a 16 year old, TBH) but just having it around is really great, I think. I remember when I was younger, I would look up stuff in more advanced books even if I couldn't understand them right away. The feeling I had was always: "Someday, I will be able to understand this..." which made me learn more physics and math.

"How to Solve it" is especially great if you do/will teach her in the future.

I would recommend the Princeton Companion to Mathematics[1]. Its got a broad range of topics in short digestible articles. (though the first section has preliminary material for understanding the later articles)

[1]http://www.amazon.com/gp/aw/d/0691118809/ref=redir_mdp_mobil...

⬐ aethertap

Thank you for that recommendation, I've been looking for a book like that for ages.

⬐ gsk

Thank you!

⬐ flatline

This is a good reference. It is basically an encyclopedia, so some sections are better than others. I'm not a big fan of Gowers' intro, it is very heavy and could have done much more in the way of explaining how to navigate the volume itself. It is generally assumed that you know integral and differential calculus.

If you are daunted by mathematical formulas and don't have a solid basis in math generally, I would recommend Pickover's "The Math Book"[1]. Very engaging; good short, non-technical descriptions of many of the same topics in Princeton; lots of pretty pictures.

[1] http://www.amazon.com/Math-Book-Pythagoras-Milestones-Mathem...

⬐ alok-g

How deep does Pickover's book go? I read in the reviews that each topic is limited to one page only, so I presume the real mathematics would not be covered. In other words, after reading this, will I understand the topics or would just understand things about them?

I also see a Physics book by the same author, but have the same question there too: http://www.amazon.com/Archimedes-Hawking-Science-Great-Behin...

⬐ flatline

IIRC there are a few things that are simple enough to actually explain in a page, but yes, one page per...artifact? Very little real mathematics, you will just understand things about them. It is, however, the best written of these pop-math type books that I've come across. If you are comfortable with undergraduate-level mathematics, the Princeton Guide is a much better book for getting an idea about a specific topic. Pickover's book is more for getting a feel for what topics are out there. They were written for two very different audiences and I like them both for different reasons.

The link was direct when I tried it. However, to answer your question, it is a chapter from The Princeton Companion to Mathematics. An amazing work covering much of mathematics.

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

Tim Gowers is also the editor of the extraordinary mathematics companion Princeton Companion to Mathematics (http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...)

The Princeton Companion to Mathematics (http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...)

The Princeton Companion to Mathematics

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

⬐ acangiano

Yes, it's #26 on the list.

⬐ tygorius

This highlights one of the weaknesses of the list -- the apparent lack of an editor. Whereas shou4577's comments explains the book's strengths and target audience, the original submission's description was so concise as to be near useless ("Coherent overview of all of pure mathematics.") This isn't intended as a slam of the original suggester, but to point out that the blog listing is only a starting point and in some regards less useful than the topic book lists some Amazon reviewers compile.

⬐ shou4577

I own this book, and I use it occasionally, but it really requires the right audience.

Firstly, it has several parts. Some of these are historical, philosophical, or otherwise interesting to a general audience, but some of them are technical (I would put most of pages 157-729, nearly 60% of the book, in this category).

For people that are interested in mathematics, but do not have a fairly extensive background (I would estimate 2-3 courses beyond the calculus series), these technical sections are probably not very useful (and certainly not entertaining).

For people (like me), who are still pursuing an education in mathematics, I would say this book is indispensable. It gives a great overview of individual branches of mathematics, including fairly rigorous explanations of important results and conjectures.

The exposition portions of the book are good, but don't warrant the high price of the book on their own. Overall, though, it is a tremendous reference.

Edit: I forgot to mention that the folks who write the articles are all outstanding mathematicians - many of them Fields medalists or other award winners. That makes for some very good reading.

⬐ omaranto

I agree with everything you said and would only add that if you have studied lots (for example, are a graduate student in mathematics), then it will probably be frustratingly shallow on anything you know reasonably well (or not mention it at all, if it's specialized), but it is still an amazing resource to get started in fields of math that are not your own.

Book recommendation: Princeton Companion to Mathematics

It's a good way to skim a lot of different mathematical topics for further exploration.

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

I'll third this recommendation. Fantastic text.

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

⬐ None

None

If you're really interested in learning a lot about mathematics, I would definitely recomment the Princeton Companion to Mathematics (see below for URL). It came out very recently, and while i'm only about 100 pages in (of well over 1000), it's down to earth writing and low prerequisites are very appealing to me.

http://www.amazon.com/gp/product/0691118809?ie=UTF8&tag=...

or if you'd rather not use my affiliate link:

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...