[boppo1]

This summary really undersells the book IMO. It's one of the more interesting books I've read in that it is not structured linearly. He introduces a few ideas which have very particular language he defines in the remainder of the book which is essentially a dictionary. If you don't come from a math-y background and you are trying to get into serious mathematics, this definitely helps to 'lift the veil' on how you might think about math. Most of the criticism I've heard about the book comes from

Paraphrased from memory: "I'm not claiming to have discovered anything special here. What I think I'm doing is describing the mental operations that lots of people perform, consciously or unconsciously, to solve different sorts of problems. You probably already do some or all of what I'm describing."

I found that he was describing some thought patterns that I am extremely familiar with because I use them so often. Making them explicit was pretty cool. I haven't gotten around to his follow-up works yet, but I look forward to them.

[sfpotter]

If you don't come from a math-y background and are trying to "get into serious mathematics", the only way to do this is to get a PhD.

I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches. These problem solving strategies and habits of mind aren't written up anywhere, and even if they were, they wouldn't be useful to read. The way you learn them is by getting the training that comes with a PhD: they are passed down as part of the mentoring process. It's not clear to me that it could happen any other way.

I think the right way to look at this is that someone getting a PhD is basically a "journeyman researcher", much like a journeyman in one of the trades. Unfortunately, this often goes sideways (particularly if an advisor is bad or if there aren't enough support systems). But much like the only way to become an electrician is to apprentice yourself, the same goes for becoming a researcher. I think this is for good reason.

[veqq]

There are a few books which admirably explain the assumptions etc.

- A Programmers Introduction to Mathematics (amazing for building intuition, explaining how to read mathematics papers, understanding beauty of proofs etc.)

- Elements of Mathematics for Economics and Finance (a quick jog through typical the high school and college math a non-mathematician would study)

- Papula's Mathematik für Ingenieure und Naturwissenschaftler (rigorous (for practitioners) and comprehensive. Starts from nothing, with sets, then teaches you fractions, then quadratic equations until you're doing vector analysis and Laplace transformations)

- Foundations and Fundamental Concepts of Mathematics by Howard Eves (wide overview of the man developments in mathematics, up to its publication)

[consilient]

> If you don't come from a math-y background and are trying to "get into serious mathematics", the only way to do this is to get a PhD.

> I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches.

There are degrees of seriousness. Sure, even a very good foundation is not enough to do novel work in say, algebraic geometry (but then again, sometimes it is enough to make progress in combinatorics) - but the strongest undergrads are still much closer to freshly minted PhDs than they are to laymen.

Mathematical maturity is the first and hardest step; after that, people will know where to go for the folklore if they want it.

[OldGuyInTheClub]

"How To Solve It" does not account for actual ability. It was recommended in high school to those of us interested in math contests. My big lesson was that if you need this book, you're going to be roadkill in math contests. If you have the chops to be a successful mathlete, you don't need this book. And so it came to pass.

There are some people who have the ability to see a problem well enough to analyze it and make progress. Some people can do this in math, some in other technical disciplines, others in the arts, and so forth. Umpteen years later, I have not seen anyone including Polya successfully /teach/ this ability. If you want to be good at math, piano, chess, sculpture, or whatever you need the talent and then it can be nurtured.

In college I took organic chemistry[0]. The exam problems would often have redundant and/or contradictory information. The student had to figure out what to NOT believe to get anywhere.

As others have mentioned, none of Polya matters for selecting and attacking research problems.

[0] It was taught well and none of the cliches of "It's all memorization" applied. I struggled but look back fondly on it because I had to think along many axes at the same time.

[ants_everywhere]

> if you need this book, you're going to be roadkill in math contests

> I have not seen anyone including Polya successfully /teach/ this ability.

This is a pretty wild take. For alternate takes, see John Horton Conway's foreword to the most recent edition of the book. There he talks about how amazing the book is for both students and teachers, including the things he learned from it as both a student and teacher. Or see Terry Tao's blog post on solving mathematical problems [0], in which he says he learned from the Polya book when preparing for Mathematics Olympiads. Conway and Tao are widely considered outside the "roadkill" category of mathematicians.

> If you want to be good at math, piano, chess, sculpture, or whatever you need the talent and then it can be nurtured.

The "either you got it or you don't" theory used to be common wisdom, but it's not supported by empirical evidence and these days it's mostly relegated to grouchy coach stock character stereotypes. You need a sustained level of interest, you need practice, and you need some amount of courage, but there's no real evidence for an innate ability that some people got and some people don't.

[0] https://terrytao.wordpress.com/career-advice/solving-mathema...

[rockpasta1]

Thank you for taking the time to debunk such comments. I have serious imposter syndrome as a minority woman in software, literally the only one among 60 others and everyday is a battle with my inner thought process. For an onlooker, I have amazing achievements behind me and there's no reason I shouldn't be confident, but I'm not and seeing such comments as before has made it worse over time, without me questioning it.

[scns]

"Genius is 1 percent inspiration and 99 percent perspiration."

https://quoteinvestigator.com/2012/12/14/genius-ratio/

[OldGuyInTheClub]

> Conway and Tao are widely considered outside the "roadkill" category of mathematicians.

Ah, but they didn't really /need/ Polya's book.

[travisjungroth]

This “need” standard doesn’t seem useful. You could say they don’t need anything besides air, water, food and shelter.

[kgwgk]

Come on, you don't really /need/ shelter.

[bigbluesax]

>If you want to be good at math, piano, chess, sculpture, or whatever you need the talent and then it can be nurtured.

No offense, but you sound like you don't really know anything about these disciplines, or about the science of learning (Peak is a good book to read). Getting a good piano teacher will absolutely accelerate your progress and being talented at piano as a kid is a pretty bad predictor of how good you'll be as an adult. Every outstanding Piano "talent" you see on stage today has been nurtured and nurtured themselves through practice, to hell and back.

[OldGuyInTheClub]

None taken but you don't know about my (adult) piano adventure with good teachers over 15+ years, either. I didn't say practice was not important. You can't squeeze blood from a stone.

[agencies]

You say that if someone has the chops to be a real mathlete they won't need Polya's _How To Solve It_

I'll say I went to college 25 years ago with people who had competed internationally in high school and who placed competitively on the Putnam, and they LOVED Polya's book.

I think whether you enjoy seeing strategies laid out well--—whether or not you've been able to figure some of it out yourself---depsnds more on your personality than on how good you are at solving creative math problems.

[mejutoco]

> As others have mentioned, none of Polya matters for selecting and attacking research problems.

While I am not familiar with research problems, and do not pretend to be, some of the strategies to tackle problems in this book are universal. Like finding a similar problem, or a simplified case, or divide and conquer, etc. etc.

[JHonaker]

Yea, these are 100% exactly how you should approach a research problem, in mathematics or otherwise.

Literally the first part of almost every research paper is outlining how this thing you're about to talk about is similar and dissimilar to other things. That's often followed by a simplified description of the idea. Then that's usually followed by the thing in full generality.

[paulpauper]

beyond the book, what contributions did he make? it would seem like the techniques should have helped him solve problems but Wikipedia does not show anything else

[bollu]

Surely you’re joking. Polya pioneered so much combinatorics I don’t even know where to begin. Heck, a lot of counting problems reduces to polya’s enumeration theorem. The page of “references” to stuff named after him should be a clue to you.

[OldGuyInTheClub]

Definitely. Whatever one may think of "How to Solve It," Polya was a renowned mathematician. The book is like the Richard Hamming lectures that often come up on HN: Great thoughts from a great mind but not easily applied by the average or even above average Joe.

[boppo1]

Richard hamming lectures? I'd google, but I wouldn't know if I'd found the ones you meant.

[OldGuyInTheClub]

Try https://news.ycombinator.com/item?id=35776480

[lporto]

https://youtube.com/playlist?list=PL2FF649D0C4407B30

[freilanzer]

But besides combinatorics, what has he done for us lately? ;)

[fuzztester]

>lately

Luckily for the world, he dodged the 'startup craze' bullet ;) - by living before it started up :)

[mathisfun123]

Every so often there's a comment on here that's simultaneously so ignorant and so arrogant that it crosses from offensive to hilarious. Congrats, this one is up there with the likes of "it could just be rsync".