# What is it like to have an understanding of very advanced mathematics?

December 24, 2011 1:07 PM Subscribe

What is it like to have an understanding of very advanced mathematics? A naive Quora question gets a remarkably long, thorough answer from an anonymous respondent. The answer cites, among many other things, Tim Gowers's influential essay "The Two Cultures of Mathematics," about the tension between problem-solving and theory-building. Related: Terry Tao asks "Does one have to be a genius to do maths?" (Spoiler: he says no.)

People think that certain skills are things "other people" do. I run into all the time as an artist, (Oh, I can't draw), but it's the same for math, computers, plumbing, cooking etc.

I don't understand it. Most of the time, they seem proud of it in some way.

posted by cmoj at 1:48 PM on December 24, 2011 [7 favorites]

I don't understand it. Most of the time, they seem proud of it in some way.

posted by cmoj at 1:48 PM on December 24, 2011 [7 favorites]

That was very interesting, thanks. In some ways, skill/expertise is in itself a skill. The first anon respondent recognizes this in much of his/her talk about acquiring the ability to use intuition to gain quick insights into the shape of problems and solutions. Skill takes practice, and practice takes time. Of course, I also believe that you also have to have certain aptitude for math; but aptitude without practice is not likely to get you very far.

posted by carter at 1:58 PM on December 24, 2011 [1 favorite]

posted by carter at 1:58 PM on December 24, 2011 [1 favorite]

Fascinating. Reads like a review of Ted Chiang's

posted by spasm at 2:01 PM on December 24, 2011 [6 favorites]

*Understand*.posted by spasm at 2:01 PM on December 24, 2011 [6 favorites]

>

While this is true, it's also true that the highest levels of mathematics are not accessible to everyone. When I was in high school I breezed through all my math classes, including calculus, and when I went to college as a math major the same was true, for a while. Group theory, topology, I ate it up. Having read and loved James R. Newman's

At any rate, thanks for the great post and links!

posted by languagehat at 2:05 PM on December 24, 2011 [32 favorites]

*"don't ever be intimidated; just follow your interests where they lead you." people who tell children that only geniuses are qualified to pursue certain interests should be shot.*While this is true, it's also true that the highest levels of mathematics are not accessible to everyone. When I was in high school I breezed through all my math classes, including calculus, and when I went to college as a math major the same was true, for a while. Group theory, topology, I ate it up. Having read and loved James R. Newman's

*The World of Mathematics*as a kid, I was confident I was going to be one of the great mathematicians written about there. Then I hit a wall: suddenly it wasn't easy any more, and between that and a quarrel with the math department I wound up going into linguistics instead. I've talked to a number of people who've had similar experiences. It's like math is a mountain that different people can climb to different levels before they start feeling the lack of oxygen.At any rate, thanks for the great post and links!

posted by languagehat at 2:05 PM on December 24, 2011 [32 favorites]

Don't worry, we happily export mathematics that you might find useful enough to learn!

As an aside, I love Persi Diaconis' result that a deck of chards requires seven riffle shuffles to obtain a high degree of randomness, the proof requires fun stuff from character theory.

posted by jeffburdges at 2:14 PM on December 24, 2011 [1 favorite]

As an aside, I love Persi Diaconis' result that a deck of chards requires seven riffle shuffles to obtain a high degree of randomness, the proof requires fun stuff from character theory.

posted by jeffburdges at 2:14 PM on December 24, 2011 [1 favorite]

lhat - I had a similar experience, although I reached my point of oxygen starvation earlier in undergrad than you did. I'm annoyed in retrospect that I didn't push harder at it, but the

I really enjoyed the essay in the OP.

posted by kavasa at 2:21 PM on December 24, 2011

*ease*with which I'd ascended the earlier peaks ended up making the shock of hard work much ruder.I really enjoyed the essay in the OP.

posted by kavasa at 2:21 PM on December 24, 2011

At some point in undergrad I hit a wall on a particular concept (Lie groups) that was seemingly insurmountable... Saw them again many years later in grad school, and they made a whole lot of sense all of the sudden. Sometimes these very abstract concepts need time to percolate in the back of the brain for a few weeks or maybe even months before they really have a chance to click. Building up more context helps quite a lot, too.

I also got a lot out of independent learning, working through some key books at my own pace. It taught me to read math independently, and take things on on my own terms. Both have proven very useful.

(I'm a new phd, now working a research postdoc somewhere in the frozen north.)

posted by kaibutsu at 2:24 PM on December 24, 2011 [4 favorites]

I also got a lot out of independent learning, working through some key books at my own pace. It taught me to read math independently, and take things on on my own terms. Both have proven very useful.

(I'm a new phd, now working a research postdoc somewhere in the frozen north.)

posted by kaibutsu at 2:24 PM on December 24, 2011 [4 favorites]

In this post people were debating whether or not one can learn how to "think like a lawyer". I wanted to point out that there is "think like a mathematician", but was having trouble explaining it. I wish I could have just pointed to that Quora post.

I also had an experience similar to languagehat (including issues with the math department!). You easily reach certain level of abstraction, and all of a sudden the next level just won't work for you.

Some people, however, seem to only live at the higher levels of abstraction. Alexander Grothendieck is perhaps the best example. Supposedly he once used the number 57 as an example of a prime number.

[If you don't see why this is funny, go here]

posted by benito.strauss at 2:24 PM on December 24, 2011 [1 favorite]

I also had an experience similar to languagehat (including issues with the math department!). You easily reach certain level of abstraction, and all of a sudden the next level just won't work for you.

Some people, however, seem to only live at the higher levels of abstraction. Alexander Grothendieck is perhaps the best example. Supposedly he once used the number 57 as an example of a prime number.

[If you don't see why this is funny, go here]

posted by benito.strauss at 2:24 PM on December 24, 2011 [1 favorite]

*Then I hit a wall: suddenly it wasn't easy any more*

But that's not the right time to leave math! The right time is when it isn't

*fun*any more; which might be before it stops being easy, or after, or, in some very happy cases, never.

posted by escabeche at 2:25 PM on December 24, 2011 [7 favorites]

Great post, it's like a glimpse into an entirely different world. As a teenager who took Advanced Placement calculus in high school and now can barely calculate tips, I found this fascinating. When I recently came across my old calc notebooks from that period, I couldn't believe that I'd ever been able to do--let alone understand--the calculations involved. There is very much this sense that you need to be a genius to be able to continue studying math into the higher levels, this becomes a self-fulfilling cycle as invariably the ones who are just okay at it will drop out.

posted by so much modern time at 2:26 PM on December 24, 2011 [3 favorites]

posted by so much modern time at 2:26 PM on December 24, 2011 [3 favorites]

I am constantly astounded by how little math educated people are able to do.

I was in court recently and witnessed the following exchange between the Judge and the Assistant District Attorney:

Judge -- So that will be the cost of court plus a fifty dollar fine. So let's see. The cost of court is 190 dollars....

ADA -- It will be 240 dollars.

Judge -- How did you just do that?

The judge reacted to the ADA's ability to add 19 and 5 as though it was some sort of parlor trick. It was simply unbelievable.

posted by flarbuse at 2:29 PM on December 24, 2011 [7 favorites]

I was in court recently and witnessed the following exchange between the Judge and the Assistant District Attorney:

Judge -- So that will be the cost of court plus a fifty dollar fine. So let's see. The cost of court is 190 dollars....

ADA -- It will be 240 dollars.

Judge -- How did you just do that?

The judge reacted to the ADA's ability to add 19 and 5 as though it was some sort of parlor trick. It was simply unbelievable.

posted by flarbuse at 2:29 PM on December 24, 2011 [7 favorites]

*The judge reacted to the ADA's ability to add 19 and 5 as though it was some sort of parlor trick. It was simply unbelievable.*

So, I'm a programmer. I do realtime 3d game graphics (and physics). I've spent the last couple of weeks designing a Lagrangian fluid dynamics simulation, written to exploit available data-parallelism (read: it'll use all your cores and your GPU). The math is as crazy as macroscopic physics gets.

I would have pulled out my phone to add 190 and 50. (Okay, not really, but make it 187 and 46, and I would have.)

I will weep tears of sweet joy the day I can have a math coprocessor installed in my skull.

posted by Netzapper at 2:49 PM on December 24, 2011 [12 favorites]

*The judge reacted to the ADA's ability to add 19 and 5 as though it was some sort of parlor trick. It was simply unbelievable.*

A facility with arithmetic is no more an indicator of any sort of important ability than is the ability to spell well.

posted by OmieWise at 2:55 PM on December 24, 2011 [7 favorites]

*You can answer many seemingly difficult questions quickly.... The trick is that your brain can quickly decide if question is answerable by one of a small number of powerful general purpose "machines" (e.g. continuity arguments, combinatorial arguments, correspondence between geometric and algebraic objects, linear algebra, compactness arguments that reduce the infinite to the finite, dynamical systems, etc.). The number of fundamental ideas and techniques that people use to solve problems is pretty small*

A friend of mine who attended a top law school expressed a similar sentiment. By his third year it was all applications of some commonly understood argumentation and litigation techniques.

posted by stp123 at 3:00 PM on December 24, 2011 [2 favorites]

Netzapper, as someone who works on Lagrangian hydrodynamics for a living and loves videogames, I am desperately curious about what game you are applying this to.

posted by FuturisticDragon at 3:18 PM on December 24, 2011 [4 favorites]

posted by FuturisticDragon at 3:18 PM on December 24, 2011 [4 favorites]

*hen I hit a wall: suddenly it wasn't easy any more...*

hi. math phd here. Having an understanding of advanced mathematics is about waking up every day and running into that wall. Most people when they hit that wall, they say "ouch" and do something less painful; mathematicians are the people who feel compelled to run right at it over and over again. The quora responder is putting a brave face on things and/or telling a Platonic story:

*You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems.*

LOL

*In listening to a seminar or while reading a paper, you don't get stuck as much as you used to...*

uhh... umm. ok. he's either putting one over the audience here or is easily impressed by advanced hand-waving techniques applied to seminars...

*You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles.*

it's easier to find a problem your machine can solve than to build a new machine to solve a particular problem. this "aesthetic preference" is driven by the desire to avoid being puzzled and stuck i.e. hit the wall.

*The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians", where _____ is something generally believed to be difficult (quantum chemistry, general relativity, pricing of derivatives, formal epistemology). Those books would be short and pithy. That's because the key ideas in those hard fields boil down to a few abstract ideas that mathematicians already understand or are well equipped to understand using their toolbox of concepts.*

nope. mathematicians often believe that subjects are trivial if the problems appear to fit within a particular set of intellectual machinery and then discover that hard problems are still hard. there are a million seminars littered with the crashed and burned remains of mathematicians who thought that subject X was going to be straightforward... i mean, it's only a little functional analysis right?

*Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being puzzled.*

umm... no.

I think from a thousand feet up, mathematics looks like one of those WWI massed infantry charges into the machine guns. Most mathematical projects never get anywhere near their goals, intuition is illusory, many advances essentially depend on something "just working," it's easy to feel like everything you've done is a failure, and your successes dwarfed by what you failed to accomplish. A regular person thinks this is rather grim, a mathematician fixes their bayonet and prepares to charge...

posted by ennui.bz at 3:22 PM on December 24, 2011 [31 favorites]

The best exploration of this topic I know of is "A New Golden Age" by Rudy Rucker. It's a short story he wrote very early in his career, when he was a grad student teaching math to undergrads.

posted by charlie don't surf at 3:24 PM on December 24, 2011 [1 favorite]

posted by charlie don't surf at 3:24 PM on December 24, 2011 [1 favorite]

*A facility with arithmetic is no more an indicator of any sort of important ability than is the ability to spell well.*

--OmieWise

Man, tell me about it, I was on the fast track in 6th grade to a doctorate in Spelling, but time goes on, and you find your dreams are crushed and the world doesn't work the way you thought it did.

posted by symbioid at 3:29 PM on December 24, 2011 [13 favorites]

"I used to be an advanced mathematician like you, but then I took an arrow to the knee..."

posted by greenhornet at 3:35 PM on December 24, 2011 [3 favorites]

posted by greenhornet at 3:35 PM on December 24, 2011 [3 favorites]

To risk stating the obvious, you don't get a PHD in arithmetic any more than you would a PHD in spelling. After a very early point math isn't about your addition and multiplication tables.

posted by idiopath at 3:38 PM on December 24, 2011 [3 favorites]

posted by idiopath at 3:38 PM on December 24, 2011 [3 favorites]

*To risk stating the obvious, you don't get a PHD in arithmetic any more than you would a PHD in spelling. After a very early point math isn't about your addition and multiplication tables.*

On the contrary!

posted by kaibutsu at 3:50 PM on December 24, 2011 [1 favorite]

I'm curious about the process of doing higher research in mathematics. You may have an intuition about something, but there's no guarantee you're going to get results, no matter how much coffee you drink. So beginning a four-year research project is a huge gamble, with a very good chance that at the end all you'll have found are numerous dead-ends. What's worse is the fact that while a mere mortal may spend four years and get nowhere, there's always some incredible freak out there who would be able to cut through the whole thing in four days. That makes a career in maths a somewhat intimidating prospect for a non-genuis.

posted by moorooka at 3:53 PM on December 24, 2011 [1 favorite]

posted by moorooka at 3:53 PM on December 24, 2011 [1 favorite]

You can answer many seemingly difficult questions quickly.... The trick is that your brain can quickly decide if question is answerable by one of a small number of powerful general purpose "machines" (e.g. continuity arguments, combinatorial arguments, correspondence between geometric and algebraic objects, linear algebra, compactness arguments that reduce the infinite to the finite, dynamical systems, etc.). The number of fundamental ideas and techniques that people use to solve problems is pretty smallA friend of mine who attended a top law school expressed a similar sentiment. By his third year it was all applications of some commonly understood argumentation and litigation techniques.

I'm guessing it's true in any line of work.

So I mean like I know nothing about fixing cars. But I know some people who

*can*talk sensibly about car trouble, and I get the sense they've got the same thing going — a small bag of widely applicable heuristics, and a good gut feel for which one to use when. So sometimes they'll ask "Does it get worse when the engine's cold?" and sometimes they'll ask "Does it still happen when it's idling?" or whatever, and the clever part is just knowing which question is appropriate in which situations.

posted by nebulawindphone at 3:55 PM on December 24, 2011 [1 favorite]

*A facility with arithmetic is no more an indicator of any sort of important ability than is the ability to spell well.*

I think this underestimates the value of a character trait I associate with spelling well: paying attention. I don't claim any special facility in spelling, but I don't often send out spelling disasters, because I heed that little voice that says "this doesn't look quite right, and you don't use that word very often. Better check it."

Paying attention, and being alert to tiny discrepancies, <Garrett Morris>been veddy veddy good to me</Garrett Morris>, both as an engineer and as a lawyer.

posted by spacewrench at 4:00 PM on December 24, 2011 [7 favorites]

*a small bag of widely applicable heuristics*

I'm not sure whether it's a bag of heuristics or something else, but I definitely sensed a meta-understanding thing when I realized I was good at programming, and I heard from an experienced psychiatrist that studies have shown physical differences between the brains of experts (in some field/activity) and non-experts. I personally believe that getting really good at

*something*is critical to success generally -- in addition to being good at whatever you're good at, you've learned

*how to be good at something*, a meta-skill involving ruthless triage and efficient identification of those pressure points where dogged determination is the best (or only) way through.

I promise I'll shut up now.

posted by spacewrench at 4:13 PM on December 24, 2011 [5 favorites]

*I'm guessing it's true in any line of work.*

Yeah, I sometimes astound people at work when I can solve networking problems that 8 people have been banging their heads against for two weeks. It's just a matter of being able to visualize how all the pieces fit together, and having a working knowledge of where the failures can be and how they manifest. I think if you focus on any skill for long enough you get that way. I think all kinds expertise are on some level, the same expertise -- breaking down problems into manageable parts, the ability to recognize problem and solution patterns, the ability to communicate clearly and to understand what other people are communicating, etc. It doesn't matter if you're a theoretical physicist or a plumber on some level.

posted by empath at 4:28 PM on December 24, 2011 [1 favorite]

Well, in programming, experience often translates to "I already made the mistake you're about to make."

posted by benito.strauss at 5:06 PM on December 24, 2011 [12 favorites]

posted by benito.strauss at 5:06 PM on December 24, 2011 [12 favorites]

*So beginning a four-year research project is a huge gamble, with a very good chance that at the end all you'll have found are numerous dead-ends. What's worse is the fact that while a mere mortal may spend four years and get nowhere, there's always some incredible freak out there who would be able to cut through the whole thing in four days.*

At the beginning of the process, you have a Ph.D. advisor. I would never set a graduate student of mine off on a project that had a "very good chance" of hitting a dead end after four years. Instead you pose them the kind of problem where, if things go badly, they'll get a minor but publishable result, and if things go well, they'll make progress that people will really care about. After enough years in the field you get a pretty good intuition for which problems are like that.

And it's very ususual for someone to have an amazing idea that turns a four-year-project into a four-day one. I'm not saying it never happens -- but it's maybe number 20 on the list of things you'd worry about as a starting researcher.

posted by escabeche at 5:15 PM on December 24, 2011 [1 favorite]

*You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems.*

See? It's not just the rest of us! (That level of candidness is unlikely to pop up in face-to-face discussions!)

posted by Twang at 5:18 PM on December 24, 2011 [1 favorite]

It is just like Will in Good Will Hunting, you hang out in empty classrooms with some dude who has a vaguely European sounding accent crossing off numbers while you laugh and laugh and laugh.

posted by Ad hominem at 5:44 PM on December 24, 2011 [5 favorites]

posted by Ad hominem at 5:44 PM on December 24, 2011 [5 favorites]

At the beginning of the process, you have a Ph.D. advisor.

Ha!

posted by benito.strauss at 5:48 PM on December 24, 2011

Ha!

posted by benito.strauss at 5:48 PM on December 24, 2011

*As a teenager who took Advanced Placement calculus in high school and now can barely calculate tips,*

I guess you've never been at a restaurant table full of mathematicians trying to deal with a single bill.

It's grim.

Arithmeticians, we're (mostly) not.

posted by leahwrenn at 8:48 PM on December 24, 2011 [4 favorites]

More seriously, I was brought up short by the quota answerer's comment about generalization. I guess I know that it's something that students struggle with, but hadn't really understood---because of course, the obvious next question to ask is "how does such and so generalize".

posted by leahwrenn at 8:51 PM on December 24, 2011 [1 favorite]

posted by leahwrenn at 8:51 PM on December 24, 2011 [1 favorite]

When I was an undergraduate math student, I had a good friend who was a few years older than the rest of us. He was a bright guy - he'd been a professional journalist and then the PR manager for a successful run for Member of Parliament (this was Canada).

He loved math and had always wanted to study it.

He did well until second year when he started to have serious issues with the proofs. Second year is hard in mathematics. The first year you basically redo high school, but right - the second year you do real math. There was a lot of work.

I worked very hard with him - partly out of altruism, partly because explaining the material helps, but mainly because he should have been able to get it, dammit! But there was a gap. He couldn't make proofs...

At some point he vanished. I called him several times - eventually his brother, who I knew a little, answered. He told me that my friend had had a nervous breakdown because he couldn't hack it. He said, "I really appreciate your call - you're the only one of his friends who managed to get through to us. But I'm not going to tell him you called - the doctor said we should not discuss anything about this time with him. I really regret it, but I'm sure you'll understand."

And what is it like being able to do advanced math (or these days, it's gnarly programming)?

Very strange. Very abstract. It's a little scary because some of it is under your control and some of it just comes or doesn't come. It's perfectly easy for you to waste a week because your mind won't do it - you get nervous. Luckily, it's always come back for me.

And there are times after having worked on these things for a day where it's hard to come back to reality... you're looking at someone and apparently chatting but diagrams are going through your head...

posted by lupus_yonderboy at 10:09 PM on December 24, 2011 [1 favorite]

He loved math and had always wanted to study it.

He did well until second year when he started to have serious issues with the proofs. Second year is hard in mathematics. The first year you basically redo high school, but right - the second year you do real math. There was a lot of work.

I worked very hard with him - partly out of altruism, partly because explaining the material helps, but mainly because he should have been able to get it, dammit! But there was a gap. He couldn't make proofs...

At some point he vanished. I called him several times - eventually his brother, who I knew a little, answered. He told me that my friend had had a nervous breakdown because he couldn't hack it. He said, "I really appreciate your call - you're the only one of his friends who managed to get through to us. But I'm not going to tell him you called - the doctor said we should not discuss anything about this time with him. I really regret it, but I'm sure you'll understand."

And what is it like being able to do advanced math (or these days, it's gnarly programming)?

Very strange. Very abstract. It's a little scary because some of it is under your control and some of it just comes or doesn't come. It's perfectly easy for you to waste a week because your mind won't do it - you get nervous. Luckily, it's always come back for me.

And there are times after having worked on these things for a day where it's hard to come back to reality... you're looking at someone and apparently chatting but diagrams are going through your head...

posted by lupus_yonderboy at 10:09 PM on December 24, 2011 [1 favorite]

Oh, and I can do a lot of practical arithmetic and physics in my head and I sometimes show off with it. :-D But I kept that active once I discovered the stereotypes about mathematicians - I didn't want to be the math guy who couldn't add the check.

posted by lupus_yonderboy at 10:11 PM on December 24, 2011

posted by lupus_yonderboy at 10:11 PM on December 24, 2011

Let's try to build bridges here. For the mathematicians here, how many numbers have you worked with personally? Would you say that you need to learn all of them in order to understand mathematics? At what level do you start working with really big numbers?

posted by Joe in Australia at 10:18 PM on December 24, 2011 [1 favorite]

posted by Joe in Australia at 10:18 PM on December 24, 2011 [1 favorite]

> At what level do you start working with really big numbers?

Oh, that one's easy. There are numbers that are so big that, even though they aren't infinite, you can't really represent them directly. Graham's Number, for example, is so large that not only can't you write down the number (even if you used one atom per digit, there just aren't enough atoms), you can't even write down

posted by lupus_yonderboy at 10:27 PM on December 24, 2011 [7 favorites]

Oh, that one's easy. There are numbers that are so big that, even though they aren't infinite, you can't really represent them directly. Graham's Number, for example, is so large that not only can't you write down the number (even if you used one atom per digit, there just aren't enough atoms), you can't even write down

*the number of the digits*in the representation of the number.posted by lupus_yonderboy at 10:27 PM on December 24, 2011 [7 favorites]

Graham's number. So large, you need new notation just to write it down. TREE(3) is bigger.

posted by BungaDunga at 10:28 PM on December 24, 2011 [1 favorite]

posted by BungaDunga at 10:28 PM on December 24, 2011 [1 favorite]

I decided early on that I didn't care about large cardinals, Joe in Australia.

Large cardinals might be described as cardinals so large they cannot be proven consistent with ZFC because chopping off the set theoretic universe at one gives a model of set theory, potentially contradicting the fact that ZFC cannot prove it's own consistency. Large cardinals are commonly characterized by their equiconsistency strength with respect other more pedestrian set theoretic axioms too.

In fact, large cardinals are so large that they make all cardinalities before them take on finite number like properties. Infinitely many Woodin cardinals imply the axiom of projective determinacy, for example. There are two jokes about the largest large cardinal, first that it's the cardinality ω of the integers because that is so much larger than every finite cardinality that came before it, and second that it's a contradiction (1=0) meaning they get "more false" as they get bigger.

Zeilberger might mock such things.

I've personally avoided working with any number other than two when I could help it. Involutions are nice. Two is as they say the oddest prime.

posted by jeffburdges at 11:13 PM on December 24, 2011 [7 favorites]

Large cardinals might be described as cardinals so large they cannot be proven consistent with ZFC because chopping off the set theoretic universe at one gives a model of set theory, potentially contradicting the fact that ZFC cannot prove it's own consistency. Large cardinals are commonly characterized by their equiconsistency strength with respect other more pedestrian set theoretic axioms too.

In fact, large cardinals are so large that they make all cardinalities before them take on finite number like properties. Infinitely many Woodin cardinals imply the axiom of projective determinacy, for example. There are two jokes about the largest large cardinal, first that it's the cardinality ω of the integers because that is so much larger than every finite cardinality that came before it, and second that it's a contradiction (1=0) meaning they get "more false" as they get bigger.

Zeilberger might mock such things.

I've personally avoided working with any number other than two when I could help it. Involutions are nice. Two is as they say the oddest prime.

posted by jeffburdges at 11:13 PM on December 24, 2011 [7 favorites]

I've always wondered what this quote by the mathematician Stefan Banach really meant:

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

Now I think I understand.

posted by rahulrg at 12:07 AM on December 25, 2011 [2 favorites]

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

Now I think I understand.

posted by rahulrg at 12:07 AM on December 25, 2011 [2 favorites]

*Graham's Number, for example, is so large that not only can't you write down the number (even if you used one atom per digit, there just aren't enough atoms), you can't even write down the number of the digits in the representation of the number*

Holy shit, that's the most astounding thing I've ever heard! In fact, I feel so elated in having understood Knuth's notation after having read the Wikipedia article about five times, that I actually want to wake my wife up and explain this in detail. It's a Christmas miracle or something; I say this in all seriousness, but just learning a new set of symbols to describe the vastness of all creation surely should count as a greater spiritual experience than attending Midnight Mass or having a

*darshan*of the shiva lingam at Amarnath.

posted by the cydonian at 12:27 AM on December 25, 2011 [6 favorites]

Jeffburdges: Here's a

Can I make a confession about my (so far) highest experience in mathematics? I felt offended when I read about Goodstein's theorem. It just doesn't seem right that a statement

posted by Joe in Australia at 12:58 AM on December 25, 2011 [4 favorites]

**really**large cardinal!Can I make a confession about my (so far) highest experience in mathematics? I felt offended when I read about Goodstein's theorem. It just doesn't seem right that a statement

**about**natural numbers cannot be proved using natural numbers. It's as though you guys corrupted them somehow.posted by Joe in Australia at 12:58 AM on December 25, 2011 [4 favorites]

I always terrible with arithmetic in elementary school. I was slow at it and I would make little errors, it was really frustrating. On the other hand, I've always enjoyed 'advanced' math where you're talking about variables rather then actual numbers. Of course I used calculators as much as possible. Since a couple years ago I've been buying math books and teaching myself more math. I taught myself linear algebra and then abstract algebra. Linear algebra is a really powerful tool for thinking about things conceptually. But what's interesting is that it basically requires a computer if you ever want to use it practically rather then prove theorems about variables.

But, there's a big difference between me and someone who does math professionally: I don't have to worry about proving anything. That makes it a lot easier to learn. I know the concepts, and I could probably write a computer that uses this math to solve problems, but I don't have to learn it in such a precise detail that I can actually prove a theorem. And, of course, if there's something I'm having trouble with, I can just skip it, go read some other book, then come back to it later. The downside, of course, is laziness. I might go a couple months without studying, so I'm not learning nearly as quickly as I would in an academic setting. Plus my understanding could be wrong. But it's just something I want to do for fun, so who cares?

The other thing: one of the books I got was this "Secrets of Mental Math", which is actually about doing arithmetic in my head. Sometimes when I'm bored I will try to do arithmetic in my head for fun, and I've been getting better at it. I've had it for I think over a year but never cracked it open. I looked at the first chapter the other day and it explained how you should actually do addition left to right, rather then right to left. I'd never thought of that, but it was actually a lot easier. Mostly I just want to see if I can get to the point where I can impress people by doing arithmetic really quickly in my head.

But really I basically just use a spell checker with everything. I remember in, I think first or second grade a teacher telling me that 'eventually' people learn to read not by looking at individual letters, but by looking at the whole word. That made a lot of sense to me at the time, and growing up I was always in the highest percentiles for reading ability and so on. So the thing is: I pretty much skipped the stage where people read by looking at each letter. So I never thought much about how words are spelled.

And of course, by the time I got to highschool, spellcheck was easily accessible, so there was never really any need for me to really buckle down and try to memorize how to spell all these words. In some cases there are words that are hard to spell check, like 'bureau'. What I'll do is memorize a 'synonym' of the word that sounds like it's spelled.

Without the little red underline, there's not much chance I would notice a misspelled word, because, like I said I'm just looking over the shape of the words and sentences generally, not looking at each individual letter.

posted by delmoi at 3:04 AM on December 25, 2011

But, there's a big difference between me and someone who does math professionally: I don't have to worry about proving anything. That makes it a lot easier to learn. I know the concepts, and I could probably write a computer that uses this math to solve problems, but I don't have to learn it in such a precise detail that I can actually prove a theorem. And, of course, if there's something I'm having trouble with, I can just skip it, go read some other book, then come back to it later. The downside, of course, is laziness. I might go a couple months without studying, so I'm not learning nearly as quickly as I would in an academic setting. Plus my understanding could be wrong. But it's just something I want to do for fun, so who cares?

The other thing: one of the books I got was this "Secrets of Mental Math", which is actually about doing arithmetic in my head. Sometimes when I'm bored I will try to do arithmetic in my head for fun, and I've been getting better at it. I've had it for I think over a year but never cracked it open. I looked at the first chapter the other day and it explained how you should actually do addition left to right, rather then right to left. I'd never thought of that, but it was actually a lot easier. Mostly I just want to see if I can get to the point where I can impress people by doing arithmetic really quickly in my head.

Well, as someone who's a terrible speller, I totally disagree. It's actually gotten a lot better over the years, just from writingI think this underestimates the value of a character trait I associate with spelling well: paying attention. I don't claim any special facility in spelling, but I don't often send out spelling disasters, because I heed that little voice that says "this doesn't look quite right, and you don't use that word very often. Better check it."

*a lot*. And words that I can't spell, even with a spell checker like 'bureau' I'll start to learn an alternative pronunciation based on how it's spelled.But really I basically just use a spell checker with everything. I remember in, I think first or second grade a teacher telling me that 'eventually' people learn to read not by looking at individual letters, but by looking at the whole word. That made a lot of sense to me at the time, and growing up I was always in the highest percentiles for reading ability and so on. So the thing is: I pretty much skipped the stage where people read by looking at each letter. So I never thought much about how words are spelled.

And of course, by the time I got to highschool, spellcheck was easily accessible, so there was never really any need for me to really buckle down and try to memorize how to spell all these words. In some cases there are words that are hard to spell check, like 'bureau'. What I'll do is memorize a 'synonym' of the word that sounds like it's spelled.

Without the little red underline, there's not much chance I would notice a misspelled word, because, like I said I'm just looking over the shape of the words and sentences generally, not looking at each individual letter.

posted by delmoi at 3:04 AM on December 25, 2011

I wonder sometime if I coulda done real math, am I/was I just lazy and didn't push through or well just not actually smart enough. I've chatted with mathematicians occasionally and seem to follow the description of oh say a recent theorem in algebraic topology but it does not seem to be something that really sticks.

I was fortunate to hear Ron Graham give a talk on "problems computers would never be able to handle" and although I was the dumbest person in the room it seemed like I was following, but other than notation with arrows that meant raising to a power a whole lot of times it's more like the memory of an amazing symphonic performance, I could describe the 'wow' feeling but could barely remember a bit of melody.

Having worked through a small introductory text in topology as a subway book I'm very impressed with delmoi's persistence an not a little concerned about lupus_yonderboy's story of the guy that pushed so hard that he had a nervous breakdown, perhaps it's not so bad I didn't go that direction, sometimes it's better not to learn the hard way.

posted by sammyo at 5:35 AM on December 25, 2011

I was fortunate to hear Ron Graham give a talk on "problems computers would never be able to handle" and although I was the dumbest person in the room it seemed like I was following, but other than notation with arrows that meant raising to a power a whole lot of times it's more like the memory of an amazing symphonic performance, I could describe the 'wow' feeling but could barely remember a bit of melody.

Having worked through a small introductory text in topology as a subway book I'm very impressed with delmoi's persistence an not a little concerned about lupus_yonderboy's story of the guy that pushed so hard that he had a nervous breakdown, perhaps it's not so bad I didn't go that direction, sometimes it's better not to learn the hard way.

posted by sammyo at 5:35 AM on December 25, 2011

This is spot on for any field: "This makes the total time you spend in life reveling in your mastery of something quite brief."

The "I'm so smart, I'm so smart, S-M-R-T" feeling is indeed not very interesting at all.

posted by Pyrogenesis at 5:39 AM on December 25, 2011

The "I'm so smart, I'm so smart, S-M-R-T" feeling is indeed not very interesting at all.

posted by Pyrogenesis at 5:39 AM on December 25, 2011

True, Pyrogenesis, but eureka moments where suddenly all the parts of a difficult problem come together and you see the solution is one specific connection of these parts--those last for a short time, but feel damn near euphoric while they last. (I'm not a mathematician, by the way.) It is not so much reveling in your own genius as enjoying the feeling of crossing over the treshold between not understanding and understanding. I've continued to read into many a problem simply out of this kind of intellectual hedonism. But maybe that's not how it works on the cutting edge of mathematics.

posted by simen at 7:29 AM on December 25, 2011

posted by simen at 7:29 AM on December 25, 2011

how many numbers have you worked with personally?

Well, my analysis prof used to say that the only numbers you had to care about were 0, 1, and ∞. I think all the others (like 2, 3, and π

posted by benito.strauss at 7:42 AM on December 25, 2011 [8 favorites]

Well, my analysis prof used to say that the only numbers you had to care about were 0, 1, and ∞. I think all the others (like 2, 3, and π

^{53}/2.781) were just special cases of '1'.posted by benito.strauss at 7:42 AM on December 25, 2011 [8 favorites]

Dolnick's "The Clockwork Universe (2011) is about the Royal Society and Issac Newton. In the book, he discussed the fact Newton's greatest mathematical insights were in his youth. He continues describing this phenomenon,

(p.229)

posted by xtian at 8:39 AM on December 25, 2011

Age is, of course, a fever chill

that every physicist must fear.

He's better dead than living still

when once he's past his thirtieth year.**'If you haven't done outstanding work in mathematics by 30, you never will,' says Ronald Graham, one of today's best-regarded mathematicians.**

The greats flare up early, like athletes, and they burn out just as quickly. Paul Dirac, a physicist who won his Nobel Prize for work he did at twenty-six, made the point with wry bleakness, in verse. (He wrote his poem while still in his twenties).The greats flare up early, like athletes, and they burn out just as quickly. Paul Dirac, a physicist who won his Nobel Prize for work he did at twenty-six, made the point with wry bleakness, in verse. (He wrote his poem while still in his twenties).

Age is, of course, a fever chill

that every physicist must fear.

He's better dead than living still

when once he's past his thirtieth year.

(p.229)

posted by xtian at 8:39 AM on December 25, 2011

xtian, according to this article the prime age is 48 nowadays.

posted by joost de vries at 9:08 AM on December 25, 2011

posted by joost de vries at 9:08 AM on December 25, 2011

I wrote in Slate about why math is no longer a young person's game.

posted by escabeche at 9:14 AM on December 25, 2011 [4 favorites]

posted by escabeche at 9:14 AM on December 25, 2011 [4 favorites]

That's reassuring.xtian, according to this article the prime age is 48 nowadays.

posted by delmoi at 10:11 AM on December 25, 2011

And I should add that the Quora answer is one of the best expositions I've seen of the

posted by escabeche at 10:15 AM on December 25, 2011

*way*that long years of experience help you do good mathematics.posted by escabeche at 10:15 AM on December 25, 2011

ennui.bz, the intuition grows on you, slowly and almost imperceptibly, until one day quite a number of years after you've finished grad school, you realize it all of a sudden while teaching a graduate course, sitting in a seminar, or talking to some colleagues at a conference about a new problem.

This is an excellent link, thanks escabeche!

posted by eviemath at 10:54 AM on December 25, 2011 [1 favorite]

This is an excellent link, thanks escabeche!

posted by eviemath at 10:54 AM on December 25, 2011 [1 favorite]

Indeed, simen, these are some of the most glorious moments that I have ever experienced, and may well be the drug that pushed me into academia. They have become much less frequent though...

"The pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings." - Alfred North Whitehead

posted by Pyrogenesis at 12:57 PM on December 25, 2011

"The pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings." - Alfred North Whitehead

posted by Pyrogenesis at 12:57 PM on December 25, 2011

xtian, according to this article the prime age is 48 nowadays.

48 is not a prime age.

[Sorry, I just couldn't leave that hanging.]

posted by benito.strauss at 8:26 AM on December 28, 2011 [5 favorites]

48 is not a prime age.

[Sorry, I just couldn't leave that hanging.]

posted by benito.strauss at 8:26 AM on December 28, 2011 [5 favorites]

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