Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How to Solve It (1945)

math.utah.edu

Post reply on HN
Print view View original

81–90 of 104 posts

Re: How to Solve It (1945)

#81

Post by marginalia_nu »

js2 wrote: post #6

Ah yes, just four steps: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. --- Compare with the Fenyman Algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution. https://wiki.c2.com/?FeynmanAlgorithm (The discussion on FeynmanAlgorithm links back to Polya's book since not everyone is Feynman.)

The Feynman algorithm is very underrated. I think some people see it as a joke, as though it's clearly not a working method. But it's how I do most of my difficult work, possibly as a result of majoring in physics where "stare at the problem for a few hours" is the standard approach.

I think my strongest asset is not understanding when a problem is difficult and staring at the problem for hours anyway.

There are classes of problems that don't trivially decompose into a series of easier problems. The only way is to punch your way through them.

Re: How to Solve It (1945)

#82

Post by kgwgk »

travisjungroth wrote: post #68
OldGuyInTheClub wrote: post #63

Earlier quoted context omitted.

> Conway and Tao are widely considered outside the "roadkill" category of mathematicians. Ah, but they didn't really /need/ Polya's book.

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

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

Re: How to Solve It (1945)

#83

Post by i_am_a_peasant »

vkazanov wrote: post #80
i_am_a_peasant wrote: post #7

Earlier quoted context omitted.

How to Prove it is pretty nice

I have both the book discussed here, and How to prove it, and back in the day the Proof book was a revelation, unlike Polya's one, which was interesting but that's about it How to prove it just made proofs click, and I wish I understood the material earlier, by high school the latest. I am convinced that all the half intuitive half ("kind of like") mechanical ("just remember this") explanations we get in school and u…

There are other books in the literature that mainly focus on how to go about proving things rather than proving specific things.

Some titles that come up are:

Mathematical Proofs: A transition to Advanced Mathematics by Chartrand, Polimeni and Zhang.

Book of Proof: Richard Hammack

Jay Cumming’s Proofs

I remember I once had a book that advertised itself as a book on Discrete Mathematics but it turned out to be more in line with the titles above, was a gem, if I find it I'll let you know.

But it's nice to see I'm not the only guy who gets a kick out of this stuff.

I wouldn't bother with things from the foundations of mathematics area, like Russel or Frege and whoever else came from that era, back when people still wrote an 800 page treatise on what a "number" even is. A lot of the material coming from that period seems to have been thrown down the drain.

Re: How to Solve It (1945)

#84

Post by fuzztester »

freilanzer wrote: post #78
bollu wrote: post #41

Earlier quoted context omitted.

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.

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

>lately

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

Re: How to Solve It (1945)

#85

Post by apocadam »

lcuff wrote: post #45

On a totally orthogonal axis: My biggest problem with tackling math problems early in life was psychological. My inner critic would say nasty things to me when I didn't get the answer right away, making it impossible to stick with exploring the problem. Once I overcame that, I got lots better at math. And working on yet another axis, the most useful tip I've gotten on how to think about problems is to "think in extre…

If you don't mind, could you share how you overcame that? I think for me at least a lot of it is/was not being able to be comfortable struggling with a hard problem for an extended amount of time, probably due to being used to only solving fairly trivial exercises and thus feeling dumb/the problem feeling impossible if it wasn't clearly solvable under a short timeframe.

Re: How to Solve It (1945)

#86

Post by jodrellblank »

mathisfun123 wrote: post #59
jodrellblank wrote: post #58

Earlier quoted context omitted.

[flagged]

the guy clearly says >but Wikipedia does not show anything else go look at the wikipedia page for george polya and tell me what state of mind you need to be in to conclude that he didn't accomplish anything outside of publishing that book. also btw >born in 1887 he lived very long and had a very long career so you're still not close >Died September 7, 1985 (aged 97)

The guy misread a Wikipedia page and you're "beyond offended" and call that "arrogant"?

> "so you're still not close"

I'm "still" not close? Still not close ... to what?

Re: How to Solve It (1945)

#88

Post by hgsgm »

ryandv wrote: post #57
sverona wrote: post #54

Earlier quoted context omitted.

Baby Rudin, despite the somewhat patronizing name, is a second analysis text. There are many things that should have explanations that he expects you to already know. Swallow your pride and start with Abbott. You'll thank me later. Now here's what I really think: I don't know your situation, but the odds are better than even that you should forget about it entirely. Doubly so if you're out of school. The bizarre math…

My question was more rhetorical and for _synchronicity's_ sake than anything; I wanted to see what sort of replies would pop out. As a recreational student, I've already committed to this text, difficulty be damned. I am no stranger to intellectual masochism. Nor do I care one iota for an "academic career," only for an understanding of results that were taken for granted in my prior CS mathematics undergraduate educa…

I used Rudin in school. I got the highest grade in my class that included several future PhDs and professors. It's an old, not very good pedagogical book. It's a useful reference, if you are living in the pre+Internet, pre color-illustration, mid 20th century. I learned analysis mainly from the guidance of my amazing professor (a postdoc who was an enthusiastic caring, clear communicator).

Like most textbooks, Rudin it has focused exercises, examples to build intuition, and proofs with little steps removed, not free-range problems. Thebanswe to a problem is based on the 20 previous pages in the text.

How to Solve It is like advice for solvimg a jigsaw puzzle (edges first, sort by shape and color and texture, ...) It is for solving problems after you've learned a whole year(s) of material, and don't know which of your knowledge contains the pieces of the answer. It's a way of searching through your knowledge and evaluating which pieces are useful for the problem at hand, and how they fit together.

Re: How to Solve It (1945)

#89

Post by hgsgm »

paulpauper wrote: post #31

this may work for simple problems. i have found it's more like: 1. study extensive background/prior art 2. have major epiphany to solve problem 3. try to get around inevitable roadblocks and complications during the process of solving it

You should read the book. It fills in about 100 more steps/options around those 3 general steps.

Getting from 1 to 2 and from 2 to 3 is the aim of the book

Re: How to Solve It (1945)

#90

Post by mathisfun123 »

consilient wrote: post #75
mathisfun123 wrote: post #44

Earlier quoted context omitted.

> Try harder problems. What part of "apostol's calculus" and "hard DP" did you not understand? I love when people are so confident in their reflexive dismissal that they're literally blinded by it. So confident that they don't even stop to consider whether their dogma might be wrong.

> apostol's calculus is hard for a calculus book , i.e. is an introductory real analysis book. If you can master analysis at the level of baby Rudin without doing hundreds of problems, you're very talented. If you still don't need to do hundreds of problems by the time you finish papa Rudin, you're a mathematical genius.

>is hard for a calculus book

Yes but I was talking about calculus as an anecdotal example. Not analysis.

> baby Rudin

This is definitely starting to look like "no true Scotsman". Anyway I did the regular amount (5-10) per section and got an A.

> you're very talented.

Trust me not the case. Just average person doing average number of pset problems.

> papa Rudin

I didn't read this, I went on to read Schilling's Measures, Integrals, and Martingales instead because I already knew enough functional analysis and wasn't interested in complex analysis.

Post reply on HN