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Mathematicians are chronically lost and confused (2014)

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Re: Mathematicians are chronically lost and confused (2014)

#81

Post by ambulancechaser »

iheartmemcache wrote: post #73
yequalsx wrote: post #61

Earlier quoted context omitted.

Going from combinatorics to algebraic geometry with formal training in the former but not the latter doesn't count in my opinion. Having formal training in pure math makes one more able to self learn other branches. I have a hard time believing that one can go from undergraduate calculus/linear algebra to algebraic geometry in anything more frequently than than extremely rarely. The first hump is understanding a bran…

No one is asserting that someone taking 100 level calculus is going to self-teach => publishing in algebraic geometry. I agree that your first hump correctly describes the first barrier. I assert that a lateral movement in 'sufficiently different' subsets of mathematics will provide an additional hump within the space of knowledge, quantified by a magnitude of eh maybe > 2/3s minimum. Going from comb. to alg. geo. is…

Yes they are. this whole thread is about mathematics being accessible to people outside academia. the greatgrandparent poster posted that they could follow mathematics despite having skipped university in a thread about mathematical maturity and being accessibility outside of formal training. someone pointed out that you could cross disciplines but this isn't what we were talking about as this builds on top of a classical training in university.

Re: Mathematicians are chronically lost and confused (2014)

#82

Post by yequalsx »

vinceguidry wrote: post #77
yequalsx wrote: post #42

Earlier quoted context omitted.

Not knowing your level of mathematical insight and knowledge makes it hard to know if your belief about your mathematical talent is a self deception. I've never encountered anyone who understood typical second year graduate level mathematics without formal training. I know such people could exist. I've just never met any. I have met people who claimed to be self taught in mathematics and it was obvious that they didn…

> I have met people who claimed to be self taught in mathematics and it was obvious that they didn't really understand the concepts though they were absolutely convinced they knew what they were talking about. This is an interesting contrast with my experience with learning programming, where, it doesn't matter how much formal education / training you get, you're still going to have to do a massive amount of self-lea…

I have programmed professionally but that was a long time ago. In my limited experience I believe one advantage programming has over mathematics in terms of self learning is that in programming one is more likely to know that a program works/doesn't work versus knowing whether your conception of an abstract definition or theorem is correct. It seems overall easier to believe a false notion in math than in programming. I might be wrong in this perception. I think your last sentence is absolutely correct.

Re: Mathematicians are chronically lost and confused (2014)

#83

Post by akeck »

laingc wrote: post #13

To me, this is about "mathematical maturity". My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the w…

> Learning Mathematics in a classical, structured way really does change the way you think

Yes! I bashed my way through an Applied Math degree in night school. It hurt my head, but it rewired my thinking permanently (for the better).

Re: Mathematicians are chronically lost and confused (2014)

#84

Post by taejo »

abecedarius wrote: post #80
daniel-levin wrote: post #64

Earlier quoted context omitted.

This is also largely a consequence of the fact that most linear algebra courses are computational by nature. They'll ask you to compute lots of things as a means of assessment. Work out this determinant. Find the eigenvalues of that matrix. "Matrices are grids of numbers". Take this matrix and write it in terms of this other basis. I only really perceived the impact of linear transformations being vector-space-struct…

I'm confused: [42] is an invertible matrix with integer coefficients, right?

I guess daniel-levin meant to say "invertible matrix with integer coefficients whose inverse also has invertible coefficients". The fact isn't too hard to see:

* If M has integer coefficients, then det(M) is an integer. * det(inv(M)) = 1/det(M) * Since M has integer coefficients, det(M) is an integer * Since inv(M) has integer coefficients, det(inv(M)) is an integer * So det(M) and 1/det(M) are both integers, so det(M) is either 1 or -1

Re: Mathematicians are chronically lost and confused (2014)

#85

Post by sixo »

abecedarius wrote: post #80
daniel-levin wrote: post #64

Earlier quoted context omitted.

This is also largely a consequence of the fact that most linear algebra courses are computational by nature. They'll ask you to compute lots of things as a means of assessment. Work out this determinant. Find the eigenvalues of that matrix. "Matrices are grids of numbers". Take this matrix and write it in terms of this other basis. I only really perceived the impact of linear transformations being vector-space-struct…

I'm confused: [42] is an invertible matrix with integer coefficients, right?

Yes, the claim is false. It's not obvious to me what it's been confused for. Edit: oh, the other commenter figured it out.

Re: Mathematicians are chronically lost and confused (2014)

#86

Post by thevardanian »

vinceguidry wrote: post #77
yequalsx wrote: post #42

Earlier quoted context omitted.

Not knowing your level of mathematical insight and knowledge makes it hard to know if your belief about your mathematical talent is a self deception. I've never encountered anyone who understood typical second year graduate level mathematics without formal training. I know such people could exist. I've just never met any. I have met people who claimed to be self taught in mathematics and it was obvious that they didn…

> I have met people who claimed to be self taught in mathematics and it was obvious that they didn't really understand the concepts though they were absolutely convinced they knew what they were talking about. This is an interesting contrast with my experience with learning programming, where, it doesn't matter how much formal education / training you get, you're still going to have to do a massive amount of self-lea…

Simple profit incentives.

You can't really make a living off of being a self-taught mathematician, whereas you can by knowing how to program, and run your own business.

The only way to make a living as a mathematician is to into academia, so that why you see all mathematician in academics. Basically it isn't that academia produces mathematician, although it helps, but that it attracts mathematicians. Just as, say, Y-Combinator attracts good startups.

Re: Mathematicians are chronically lost and confused (2014)

#87

Post by lawpoop »

yequalsx wrote: post #82
vinceguidry wrote: post #77

Earlier quoted context omitted.

> I have met people who claimed to be self taught in mathematics and it was obvious that they didn't really understand the concepts though they were absolutely convinced they knew what they were talking about. This is an interesting contrast with my experience with learning programming, where, it doesn't matter how much formal education / training you get, you're still going to have to do a massive amount of self-lea…

I have programmed professionally but that was a long time ago. In my limited experience I believe one advantage programming has over mathematics in terms of self learning is that in programming one is more likely to know that a program works/doesn't work versus knowing whether your conception of an abstract definition or theorem is correct. It seems overall easier to believe a false notion in math than in programming…

This is the problem I had learning math in school, and why I believe I'm doing okay as a self-taught programmer. I always thought I was doing okay until I got my graded test back.

Did I mix up greater-than, less-than again at work? Well, I'm getting opposite behavior. The errors make themselves apparent. I can play around with it, try things, and get basically real-time feedback as to whether I'm doing it right. What the hell is a monad? Well, here's a problem that is basically crying out for a monad solution.

Re: Mathematicians are chronically lost and confused (2014)

#88

Post by abc_lisper »

riazrizvi wrote: post #60

Beautiful article. Love the advice at the end!: "What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion."

I can't think of a better argument to goad programmers into writing code comments, especially for elegant code.

For some reason, programmers think good code should be self explanatory. It doesn't have to - if your insight is outside the flow (or the argument) of your code.

I see a lot of Java code written like this; if the programmer's job is to multiply two numbers, they add number a number b times, and say comments are not needed!

Re: Mathematicians are chronically lost and confused (2014)

#89

Post by sudoscript »

agentultra wrote: post #30
laingc wrote: post #13

To me, this is about "mathematical maturity". My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the w…

I disagree that mathematics itself is difficult to engage with outside of academia. There are some academic mathematicians who think mathematics has to be hard because it was difficult for them to understand. There are others for whom sharing their insights is more important than searching for new ones. I think that the set of mathematicians together have done a rather good job in the 20th century of producing mathem…

Programming has an instant feedback loop -- you know almost immediately if what you wrote works or doesn't. Math is more abstract, and doesn't have that. As humans, it's much easier to learn something on your own when you have immediate feedback. I think that's why programming is an outlier for people who are self taught.

Re: Mathematicians are chronically lost and confused (2014)

#90

Post by selimthegrim »

jonstokes wrote: post #75

A mathematician was walking home from campus one day, and as he walked he was pondering a particularly thorny problem. At one point, he snapped out of his reverie and looked around and realized that he had no idea where he was. He saw a young boy playing with a ball in a yard, and figured maybe the boy could tell him the way home. So he says to the boy, "young man, do you know where Prof. So-and-so lives?" The boy lo…

This story is usually told about Norbert Wiener and his daughter.
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