HN Books @HNBooksMonth

https://www.3blue1brown.com has you covered for linear algebra. Now as far as the math needed for physics, I highly suggest Roger Penrose's 'Road to Reality' (https://www.amazon.com/Road-Reality-Complete-Guide-Universe/...). As the reviews say it's not an easy read but what it does provide you with is all the mathematics you're going to need to learn to understand today's physics. The book provides a high-level overview of the mathematics - which is technically complete but so concise that it's difficult to learn from. So use that to take a deeper dive into a mathematical subject. What the book is really providing is a roadmap: you need to understand these concepts from these mathematical disciplines to understand this area of physics and then proceeds with the high-level description of those concepts. Take the deep dive as needed and you'll be amply rewarded.

⬐ MikkoFinell

Covered for linear algebra? Yes those videos have some nice visuals but the material is just scratching the surface.

I've not read this yet, but have paged though it a bit at a bookstore, and it looks like it has potential: "The Road to Reality: A Complete Guide to the Laws of the Universe" by Roger Penrose [1].

[1] https://www.amazon.com/Road-Reality-Complete-Guide-Universe/...

⬐ tzs

Can anyone explain why this is getting down voted? As far as I can tell from examining it quite a bit at the bookstore, and from several reviews, it looks like it satisfied the request.

Is there something about it that I have missed?

The Road to Reality, by Roger Penrose:

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

This has been a really refreshing book for me to read. I studied mathematics in college but haven't really exercised that part of my brain since graduating and working as a programmer. Although this book is probably inaccessible to someone without some formal mathematical training, it's still one-of-a-kind. Nobody else of Penrose's stature has ever attempted to go from zero to string theory in a single volume, with all the physics and mathematics explained and very little left out.

For me, it's really been nice to finally satisfy a lifetime of curiosity that had built up about quantum theory. My fascination with it has never been enough to drive me to be a physicist, but it was enough for me to feel uneasy about not really knowing the underlying mathematics.

⬐ gfodor

Did you get through the book? I've tried twice so far in my life, I've stalled out each time. Any tricks?

⬐ brg

One trick is to stop worring about completely understanding the maths on your first pass through the material. Instead make note of what you were confused about and continue on.

If for instance you find that you need a refresher on complex integration, don't stop reading and do quarter's worth of review of complex analysis. Don't expect to complete a year of differential geometry in the few chapters that lead to Gravity.

After you finish through the a first pass, definitely go back to sections that you want to understand better and do the necessary work to improve your understanding.

⬐ nilkn

I finished it but certainly can't claim to have understood everything in it. I read it on and off for several weeks so it definitely took some effort for me.

It's an incredibly dense book. There's nothing else out there quite like it. Penrose also has a pretty idiosyncratic method of presenting ideas. This means that sometimes you can't cross-reference his explanations even if you want to. He covers some exotic and nonstandard topics as well. I haven't personally seen his diagrammatic notation for tensor algebra anywhere else, for instance. And the book was my first and only exposure to hyperfunctions.

The "trick", honestly, is to already have some prior exposure to most of the tricky mathematical ideas, like differential forms and fiber bundles. I was by no means an expert on differential geometry but I'd been exposed to it a few times as an undergraduate.

The other "trick" is simply not to skip around. I was tempted at first to browse and read whatever topics interested me; this book is not structured that way. He constantly references backwards in the book, and even if you know the mathematical material some of his references can be confusing. This is probably the greatest weakness of the book. For instance, gauge connections aren't really covered fully in any single section of the book. They're discussed briefly in the chapter on fiber bundles, and then again in the chapter on electromagnetism, and then again I believe in the chapter on quantum mechanics, and none of these discussions is complete, but they are if taken together.

Once again ColinWright graces the front page of HN on a weekend by submitting a story on mathematics education. The blog post submitted here, by an undergraduate physics major at the University of Texas at Austin, prompted me to read some of the author's other writings. The author's perspective on the importance of mathematics as a tool for understanding physics immediately reminded me of some good reads by older authors on physics. "How to Become a Good Theoretical Physicist" (HTML title "Theoretical Physics as a Challenge") by Nobel laureate Gerard 't Hooft

http://www.staff.science.uu.nl/~hooft101/theorist.html

lists essential knowledge that everyone should possess who desires to advance theoretical physics, and included in that knowledge is much mathematics. There is a whole book, The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose,

http://www.amazon.com/The-Road-Reality-Complete-Universe/dp/...

that is marketed as a book about physics but includes a huge section reviewing secondary school mathematics as an essential background to physics.

The blog post submitted here has a title that is an homage to the article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by Eugene Wigner in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).

https://dtrinkle.matse.illinois.edu/_media/unreasonable-effe...

People who know physics have long been delighted to find in physics applications for the mathematics they learned in mathematics courses without a hint of how useful the mathematics would be. The blog post author, however, goes beyond that perspective to urge, "Let’s think of mathematics in the abstract. Mathematics, at its most basic, is a very simple set of very well-defined rules. The rules describe the behavior and interaction of certain completely imaginary objects. Upon these rules, mathematicians have built others." And that brings to mind Paul Halmos's article (with its intentionally provocative title, an example of Halmos's spicy style in expository articles about mathematics) "Applied Mathematics Is Bad Mathematics" Halmos, P. "Applied mathematics is bad mathematics." Mathematics tomorrow (1984).

http://books.google.com/books?hl=en&lr=&id=FcgB818WA...

Halmos claims that mathematics is interesting and beautiful whether or not it has an apparent application.

Other replies already posted to this submission have helpfully mentioned the issue of empirical tests of what method of teaching mathematics may best help young learners appreciate (and later apply) mathematics. I have been deeply interested in cross-national comparisons of educational practice since living overseas beginning in 1982. In those days, one way in which school systems in most countries outdid the United States school system, economic level of countries being comparable, was that an American could go to many different places and expect university graduates (and perhaps high school graduates as well) to have a working knowledge of English for communication about business or research. I still surprise Chinese visitors to the United States, in 2012, if I join in on their Chinese-language conversations. No one expects Americans to learn any language other than English. Elsewhere in the world, the public school system is tasked with imparting at least one foreign language (most often English) and indeed a second language of school instruction (as in Taiwan or in Singapore) that in my generation was not spoken in most pupils' homes, as well as all the usual primary and secondary school subjects. At a minimum, that's one way in which schools in most parts of the world take on a tougher task than the educational goals of United States schools.

It was on my second stay overseas (1998-2001), that I became especially aware of differences in primary mathematics education. I began using the excellent Primary Mathematics series from Singapore

http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....

for homeschooling my own children, and I browsed Chinese-language bookstores in Taiwan for popular books about mathematics as my oldest son expressed an avid interest in mathematics. I discovered that the textbooks used in Singapore, Taiwan (and some neighboring countries) are far better designed than mathematics textbooks in the United States. (During that same stay in Taiwan, I had access to the samples United States textbooks in the storeroom of a school for expatriates, but they were never of any use to my family. I pored over those and was appalled at how poorly designed those textbooks were.) I discovered that the mathematics gap between the United States and the top countries of the world was, if anything, deeper and wider than the second-language gap.

Now I put instructional methodologies to the test by teaching supplemental mathematics courses to elementary-age pupils willing to take on a prealgebra-level course at that age. My pupils' families come from multiple countries in Asia, Europe, Africa, and the Caribbean Islands. (Oh, families from all over the United States also enroll in my classes. See my user profile for more specifics.) Simply by benefit of a better-designed set of instructional materials (formerly English translations of Russian textbooks, with reference to the Singapore textbooks, and now the Prealgebra textbook from the Art of Problem Solving),

http://www.artofproblemsolving.com/Store/viewitem.php?item=p...

the pupils in my classes can make big jumps in mathematics level (as verified by various standardized tests they take in their schools of regular enrollment, and by their participation in the AMC mathematics tests) and gains in confidence and delight in solving unfamiliar problems. More schools in the United States could do this, if only they would. The experience of Singapore shows that a rethinking of the entire national education system is desirable for best results,

http://www.merga.net.au/documents/RP182006.pdf

but an immediate implementation of the best English-language textbooks, rarely used in United States schools, would be one helpful way to start improving mathematics instruction in the United States.

The blog post author begins his post with "In American schools, mathematics is taught as a dark art. Learn these sacred methods and you will become master of the ancient symbols. You must memorize the techniques to our satisfaction or your performance on the state standardized exams will be so poor that they will be forced to lower the passing grades." This implicitly mentions another difference between United States schools and schools in countries with better performance: American teachers show a method and then expect students to repeat applying the method to very similar exercises, while teachers in high-performing countries show an open-ended problem first, and have the students grapple with how to solve it and what method would be useful in related but not identical problems. From The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999): "Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x

"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77

A great video on the differences in teaching approaches can be found at "What if Khan Academy was made in Japan?"

http://www.youtube.com/watch?v=CHoXRvGTtAQ

with actual video clips from the TIMSS study of classroom practices in various countries.

⬐ Evbn

Why did you open with an ad hominem and ad datum reference?

⬐ mquander

I assume the answer is because all comments must begin with words, and those words seemed like a nice beginning. Maybe you should say what seemed notable about that sentence?

⬐ WildUtah

Ad datum means "to (toward) the given (item)." You probably meant ad diem, "to the day (or date)."

For algo/datastructures, CLRS (http://www.amazon.com/Introduction-Algorithms-Thomas-H-Corme...) is the gold standard.

For OS's, Tanenbaum (http://www.amazon.com/Modern-Operating-Systems-Andrew-Tanenb...) is popular.

'Math' is broad - if I can recommend only one book to cover all of Math I'd probably say 'The Road to Reality' (http://www.amazon.com/The-Road-Reality-Complete-Universe/dp/...). More practically (for the subset of math most programmers are likely to care about), you'll do fine with one good discrete math book and one linear algebra book. Throw in one each on Stats, Abstract Algebra, Calc (up to ~diffeq), and Real Analysis (in roughly that order) if you're a bit more ambitious ;-)

⬐ mvanga

I just wanted to add that I felt Tanenbaum's Modern Operating Systems was a tad light on some of the core topics (especially when trying to get a good grip on the issues behind concurrency). I personally recommend the book by Silberschatz: http://www.amazon.com/Operating-System-Concepts-Abraham-Silb...

I also think for understanding how operating systems work, nothing beats writing your own! I learned most of the concepts by building a toy OS during the better part of my undergraduate studies. I highly recommend this for people who like coding and are afraid of jumping into the theory too quickly. For example, analyzing memory allocation algorithms is never as interesting as when you have to pick one for your own kernel!

I second these recs: all three are fantastic sources. MTW is very imposing (the damn thing must weigh twenty pounds), but don't be threatened, it is a crystal clear exposition that you really should not miss.

I'm not quite sure, though, if tel was asking for full mathematical developments of the theory (in which case the combo of MTW and Wald are, IMO, indisputable must haves), or something that just gets to the point quickly, assuming that you don't need help on the math.

In any case, anyone and everyone should also read Penrose's Road To Reality (my own shameless affiliate link: http://amzn.com/0679776311?tag=gubbins-20) for a very different take on...well, pretty much everything. The book is a complete failure at its stated goal of making mathematical physics accessible for a lay audience (I suspect when you're as smart as Penrose it's hard to figure out what an average Joe is capable of grokking...), but as a casual and wildly different sweep through a lot of interesting topics for someone that already knows math, it's fantastic.

Maybe more on topic than you suggest, as some Physics books double as pretty good books about math.

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

OK, I haven't actually read it, but it looked like a really good book for learning a lot of interesting math when I thumbed through its contents at the library. :)

⬐ CamperBob

I agree with hyperbovine -- Road To Reality is a downright terrible book for the casually-interested nonprofessional, and I don't understand why it gets recommended so frequently. Roger Penrose is a bright fellow and a good writer but this book is not for the person who did OK in high school math and physics and now wants to take it to the next level.

⬐ MaysonL

How would it be for a guy who aced second semester freshman physics and the math GRE (40 years ago), and top 100 on the Putnam?

⬐ CamperBob

I'd guess you'd be in his target audience. Someone who's already comfortable with complex causality and relationships between seemingly-random facts.

⬐ hyperbovine

This is not a good book for someone who doesn't know math. He's on to hyperbolic geometry by ch 2. (unsuccessfully) reading this book was one of the things that convinced me to back to school to study more math :-)

You should take a look at a book called The Road to Reality by Roger Penrose. While it's geared more towards physics, this book has proven to me to be the most enlightening mathematics text I've ever read. Admittedly I'm only about 10 chapters in - it's a very dense book, and you'd do well to go through it slowly. But, if you're interested in math, this book will blow your mind.

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

none ref link for those inclined to make a purchase

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

Roger Penrose's A Road to Reality :

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

⬐ carterschonwald

i've looked at it, and i'm not convinced that it does a good job explaining the math the reader doesn't a priori know.

Its one thing to go from mathematical physics to theoretical physics, its another thing to go from pure math with a theoretical computer science bent to mathematical physics and theoretical physics. Does this difference make sense, or am I missing some observation? (i don't know enough to know which is more likely)

⬐ slackenerny

You need this: http://mitpress.mit.edu/SICM augmented by e.g. Lanczos Variational Principle (the closest to a platonic ideal of exposition for this didactically elusive topic) and various lecture notes for classical mechanics courses http://www.damtp.cam.ac.uk/user/tong/teaching.html. From there move to Byron & Fuller Mathematical Physics and only then you may question where to next, that's the bare minimum of theory in physics. Baez & Muniain "Gauge Fields & bla bla" has very good narrative on one offer what to do after, Sethna Statistical Mechanics on another. Speaking of bare minimums there's a youtube lecture series by Susskind "Theoretical Minimum of Physics" or sth.