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Mathematicians are chronically lost and confused (2014)

j2kun.svbtle.com

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Re: Mathematicians are chronically lost and confused (2014)

#71

Post by j2kun »

laichzeit0 wrote: post #12
kinai wrote: post #2

Does anybody know a good guide on where to begin? Resources are not the issue here, but usually the overwhelming amount of information regarding all those topics and areas of mathematics. I was always very interested but got discouraged rather quickly, even after a semester at university. So far my favorite access to math was through philosophy.

Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament.... Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book). Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, w…

Caveat Emptor: Aluffi and Axler are texts meant for vastly different levels of maturity. The former for first-year graduate students, and the latter for first/second-year undergraduate students.

Re: Mathematicians are chronically lost and confused (2014)

#73

Post by iheartmemcache »

yequalsx wrote: post #61
iheartmemcache wrote: post #53

Earlier quoted context omitted.

Define 'good'. I've seen people in their early 20s go from one branch (say, combinatorics) to another branch (say, algebraic geometry) without taking any formal courses [one could argue this is equivalent to going from being a great neurosurgeon then 6 months later publishing papers at the forefront of pancreatitis research]. They end up out performing post-docs who've spent a decade solely in that field, often withi…

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 amazingly hard; hard enough, I suggest, that it's the equivalent of modern day 'self-teaching', given the large corpus of knowledge one has to attain[1]. I assert it'd take at least ~2 years for one to move from combinatorics to an understanding of say, Sheaf Theory circa 1950 (up to Serre) unless you're one of those rare guys like Tao who can jump from field to field (ugh pun not intended).

[1]http://matt.might.net/articles/phd-school-in-pictures/ Getting published in the J. of Top. is being on the arc (perhaps, even deforming the disc in R^2 heh heh). Being able to move laterally s.t. your findings substantial enough they are accepted into the J. of Alg. Geo. requires a requires such an absurdly large lateral movement along the arc that it's a feat analogous in difficulty to self-teaching oneself up until say the 1920s.

Re: Mathematicians are chronically lost and confused (2014)

#74

Post by stepvhen »

Koshkin wrote: post #59
stepvhen wrote: post #38

Earlier quoted context omitted.

I think math is "easier" to learn while it still holds a clear practical value, up to calculus and linear algebra. Past that, when you start to enter the world of "pure" mathematical s, it can be muh more difficult. The practical value of earlier subjects allows a student or what have you to draw connections between what they already know and this new concept. Something like group theory however is more difficult to…

Actually, group theory is not a very good example of a theory that has no or few practical applications; see https://en.wikipedia.org/wiki/Group_theory#Physics . In general, you would probably be surprised to learn how much of the modern mathematics (including category theory) has already made its way into theoretical physics and other sciences.

Ha, well then my class/texts have failed to mention these practical uses.

Re: Mathematicians are chronically lost and confused (2014)

#75

Post by jonstokes »

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 looked at him and said, "Dad, what's wrong with you?"

Re: Mathematicians are chronically lost and confused (2014)

#77

Post by vinceguidry »

yequalsx wrote: post #42
agentultra wrote: post #30

Earlier quoted context omitted.

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…

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-learning before you can hope to achieve excellence.

All good programmers are self-taught. No good mathematicians (currently) are. I wonder why this is. Probably boils down to what 'math' is trying to accomplish vs. what 'programming' does.

Re: Mathematicians are chronically lost and confused (2014)

#78

Post by justifier »

justifier wrote: post #66
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…

for me your comment is so sad to read mathematics is unconstrained by any desired artificial 'engagement' requirement engagement is as easy as acting on and developing an interest and i encourage everyone to engage with mathematics your attempts to create an artificial toll or make a case for one, especially in lieu of your argument being unsolicited from the contents of the linked article, is suspicious at best, and…

shame.. i am passionate about discussing this topic but it is hard to 'engage' with unaccompanied downvotes

Re: Mathematicians are chronically lost and confused (2014)

#79

Post by yequalsx »

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…

The person I responded to seems to think that mathematics is not difficult to engage with outside of academia. You seem to be under the impression that even making a lateral move post professional training is amazingly hard. It appears that you agree with my first post that grasping second year graduate level mathematics does require formal training for all but a very few.

Re: Mathematicians are chronically lost and confused (2014)

#80

Post by abecedarius »

daniel-levin wrote: post #64
vecter wrote: post #16

I have a great personal story that highlights how long the journey of understanding mathematics is. I took linear algebra my freshman year of college. It was the non-math major course, so it didn't require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to m…

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?
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