Stuck

October 20, 2017 8:51 AM

If you hold one mental image of [mathematician] Andrew Wiles, he wants it to be this: not the triumphant scholar with the medal around his neck, but the child learning to glory in the state of being stuck. Wiles is famous for cracking Fermat's Last Theorem. He was asked by blogger Ben Orlin what themes he would like to share with a broad public audience. So he talked about being stuck.

About Andrew Wiles.
About Fermat's Last Theorem. (The famous equation made an appearance on The Simpsons.)
Why has Fermat's Last Theorem been so difficult to prove?
Andrew Wiles on getting stuck and on creativity (8 minute interview).

via StudyHacks

posted by storybored (22 comments total)

r/math had a good discusion on this blogpost.

just wish I could at least be stuck on something that's not utterly trivial, what the heck is a de Rham Cohomology?

and that's just being snarkotistical, just what a neighborhood means is really where I guess I flounder (set of points inside an n-ball with center x and radius epsilon>0.)
posted by sammyo at 9:27 AM on October 20, 2017

GF is a pure mathematician, so I get to witness this cycle (of getting stuck, frustration, teeth-gritting-determination and eventual breakthrough) quite a bit without ever truly understanding the problems she's solving. It's still pretty awesome, though.
posted by Navelgazer at 9:44 AM on October 20, 2017 [2 favorites]

It's very much a familiar process for me as part of my own research. Getting stuck like this happens several times a year, for weeks or months of time. Sometimes we can resolve it, other times you have to punt and just put the frustration out into the literature along with everyone else. Some things are solved by a whole bunch of people and groups chipping away at problems for decades even. OP is a really good description of what it's like and why it's necessary.
posted by bonehead at 9:51 AM on October 20, 2017 [1 favorite]

Our students lack persistence. Give them a recipe, and they settle into monotonous productivity; give them an open-ended puzzle, and they panic.

And you're surprised? Their grade is contingent on their performance, ergo if they don't know how to get the answer they'll panic and hate this subject.

If you want kids to be okay with being stuck, you have to be okay with their being stuck, then you have to teach them that you're okay with their being stuck. Standing there and saying "math is about exploration!" does fuck-all to change things.
posted by disconnect at 10:01 AM on October 20, 2017 [24 favorites]

What I'll always associate Wiles with is how when he was doing the lectures announcing his Fermat's Last Theoreom proof, he didn't actually say what he was doing, so people in attendance only slowly realized what he was heading for, until his final mic drop. That's mathamatin' in style.
posted by tavella at 10:02 AM on October 20, 2017 [17 favorites]

A family friend is a lofty mathematician (knot theory is his focus), and when he was doing graduate level courses, he would make bread while tackling problems. That way, even if he didn't make progress on a problem, he'd have fresh bread.
posted by filthy light thief at 10:09 AM on October 20, 2017 [13 favorites]

Standing there and saying "math is about exploration!" does fuck-all to change things.

Also, my wife, a high school math teacher, likes to say learning (and teaching) math is about learning logic and ways to figure things out. You might never use the math you use in school, but you've learned problem solving, which is very useful in the real world, where nothing comes with clear instructions and it's not always obvious when you've reached the correct conclusion.
posted by filthy light thief at 10:13 AM on October 20, 2017 [3 favorites]

Just simply being stuck kind of sucks, if you don't have some tools and some experience (that you probably acquired through good guidance) about how to sense what directions might eventually lead to progress, if you're patient.

Good mathematicians generally have come to terms with the feeling of being stuck, but their comfort is generally predicated on lots of experiences being stuck in the past, which eventually led to some success.

Without a lot of experience, it's harder for younger students to judge what kind of problems will reward their patience in the being stuck phase, and that's where good instruction helps a lot.
posted by Wolfdog at 10:50 AM on October 20, 2017 [3 favorites]

Give them a recipe, and they settle into monotonous productivity; give them an open-ended puzzle, and they panic.

Well, for a lot of us we panic because our brains simply are not wired in a way that enables us to handle opened ended puzzles of this type. I'm certainly not. My son is highly gifted in math, and he is already, at 11, doing work that simply baffles me -- even when shown the solution I have know idea how one would begin to get there.

"math people" (sorry) frequently tell me "oh, you just have to learn how to do it" but I would strenuously disagree. For a lot of us, we panic at the open ended puzzled because we are not and never will be equipped to do them. I can't conceptualize my son's math any more than I could paint a great masterpiece. Oh sure, I might be able to learn how to paint, even paint realistic figures, through training, but there is a gulf between understanding the mechanics of the thing and making the leap necessary to be an artist.
posted by anastasiav at 11:30 AM on October 20, 2017 [1 favorite]

can't conceptualize my son's math any more than I could paint a great masterpiece

just think of math as paint by numbers -- rim shot

 
posted by sammyo at 11:48 AM on October 20, 2017

As I recall from a random biographical snippet, when asked what distinguished him from other researchers, Newton replied something to the effect of being able to hold a problem in mind for a long period of time.
posted by jamjam at 12:04 PM on October 20, 2017

I should actually add Wiles's mic drop, for those who haven't seen it:

Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D.

posted by tavella at 12:45 PM on October 20, 2017 [7 favorites]

(Though someone did subsequently point out a gap in the proof, which he and one of his doctoral students had to spend a year or two patching before the final, final proof. But that doesn't make the initial presentation less stylish.)
posted by tavella at 12:50 PM on October 20, 2017 [2 favorites]

Just simply being stuck kind of sucks, if you don't have some tools and some experience (that you probably acquired through good guidance)

I don't know; I had an awful lot of fun mucking around with Fermat's Last Theorem and noodling my way through Edwards's mathematical history of the subject, in late adolescence. I was definitely stuck, but enjoying myself.
posted by Coventry at 2:35 PM on October 20, 2017

what a neighborhood means is really where I guess I flounder (set of points inside an n-ball with center x and radius epsilon>0.)

Where do you flounder with this?

For what deRham cohomology is, the best answer will depend on why you want to know.
posted by Coventry at 2:46 PM on October 20, 2017

Well, for a lot of us we panic because our brains simply are not wired in a way that enables us to handle opened ended puzzles of this type.

You should read "Mathematical Mindsets" by Jo Boaler! It will help you, and it will help you help your child. (Also it's super approachable: friendly and non-technical, and although it has math examples in it it's a book on education theory (as applied to math), not on math itself.)
posted by eviemath at 5:53 PM on October 20, 2017 [1 favorite]

I haven't read much of Mindsets, but I like the general philosophy very much, and it lines up closely with my experience, in all three roles as student, teacher, and observer.
posted by Coventry at 9:21 PM on October 20, 2017

As a point of order, it's technically not Fermat's Last Theorem anymore (and really never was, being rather Fermat's Last Conjecture); it's Wiles' Theorem. I'm not sure they're going to formally rename it, but my understanding is that the usual case in mathematics is for nomenclature to follow the prover.
posted by adrienneleigh at 11:47 PM on October 20, 2017

I've always wondered myself, so the big question: what do we think was the case in Fermat's time? Do we think he...

A. Actually had a proof using the mathematics available to him at the time (rather than the Crazy Future Math that Wiles used)
B. Thought he had a proof but actually didn't
C. Was lying in order to...

C1. Encourage others to seek a proof
C2. Make himself look cool
C3. Troll us all
posted by BiggerJ at 5:06 AM on October 21, 2017 [1 favorite]

I'm going to vote for B. Hasn't there been a series of mistaken proofs submitted over the years? Particularly likely if he never wrote it down.
posted by 92_elements at 5:16 AM on October 21, 2017 [3 favorites]

There's a nice proof for n=4 which he might have thought he could generalize. It can all be done with tools he was fluent with. I think that's what happened.
posted by Coventry at 6:49 AM on October 21, 2017 [3 favorites]

I always liked how he ended the lectures presenting the proof - "I think I'll stop here".
posted by madcaptenor at 6:57 AM on October 21, 2017 [2 favorites]