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‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

quantamagazine.org

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Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#21

Post by yodsanklai »

bllguo wrote: post #4

Great story that reminds me of something I've had. There's just so much knowledge out there - and growing - that researchers have to be increasingly specialized. Isn't it inevitable that 4 years of undergrad + 1-2 yrs of grad school classes won't prepare you sufficiently to do novel research? Imagine when people have to be in school for something like 10 years before they know enough to do "real" work! And people wil…

> Isn't it inevitable that 4 years of undergrad + 1-2 yrs of grad school classes won't prepare you sufficiently to do novel research?

I think it depends what you call "novel research". Most of the time, novel research means making a small contribution to an existing paper (and for a graduate student, the paper to extend and the contribution are usually suggested by their supervisor).

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#22

Post by haZard_OS »

bllguo wrote: post #4

Great story that reminds me of something I've had. There's just so much knowledge out there - and growing - that researchers have to be increasingly specialized. Isn't it inevitable that 4 years of undergrad + 1-2 yrs of grad school classes won't prepare you sufficiently to do novel research? Imagine when people have to be in school for something like 10 years before they know enough to do "real" work! And people wil…

I was always taught that a Masters wasn't intended to prepare one to be competent at novel research; that's what the PhD is for.

A Masters, based on this view, is designed to instill expertise in a field's main body of theory and techniques.

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#23

Post by chatmasta »

Spielman was my algorithms Prof in college and legitimately one of the smartest guys in terms of raw intelligence I’ve ever met. He was a great teacher, able to explain the most complicated of problems in language that I could intuitively understand and get excited about. Before that class I had a lot of problems with the math sides of CS (still definitely do), but he really made the subject approachable for me. I started the class getting C’s, but after visiting his office hours nearly weekly, by the end of the semester I was getting 100s on my problem sets.

That class, and the way he taught it, convinced me that advanced math, whilst hard for me, is not beyond my grasp or really that of anyone. It’s just a matter of breaking problems down into manageable chunks and recognizable sub-problems, a skill at which Spielman is especially adept.

I knew he was smart then (he had just received a $500k McArthur “genius” grant), but reading this article really put into perspective how effective he is and how lucky I was to have him as a professor. His same skill for breaking down problems that helped me succeed in his class seems to be the same skill that enabled him to solve this problem.

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#24

Post by JadeNB »

garmaine wrote: post #10
ericand wrote: post #2

"He guessed that the problem might take him a few weeks. Instead, it took him five years" Perhaps the key to their success was typically programming work estimations ;) More seriously, I think that if you think something is easy for you, or that you are almost there, you may be more willing to put in the necessary time, than if you knew it would take 5 years from the outset. "Spielman realized that he himself might b…

Related but different advice: build the habit of learning things by trying to figure it out yourself. Richard Feynman did this when learning quantum mechanics, and invented the Feynman diagram as a result. While not a different model or theory per se, the diagram made apparent the connection to functional integrals and a different method for solving that made previously intractable problems (relatively) simple. Funny…

> Related but different advice: build the habit of learning things by trying to figure it out yourself.

This can be a wonderful habit, or it can create cranks. As you mention, Feynman used it to great success, but he was Feynman. I had a few classmates in graduate school—I nearly was one, until my advisor set me straight—who couldn't make any progress because they couldn't take any results for granted; they wouldn't do anything until they could prove everything. This is intellectually admirable, but a sure recipe for stagnation (at least for me and these classmates).

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#25

Post by zentiggr »

JadeNB wrote: post #24
garmaine wrote: post #10

Earlier quoted context omitted.

Related but different advice: build the habit of learning things by trying to figure it out yourself. Richard Feynman did this when learning quantum mechanics, and invented the Feynman diagram as a result. While not a different model or theory per se, the diagram made apparent the connection to functional integrals and a different method for solving that made previously intractable problems (relatively) simple. Funny…

> Related but different advice: build the habit of learning things by trying to figure it out yourself. This can be a wonderful habit, or it can create cranks. As you mention, Feynman used it to great success, but he was Feynman. I had a few classmates in graduate school—I nearly was one, until my advisor set me straight—who couldn't make any progress because they couldn't take any results for granted; they wouldn't…

I think we can generalize the last two comments to: when you look at a problem, learn about it, it's structure, until you think you can resolve part of it. Keep repeating that until you get there. Assume that what everyone knows is good until your next steps push you to reexamine assumptions, and don't be reluctant to do so.

Happy ground between 'must build stack myself' and 'too orthodox to ever question the masters'.

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#26

Post by gmt2027 »

JadeNB wrote: post #24
garmaine wrote: post #10

Earlier quoted context omitted.

Related but different advice: build the habit of learning things by trying to figure it out yourself. Richard Feynman did this when learning quantum mechanics, and invented the Feynman diagram as a result. While not a different model or theory per se, the diagram made apparent the connection to functional integrals and a different method for solving that made previously intractable problems (relatively) simple. Funny…

> Related but different advice: build the habit of learning things by trying to figure it out yourself. This can be a wonderful habit, or it can create cranks. As you mention, Feynman used it to great success, but he was Feynman. I had a few classmates in graduate school—I nearly was one, until my advisor set me straight—who couldn't make any progress because they couldn't take any results for granted; they wouldn't…

What I took from Feynman's biography is that he would periodically attempt to rebuild his understanding of all of physics from first principles i.e. basic arithmetic, through algebra to calculus, mechanics etc. Looking for new insights and simplifications, he would apply new tools to reformulate and solve older problems. His seminal work in Quantum Electrodynamics used the Principle of Least Action to great effect. Despite initial scepticism, he came to appreciate the power of LA after seeing that it could be used to elegantly reformulate Classical Mechanics.

Doing this even after becoming a world famous physicist led to the Feynman Lectures in Physics - written as much for his own benefit as for his students. However, I am not sure he used this strategy when learning entirely new concepts the very first time.

It makes sense to do this as practice. To build new theoretical structures, building new constructions of the stuff we already know is a great way to sharpen a toolbox of problem-solving techniques and reveal potential shaky theoretical foundations.

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#27

Post by JadeNB »

zentiggr wrote: post #25
JadeNB wrote: post #24

Earlier quoted context omitted.

> Related but different advice: build the habit of learning things by trying to figure it out yourself. This can be a wonderful habit, or it can create cranks. As you mention, Feynman used it to great success, but he was Feynman. I had a few classmates in graduate school—I nearly was one, until my advisor set me straight—who couldn't make any progress because they couldn't take any results for granted; they wouldn't…

I think we can generalize the last two comments to: when you look at a problem, learn about it, it's structure, until you think you can resolve part of it. Keep repeating that until you get there. Assume that what everyone knows is good until your next steps push you to reexamine assumptions, and don't be reluctant to do so. Happy ground between 'must build stack myself' and 'too orthodox to ever question the masters…

> Assume that what everyone knows is good until your next steps push you to reexamine assumptions, and don't be reluctant to do so.

This is a fantastic summary. I like it a lot, and may steal it for future use.

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#28

Post by anitil »

chatmasta wrote: post #23

Spielman was my algorithms Prof in college and legitimately one of the smartest guys in terms of raw intelligence I’ve ever met. He was a great teacher, able to explain the most complicated of problems in language that I could intuitively understand and get excited about. Before that class I had a lot of problems with the math sides of CS (still definitely do), but he really made the subject approachable for me. I st…

I remember watching a (maybe MIT?) youtube lecture where someone described talking to Claude Shannon the same way. You'd talk about some idea and he'd simplify and throw things away, until you had just a kernel of an idea that was so simple it was just obvious.

Re: ‘Outsiders’ Crack 50-Year-Old Math Problem (2015)

#29

Post by jfoutz »

ggggtez wrote: post #18
jfoutz wrote: post #14

Earlier quoted context omitted.

That sounds like an abstraction problem. I’m totally willing to accept I’m crazy. But my gut feeling is there is a lot of technical debt in representation, so it’s hard to move to the frontier.

In this case, you're probably just wrong. Consider for half a second how many programmers fail Fizz Buzz, and then think about how much work it takes to get from that to the frontier of computational science (i.e. not just installing another javascript library, but actually computation). There's a lot of low hanging fruit out there, but you still have to actually know how to find it.

I want to point out that Maxwell had like 40+ equations for electricity that wound up being 4. I suspect that many fields have rules, but those rules aren't the most concise representation.

it will always be hard to get to the frontier. but that path should be easy. we can get years and years of explorers by building good roads.

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