It's hard to explain the experience of expertise. That's why one of the first things they teach you in journalism school is to avoid questions like, "What's it like to be a mathematician?" It's hard for your interview subject to know how to respond and you seldom get a useful answer.

But not never.

On Quora, someone* asks, "What is it like to have an understanding of very advanced mathematics?" And the responses are surprisingly interesting. Especially the first, wherein an anonymous mathematician lays out a detailed account of how advanced mathematics have altered his/her view of the world and of being a mathematician.

• You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.

• You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion.

These are only two bullets on a multi-bullet post. You really should read the whole thing.

Great find, noggin!

*I couldn't tell who had asked the question. Maybe I'm just not familiar enough with Quora. If you can see a name for the thread's original author, let me know.

“It’s not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.”

That’s a pretty cool and accurate description of what it is like to have an advanced understanding of other things as well, it seems to me.

Yes, that is basically what being a scientist is about.

No, that’s what being a magical elf is all about.

While I’m nowhere in the same league as these guys, I commonly find calculus useful. I certainly don’t remember all the integral or derivative laws beyond dx/dt x^2 = 2x, but that’s what Wolfram is for. It’s just the basic concepts – if you known those you find uses everywhere. If you don’t know it then you wonder how it could be useful – and then you badly reinvent integration. So I’ve got my socket wrench set, and these guys have a whole toolbox.

Interesting read! I do have to admit my relief that supermathpeople do not possess “some mysterious mental faculty that is used to crack a problem all at once”… like you know, ‘magic’ or something. Most of the concepts described can be extrapolated from and applied to other disciplines and situations of critical thinking, too, as Blueelm said.

As a mathematician, I find that most people think I spent my college years in courses like “Super Advanced Arithmetic 401,” and I can now multiply two six-digit numbers in my head. Thanks a lot, “Rain Man.”

More like, “thanks hollywood”. There are a number of stereotypes regarding savants and such that hollywood have helped supersize…

Also, when you tell people what you do at parties, they tell you how much they hated math. i.e. the thing which you have devoted your life to.

The question was added by Brad O’Neill: http://www.quora.com/log/revision/5657872

According to xkcd this is how it feels:

Purity.

I’ve known that joke for 20 years, but of course there is an xkcd for it.

Fascinating.

I have an undergraduate degree in physics, and I’m working on a Ph.D in biomedical engineering. So I intersect with mathematicians regularly. I’m not one of them, though, and some of these bullet points get at why that is.

The biggest one is that I have serious trouble handling mathematics that describes something I can’t visualize. If I can visualize it, I can come to understand it more or less intuitively. Otherwise, I get lost in the analysis and have no way to check myself to make sure my work makes sense.

This is generally not a problem in my BME work. While I work with systems that have a whole ton of state variables, I generally don’t have to look at the relationships between more than two or three of them at once. So the visualization is possible, if not always trivial. And things are generally not that abstract — I’m dealing with variables that have concrete physical meanings.

But it does mean I’ll never be a theoretical physicist.

I admit that I’m still learning how to work comfortably and productively in a state of confusion. I’m a lot better at it than I used to be, but I still wouldn’t say I’m ever really comfortable. But that might have more to do with the experience of being a student: still not being entirely sure how much of what I don’t know/understand is okay not to know/understand, and how much is stuff I

shouldknow and will be judged negatively for not knowing.I guess at a certain point, you get confident enough to believe that the stuff you don’t know or understand must be objectively difficult, and you’re not stupid for not already knowing it, and nobody is going to say “WTF, what do you mean you don’t understand this? This is baby stuff. How did you get here without knowing this? You’re fired. Get out.”

Probably this happens around the time you get tenure.

[Edited to fix typo]

One of my roommates in college was a math major, and in general one of the (intellectually speaking) smartest people I’ve ever met. In junior year, he saw my girlfriend knitting a scarf. He watched for a minute or so and then said, “Oh, so that’s how it works.” Apparently he had a sufficient understanding of groups, knots, and geometry to abstract how the process worked. That’s my usual go-to example of a) what it’s like to be a mathematician and b) what it is like to be much smarter than me.

my genius math roomate taught me functional calculus with two pictures on one piece of paper in 15 minutes. I still have that understanding, while my coursework faded. He currently teaches math teachers. Go future!

I once sat at dinner with a group of mathematical physicists and a after a few minutes of thinking about it we all agreed that sewing machines were a topological impossibility.

Sounds like critical thinking to me…

Over 1600 people earn PhDs in mathematics every year from US institutions alone (1100+ if you don’t include statistics/biostatistics), so we are not exactly a tiny class.

1600/year out of 307,000,000 people is a pretty tiny class.

Well, OK. It’s all relative, I guess; for example, only around 1000 people get PhDs in History every year, 600 in Sociology. Maybe we need a thread on “what it’s like for an Anthropologist” (500 PhDs/year). Or maybe just “what it’s like for an expert at something,” since neither an advanced degree nor special training in an academic field are prerequisites for clarity of thought or the ability to systematically attack a problem, even an abstract one.

The exact same question was posted to MathOverflow a little while ago by someone going by “Kris Haamer”.

Did you hear about the mathematician that had a case of constipation?

She/he worked it out with a pencil.

Buwahahahahaha.

Thank you. Carry on.

Read up on Paul Erdos, then you’ll know; it’s basically a trade off in mental faculties.

My father asked – in a surprisingly disparaging tone – what I might do with my degree in pure mathematics. “What are you going to do, Listener 43, open a theorem store?” he asked.

“Ha, that’s a good one, Dad,” was my witty reply.

So where’s your store located?

Brilliant find, thanks!

It’s great for illustrating Brown, Duguid & Collins (1988) seminal report on “Situated Cognition and the Culture of Learning” (full text on the web), which deals specifically with mathematics being more than just a set of abstract rules but a way of perception:

“(…) students need much more than abstract concepts and self-contained examples. They need to be exposed to the use of a domain’s conceptual tools in authentic activity — to teachers acting as practitioners and using these tools in wrestling with problems of the world. Such activity can tease out the way a mathematician or historian looks at the world and solves emergent problems.”

I think a lot of this list covers things many of us already do. The best example was given by Wiles, when he described bumbling around in his rooms for six months (or so) at a stretch before finding the light switch for that particular room. But it’s not exclusive to maths — think of any problem you’ve had in any domain of knowledge, and it’s a case of solving the problem by first being a bit confused about it, then examining your collection of mental tools and finding the tool you think most likely to fit, testing it, and refining your solution. Lather, rinse, repeat.

Maybe that means that the question of what it’s like to be a mathematician is best answered as “human”.

I agree with a lot of this. I remember a group discussion on the value of the humanities where I ended up arguing that pure mathematics is a humanities discipline par excellence for similar reasons.

I think the point about intuition is probably true of any kind of expertise, and also a damn fine way to explain it. “The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true.” Yes. That.