Mathematicians are chronically lost and confused (2014) • Hacker News Board

Re: Mathematicians are chronically lost and confused (2014)

#51

Post by Nimitz14 » Fri, Jun 10, 2016, 2:58 PM UTC

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…

Yup. He's probably just unaware of how difficult and vast math actually is. Reminds me of comments I've read from programmers talking about learning 'advanced mathematics': Linear Algebra. Lol.

Re: Mathematicians are chronically lost and confused (2014)

#52

Post by bglazer » Fri, Jun 10, 2016, 3:03 PM UTC

wmsiler wrote: post #35

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…

Linear algebra is the sort of topic where it seems like you can always go deeper. I also got an A in my linear algebra class in college. When I got to grad school for math, I took the first year graduate course on algebra and saw linear algebra in terms of algebraic things like modules and representations. I decided that I hadn't really known linear algebra before, but that I did then. Then I took differential geomet…

I've been learning (trying to learn) linear algebra by studying Gilbert Strang's "Introduction to Linear Algebra" book. There are some really mind bending sections in which he explicitly treats a concept like linear regression in terms of geometry and then algebra and then calculus and then probability and then ties them all back together.

What's really struck me is when I dive into a Wikipedia rabbit hole of linear algebra, following links for terms I don't know. I start on the Topology page, read the introduction section of 10 articles that start with "x is a generalization of y" and somehow end up back on the Topology page. Shallow exposure to the sheer volume and conceptual depth of stuff like topology and algebra has really made me respect modern math.

Re: Mathematicians are chronically lost and confused (2014)

#53

Post by iheartmemcache » Fri, Jun 10, 2016, 3:07 PM UTC

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…

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 within months. In my mind that's just as impressive as the Ramanujans and all (who was 'discovered by Hardy' but still had quite a base line of formal training prior to being invited to England).

Mathematics has developed so extensively that there's going to be a base corpus of knowledge one generally has to acquire before they're at the caliber of being able to publish. Take a tenured topologist and place him in at a conference where people are presenting their findings in an entirely different field and he'll more often than not struggle to keep up. Combined with the fact that higher level education is a lot easier to attain than it was say, 50 years ago (and definitely 100 years ago) allowing a larger percent of population who those who want to enter into pure mathematics to do so, it's fairly understandable why you won't see many 'good' self-taught mathematicians. (Though, I argue that recluses like Perelman who go off the grid for years at a time after being inspired by something like Ricci flow to solve Poincare are the analogues of yesteryears self-taught mathematicians, as they are no longer collaborating with institutional academia.)

Re: Mathematicians are chronically lost and confused (2014)

#54

Post by mietek » Fri, Jun 10, 2016, 3:11 PM UTC

kinai wrote: post #39

laichzeit0 wrote: post #12

Earlier quoted context omitted.

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…

I don't find calculus and lin alg very interesting, school ruined those for me forever I guess. I am mostly interested Logic and Information Theory but I guess the latter doesn't go with a good calculus base. Thanks for the links, I will read the book of proof and try the exercises, though mostly studied those topics already.

You may be interested in Paul Taylor’s “Practical foundations of mathematics”.

Re: Mathematicians are chronically lost and confused (2014)

#55

Post by tuckerman » Fri, Jun 10, 2016, 3:17 PM UTC

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…

(warning: studied Math in college, so don't know how that biases me in this discussion)

I wonder if some of this is because of the number of jobs in industry for each discipline? There are a great many jobs available which nurture CS skills but I can't quite think of a job besides academia off the top of my head where that lecture in Toric Varieties I took would come in handy.

Maybe its that there is a career/financial incentive to go deeper into CS but not mathematics?

Re: Mathematicians are chronically lost and confused (2014)

#56

Post by pm24601 » Fri, Jun 10, 2016, 3:26 PM UTC

All they have to do is read this article: https://news.ycombinator.com/item?id=11874395

Re: Mathematicians are chronically lost and confused (2014)

#57

Post by alistproducer2 » Fri, Jun 10, 2016, 3:37 PM UTC

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…

Similar theme:

I was randomly thinking about logarithms and it just "clicked" that what a logarithm really tells you about a number is it's order of magnitude relative to some base.

I had never seen it explained that way, but once I thought about it it seemed so simple. It's funny, programming has actually helped me understand math. It helps me to look at a formula or function as a programming function. I try to read maths like I read source code. I ask the same question: what dynamic system is this jumble of symbols trying to represent? How does an object behave as it moves through this system?

Once I began to think about math like that, a lot of things began to "click" for me.

Re: Mathematicians are chronically lost and confused (2014)

#58

Post by whynotqat » Fri, Jun 10, 2016, 3:38 PM UTC

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…

Certainly some that were active in the last 50 years, though it's hard to think of any that were trained in that time-frame: Gelfand and Ian MacDonald are the first pair that come to mind, though Gelfand had some great mentors and Macdonald did an undergraduate degree.

Re: Mathematicians are chronically lost and confused (2014)

#59

Post by Koshkin » Fri, Jun 10, 2016, 3:42 PM UTC

stepvhen wrote: post #38

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 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.

Re: Mathematicians are chronically lost and confused (2014)

#60

Post by riazrizvi » Fri, Jun 10, 2016, 3:44 PM UTC

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."