Andrew Wiles: what does it feel like to do maths? | Hacker News Reader

Andrew Wiles: what does it feel like to do maths?

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255 pointsjseliger71 comments

Its well worth watching one of the best documentaries ever made:

http://www.dailymotion.com/video/x223gx8_bbc-horizon-1996-fe...

He explains the experience of a genuine Eureka moment after herculean effort and false dawns and its incredibly moving. I also enjoy when the other mathematicians are asked to explain modular forms... and struggle a little :)

I just watched, first the first two minutes and then the whole thing. Just... beautiful. Makes you want to get up and actually put effort into something, whatever that might be!

>I just watched, first the first two minutes and then the whole thing. Just... beautiful. Makes you want to get up and actually put effort into something, whatever that might be!

The look in his eyes when he says "nothing I'll ever do will be as important" is simply inspiring.

The first two minutes are absolutely extraordinary, both for the insight, and for the emotion revealed.

Oh wow it's Michael Nielsen. Your neural network book helped me get through the last year of undergrad. Thanks!

Very nice film. I adore Horizon when it is pure, raw and serious like that. Thanks for the post.

Donut. Bridge. Modular Form.

..Now what you have to handle when you start doing mathematics as an older child or as an adult is accepting this state of being stuck. People don't get used to that. Some people find this very stressful. Even people who are very good at mathematics sometimes find this hard to get used to and they feel that's where they're failing. But it isn't: it's part of the process and you have to accept [and] learn to enjoy that process. Yes, you don't understand [something at the moment] but you have faith that over time you will understand — you have to go through this.

Well then.. found my confidence booster for 2017. Thanks!

Great quote. This feeling translates well into many fields and disciplines.

100% agreed, as I posted this not because of getting stuck in just mathematics but in computer programming, physics, biology, economics, linguistics, adulthood.. to name a few.

"What I fight against most in some sense, [when talking to the public,] is the kind of message, for example as put out by the film Good Will Hunting, that there is something you're born with and either you have it or you don't. That's really not the experience of mathematicians. We all find it difficult, it's not that we're any different from someone who struggles with maths problems in third grade. It's really the same process. We're just prepared to handle that struggle on a much larger scale and we've built up resistance to those setbacks."

I disagree. I understand the purpose of this statement, i.e. not to discourage people, etc. but clearly there are differences between Terry Tao, Wiles, Feynman, and many others and your typical PhD in these fields. This is like saying anyone can become Phelps with practice (although pinpointing his unique advantages can be complicated: https://www.scientificamerican.com/article/what-makes-michae...). Not everybody can be above average! And of course, there are the rare cases such as Ramanujan.

I think a better explanation is the one that Stephen gives in On Writing:

"I don’t believe writers can be made, either by circumstances or by selfwill (although I did believe those things once). The equipment comes with the original package. Yet it is by no means unusual equipment; I believe large numbers of people have at least some talent as writers and storytellers, and that those talents can be strengthened and sharpened."

As a disclaimer, I've a Ph.D. in mathematics and work professionally as a mathematician. That said, this sentiment is something that I'm very sensitive to.

I've a problem with your phrasing because I feel it implies that to be a successful mathematician someone needs to operate at the level of someone like Andrew Wiles or Terry Tao. More generally, the question to me is whether mathematics should be treated differently than any other field like engineering, law, or cooking. To me, the answer is no. I believe that becoming a professional mathematician is primarily about hard work and long hours. Of course, natural talent helps, but it helps in every field. Very specifically, even if someone is not born with some kind of natural talent for math, with sufficient training I believe that most people can be successful professionally as well as provide new results that aid everyone in the field. And, again, I don't believe that this is any different than any other field. With enough training, someone who can figure out a way to burn a bowl of cereal can become a wonderful cook. They may not become Julia Childs, but they don't need to be. Further, they can create new dishes that everyone can enjoy and benefit from.

Again, the core of my point is that most people I interact with believe that mathematics is special from all other fields. It's something that I strongly disagree with. I believe that this sentiment discourages people from entering the field. On a personal level, I find it alienating because it creates an artificial social separation.

In any case, I'm glad people are talking about this and, certainly, these are just my thoughts.

I find that people have a very thick illusion of understanding when it comes to academic work. It is very difficult to understand that everyday progress in say, developmental neurobiology, happens largely by dint of professionals working 45-50 hour weeks. It's hard work, it requires a great deal of training, intelligence, and diligence. It doesn't require genius, as much as it requires hard work and a little good fortune. The illusion of understanding leads us to boil down knowledge creation to "aha" moments, when it is in reality a long slog of reading, managing a lab, reviewing articles, doing lab work, writing, teaching, debugging stats or code, thinking, shopping for lab equipment, etc. all of which are messy human tasks that actually constitute advancing knowledge.

> Very specifically, even if someone is not born with some kind of natural talent for math, with sufficient training I believe that most people can be successful professionally as well as provide new results that aid everyone in the field.

You sum it up very well - I completely agree. I might not have the genes to be Einstein, but I can be a professional scientist. A very important point that is often missed.

And to turn it around, you might actually have the genes to be the best. But if you never put in the thousands of hours of quality work necessary to brush up against that genetic ceiling, you'll never know.

I find that this is case in >99.9% of the people who say they don't have the genes for something.

This is like saying anyone can become Phelps with practice.

Aside from the objections that others have made, that the main point is being a professional mathematician and not a unique genius, I would mention that "greatness" in mathematics (and other professional fields) isn't comparable being the best in an arena sport (or even in a mental sport like chess or go). The greatness of each great mathematician can be different - it's not solving some particular assigned problem all mathematicians face, it's choosing which problems to tackle, it's finding effective ways to frame problems, it's choosing who to associate with and get ideas from and give ideas to, it's teaching with extraordinary clarity or challenging your students in a unique way or whatever. The "unique genius" perspective closes-off the need of each mathematician to find the pursuit which them best.

And your Stephen quote is equally unfortunate. The world is more evolving out of the view of there being singular writers moving more towards a place which welcomes those who effective use their particular gifts in whatever kinds of writing and storytelling suits them.

You might be misinterpreting what he's saying.

He's not saying if you work hard and practice you can be Phelps.

What he's saying is that if you work hard and practice you can become a very good swimmer, even if not world class.

You jumped to that conclusion on your own - studying math hard doesn't mean you'll end up an Andrew Wiles, it means you too can gain an understanding of mathematics.

Not all mathematicians are celebrity scholars, there are plenty of average mathematicians that are moving the field forward inch by inch.

I disagree with your disagreement.

When people say that there is not so much something special about themselves but that the work they put in enabled their success, they are not claiming the reverse is true: that hard work results in success.

They are saying that success is the result of hard work. Not that hard work results in success.

It seems obvious to me that success at the highest levels is the result of hard work and talent and luck.

I'm not clear what distinction you're making. It's a bit of a truism (to the point of it being a cliche) for a successful person to say "I'm successful because I worked hard."

Of course that isn't an identical statement with "hard work results in success." Nor should anyone with a fluent grasp of language conflate the two statements.

So what are you disagreeing with, really? That's what I'm not clear about. Successful people, for diplomatic reasons, emphasize their work ethic instead of their talent. Certainly they have both. Did anyone claim differently?

> what are you disagreeing with

That saying "success is due to hard work" is the same as saying "you can become <famous successful person> by hard work". The latter is not true, and pointing that out is not a refutation of the first.

One famous successful person, Thomas Edison, would have disagreed with you: "Genius is one percent inspiration, ninety-nine percent perspiration."

Conversely. Einstein once said that if he had one hour to save the world, he would spend fifty-five minutes defining the problem and only five minutes finding the solution.

And of course each would think this way, given their backgrounds. Edison was an inventor and a CEO; Einstein a theoretical physicist. Perhaps HN is familiar with the Parable of the Elephant?

The point the person was making is that "a implies b" is quite different from "b implies a" - the converse of a statement is a different statement.

Not all hard working people are successful, but all successful people are hard working in math is the point.

Everyone I know with a PhD has asserted that it is 99% about being willing to slog through the years required to get the work done. That seems to suggest that the only difference (or most of the difference) between an average person and someone with a PhD is that the PhD just... took the time to get a PhD.

You need to get onto the PhD programme in the first place - meaning you'd probably need a good degree and some sort of funding - which does select reasonably well for ability. There's gonna be exceptions however (people who are otherwise capable but lack either of those, or people who are not the smartest but have managed to secure both)

that's what they assert when discussing it in polite company. perhaps there's a difference between what people say to others and what is really true?

Then I'll say it anonymously as a former graduate student who left academia? For getting a PhD at any tier 2 or lower school, it is sufficient to work hard and be reasonably intelligent.

getting a PhD is just a credential. that's a pretty weak measure of success. I was referring to public statements made by "famously successful" people.

Take every non-trivial side project you ever worked on and calculate the percentage you completed. Not half ass completed, but actually completed completed. Like you commented everything, wrote documentation, put it on Github, have decent test coverage, etc. Or how many Coursera courses did you sign up (with the real intent of actually finishing) vs how many did you _actually_ finish, doing all the homework, getting a really good grade? Employers look for people with degrees because it's very good estimate for "this person can finish shit" for this very reason.

To say PhD is "just a credential" and has no predictive capabilities in terms of being "successful" I think is being disingenuous.

I didn't say it has no predictive capabilities. You said that. Is that what you think? Because it's not what I think.

I said a credential alone is a pretty weak measure of success. It might be a very strong measure of potential for future success. Do you see the difference?

Almost everyone can do some kind of math. For the same amount of discipline and effort, some people can prove Fermat's Last Theorem, and some people can prove Fermat's Little Theorem.

And everyone can coast. Some people can get as far as Calculus without really trying, and some people can only get to basic arithmetic without really trying.

But no one can just wander up to a whiteboard and solve a unproven theorem from intuition alone.

>But no one can just wander up to a whiteboard and solve a unproven theorem from intuition alone.

...except von Neumann. But most would agree he really did have something special about him. (Especially if you include his frequent desires to nuke Russia.)

The post schows a good example of someone who achieved mastery in his field of interest and solved a really hard problem. The traits/habits he describes remember me of the lessons I learned from the book Mastery by Robert Green. Just one line from a short summary [1]:

>> Desire, patience, persistence, and confidence end up playing a much larger role in success than sheer reasoning powers.

[1] https://sivers.org/book/Mastery2

Do you recommend reading the full book or the summary?

I like the works of Robert Green so my biased opition is that the book is worth reading. He uses a lot of different examples for describing a principle and generally one or two of them get stuck in my mind making easier to remember the principles. It is kind of "learning by example". His books are also easy to read for me despite of having English as my third language.

I highly recommend the full book. Robert Greene is killer at providing examples, proving his points then boiling down those examples into action you can take. The persuasive force of hearing the patterns of mastery in the lives of people we all admire, and seeing the variations on the theme that can apply to our own lives, is lost in a summary.

Also the audiobook is excellently read, if you're into that sort of thing.

What I'd like to know is what his routine is like. I think I recall watching the FLT documentary and he talks about retreating to his home office for hours at a time where he was not to be interrupted. I wonder how he or other mathematicians organize their time and track their progress.

For most of my life I've struggled with procrastination, time management issues, and concentration problems. I feel like the only time I can concentrate is after 10pm, for a narrow window of an hour or 2 before I go to sleep. I'd guess mathematicians must not suffer from that problem or have learned to overcome it to be able to wrangle their minds around abstract mathematical constructs.

If you are like me -- It's something you have to chip away at slowly, because your brain is probably so used to having sudden, increased rushes of pleasurable activities -- whether that's habitually checking HN, facebook, reddit or whatever quickly, or reading an interesting topic or checking out an interesting book -- yes all these things are great, but if they aren't directly, tangibly helping you towards achieving your goals, however insightful or educational they might be, you have to be able to restrain. You can actually learn to concentrate again. It will take time and work, just like anything else in life. If you want to get in shape and start running, you don't start running 3 miles right out of the gate, no you start with maybe one mile, where you split between jogging and walking. Then you jog a full mile, then run a mile, then 1.5, etc. Same with being able to intently concentrate for long periods of time -- you have to learn how to do it. How? By trying to concentrate intently on smaller, tangible things. The better you get at it, the more you will be able to do it. That's why I like things like the pomodoro timers. On my good days, I am able to just sit down and blaze through a couple of pomodoros and not even notice the timer went off long ago. Most important thing I have found is setting tangible, short, achievable goals that all point to a larger goal, and then setting aside a reward. We are so used to having our reward now, now, now, but we need to retrain our minds that the reward comes after. There are more aspects to procrastination I have found -- fear of failure, fear of success, fixed mindset, all these concepts are great, but at the end of the day, in my opinion, I treat my brain like it is a muscle, and you can train it to do things just like anything else in life.

Thank you for the tip. I will give it a try. I see procrastination/concentration impairment coming from so many directions: caffeine, information bombardment, interruptions from coworkers, noise, stress, meal-planning, simply being human and needing to rest, staying at home too much. It's overwhelming to manage.

I will try pomodoro and just setting a small concentration goal and building on it.

Thanks I am glad you found it helpful.

Write something down on actual paper and hang it up on your refrigerator or bathroom mirror.

Write the steps down. Focus on single steps. Reward yourself in proportion to the size of the task, don't do one little thing then go on a netflix binge.

Yes distractions are always going to be there. Sadly, and I really mean sadly, the open office concept won, and even people my age (millennials and younger) are starting to become indoctrinated with the concept. In any case that's another story for another day -- focus on what you can control. Caffeine doesn't impair your concentration, but overuse can, and it isn't a replacement for sleep. Information bombardment is a problem, stop it! Is it helping you achieve your goals? Set it aside for another time, or use it as a reward. Coworkers interrupting? Place headphones on. That doesn't work? Drag a whiteboard over to block off your desk from the main traffic. If you are getting work done and knocking things out, people don't have a problem with someone focusing on getting things done. It's only when you are unproductive, do the petty things start coming in "Oh well, Jimbob is just not a good team player, he isn't making himself available enough" etc... don't focus on what you can't control and what people will say, only focus on what you can do and control, and the rest will work itself out, I promise.

This is great advice. Most of the ideas you mentioned are also explained (and expanded upon) in "The Now Habit" by Neil Fiore – which I wholeheartedly recommend OP to read.

> I'd guess mathematicians must not suffer from that problem

In my experience (both myself and what I've seen in other successful mathematicians), there is a lot of truth to your guess. I'm also a number theorist (and know Andrew, for what it's worth). I enjoy concentrating on something for a very long time, to the exclusion of everything else. I try to organize my life so that I can "retreat to my home office" and stay there and work as long as possible. The thing I struggle with is forcing myself to take breaks and stopping concentrating, since I can get way too obsessed with finishing one thing, to the exclusion of everything else. Balance is hard. An extreme fictional example of too much concentration is featured in the recent movie "The Accountant", in which the (autistic) protagonist very obsessively concentrates on an accounting situation, and is incredibly frustrated when real-world circumstances distract him from doing so.

It is funny you mention you have the narrow window of two hours before you sleep because if I remember correctly from Simon Singh's book, "Fermat's Last Theorem," that is exactly how much time he spent regularly on Fermat's Last Theorem (I can't remember if it was every day or less frequently). I think it took him seven years of commitment though (not counting the problems found in his paper and his fix in the subsequent year).

Terence Tao has written about his routines - https://terrytao.wordpress.com/2008/08/07/on-time-management...

I also found http://calnewport.com/books/deep-work/ has some pretty good suggestions.

Any other big proofs / theorems that came seemingly out of nowhere? Yitang Zhang's bounded prime gaps comes to mind. And of course that physics thing from that Swiss patent clerk guy, can't remember his name offhand.

> that physics thing from that Swiss patent clerk guy

I think you're thinking of Carl deGrasse Feynman.

Grigori Perlman dropped a proof of the Poincaré conjecture with little warning.

Yes. And Masha Gessen wrote an excellent book called Perfect Rigor about the proof and the man: https://jakeseliger.com/2016/11/29/perfect-rigor-a-genius-an.... Highly recommended.

Hottest mixtape of all time

Granted he was one of the world's best two or three mathematicians at the time.

Sphere packing in dim 8 and 24 (https://en.wikipedia.org/wiki/Maryna_Viazovska) and the "Set" game limit (https://www.quantamagazine.org/20160531-set-proof-stuns-math...) are two more recent examples.

I loved the section on "What do you do when you get stuck?"

Then you have to stop, let your mind relax a bit and then come back to it. Somehow your subconscious is making connections and you start again, maybe the next afternoon, the next day, the next week even and sometimes it just comes back. Sometimes I put something down for a few months, I come back and it's obvious. I can't explain why. But you have to have the faith that that will come back.

I'm not a mathematician, but yup, I think that's true in so many ways. The number of times I just take a break, come back..."Oh I know!".

That's the part that stood out to me, too. It's the interval between hyperproductivity and just scraping by... It took me a long time to understand my own process: that there is a pattern, and how to take advantage of the high and cope with the low.

For those who don't know, Andrew Wiles proved Fermat's last theorem.

Regarding the last question, "Do you think maths is discovered or invented?"

Couldn't math be considered both? A process of invention as far as deciding which axioms to assume and a process of discovery when finding the repercussions of those assumed axioms

> A process of invention as far as deciding which axioms to assume (...)

Possibly, but I think what mathematicians find is that they end up having little choice about what sets of axioms (when taken together) are useful, fertile, and/or interesting and which aren't - when studying various mathematical systems. Which leads us to consider that trip down to what those essential various groupings of axioms to once again feel more like discovery than invention.

Further, abstract math tries to shave concepts down to only what is logically essential. Arriving at what's left at that point is more akin to discovering a rare diamond or gold nugget buried deep beneath the surface, and inherent to the basic "construction" of the universe (philosophically, that which might exist beyond/before/after/without us).

This was so so encouraging and reminded me of why I started this math thing in the first place.

This post made me better today, because it reminded me it's okay to be stuck. You just have to keep going.

"Do you think maths is discovered or invented?

To tell you the truth, I don't think I know a mathematician who doesn't think that it's discovered."

Anyone else struck by this? It has really never occurred to before that I've always assumed that we were simply discovering math versus creating it.

Well, I suppose it's a philosophical thing, but I don't subscribe to the discover point of view. However, to be clear, I don't think there's anything wrong with this viewpoint.

For me, math is a set of rules that we know to be consistent. Based on these rules, we put together new constructions that obey this framework. When I prove a theorem, I don't really internalize it as discovering something that was already there, but as putting together a new creation based on a set of tools that I already have. As the author of the result, I have the flexibility to be as creative as I want in how I prove the result and that creativity has an affect on how people view and internalize the theorem. I mean, if someone writes a narrative story we could say that the story was always there and that they just discovered it in a sea of words. Again, there's nothing wrong with that point of view, but I prefer to say that the person created the story.

Opinion isn't as clear cut as Wiles makes out. There are definitely opposing viewpoints.

This would be a good place to start: https://en.wikipedia.org/wiki/Philosophy_of_mathematics

Perhaps it would be correct to say that we discover mathematical objects but invent theorems about them.

Gödel would have probably answered the opposite way, as would I think many logicians.

Gödel was a platonist.

So are logic frameworks discovered or invented? Are true/false values discovered or invented?

I think the question whether math is discovered or invented is silly. It is both.

We create the language and methodologies while we discover the relationships. It's both, as mathematics requires both.

This makes me wish I became a mathematician. Anyone know of modern day self-taught mathematicians that contributed to research? I'd like to learn maths on my own, but if there is no possibility of research contribution, then I guess my time would be better spent with something else.

Very few self-taught people contributed to research in the 20th century. I can't speak with authority about mathematics, but in my own field (physics) the only self-taught researchers I can think of were people trained in allied fields (math, chemistry) who ran into physics problems in the course of work in their own discipline.

The root cause, I suppose, is that it takes much less time and effort to just get a PhD than to make an original research contribution, so most people get credentialed along the way. Academic programs also immerse you in current research work, making it possible to figure out where you could make contributions in the first place.

Interestingly, I never thought of math as an art, but reading this quote in the article makes sense:

Well, mathematicians are not that philosophical. (Laughter) We're artists, we just enjoy it and we leave it.

Fascinating short interview. I wish they would have asked him what he thinks of theorem provers and if he maybe uses them in his workflow in some way.

I never expected this interview to be motivational.

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