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Mathematicians Are Chronically Lost and Confused

Soulskill posted about 8 months ago | from the dude-where's-my-cartesian-plot dept.

Math 114

An anonymous reader writes "Mathematics Ph.D. student Jeremy Kun has an interesting post about how mathematicians approach doing new work and pushing back the boundaries of human knowledge. He says it's immensely important for mathematicians to be comfortable with extended periods of ignorance when working on a new topic. 'The truth is that mathematicians are chronically lost and confused. It's our natural state of being, and I mean that in a good way. ... This is something that has been bred into me after years of studying mathematics. I know how to say, “Well, I understand nothing about anything,” and then constructively answer the question, “What’s next?” Sometimes the answer is to pinpoint one very basic question I don’t understand and try to tackle that first.' He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ... But more often than not you'll find that by the time you revisit a problem you've literally grown so much (mathematically) that it's trivial. What's much more useful is recording what the deep insights are, and storing them for recollection later.'"

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That settles it (4, Funny)

immaterial (1520413) | about 8 months ago | (#46412715)

I was the best mathematician in my university math classes. Who knew?

Re:That settles it (0)

Anonymous Coward | about 8 months ago | (#46412755)

"Lost and confused"? You're in a balloon!!

Re:That settles it (2)

ackthpt (218170) | about 8 months ago | (#46413037)

I was the best mathematician in my university math classes. Who knew?

Genius ahead of its time.

I know I was hard at work involving the subtraction of beer from a case, addition to empty bottles, dividing time between drinking and the necessary room and multiplying the number of pink elephants surrounding me.

Heady times.

Re: That settles it (1)

ShieldW0lf (601553) | about 8 months ago | (#46414929)

I got in a serious car accident and spent the second half of the year recovering in the residence I'd already paid for without going to class. Then, for a lark, I got totally hammered and wrote the Calculus exam with my mates.

Aced the exam, passed Calculus despite having not gone to class at all or done a single assignment.

I still find it hilarious, but my mom was not particularly proud of me.

Learning != linear? (1)

Clyde Machine (1851570) | about 8 months ago | (#46412743)

Sounds like learning is not necessarily a linear process. Makes me feel better about my learning experience!

Re:Learning != linear? (3, Interesting)

CapeDoryBob (204240) | about 8 months ago | (#46412841)

Math should not be taught as a linear process, but as a spiral. Visit the topics at first, so the student can understand why something is important when it is presented rigorously.

Re:Learning != linear? (0)

Anonymous Coward | about 7 months ago | (#46419051)

Maths I think is not a spiral but a case of leaps and bounds as somethings do not make sense when you first see them ,leave them and then come back after looking at something else and then it makes sense ,not as he points out in his article that you have matured but you are looking at the problem from another angle

So the answer is "42" (1)

stedlj (62084) | about 8 months ago | (#46412795)

Now to learn what the question is...

"Trivial" (1, Interesting)

oldhack (1037484) | about 8 months ago | (#46412807)

Everything, and only things, that math people do is "trivial".

Re:"Trivial" (3, Funny)

Jeremy Erwin (2054) | about 8 months ago | (#46412979)

I suppose you can reduce arithmetic and geometry (both quadrivia) to logic (trivia), but the liberal arts are seven in number for a very good numerological reason.

Re:"Trivial" (1)

K. S. Kyosuke (729550) | about 8 months ago | (#46413927)

the liberal arts are seven in number for a very good numerological reason

I thought the number of the counting shall be three? Well, at least since five is right out, you can't just get rid of arithmetic and geometry, you either have to get rid of something else as well, or keep one of those.

Oh maths (0)

Anonymous Coward | about 8 months ago | (#46414789)

Oh Maths, your just the tool physicists use. Like a hammer, a bath tub or a Large Hadron Collider.

Becoming a mathematician is like becoming someone who is fascinated in shoes, or briefcases or watches or hammers. Not the application of shoes, briefcase or watches, but the objects themselves. People need to never use them, never get them dirty, never do anything with them, leave them and their nice neat proofs alone, vacuummed packed and hidden away. Black and white, so pure, its like a wacky religion thats out of touch.

  These stupid silos. I work at a university where they abolished the physics department to keep the mathematics department. Physics was part of Science, where Maths was part of Law and Accounting (which is where it should obviously be, because you know, its not a science). Law and accounting had so many graduates, and they all had to do these maths units, you know, a whole subject on percentages, a whole subject on addition.

  So while the physics guys were going out to high schools, teaching physics and maths, and inspiring minds, Maths were teaching basic addition and subtraction to uni people. It was in their interest to keep high schools stupid, because if students could do the maths, they wouldn't need these stupid feeder units and they wouldn't exist.

  So now everyone wants to be a lawyer, business graduate (you know, to do the business) and none of them can do maths. No one can of course apply maths, not even the mathematicians. I ask them all the time, where might you use that? And they say with an abstract question like this. Give me an applied example, in any field, they scratch their head and say don't be silly I'm not a trades person, I'm a mathematician. Why are you teaching that then? Because they need to know it.

  Of course I'm using the same maths to solve lots of problems, and because I don't rote teach it, my student can apply it to stacks and stacks of problems even outside the field I'm teaching them in. Some mathematicians think the same way I do, until they become the head of the mathematics department, which they then think this is an awesome job, don't blow the case.

  Mathematicians aren't confused, well they shouldn't be, there entire world is fabricated and unrealistic.

Re:Oh maths (0)

Anonymous Coward | about 8 months ago | (#46414991)

but they can spell and use grammar correctly

Re:Oh maths (1)

khallow (566160) | about 8 months ago | (#46415223)

Becoming a mathematician is like becoming someone who is fascinated in shoes, or briefcases or watches or hammers.

An important class of people inordinately fascinated by shoes, briefcases, watches, and hammers are manufacturers.

Re:Oh maths (0)

Anonymous Coward | about 8 months ago | (#46415231)

Electrical engineering makes extensive use of many of the first and some of the second year level mathematics.
Things like; designing antennas.

Some of the mathematics I learned in second year was directly related to optimisation problems. Which are extremely useful for all things involving logistics.

Finally, you appear to have a high regard for physics, which is great. All of the physics subjects I took had Mathematics subjects as pre-requisites, or co-requisites.
The physics subjects needed the content of the mathematics subjects.

Finally, my mathematics subjects never taught me by rote. If yours did, then you went to a shit university. (Also; if your school didn't teach you very good maths skills, then you chose shit subjects, or went to a shit school - which again, whilst perhaps outside of your direct control, is still your problem).

Just because *you* hated your maths subjects. Doesn't mean they are bad subjects. It means the institutions you went to were shit.

Failing as a math teacher (5, Insightful)

Anonymous Coward | about 8 months ago | (#46412849)

All too often I've encountered math teachers who failed to properly explain advanced mathematical concepts because to them it was obvious and trivial.

Gee, thanks.

Re:Failing as a math teacher (4, Insightful)

jones_supa (887896) | about 8 months ago | (#46412941)

I have sometimes thought that the best teacher might be another student who has just a moment ago barely (but still correctly) grasped the concept.

Another student? (0)

Anonymous Coward | about 8 months ago | (#46413107)

Those are called teaching assistants. I mastered a lot of stuff when I spent my senior year being a teaching assistant. The review was great and I understood all the stuff I had just hurried over in previous years.

Re:Another student? (2)

the phantom (107624) | about 8 months ago | (#46414221)

There is nothing like teaching a topic to force you to learn it.

Re:Failing as a math teacher (2)

esldude (1157749) | about 8 months ago | (#46413467)

I think you are right. Psychology learning investigations even with toddlers found in a natural environment (non-classroom) people learn by watching or interacting with those just very slightly more advanced than them. People who know just a little something they don't. This is actually obvious. Two people like that can communicate very easily as they are at a similar point of learning. The person knowing the extra thing or two learned it recently. Making it easy to help someone else replicate the aha! experience with new concepts.

Re:Failing as a math teacher (2)

the phantom (107624) | about 8 months ago | (#46414211)

That is a good argument in favor of instructors giving time for students to work together in class (though this is often difficult to do in large lecture sections in a university setting, where contact hours are limited), and for students to form study groups outside of class (something that I strongly encourage my students to do whenever possible). I remember a time when the concepts that I am teaching were difficult to understand, but, frankly, that was 15 or 20 years ago, and I have forgotten what I had to do to make it click. I do what I can, but peer interactions are often far more productive than anything I can do.

Re:Failing as a math teacher (2)

ldobehardcore (1738858) | about 8 months ago | (#46415523)

When I was in high school, I was very good at math but couldn't be bothered to actually apply myself and take the highest level classes. I picked up everything pretty quickly, and I remember it being hellish when the teacher would say "break into small groups." Nobody cared to actually do the work. Most of the time it would be me and 1 or 2 other people in the class who actually got the concept, and everyone else would beg us to let them just copy off of me. I never consciously let anyone copy off of me. Most of the time, the other people who got the concepts would let everyone copy off them though. To this day that kind of laziness sticks in my craw, and I simply refused to let people copy. I offered to re-teach the concepts one on one, with the stipulation that they teach what they learned one on one to the others. This made it so I only had to re-teach the class once, then I could do my own homework. It was nice in that I was recognized for my abilities, and I was a decent teacher. I'll be damned if my students just copy off of me, so I did my best to make them prove they could do the math along the way. I think the biggest problem was probably that none of my math teachers actually gave a damn about math. They were all football/basketball coaches, and teaching math was just their night job, really. So the jocks and the cheerleaders always got passing grades, (at least C-) and everyone was left to flounder, since the coaches were too busy chatting with their favorite quarterbacks, pointguards, and cheerleader captains.

Re:Failing as a math teacher (1)

jandersen (462034) | about 7 months ago | (#46418837)

I have sometimes thought that the best teacher might be another student who has just a moment ago barely (but still correctly) grasped the concept.

I'm not convinced. What you really need, in order to teach a difficult subject, is understanding of why it is difficult to understand for an initiate, and I suppose somebody who has just learned may be much closer to that understanding, but on the other hand, you also need a very thorough understanding of what the subject is all about, and you probably only get that with experience. I think what would really make a great teacher is somebody who has long, practical experience of what the subject is used for and who really enjoys answering questions.

Re:Failing as a math teacher (1)

GerbilKor (2926575) | about 7 months ago | (#46419517)

I think the same. Accordingly, math teachers may not teach as well because they are the select few who naturally understand and enjoy the subject. Most of their students have a very different perspective.

Re:Failing as a math teacher (2)

93 Escort Wagon (326346) | about 8 months ago | (#46412983)

An alternative explanation is those math teachers didn't actually understand the concept, and therefore were unable to properly explain it.

Re:Failing as a math teacher (3, Insightful)

techno-vampire (666512) | about 8 months ago | (#46413311)

What's worst is a teacher who defines a new term in a way that only makes sense if you already understand the concepts behind it. As an example, Rudy Rucker once defined a cardinal number (in a book) as, "A number is a cardinal number if it doesn't share its cardinality with any other number." Now, if you know what a cardinal number is, and what "cardinality" means, that's true. If you don't, as most of the readers of that book wouldn't, it's useless.

Re:Failing as a math teacher (3, Funny)

ColdWetDog (752185) | about 8 months ago | (#46413603)

Wow. He must write a lot of computer documentation.

Re:Failing as a math teacher (0)

Anonymous Coward | about 8 months ago | (#46413763)

Wow. He must write a lot of computer documentation.

Judging from the code I've seen, no one writes a lot of computer documentation

Re:Failing as a math teacher (1)

Chris Mattern (191822) | about 8 months ago | (#46414733)

One assumes that he then defined cardinality as "the quality that uniquely defines a cardinal number."

Re:Failing as a math teacher (1)

techno-vampire (666512) | about 8 months ago | (#46414841)

It's been decades since I read that book, but unless my memory's worse that I think, he didn't define it at all.

Re:Failing as a math teacher (2)

Vyse of Arcadia (1220278) | about 8 months ago | (#46414883)

This is how definitions work. Definitions would get absurdly long and difficult to read if we defined everything in terms of first principles. I could concisely describe a solvable group as a group having a subnormal serious whose factor groups are all abelian. If I have to go back and explain group and subnormal series and factor groups and abelian it ballloons to a page in length, and those are all concepts that are useful elsewhere is well.

Presumably that author wasn't just defining things cyclically and had defined cardinality elsewhere. You'd just have to go back and look it up.

Re:Failing as a math teacher (1)

techno-vampire (666512) | about 8 months ago | (#46415019)

Presumably that author wasn't just defining things cyclically and had defined cardinality elsewhere.

One would think so, but no. When I came across that book I was trying to learn about such things and I'd think that would have remembered it if he had.

Re:Failing as a math teacher (1)

royzeng (3157739) | about 7 months ago | (#46416405)

I remember when I was studying in Ireland, one of my maths lecturer created a question that he can not answer.....and finally he asked help from students... http://www.szwelder.com/ [szwelder.com]

Re:Failing as a math teacher (1)

Galilee (90424) | about 7 months ago | (#46418733)

That comment brought back a bad memory from a calc 2 course I took in college. The professor was ancient. On the first day of class he refered to a silent film actress who never was able to make it in the talkies.

One day he was going over a problem on the chalkboard. A student asked how the professor got from one line to the next. The professor threw up both hands and exclaimed that he'd never be able to cover the course material if he had to go over every trivial detail. He then angrilly filled up the entire chalkboard at the front of the classroom and half of the board on the side wall with the "trivial detail". You could have heard a pin drop, the entire class was stunned into silence. The professor clearly knew math, but he sure didn't know how to teach it to freshman. He was much too advanced.

An old mathematicians' joke (5, Funny)

Anonymous Coward | about 8 months ago | (#46412889)

There are two types of theorems: trivial and unproven.

Re:An old mathematicians' joke (2, Funny)

Anonymous Coward | about 8 months ago | (#46413227)

There are two types of theorems: trivial and unproven.

Actually, there are 3. Proof left for the reader.

Re:An old mathematicians' joke (0)

Anonymous Coward | about 7 months ago | (#46416193)

There are two types of theorems: trivial and unproven.

Actually, there are 3. Proof left for the reader.

"I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." -- Pierre de Fermat

Re:An old mathematicians' joke (1)

Chrisq (894406) | about 7 months ago | (#46417211)

There are two types of theorems: trivial and unproven.

Actually, there are 3. Proof left for the reader.

Does that include proofs that are too big to fit in the margin?

Nonlinear differential equations (1)

TempleOS (3394245) | about 8 months ago | (#46412945)

I got an A in nonlinear differential equations. That's as high as math gets.

Re:Nonlinear differential equations (1)

Teun (17872) | about 8 months ago | (#46413637)

And yet, according to your other postings you're quite a nutcase...

bred (0)

Anonymous Coward | about 8 months ago | (#46412973)

"This is something that has been bred into me after years of studying mathematics."

He must be a mathematician if that's how he thinks breeding works.

Sound Like Software Development (1)

avandesande (143899) | about 8 months ago | (#46412989)

EOM

Re:Sound Like Software Development (1)

Tailhook (98486) | about 8 months ago | (#46413445)

The two are basically the same [slashdot.org] , or so the physiologists tell us.

Bizarre advice (4, Insightful)

AthanasiusKircher (1333179) | about 8 months ago | (#46412995)

He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ...

While I agree that one shouldn't waste time questioning every statement you encounter, there's a very ancient and useful tradition in math pedagogy that emphasizes these sorts of things. See, for example, gradually building up geometrical theorems from a few axioms, a la Euclid.

Often, the process of working out complex proofs for yourself is crucial to understanding why things work, not to mention developing and practicing logic skills that are essential in math and elsewhere.

I'm not saying one should waste time trying every exercise or redoing every proof, but some of my greatest insights into the inner workings on math have come from exercises that took me a couple hours to work out or textbook passages I went over a number of times and really dug into how the details worked. If I skipped everything I couldn't do in 5-10 minutes, I doubt I'd ever have developed the more advanced skills and intuitions that would be necessary to see why some results are "trivial."

Re:Bizarre advice (5, Interesting)

vdorie (1106873) | about 8 months ago | (#46414053)

I came here to post a similar sentiment. I think it is a terrible idea to just blow ahead every time an assertion is too confusing. Getting the big picture and developing mathematical intuition is great but it doesn't mean that you'll actually be able to do math. For that, practice.

I actually advise the opposite, to bang your head against a problem over and over until it breaks (the problem, that is). I don't think that we as a society or a species or whatever deal with confusion very well, and tend to take it as some sort of personal deficiency. We've also done a great deal of dumbing math down, so that when someone tries to make the jump from, say, AP calculus to real analysis, minds get blown and souls shattered. It's probably not that mathematicians enjoy crushing students, but rather that higher levels of math are just plain confusing for most people. They're based on abstractions that are pretty far removed from the human experience. None of this is to say that people who are good at math are better somehow, but it usually means that they put in a lot of time. I suspect that a lot of people who are math-phobic would get over it if you locked them in a room with nothing but math books to keep them busy.

One that is clear, however, is that most mathematicians have no fscking clue what the word "obvious" means. There are some brilliant, dead authors that I would love to punch in the face.

Re:Bizarre advice (1)

lgw (121541) | about 8 months ago | (#46414493)

I actually advise the opposite, to bang your head against a problem over and over until it breaks (the problem, that is).

But most people actually give up first, and quit studying. People "turning themselves off" to the study of math is a very common problem. Most people can only take so much frustration.

Re:Bizarre advice (0)

Anonymous Coward | about 8 months ago | (#46414501)

None of this is to say that people who are good at math are better somehow

We are, though.

Re:Bizarre advice (2)

Zero__Kelvin (151819) | about 8 months ago | (#46415255)

Perhaps, but those of us who can figure out how to create an account and log in are even better!

Re:Bizarre advice (1)

smallfries (601545) | about 7 months ago | (#46417417)

One that is clear, however, is that most mathematicians have no fscking clue what the word "obvious" means. There are some brilliant, dead authors that I would love to punch in the face.

.

I think that they know exactly what it means, but that you are confusing it with the non-technical meaning. In maths it generally means "I have managed to work this out, and I suspect that you will be able to (eventually) without my help. If you cannot, that I presume that you are an idiot and that you do not deserve my help". Contrast the meaning with the technical use of non-obvious: "Oh fuck, we're boned".

In general you should treat obvious things with care, and only skip past the trivial.

Re:Bizarre advice (0)

Anonymous Coward | about 8 months ago | (#46414143)

Depends what you are trying to teach. A lot of early math courses are trying to teach the mechanics of established principles so that you can solve specific, common problems and situations. There is a lot of basic stat you can do if you don't know why the standard deviation formula is the way it is, or a lot of practical calculus you can do without knowing how to do a delta-epsilon proof. There is potentially a lot more you can do if you know how to derive things, especially when forgetting bits and pieces but sometimes that either comes later or is considered a different topic. In other words, a lot of low level courses are intended as applied math and not an intro to pure math.

Re:Bizarre advice (0)

AthanasiusKircher (1333179) | about 8 months ago | (#46414737)

A lot of early math courses are trying to teach the mechanics of established principles so that you can solve specific, common problems and situations.

If the only point is to teach someone how to apply some precise algorithm to a specific type of problem, why bother teaching math at all? Just say --if problem type X, type numbers into computer and run program A, if type Y, run program B, etc. There's no point teaching someone how to act like a glorified calculator anymore... this thinking is a couple decades out of date.

Re:Bizarre advice (2)

AthanasiusKircher (1333179) | about 8 months ago | (#46414851)

Sorry -- accidentally hit submit before finishing my post.

A lot of early math courses are trying to teach the mechanics of established principles so that you can solve specific, common problems and situations.

If the only point is to teach someone how to apply some precise algorithm to a specific type of problem, why bother teaching math at all? Just say --if problem type X, type numbers into computer and run program A, if type Y, run program B, etc. There's no point teaching someone how to act like a glorified calculator anymore... this thinking is a couple decades out of date.

There is a lot of basic stat you can do if you don't know why the standard deviation formula is the way it is,

For frack's sake, no! If you don't know how standard deviation actually works, you are doing more harm than good by using it. There's more nonsense propagated by people using statistical measures without knowing what they are doing than just about anything else. I'd go so far to say it's the biggest problem in science today, aside from too much corporate influence in some areas. The derivation of statistical formulas often tells a lot about the assumptions each makes... which are essential to understand if the conclusion drawn are to be meaningful.

or a lot of practical calculus you can do without knowing how to do a delta-epsilon proof.

Those are only one way to derive basic calculus. And besides, there is a lot of room between deriving all of calculus from the fundamental axioms of the real number system rigorously, and not knowing anything other than some algorithmic symbolic manipulation without knowing what goes on "under the hood." I'm not saying everyone needs to do the former, but we should not assume the latter is fine either. I've seen a lot of crap done by people using basic math (or even educated folks using calculus or diff eqs) without realizing the assumptions of what they are doing.

Re:Bizarre advice (0)

Anonymous Coward | about 8 months ago | (#46415635)

. And besides, there is a lot of room between deriving all of calculus from the fundamental axioms of the real number system rigorously, and not knowing anything other than some algorithmic symbolic manipulation without knowing what goes on "under the hood."

I don't see how that goes against what I posted in the previous post, and you seem to either support what I was saying or trying to insist I was making a false dichotomy I was not. Nonetheless, there is a lot that can be done to teach an intuitive understanding of the mechanics of math without getting into expecting people to be good at proofs. The type of learning and thinking needed for proofs is kind of a special way of thinking that, while helpful for expanding math understanding, are not needed in a lot of basic cases and can become an anchor holding some people back.

If you don't know how standard deviation actually works, you are doing more harm than good by using it. T

There is a huge difference between knowing how the standard deviation is a description of of the width of a normal distribution, which has plenty of caveats, and being able to understand how to derive maximum likelihood estimators and deal with bias correction.

If the only point is to teach someone how to apply some precise algorithm to a specific type of problem, why bother teaching math at all? Just say --if problem type X, type numbers into computer and run program A, if type Y, run program B, etc. There's no point teaching someone how to act like a glorified calculator anymore... this thinking is a couple decades out of date.,

The same reason you teach any subject today: because some one needs to make the computers work, sometimes you don't have a computer, for the sake of learning, because it leads into understanding something more complex, because it is more general implications than just getting a number out of a computer, etc. At all levels of courses, you have to draw a line at what to teach, and at the low level a lot gets left out.

Re:Bizarre advice (0)

Anonymous Coward | about 8 months ago | (#46414157)

I totally agree with what you are saying, but what I see here is a bigger problem which is a side-effect of global capitalism. The global economy is increasingly more competitive, and this has a trickle down effect on the cost effectiveness of every product and service. It seems to me that we are approaching a brave new time when only the skills and knowledge which are economically valuable will be taught. Educational insitutions across the globe must collectively resist this influence.

Re:Bizarre advice (1)

khallow (566160) | about 8 months ago | (#46415185)

It seems to me that we are approaching a brave new time when only the skills and knowledge which are economically valuable will be taught.

You can drop the qualifier, "economically". The only sort of value is economic value because if something has value, then that means you're willing to sacrifice something of significance for it. And that's all economics really is about, making choices with tradeoffs in order to get the stuff you value more.

This paragraph gives me the impression that you advocate educational institutions should resist giving what students and society wants out of education and instead deliver what some intellectual elite thinks is more valuable.

Re:Bizarre advice (1)

mr3038 (121693) | about 7 months ago | (#46417699)

It seems to me that we are approaching a brave new time when only the skills and knowledge which are economically valuable will be taught.

This paragraph gives me the impression that you advocate educational institutions should resist giving what students and society wants out of education and instead deliver what some intellectual elite thinks is more valuable.

I read that as "we should only teach skills and knowledge that provides more monetary value for the society in the long run, compared to the resources spent on education". As a whole, I agree. However, we should improve on detecting childs clearly above average and using extra resources on them. I believe that everybody should have basic education but there's no reason to spend huge amount of education resources on everybody.

Re:Bizarre advice (0)

Anonymous Coward | about 7 months ago | (#46418701)

The only sort of value is economic value because if something has value, then that means you're willing to sacrifice something of significance for it. And that's all economics really is about, making choices with tradeoffs in order to get the stuff you value more.

There are other values. Moral values for example. Morals have no economic value. In fact, it often has negative economic value, as the economy is harmed by people trying to pursue morals.

Bread and circuses is another example of people valuing non-economic things, such as security. They sacrifice their freedom for security, and they get neither.

Re:Bizarre advice (1)

Anonymous Coward | about 8 months ago | (#46414843)

The article says explicitly that there are times when digging your heels in is necessary (for the more important stuff). But a lot of math textbooks will leave oodles of exercises to the reader. If each one takes three hours you'll never get anywhere. I think his point is not to take the "do everything" attitude to the extreme, which he observes people doing.

Re:Bizarre advice (1)

AthanasiusKircher (1333179) | about 8 months ago | (#46414945)

The article says explicitly that there are times when digging your heels in is necessary (for the more important stuff).

Yes, you're right. I was mostly responding to the quotation in the summary. But I still have a couple problems with this: (1) how exactly is a beginner to know what is "important"? And (2) the most insightful things that have happened to me were doing random exercises that interested me, rather than necessarily something "important."

I'd say his attitude is right that you needn't do every exercise or proof (and I actually already said this in my first post), but if you are interested and motivated to solve a problem, you often gain something by sticking with it, regardless of whether it takes more than 10 minutes or is "important."

Any Good Scientist Knows This (2, Insightful)

Anonymous Coward | about 8 months ago | (#46412997)

We must all learn to exist in that exquisitely uncomfortable place where everything we know is always up for reassessment. Otherwise, we miss change, and change is the only constant.

Re:Any Good Scientist Knows This (1)

Alioth (221270) | about 7 months ago | (#46417741)

Change is only constant if the degree of the leading term is 1 :-)

"on a roll" (1)

lkcl (517947) | about 8 months ago | (#46413015)

i had to be woken up at around 9:20am for a 3 hour A-Level Maths exam that had started at 9am and was to end at 12. starting at around 9:25 on the first question, after around 25 minutes i gave up and went onto the 2nd question. this one i did in around 15 minutes. from there i accelerated, completed *every* question, returned to the first and completed it in a few minutes. i then sat back for a while, then got some coloured pens and coloured in one of the graphs. i might even have been bold enough to have left 10 or 15 minute early.

when the results were in i learned i'd got an A. on an exam that was supposed to be 3 hours and i'd completed every question in a little over 2. that was 1987 and i've never forgotten what happened. the point is: i know that once you get started, and get into the mindset, anything is possible: questions you couldn't answer suddenly become easy.

Re:"on a roll" (1)

Anonymous Coward | about 8 months ago | (#46413267)

Compare and contrast your amazing story with your granddad's from WWII and think about which one you'd rather hear.

And people say war is a bad thing!

Re:"on a roll" (0)

Anonymous Coward | about 8 months ago | (#46413621)

Douche-bag!

Re:"on a roll" (0)

Anonymous Coward | about 8 months ago | (#46413947)

There's two things I take away from your story:

1) NOTHING is as important as a good night's sleep before an exam.

2) Nobody apparently told you about the rules of capitalization. Perhaps they simply shrugged an said "Ah well, he's good at Math". Or maybe you were a victim of the early specialization of the British system.

Re:"on a roll" (0)

Anonymous Coward | about 8 months ago | (#46414807)

I had a similar experience. 3.5 hour exam, completed in 20 minutes---because I drank a soda right before the exam and *really* needed to go (and going for a restroom break during an exam isn't my thing).

Can't say I got an A, but I passed (and it was one of those tough pass/fail exams).

Re:"on a roll" (0)

Anonymous Coward | about 8 months ago | (#46415455)

I had a similar experience on the AIME (15 questions in 3 hours). After two hours, I had only solved 3 questions. I panicked for about 5 minutes, then shit just started clicking: I completed 11 of the 15 by the end, and got 10 correct, which allowed me to advance to the USAMO. I still think about it from time to time. I'm posting anonymously so I won't be accused of bragging by the mathematically disinclined.

Mathematician or politician? (1)

Smoky D. Bear (734215) | about 8 months ago | (#46413045)

I think the "lost and confused" applies to both...

Post 1! Mathemetician FTW! (0)

Anonymous Coward | about 8 months ago | (#46413081)

At least it would have been if I hadn't been lost and confused.

you heard the one about ... (1)

swell (195815) | about 8 months ago | (#46413101)

the constipated mathematician ?

He worked it out with a pencil.

Re:you heard the one about ... (2)

ClickOnThis (137803) | about 8 months ago | (#46414269)

the constipated mathematician ?

He worked it out with a pencil.

Old School.

Nowadays, he'd work it out ... [*dons sunglasses*] ... digitally.

Re:you heard the one about ... (0)

Anonymous Coward | about 8 months ago | (#46414511)

Maybe but I'm not sure anyone other than the gotse dude could use an iphone to clear his bowels

Re:you heard the one about ... (1)

ClickOnThis (137803) | about 8 months ago | (#46414583)

Maybe but I'm not sure anyone other than the gotse dude could use an iphone to clear his bowels

I was not referring to electronics. [reference.com]

Re:you heard the one about ... (0)

Anonymous Coward | about 8 months ago | (#46414621)

that's a serious build-up of calculus!

Math Professor here... (0)

Anonymous Coward | about 8 months ago | (#46413173)

I tell my students this regularly. Mathematics is confusing. The stuff you know already is "simple" but at every level, the stuff you are learning is "hard". Here's a wonderful quote I read in a book by Norwegian mathematics Berndt Oksendal, who apparently found it posted around the Tromso University math department:

We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things.

Comfort with not knowing (1)

dlenmn (145080) | about 8 months ago | (#46413223)

I'm a physics graduate student, and while I'm not quite in the same boat as the mathematicians, I'm familiar with the problem. You spend a lot of time trying and failing to figure out what's going on. You have to be comfortable with not knowing things you want to know. I think that's a really useful ability because you don't demand easily digestible answers for everything. Such answers rarely exist, although many people seek them from short articles and soundbites.

It think it also has larger philosophical implications.. Forgive me for bringing up religion, but most (albeit not an overwhelming majority of) physicists do not believe in any higher power. If you're comfortable with not knowing things, then the answers provided by belief in a higher power doesn't provide additional comfort. You have no need for that hypothesis. (I'm not saying that people like religion simply because it provides easily digestible answers -- although religious groups *cough* young earth creationists *cough* certainly go for that. Many religious Jews spend their lives studying and debating the meaning of the bible; it's anything but easily digestible.)

Re:Comfort with not knowing (1)

RightwingNutjob (1302813) | about 8 months ago | (#46414387)

See, there's a difference between knowing what you don't know and living in a sea of ambiguity the way the OP seems to imply. In mathematics especially, there is a very tall and elaborate edifice of deductions and axioms from which all exploration takes place.

For example, one of the more mind-bending exercises in undergrad abstract algebra is proving Peano's axioms for integers. On the one hand you could say "well, I thought I knew basic arithmetic, but now I have to question even that: I'm lost!" But on the other hand, when you go through that exercise, you have very powerful tools in your toolbox: deduction, group theory, ring theory, etc, which you spend time building up and exercising exhaustively before you attack the natural numbers. So you're not really "lost" as in at sea without a clue, but you're just approaching something from a new direction with very well-defined assumptions and rigid reasoning.

And if I can hope to contribute to the religious debate without sparking too big of a flame war: maybe this same conflation between being completely lost and working in an unfamility coordinate system may be at play when Skeptics and scientists describe why they're athiests. Empirical evidence and deductive reasoning can peel away some scripture as obviously false, but when you're denying a higher power by an appeal to logic/reason/etc, you're still assuming the presence of this abstract thing called mathematical/empirical truth, and perhaps even Order with a capital 'O'.

I'm sure I'm not at all speaking for any sort of majority view of believers or skeptics or deists, but why is it not valid to call that God and be comforted by its existence, as opposed to say chaos?

Math is great. (0)

Anonymous Coward | about 8 months ago | (#46413269)

Math is great.
Proof is left as an exercise to the reader.

all thinkers are confused (1)

swell (195815) | about 8 months ago | (#46413277)

Never trust the one who has the answers. The politician. The Preacher. The grammar school teacher. Seek those who have questions.

I'm a writer and inventor, I hope to come to understand things with my writing. I may draw a concept in an attempt to understand it better. I have written programs to unravel mysteries (you've seen the 'game of life'?) I try to reserve judgement when presented with an obvious 'truth' on Slashdot (as most of you do !).

Here's my email sig, feel free to share it:
"Your life is not going to be easy, and it should not be easy. It ought to be hard. It ought to be radical, it ought to be restless, it ought to lead you to places you'd rather not go." - Henri Nouwen

RIP Philosophy of Mathematics (2)

Myu (823582) | about 8 months ago | (#46413479)

If this is how things stand, then the Philosophy of Mathematics to date is a catastrophic failure. When there is no better methodology than "fumble around in the dark a bit until suddenly you're convinced" then the project of attempting to guide students in understanding maths has done no work at all.

Is this the fault of the philosophers or the mathematicians? I'm inclined to think that the philosophers have at least failed in their advocacy, if not in their actual subject.

Re:RIP Philosophy of Mathematics (0)

Anonymous Coward | about 8 months ago | (#46414761)

There are all sorts of standard methodologies to solving new problems in mathematics Things that work too well with standard methods though become less interesting and at best a tedious process of applying something you know will likely work without learning something new about math. In that case, such things get relegated to applied math, where they can be used as a tool to look at other fields. But within math, if you come up with a standard methodology, then anything that applies to becomes less interesting and essentially solved already, with many mathematicians trying to work on questions that are unsolved. There are still common things to try depending on exactly what problem you are working on, but chances are if you are working on a problem long enough, it is because you've exhausted those things and are treading into new territory.

Re:RIP Philosophy of Mathematics (0)

Anonymous Coward | about 8 months ago | (#46415241)

You really don;t know anything about which you speak (IAAM).

Lots of us are confused (1)

140Mandak262Jamuna (970587) | about 8 months ago | (#46413549)

I don't see what is so special about mathematicians skimming over stuff and not sweating the small stuff. Many project planning meetings we treat lots of stuff as black boxes and proceed with some simple assurance that it would do what the team says it will do. The post processing guy would have a very nebulous idea of the geometry core team's claims. Nobody understands what the mesh maker says anyway. Then there are the mathematicians from the solver group. Upside down triangles, dots crosses some time three integral signs lined up like sails of some old ship .. Eventually we understand enough of it to make it work most of the time. Even after the project is done and the feature has been shipped and the user story has been marked complete and independently verified by the user proxy, nobody understands how the mesher meshed it nor how the solver solved it.

Re:Lots of us are confused (0)

Anonymous Coward | about 7 months ago | (#46417491)

Then there are the mathematicians from the solver group. Upside down triangles

Are you sure they aren't downside up? A triangle doesn't have an upside.

This is how mathematics really works... (0)

Anonymous Coward | about 8 months ago | (#46413611)

http://en.wikipedia.org/wiki/Proof_by_intimidation

Confused is the goal (0)

Anonymous Coward | about 8 months ago | (#46413677)

It's a good thing to be confused, reach understanding, then rinse and repeat... If ignorance isn't part of your day, then your not pushing yourself hard enough...

Sleep on it (0)

Anonymous Coward | about 8 months ago | (#46413735)

I find the advise to "sleep on it" a great way to solve problem. It was actually start from a completely different angle to attack the same problem. I also developed the habbit of going to bed early before exams, so that my brain wasn't tired from cramming the night before, and had my wits when I need them most. So, when in doubt, and getting in to a bind, sleep.

Similarly with Engineering and Programming. (2)

Michael McClary (1158729) | about 8 months ago | (#46413823)

Some similar effects occur with engineering and programming. For instance:

An engineer is ALWAYS working on something that's broken. That's because, when he gets it fixed, he moves on to the next thing that's broken. (It's like the thing you're searching for always being in the last place you look. It's not Murphy's law, It's becaue, when you find it, you stop looking.)

A good programmer doesn't come to a problem with all he needs to solve it. Instead he comes to it with a big toolbox, SOME domain knowledge, and the skills needed to learn the rest during the project. This will be mostly stuff related to the project, but may include more programming tools as well.

Designing/architecting a program or system is like handling a black bag with the solution inside, in the form of blocks connected by strings. You squeeze it around until you get it into two lumps with very little string running through the thin neck. Then you it into two bags and document all the strings that went through the cut. Repeat unti the bags are small enough to understand easilyj and keep the entire explanation in your head. (In the case of a program that means the code itself fits on a page, with over half of the page being comments.) Then you can open the little bags and grok each one - which by now will be either trivial or maybe embody a single deep concept or "neat hack". (But avoid "neat hacks" if they're not obvious or if something straightforward does the job just fine.)

Not following you... (1)

Anonymous Coward | about 8 months ago | (#46414117)

A good programmer might know the exact solution immediately upon seeing the problem. It could have been something they've solved many times before. There are times when I'm quite bored implementing solutions, because they are the same (general) solutions I've implemented many times before.

Programming is really *nothing* like math, in my opinion. Programming is nearly always *pragmatic* in nature. Many mathematicians study things with no obvious practical application. Programming is answering the question: "how?". Math is answering the question "what is?".

Re:Not following you... (0)

Anonymous Coward | about 8 months ago | (#46414913)

I would call that programmer quite a bad programmer. A good programmer would have implemented the general solution once, and reused it all those other times.

There is no need to over-generalize your code (Greenspun's tenth rule), but the moment you can say 'implemented many times before' is way past the moment your creations should have morphed into an automated replacement for yourself.

(PS: A 'great' programmer does this without falling prey to the Inner-platform effect.)

Re:Similarly with Engineering and Programming. (1)

Agent0013 (828350) | about 7 months ago | (#46419187)

(It's like the thing you're searching for always being in the last place you look. It's not Murphy's law, It's because, when you find it, you stop looking.)

That is a nice trite little phrase, but when one thinks about if for a minute you can see situations where it is not always true. Sometimes the thing was in the first place you looked for it, but you failed to notice it or overlooked it for some reason. I find this happens to me often when hunting a geocache. I have developed good instincts to where they may be hidden and look in the most likely places. When the cases come up where I still haven't found it and have to keep looking over the object or area at ground zero I will start to look in other more creative places. Then sometimes it turns out it was where I looked in the first place but it was disguised or camouflaged enough to not be noticeable. So it wasn't the last place I looked, but the first or second or whatever.

Bait headline (1)

jasonla (211640) | about 8 months ago | (#46415063)

Cute headline, anon, but it's kinda misleading and reeks of BuzzFeed's desperate "read me!" attempts.

Andrew Wiles on exploring the dark (1)

jnana (519059) | about 8 months ago | (#46415677)

Andrew Wiles [wikipedia.org] made the following comment [pbs.org] that has always stuck with me:

Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them.

Re:Andrew Wiles on exploring the dark (1)

jnana (519059) | about 8 months ago | (#46415695)

Oops, that's what I get for just reading the comments before commenting and not looking at the article until after posting! The exact quote is prominently mentioned in the article.

ADHD (1)

denisbergeron (197036) | about 8 months ago | (#46415775)

[satire] What, math researcher have ADHD, what a surprise, maybe we will some Asperger among them, who know. [/satire]

Frontiers aren't easy (0)

Anonymous Coward | about 7 months ago | (#46416425)

It's difficult sometimes trying to throw yourself into a direction with some Math to guide you I suppose. I wouldn't know but I suspect software developers have similar problems too, especially for cutting edge software. I wouldn't know about that yet either. Some day.. lol

Sounds Like Any Advanced / Graduate Research (1)

WhoBeDaPlaya (984958) | about 7 months ago | (#46416689)

Felt much the same throughout my sojourn as a doctoral candidate in Electrical Engineering.

"Lost and confused" is about mathematics (0)

Anonymous Coward | about 7 months ago | (#46416907)

It's the standard state of being for a Ph.D student.

Wall touchers (0)

Anonymous Coward | about 7 months ago | (#46418579)

I once worked at a large computer manufacturer and one day a tech told me that a new guy was a "wall toucher". I asked him what he meant and he said the guy was a mathematician and all mathematicians run their fingers against the wall as they walk down the halls. This place had many mathematicians on staff and I noticed it was true. You could tell if a guy was a mathematician because he would touch the wall as he walked down the hall. The physicists, mech engineers, comp sci engineers, electronics techs didn't do it but every mathematician did.

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