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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 w

by boppo1 3y ago

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 3y ago
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 3y ago
  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 3y ago
  > 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 3y ago
"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.
- agencies 3y ago
  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.
- paulpauper 3y ago
  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 3y ago
    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.

by boppo1 3y ago

sfpotter 3y ago

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 3y ago
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 3y ago
> 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 3y ago

"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.

agencies 3y ago
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.
paulpauper 3y ago
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 3y ago
  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.