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Comment by vladislav | Hacker News Reader

In my experience, being an amateur mathematician is more fun that doing it professionally, and potentially just as productive. You don't get nearly as much time for math, so you make the most of it and work only on the most interesting problems as opposed to just writing another paper. There's no time for beating your head against the wall, so you just do it when you're inspired, which is how problems get solved anyway.

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People who only do something when inspired are reliably and overwhelmingly bested by people who do it consistently.

Is that a generic claim? I'm making a specific, counterintuitive, and empirical claim about productivity and work-style of mathematicians, having been on both sides. Even for professional mathematicians, the solution to a problem can come all at once, while one is not even actively pursuing it, which begs the question of the purpose of beating your head against the wall when doing math. For just one example amongst many, see the story of Yitang Zhang, who made his breakthrough in establishing the first finite bound on gaps between prime numbers, while out at a barbeque and having largely given up on the problem after two years of effort.

It seems likely that the two years of effort were essential in getting him to the point where he could have the breakthrough.

Are you confident the majority of the gains weren't made in the first year? It's exactly my point that knowing when to give it a rest is hard to estimate, and those who work on these problems professionally are likely to just keep pushing, potentially just spinning their wheels. Interestingly, the brain keeps working on these problems subconsciously, so you can actually get more done if you work on more things and tend to give up a bit earlier. I've solved hard problems before in an afternoon, a year after being hopelessly stuck on them, and my sincere estimate is that I would have made exactly zero progress had I kept pushing.

It seems unlikely that he would have had the breakthrough without those two years of effort. Step one of the Feynman algorithm is necessary for step two.

I think the counterintuitive aspect of problem solving at that level is how nondeterministic it is. It is no doubt that learning everything you can about a problem is critical to setting yourself up for being able to solve it, but this can happen a lot earlier than when someone burns out on a problem.

If that's so, why are most and the strongest results achieved by professionals?

I would imagine there are more man-hours of professional mathematics research than amateur.

Most mathematical results would be unreachable by amateurs due to the sheer amount of background knowledge, especially of existing literature, required to make progress on a problem (and in some cases to even understand the statement of the problem, e.g. https://en.m.wikipedia.org/wiki/Hodge_conjecture).

Elementary graph theory and combinatorics are somewhat outliers in this regard, as there is not so much "theory" per se that one has to build on or work with. Tim Gowers's "Two Cultures" essay is an interesting read on this topic: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

I think for a large number of people, myself included, the "Eureka!" moment often occurs when you're least focused on the problem at hand. I remember in college, studying EE, I was struggling to make sense of how a flip-flop worked (the basis of a register). My ah-ha moment was literally in the shower.

What was it?

If I've been working on a problem at work for too long, I'll often stop working on it and just work on something else, because I know that tomorrow when I take a shower or when I'm walking to work the next day I'll figure it out in my head.

For more on this subject, see the 1945 essay by French mathematician Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field. https://archive.org/details/eassayonthepsych006281mbp

Thank you for sharing. One thing that impressed me about Hamming's essay on how to produce world-class research [1] is that it suggests a necessary, but not sufficient prerequisite is to spend your waking life thinking hard about some problem. This is necessary in order to provoke the unconscious mind into doing the heavy-lifting required for a "Eureka!" moment, so I was pleasantly surprised when Hadamard's book has a chapter titled "The Unconscious and Discovery". It amazed me that even in the modern world we do not understand the processes that led to these breakthroughs, so we can only leave them in the realm of the otherworldly and the mystical.

[1] http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

Another example: Kekule 'said that he had discovered the ring shape of the benzene molecule after having a reverie or day-dream of a snake seizing its own tail.'

https://en.wikipedia.org/wiki/August_Kekul%C3%A9#Kekul%C3%A9...

Musicians commonly report waking up with song ideas fully formed. We shouldn't be too surprised, really; the complex sentences out of our mouths (more often than not) arrive without any conscious thinking. Jung said that our egos are like planets orbiting a Sun they're unaware of.

where do people have these discussions and can I join? I wake up almost every morning with music in my head from the dream that is (more often than not) original, and IMO sounded like any hit on the radio.

And then within a few minutes I quickly forget the music. And there’s no way to remember it because it’s not anything that I can just find and listen to.

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