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

Re: Mathematicians are chronically lost and confused (2014)

#101

Post by mindcrime » Fri, Jun 10, 2016, 7:22 PM UTC

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…

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.

I've often wondered if automated theorem provers (Coq and the like) could be useful for this. Sure, it requires learning Coq (or Agda or ACL2 or whatever), but it seems that it might be worthwhile just as a way to sanity check one's work - somewhat akin to the way a compiler "checks your work" when programming.

Re: Mathematicians are chronically lost and confused (2014)

#102

Post by nilkn » Fri, Jun 10, 2016, 7:39 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…

> I've never encountered anyone who understood typical second year graduate level mathematics without formal training.

I've known one such person. He skipped Math 55 at Harvard as a freshman because it was too basic for him.

He's such an extreme outlier though that there's almost nothing to learn from his case other than to marvel at it.

Re: Mathematicians are chronically lost and confused (2014)

#103

Post by wolfgke » Fri, Jun 10, 2016, 7:44 PM UTC

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…

> I've recently been going through temporal logics. If only more programmers were aware of the beauty of the proof of weak fairness. And its implications in system specifications.

I'm a mathematician and have read about temporal logic. What beautiful proof of weak fairness do you mean here?

Re: Mathematicians are chronically lost and confused (2014)

#104

Post by ivansavz » Fri, Jun 10, 2016, 7:47 PM UTC

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.

I am totally biased, but I think my book is a good introduction to math fundamentals (high school) and calculus: https://minireference.com/

preview: https://minireference.com/static/excerpts/noBSguide_v5_previ...

As an added bonus, the book also covers mechanics (Physics 101), which is really important to understand to build up you general modelling skills. (Physics is all about coming up with math models to describe reality.)

Re: Mathematicians are chronically lost and confused (2014)

#105

Post by arghbleargh » Fri, Jun 10, 2016, 7:50 PM UTC

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 think this highly depends on what you mean by self-taught. Probably a majority of good mathematicians are "self-taught" in the sense that they learned most of the things they know from reading books or thinking on their own rather than taking classes. What you won't see commonly is someone who has a day job but becomes really good at mathematics on the side. This is probably because of the sheer amount of training it takes to become what is considered "professional-level" in math. It takes quite a bit of background before you can even hope to do something that hasn't been done before. The same is not true for programming.

Re: Mathematicians are chronically lost and confused (2014)

#106

Post by throwawaysocks » Fri, Jun 10, 2016, 7:57 PM UTC

Nimitz14 wrote: post #51

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…

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.

As someone who has done research in the area, I'd argue that a lot of work on temporal logics is at about the same level of difficulty as a better-than-average undergraduate linear algebra course. Certainly none of the standard results about various temporal logics even come close to approaching the difficulty of a first or second year graduate mathematics course.

Re: Mathematicians are chronically lost and confused (2014)

#107

Post by shasta » Fri, Jun 10, 2016, 7:57 PM UTC

thevardanian wrote: post #86

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…

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-…

I think this is it. To be good at mathematics you need a considerable amount of time to sit and do mathematics. The only way someone is going to fund this as part of a job is if you're already a mathematician.

Re: Mathematicians are chronically lost and confused (2014)

#108

Post by chipsy » Fri, Jun 10, 2016, 8:01 PM UTC

jondubois wrote: post #49

pixl97 wrote: post #44

Earlier quoted context omitted.

>Any uncertainty can be quickly resolved by simply running some code. Until you have to interact with a black box of someone else's code. You can only be certain that the data you sent that particular time works, not that all possibilities of valid data work.

I guess with programming, you can never get 100% certainty when it comes to correctness of your code - Unless you perform formal analysis which is impractical for most use cases.

You might not succeed at 100%, but you can remove "classes of error" by adopting particular styles or techniques backed by formal analysis. That's one of the biggest appeals of compiler technology - it can encode an understanding of patterns proven to detect failure, and in so doing lower your resulting bug count.

Re: Mathematicians are chronically lost and confused (2014)

#109

Post by p4wnc6 » Fri, Jun 10, 2016, 8:07 PM UTC

thevardanian wrote: post #86

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…

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-…

Just chiming in to add another vote that this is the explanation. In most business settings, business types don't like mathematical maturity or really even care unless it is fundamental to the bottom line -- and even then, it's usually only fundamental to one tiny corner of the product or business, and after that everything else is reporting software, client analytics, customer support, websites, etc.

Businesses interface socially with other entities (individuals, governments, corporate customers). That social interface will always be 99% report/client service/cursory analytics and 1% hard science domain expertise. That's just the nature of business.

It could be different in a government lab, a think tank, or a boutique consultancy, and it's definitely different in organizations like prop trading firms that do not interface with outside entities in the way that almost all businesses do. But all of those things also place a much higher value on employing people with significant math maturity and hard science expertise.

Re: Mathematicians are chronically lost and confused (2014)

#110

Post by throwawaysocks » Fri, Jun 10, 2016, 8:22 PM UTC

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…

> ...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.

So I'm a self-taught programmer, and this attitude annoys the hell out of me.

1. EVERYTHING requires self-learning to be more than mediocre. In fact, the more I learn about other fields -- engineering, medicine, law -- the more I realize those fields probably actually require MUCH MORE self-directed learning than programming. It's just that this self-learning is much harder to do in isolation without at least a college education. E.g., try teaching yourself any of the traditional fields of engineering with only what you learned before 10th grade. You'll find it's much harder than teaching yourself programming. Whereas you can easily land a developer position with things that the average pre-calculus 11th grader can pretty easily teach themselves.

2. College-educated people who didn't start programming until 18+ ALSO do a lot of self-learning. The difference between them and me (or anyone else who self-taught as a kid) is that they could learn in 1-2 years during their 20s what it took me 4+ years to learn in my pre-teens/teens. Which there's absolutely nothing wrong with. I called it reading tutorials and messing around at night while they called it reading a textbook and doing course projects/homework assignments. I called them high school friends and IRC budddies while they called them TAs. But the actual content of the work and the amount of external direction provided is about equivalent.

Just because programming is simple enough that you can learn how to do enough to get a well-paying job before hitting 16 doesn't mean that people who go through a college degree aren't also doing a ton of self-learning alongside lectures.