How do these punks do it?

For the discussion of math. Duh.

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samspotting
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How do these punks do it?

Postby samspotting » Tue Aug 05, 2008 2:39 pm UTC

Theres a section of students at my university called advanced math, where they take advanced linear algebra and advanced calculus which is like the regular stuff on cocaine (rigourous and proof based). They own up contests like the putnam which most of them less than 3 months into university. How do they do it? I am considering switching from actuarial science (everyone tells me its the most boring thing in the world, screw the high pay) to pure math since it seems so interesting. I am just wondering what they did apart from high school to obtain such a high intuition of mathematics. I had no idea people with this good at math from high school existed. Anyone have recommened books to improve problem solving and intuition? Should I be looking at a specific topic such as number theory or logic? I have a coop term in the fall so I have a lot of free time to learn.

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Re: How do these punks do it?

Postby gorcee » Tue Aug 05, 2008 3:00 pm UTC

samspotting wrote:Theres a section of students at my university called advanced math, where they take advanced linear algebra and advanced calculus which is like the regular stuff on cocaine (rigourous and proof based). They own up contests like the putnam which most of them less than 3 months into university. How do they do it? I am considering switching from actuarial science (everyone tells me its the most boring thing in the world, screw the high pay) to pure math since it seems so interesting. I am just wondering what they did apart from high school to obtain such a high intuition of mathematics. I had no idea people with this good at math from high school existed. Anyone have recommened books to improve problem solving and intuition? Should I be looking at a specific topic such as number theory or logic? I have a coop term in the fall so I have a lot of free time to learn.


Simple. There's a character difference. It's very evident in your post.

"They own up contests like the putnam which most of them less than 3 months into university"

"I am considering switching from actuarial science (everyone tells me its the most boring thing in the world, screw the high pay)"

They are there because they want to learn the math. They are there for the stringent pursuit of knowledge. You are there to gather the qualifications for a job.

Neither of those things are negative. But the motivations are present in the student long before the student is present in the university. Good news is, it's not too late to change!

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Re: How do these punks do it?

Postby samspotting » Tue Aug 05, 2008 3:50 pm UTC

Aye, I wanna be on a knowledge quest too!

So my question is whats the best way to start? I have a feeling its number theory but im not sure. Theres so many methods and so much depth of math, and theres quite a few people here like me so I thought this was the best place to start.

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Re: How do these punks do it?

Postby antonfire » Tue Aug 05, 2008 5:10 pm UTC

A lot of people did a lot of preparation for mathematics competitions, which does a good job of hammering certain key concepts into one's head. This is probably connected to their apparent (and perhaps actual) mathematical intuition. A lot of people went to math camps (PROMYS, Ross, Mathcamp, and others), which definitely does a lot of good. These options aren't really open to you now that you're in university, but there might be similar things.

Number theory is a good way to get introduced to proof-based mathematics, because the way it's usually taught the proofs aren't too hard, aren't too trivial, and aren't too tedious. Most universities have a course that's designed primarily to teach people how to prove things, and a lot of the time it's a number theory course. You should look into that.

Also, recreational mathematics (as the name implies) can be a lot of fun. Buy or borrow some books by Martin Gardner.
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Re: How do these punks do it?

Postby gorcee » Tue Aug 05, 2008 6:37 pm UTC

samspotting wrote:Aye, I wanna be on a knowledge quest too!

So my question is whats the best way to start? I have a feeling its number theory but im not sure. Theres so many methods and so much depth of math, and theres quite a few people here like me so I thought this was the best place to start.


Also, don't forget you can start applied and go pure!

I'm an applied mathematician, and so I get a lot of flak from some of the purists, but the truth is Pure math is necessary in applications, even moreso nowadays. The simple stuff is old and busted. The new hotness requires advanced mathematical concepts.

So find something you think is cool, but doesn't have to be related to mathematics. Then, uncover the mathematics behind it. Hell, I even wrote a paper describing the attack table model in World of Warcraft as a nonstationary discrete Markov chain. Math can be fun, and other things can be fun, and the math in other fun things can be fun.

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Re: How do these punks do it?

Postby Yakk » Tue Aug 05, 2008 7:07 pm UTC

When you where 12, did you spend time at a hockey game examining the successive differences between various equations when evaluated on the integers?

Ie:
x^1
1 2 3 4 5 6 7..
1 1 1 1 1 1 1..
0 0 0 0 0 0 0..

x^2
1 4 9 16 25 36...
3 5 7 9 11...
2 2 2 2 2...
0 0 0 0 0...

x^3
1 8 27 64 125 216...
7 19 37 61 91...
12 18 24 30...
6 6 6...
0 0...
instead of watching the hockey game?

That kind of behavior, together with pushing oneself to take more mathematics (math camps, olympiads, enrichment material, solving exams multiple ways, reading math for recreation, etc etc), is how someone got good at math before arriving at university. Some of them might even have interacted with older people who taught them, like older siblings, parents, etc.

Can you catch up? Certainly! Presuming you have the talent, the willingness to spend as much or more time thinking about mathematics, and the inclination to follow through.

The fact is, courses are not an ideal way to master something. They are a way to get exposed to it. If you have spent your entire life only doing the course work assigned to you by your teachers in a given subject, then you are not learning how to master a subject -- you are learning the bare minimal requirements, at best.

There are people who really enjoy working on the subject. They spend large amounts of time thinking about it. And this makes them better at it. Much like a football player gets better by constant practice.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: How do these punks do it?

Postby gorcee » Tue Aug 05, 2008 7:14 pm UTC

Yakk wrote:The fact is, courses are not an ideal way to master something. They are a way to get exposed to it. If you have spent your entire life only doing the course work assigned to you by your teachers in a given subject, then you are not learning how to master a subject -- you are learning the bare minimal requirements, at best.


Aye. You don't go to engineering school because you will have to compute beam bending every day. You go to engineering school to give you the tools to know how to think, so that when it comes to computing how a beam bends, you are armed with not only the knowledge, but also the process.

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Re: How do these punks do it?

Postby Ended » Tue Aug 05, 2008 9:05 pm UTC

gorcee wrote:The new hotness requires advanced mathematical concepts.

You speak the Truth. (sigg'd).
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Re: How do these punks do it?

Postby 3.14159265... » Tue Aug 05, 2008 11:02 pm UTC

I am angered at most of these posts.

I didn't see what "real" math was till about the end of first year. I loved it, I learned it.

The reason those kids are good at it is because when they showed interest there was someone who taught them math. For some of us that didn't happen (parents wanted a doctor).

I refuse to believe there is such thing as "talent". If you love it, and work at it anything is possible.
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Re: How do these punks do it?

Postby Yakk » Wed Aug 06, 2008 4:30 am UTC

I know people who love abstract mathmatics, and worked really hard at it.

But, they sucked at it. Hard.

Meanwhile, they where good at computer science. With very little effort, they could get 90%+ in their CS courses, while spending tonnes of time working on their Math courses manage to pull off 60%s.

They graduated with a double degree in Pure Math and Computer Science because the double-major rules allowed someone to average their CS and Pure Math courses for their "Major Average". That is it.

If you claim there is no such thing as talent, then you'd have to find where that person didn't work really damn hard, and didn't really love the subject. He just found it extremely hard.

Others, with far less effort, could get higher marks in that subject.

Talent exists (it doesn't have to be "innate", it just has to be really hard to mutate at the current time for it to act like "Talent") -- but working hard often matters more.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: How do these punks do it?

Postby 3.14159265... » Wed Aug 06, 2008 3:40 pm UTC

Yakk wrote:Talent exists (it doesn't have to be "innate", it just has to be really hard to mutate at the current time for it to act like "Talent") -- but working hard often matters more.
I have never experienced something where working hard didn't matter far more.
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Re: How do these punks do it?

Postby samspotting » Wed Aug 06, 2008 7:10 pm UTC

I am actually surprised how hard I am working now just because the material is interesting. I was quite the slacker in high school. The downside is my social life has suffered, but I'm finding more and more joy delving into math, which I am sure is the same for many people here.

I did a course on introductory number theory and proofs. We covered induction and strong induction, different ways to prove things, linear diophantine equations, polar coordinates, primality, divisibility, and modular arithmetic, and applied those concepts to RSA encryption. It was quite a fun course.

Applied math seems to be just a load of differential equations classes one after another. I don't really have a physics background (a slew of bad high school teachers put me off that subject). Is there any theoretical math combination you can do with computer science? Combinatorics seems interesting. How well does pure math and computer science merge? Theres also a major offered called computational math which uses a lot of linear programming which seems interesting as well.

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Re: How do these punks do it?

Postby Yakk » Wed Aug 06, 2008 7:24 pm UTC

Information theory, computability theory, and similar issues are relatively "pure math"ish parts of Theoretical Computer Science.

Some fun results from that sub-field: "can you determine if a given computer program halts", "is there a compression algorithm that always reduces the amount of data needed", "how many bits of this special number X can a given axiom system deduce", "can a system of reasonably powerful logic be both consistent and complete" (yes, you can prove that within the theory of computer science).
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: How do these punks do it?

Postby Marbas » Wed Aug 06, 2008 11:59 pm UTC

Yakk wrote:I know people who love abstract mathmatics, and worked really hard at it.

But, they sucked at it. Hard.


Well...this doesn't bode well for me. On every aptitude test ever I score higher marks in the "Verbal" section. Or whatever section involves more words.
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Re: How do these punks do it?

Postby Buttons » Thu Aug 07, 2008 2:35 am UTC

"Abstract math" is generally wordier than, say, calculus. So I wouldn't really be worried, yet.

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Re: How do these punks do it?

Postby Alpha Omicron » Thu Aug 07, 2008 3:43 am UTC

samspotting, do you go to the University of Waterloo?
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Re: How do these punks do it?

Postby Frimble » Thu Aug 07, 2008 10:53 am UTC

samspotting wrote:How do these punks do it?


They went to a secret room where they had a metal cable plugged into the back of their head. After sitting there a few hours they declared "I know linear algebra!" to which the examiner replied: "show me.". 8)
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Re: How do these punks do it?

Postby samspotting » Thu Aug 07, 2008 2:26 pm UTC

Yeah I got to Waterloo. Doing the regular non advanced calc and algebra classes, cept next semester where im taking calc 3 advanced and seeing how i do. Doing CS 136, Scheme + C as my cs stream.

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Re: How do these punks do it?

Postby Yakk » Thu Aug 07, 2008 3:08 pm UTC

Oh crap, you are skipping from 138 (Honors Calc II) to 247 (Advanced Calc III)? That will be hard. 247 is basically a slightly more applied version of Real Analysis, the 3rd year pure-math majors course (focusing more on R^n than Real Analysis does, humorously).

Do you know anyone in those classes? See if you can photocopy their notes.

Go to MathSOC, and see if you can find 147/148 exam bank (Advanced Calc I and II) -- there are certainly multiple copies. Do the exams. They may also have copies of old assignments.

Do not presume just because it "looks familiar" that you can do it: actually do the assignments/exams. (It should only take a few hours to do the exams, right?)

See if you can borrow a friends copy of the 145 course notes (Advanced Classical Algebra). Do the exercises to practice some proof techniques. See if you can get a friend who took those courses to look over your proofs, and point out where you are screwing up.

(Course numbers may have changed).

Btw, who is teaching Calc 247? Some instructors are less draconian than others. :-)

Remember: the people in 145/146/147/148 did an assload more proofs, and where expected to be able to generate them quite often. If you want to "get up to speed", you really should start now and work hard at catching up.

Hmm. Looks like the mathSOC exam bank is online:
http://www.mathsoc.uwaterloo.ca/exambank/
with U(W) credentials. There might be an online exam. :-)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: How do these punks do it?

Postby Alpha Omicron » Thu Aug 07, 2008 5:32 pm UTC

EDITed out for offensive.

Re-post: Some of these students may be the products of an unfortunately common parenting style which pushes academic success at the cost of development as a person.
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Re: How do these punks do it?

Postby Yakk » Thu Aug 07, 2008 5:35 pm UTC

Alpha Omicron wrote:Given what I know about UW, I would say at least part of it involves these kids' parents pushing them like crazy. The stereotypical Asian parenting style is a little weird, as I see it.

The stereotypical racist parenting style is also a little weird.

The Advanced Math classes at U(W) where not Asian-Only clubs, not by any stretch of the imagination. Your prejudices are leaking.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: How do these punks do it?

Postby samspotting » Thu Aug 07, 2008 6:03 pm UTC

I have found it that most of the advanced sections consist of a mix of asian and eastern european. Those guys are crazy. The ones I met were actually lacking in social skill but you could tell they could pwn so hard at math. Actuarial science however is almost completely asian, more of us care more about the money than the actual profession. People in it are too money centered, and it seems like they foam at the mouth over actuarial salaries, something I want to avoid.

I will probably get a book on analysis and calculus 3, I have a lot of time over my work term. I know Calc 3 Advanced is brutal, I took a look at the Calc 2 Advanced midterm and having just finished calc 2 I couldn't do half of it. But I can't play catch up by doing easier classes eh...

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Re: How do these punks do it?

Postby Alpha Omicron » Thu Aug 07, 2008 6:07 pm UTC

Yakk wrote:Unfair racial stereotyping

Apology.
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Re: How do these punks do it?

Postby Yakk » Thu Aug 07, 2008 7:57 pm UTC

Have a copy of the Yellow Pig book (Calculus by Spivak)? It covers the basics of an introductory approach to Calculus from a reasonably rigorous direction.

http://www.amazon.com/Calculus-Michael- ... 0914098896

Ug, the 3rd edition cover is ugly.

http://www.amazon.ca/Calculus-Michael-S ... 762&sr=8-1

If you manage to do every exercise in that book, especially the ones with *s and **s, you'll be prepared for 247.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: How do these punks do it?

Postby darren » Fri Aug 08, 2008 3:32 am UTC

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Re: How do these punks do it?

Postby z4lis » Fri Aug 08, 2008 6:22 am UTC

darren wrote:Agreed on Spivak. Part of succeeding in math is seeking out books supplemental to the required text... I almost failed real analysis because I found Rudin unreadable, until I came across spivak.


I'll keep this little quote in mind. I did a "math audition" for a group of professors, the head of which feels that I'm ready to basically skip through the "Intro to Math" class (proof writing and basic set/algebraic/number theory) and an analysis class straight into a rigorous calculus class. While I trust his decision, I'll admit I'm a little shaky about this upcoming semester. I might spend a week before classes start stalking the library for a book to read through to make sure I'm actually ready for the class. In any case, I'll keep your advice in mind and seek outside sources when possible. I learned more about vector calculus from watching some MIT lectures about electromagnetism than I did my professor. Perspective matters, eh?
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Re: How do these punks do it?

Postby Alpha Omicron » Fri Aug 08, 2008 6:51 pm UTC

I keep hearing good things about Spivak. Will keep an eye out, I suppose.
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Re: How do these punks do it?

Postby Anthemyst » Fri Aug 08, 2008 9:31 pm UTC

I took a proof-based version of Linear Algebra (Infinite Dimensional Linear Algebra), and it was exhilirating. No textbook, just every week was, "Here, prove that a matrix A is injective if and only if Ar=0 has only the trivial solution" (that was the second week). I had to work harder at it than the rest of my classes put together that semester, but I loved every second. It was the kind of class where you might not survive, but IF you survived, you probably got an A. So what I'm saying is, those students might not have been AMAZING math students before, but the class you described probably turned them into ones. After a class like that, not only do you know Linear Algebra inside and out, but you also know how to think up proofs for challenging questions, and how to write proofs.
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Re: How do these punks do it?

Postby samspotting » Sat Aug 09, 2008 12:48 am UTC

That does remind me, my linear algebra course this semester was less than subpar, I understood the material but it seemed like a very weak introduction.

Does anyone have a good linear algebra text or resource?

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Re: How do these punks do it?

Postby kgrizzly » Sat Aug 09, 2008 8:21 pm UTC

There must be a thousand texts on linear algebra, and even among the non-crappy ones there is a wide variety, some of which will appeal to you more than others all depending on your particular tastes. Of the ones that I have personal knowledge of, I really like Serge Lang's "Linear Algebra" as an undergraduate text. It takes the high road (read: abstract approach, real proofs) from page 1. But if you would prefer a somewhat more applied approach, a good alternative is Lipschutz's "Linear Algebra", part of the Schaum's Outline Series.

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Re: How do these punks do it?

Postby darren » Sun Aug 10, 2008 4:49 am UTC

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Re: How do these punks do it?

Postby samspotting » Thu Aug 14, 2008 10:56 pm UTC

I talked to the prof teaching advanced calculus, he said that he didnt really like spivak's book and instead recommended "An Introduction to Analysis" by wade. I looked on amazon and it seems to be hated by most reviewers. Does anyone have any opinions on the book? It is very expensive.

Rudin's book seems to be very praised, as is apostols book.

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Re: How do these punks do it?

Postby z4lis » Fri Aug 15, 2008 5:47 am UTC

samspotting wrote:I talked to the prof teaching advanced calculus, he said that he didnt really like spivak's book and instead recommended "An Introduction to Analysis" by wade. I looked on amazon and it seems to be hated by most reviewers. Does anyone have any opinions on the book? It is very expensive.

Rudin's book seems to be very praised, as is apostols book.


Keep in mind some of the reviews might be from people who weren't anticipating a rigorous approach. Granted, I've never used the book; all I know is that I actually have a class with Wade this semester, and another professor said he was good. Perhaps he's better at teaching from his own book. :P
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Re: How do these punks do it?

Postby DavCrav » Fri Aug 15, 2008 5:05 pm UTC

3.14159265... wrote:I am angered at most of these posts.

I didn't see what "real" math was till about the end of first year. I loved it, I learned it.

The reason those kids are good at it is because when they showed interest there was someone who taught them math. For some of us that didn't happen (parents wanted a doctor).

I refuse to believe there is such thing as "talent". If you love it, and work at it anything is possible.


Be angered away. Doesn't mean it isn't true. I showed an interest in maths, but there was nobody qualified enough around me to really teach me stuff until I got into university, when I'd already demonstrated a clear aptitude towards the subject. Some people are just good at it. I don't mean people who are OK at maths, who will come away with good degrees; I mean people who go on to become Fields medallists. These people are just good at it, and didn't have to work at it. (They had to work to get the Fields medal, but not to get their degree...)

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Re: How do these punks do it?

Postby Yakk » Fri Aug 15, 2008 5:38 pm UTC

They do "have" to work at it to be as good as they are. That doesn't mean doing the same amount of work will result in the same results, or that work isn't important -- working at it is very important.

Much of the "talent" does come from spending a reasonably large chunk of your life thinking about this stuff. But if you try to work "just as hard" on a mathematical problem during (say) the duration of a course, you won't get the same results as someone who has been thinking about math in a deepening and deepening sense since they where 10 years old.

And then there is talent -- with a program that attracts the top fraction of a fraction of a percent of the best math students in Canada, you will have some people who are really good at it.
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Re: How do these punks do it?

Postby 3.14159265... » Fri Aug 15, 2008 5:57 pm UTC

DavCrav wrote:
3.14159265... wrote:I am angered at most of these posts.

I didn't see what "real" math was till about the end of first year. I loved it, I learned it.

The reason those kids are good at it is because when they showed interest there was someone who taught them math. For some of us that didn't happen (parents wanted a doctor).

I refuse to believe there is such thing as "talent". If you love it, and work at it anything is possible.


Be angered away. Doesn't mean it isn't true. I showed an interest in maths, but there was nobody qualified enough around me to really teach me stuff until I got into university, when I'd already demonstrated a clear aptitude towards the subject. Some people are just good at it. I don't mean people who are OK at maths, who will come away with good degrees; I mean people who go on to become Fields medallists. These people are just good at it, and didn't have to work at it. (They had to work to get the Fields medal, but not to get their degree...)


Consider for a second that no one was qualified to teach me math until I got to university either, at which point I did most of the undergrad in first year. Consider also that I did most of undergrad physics by the end of second year.

Now, consider that you are still wrong. That those kids without "talent" just found the things you do incredibly boring, and that they are way better than you at some other stuff that maybe "sitting around doing nothing" for all it matters.

Consider that I understand what you are saying, and still think you are wrong. If you can, then you are one of the "talented" ones. Now consider, that some of those without "talent" find the above intuitive.

Your position is non-falsifiable, because anyone who does something they didn't have the "talent" for... well, you didn't measure their "talent" properly originally.

It does make one feel better to think one is smarter though. Perhaps others feel that way about sex... *goes off to wonder*
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Alpha Omicron
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Re: How do these punks do it?

Postby Alpha Omicron » Fri Aug 15, 2008 6:51 pm UTC

@3.14...: I refute your argument thus.
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Yakk
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Re: How do these punks do it?

Postby Yakk » Fri Aug 15, 2008 7:05 pm UTC

~Pi, I know people who find mathematics very interesting, who spend lots of time trying to understand mathematics, and who managed to get a Pure Mathematics degree. They would spend lots and lots of effort, finding the subject extremely interesting, and get a 60% in a course.

(The only way they graduated with a degree was by doing a double-major in-faculty, in which your major average was the average of all courses in the faculty, and not just courses from your discipline.)

So we have someone who spent lots of work on a subject they found interesting, and still sucked at it. Do you have an alternative explanation?

The position that "talent" exists is falsifiable -- if there is a nearly perfect correlation between effort spent on learning something, and results from the learning, then there is almost certainly no "talent" factor.

To an extreme example, humans clearly have a "talent" for learning languages. Attempting to teach a Chimp to learn a language results in a chimp that can speak very poorly (in sign language) if measured by human aptitude levels. Similarly, there are a number of injuries, birth defects, etc that generate people who are clearly worse at entire spectra of human activities.

On the other hand, there seem to be people who are extremely good at many subjects, who perform far and above their rivals in the same field. Either they are somehow spending much much much more effort learning the subjects, or their learning is somehow amazingly more efficient in a way that we cannot distill, or ... they are talented.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

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Re: How do these punks do it?

Postby DavCrav » Fri Aug 15, 2008 8:00 pm UTC

3.14159265... wrote:Consider for a second that no one was qualified to teach me math until I got to university either, at which point I did most of the undergrad in first year. Consider also that I did most of undergrad physics by the end of second year.

Now, consider that you are still wrong. That those kids without "talent" just found the things you do incredibly boring, and that they are way better than you at some other stuff that maybe "sitting around doing nothing" for all it matters.

Consider that I understand what you are saying, and still think you are wrong. If you can, then you are one of the "talented" ones. Now consider, that some of those without "talent" find the above intuitive.

Your position is non-falsifiable, because anyone who does something they didn't have the "talent" for... well, you didn't measure their "talent" properly originally.

It does make one feel better to think one is smarter though. Perhaps others feel that way about sex... *goes off to wonder*


Ooh, you did all of undergrad maths in a year. Good for you. Makes two of us. (Although given the way you spell 'maths', my guess is that your entire undergrad degree is what most people here do in their first year anyway...)

Actually, I can't be bothered to argue, because as somebody who is employed to do mathematics by a university, I've seen quite a few talented mathematicians, worked with quite a few as well. I don't need arguments because I have my own evidence, which is the mathematicians themselves.

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Re: How do these punks do it?

Postby Yakk » Fri Aug 15, 2008 8:11 pm UTC

DavCrav, 'maths' is British English, and not at all standard in the rest of the English-speaking world.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.


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