
Andrew Wiles: what does it feel like to do maths? - jseliger
https://plus.maths.org/content/andrew-wiles-what-does-if-feel-do-maths
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duncanawoods
Its well worth watching one of the best documentaries ever made:

[http://www.dailymotion.com/video/x223gx8_bbc-
horizon-1996-fe...](http://www.dailymotion.com/video/x223gx8_bbc-
horizon-1996-fermat-s-last-theorem_shortfilms)

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 :)

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

~~~
stuxnet79
Oh wow it's Michael Nielsen. Your neural network book helped me get through
the last year of undergrad. Thanks!

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spynxic
_..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!

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

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spynxic
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.

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Jun8
"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...](https://www.scientificamerican.com/article/what-makes-michael-
phelps-so-good1/)). 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."

~~~
kxyvr
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.

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b_emery
> 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.

~~~
sn9
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.

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NumberCruncher
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](https://sivers.org/book/Mastery2)

~~~
Numberwang
Do you recommend reading the full book or the summary?

~~~
NumberCruncher
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.

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digler999
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.

~~~
treehau5
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.

~~~
digler999
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.

~~~
treehau5
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.

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daxfohl
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.

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

~~~
jseliger
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...](https://jakeseliger.com/2016/11/29/perfect-rigor-a-genius-and-the-
mathematical-breakthrough-of-the-century-masha-gessen). Highly recommended.

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socmag
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!".

~~~
ibejoeb
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.

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

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spynxic
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

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j1vms
> 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).

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cyberfunk
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.

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klt0825
"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.

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

~~~
fmoralesc
Gödel was a platonist.

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LeicesterCity
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.

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tpudlik
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.

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ktRolster
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._

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kriro
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.

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reefwalkcuts
I never expected this interview to be motivational.

