Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#51

Post by pcmonk » Wed, Mar 05, 2014, 2:52 PM UTC

hessenwolf wrote: post #40

pcmonk wrote: post #30

Earlier quoted context omitted.

I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating…

I regularly explain my PhD thesis on napkins. The explanation involves twins, one with a broken nose; fish; GPS satellites; and a tractor driving in a field. Somehow this holds a crowd better than non-linear 72-dimensional space, and isometric and rigidity matrices.

But could they (1) extend your results or (2) have enough of an understanding of it to apply it to their work? The benefit of a napkin explanation is that they understand it well enough to know whether it would be useful/interesting to learn the issue better. If they really want to apply your results, though, then they'll need a better understanding. It's true that math papers aren't good for giving a surface-level intro to a subject, but they're not made for that. They're made so that if someone really wants to learn the subject, then they can.

Re: Mathematicians are chronically lost and confused

#52

Post by egdelwonk » Wed, Mar 05, 2014, 2:52 PM UTC

11001 wrote: post #42

egdelwonk wrote: post #32

What's the best way to relearn math?

Textbooks and exercises. Avoid shortcuts such as online tutorials or anything with "for hackers" in the title.

I've been using Khan Academy to refresh my skills and it's been really helpful, but do you have any recommendations for textbooks?

I'm 10 years out of school and haven't really needed to flex my math muscles in years.

Re: Mathematicians are chronically lost and confused

#53

Post by alex-g » Wed, Mar 05, 2014, 2:53 PM UTC

nnq wrote: post #17

michaelochurch wrote: post #4

Earlier quoted context omitted.

I'm of the mind that there is immense value in being able to figure out difficult proofs. Absolutely. However, the rabbit hole is very deep. Many papers make leaps from one sentence to the next that, if you're not familiar with the field, can take a couple days to figure out. Even then, real world proofs are informal and therefore not air-tight. They're close enough, almost always, but there's a reason why a mathemat…

> most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight , it has to be or the code won't compile or will throw a runtime exception, th…

This is a very good thought. Some current projects are trying to develop computable mathematical foundations in a more structured way. Homotopy type theory (http://homotopytypetheory.org/) is one example that has a lot of buzz around it just now, but automated theorem proving has been trying to work with higher-order concepts for ages now.

In the classical approach of "compiling" everything into sets/logic/etc., you end up with just the assembly language problem that's being discussed, where all the high-level structure vanishes. In order to do your bottom-up approach instead, one of the things that needs to happen is to make the theory really compositional, so that once you've defined some abstraction or higher-level concept, you can use it in constructions and proofs without having to break the abstraction. You don't need to know - and in fact you shouldn't be able to find out - just how the natural numbers were constructed, as long as they work by the right rules. This motivates the use of type theory to describe mathematical objects, and say which operations are allowed. We want to be able to add two numbers and get another number, but we don't want to be able to intersect two numbers as if they were sets, even if they happen to have been built out of sets.

So I think you are right - or at least, there are plenty of people who agree with you that this is a good idea. It is difficult to actually do, of course, but that's life.

Re: Mathematicians are chronically lost and confused

#54

Post by jmnicolas » Wed, Mar 05, 2014, 2:53 PM UTC

Before a few articles of this kind I never suspected there was such depth in Maths.

There's already so much to learn in programming, but I'm sure I'd love to dive in Maths (without the pressure of school like "understand this or you're an idiot").

Re: Mathematicians are chronically lost and confused

#55

Post by makmanalp » Wed, Mar 05, 2014, 2:54 PM UTC

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

I've argued the same with a mathematician friend of mine. I hate academic papers because of their seemingly convoluted and backwards way of explaining things. His answer was that papers were not made to convey thoughts to laymen, they were made to communicate facts and proofs with as little ambiguity as possible, optimized for reading by other mathematicians. It's meant to be high bandwidth (hence the terse style and lack of intuitive explanation) and low ambiguity (hence the seemingly backwards order of explaining things and the pages of "a is blah, b is foo").

It's an interesting discussion because from the mathematician's perspective, they don't see why they should cater to anyone who doesn't bother absorbing the lingua franca and the method of delivery. Countering that is the philosophical argument that information should be as available as possible. Countering that is how practical and useful that is, and whether the cost / benefit would be worth it.

I still think we can have our cake and eat it too, but I'm not sure. I think if the purpose is merely to transmit proofs and axioms unambiguously, I think we can have a language that performs just that and nothing else. I think stuff like this exists, but I don't know why it isn't the standard to publish with it.

Then the explanation can reside alongside this unambiguous description, and can take whatever liberties it pleases.

Re: Mathematicians are chronically lost and confused

#56

Post by helicon » Wed, Mar 05, 2014, 2:57 PM UTC

napowitzu wrote: post #19

This is true with many, many things. Very often it is the connections between ideas that yields the deep understanding, not the ideas themselves. Focusing too intensely on a single idea or subject results in not making connections and, consequently, not really understanding.

I agree. I think most if not everything he said can be applied to learning almost any discipline. I studied physics at university and I definitely recognize the authors description.

Re: Mathematicians are chronically lost and confused

#57

Post by chriswarbo » Wed, Mar 05, 2014, 3:01 PM UTC

japhyr wrote: post #35

acjohnson55 wrote: post #28

I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety…

as the math truck barrels on ahead I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now. What most of my students think: "I could never do m…

> I would like to see every elementary school have a math specialist, who knows advanced math, to help students with their overall understanding when they get off track. Helping a kid master some mechanics does a little to get them back on track, but diagnosing misunderstandings takes more math expertise than most elementary teachers have.

The classic example of this is "Benny's Rules", eg. http://math-frolic.blogspot.co.uk/2012/11/bennys-rules.html

Re: Mathematicians are chronically lost and confused

#58

Post by jorleif » Wed, Mar 05, 2014, 3:04 PM UTC

makmanalp wrote: post #55

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

I've argued the same with a mathematician friend of mine. I hate academic papers because of their seemingly convoluted and backwards way of explaining things. His answer was that papers were not made to convey thoughts to laymen, they were made to communicate facts and proofs with as little ambiguity as possible, optimized for reading by other mathematicians. It's meant to be high bandwidth (hence the terse style and…

I think the process of peer review often causes papers to be optimized for being hard to critique rather than easy to understand. I have myself participated in writing papers where we decided to leave out some non-crucial but very useful detail just because it opens up too many questions and opportunities for critique.

Re: Mathematicians are chronically lost and confused

#59

Post by applesinjuice » Wed, Mar 05, 2014, 3:06 PM UTC

coldpie wrote: post #25

cs702 wrote: post #22

Earlier quoted context omitted.

Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. This is SO TRUE. The sa…

Just as a specific example, I had this experience with Bayes' Theorem " rel="nofollow">http://en.wikipedia.org/wiki/Bayes%27_theorem> . As an informal paper for my computer security class, we used Bayes' theorem to implement aimbot detection in a simple FPS. It sounds like a big, complicated theorem with a special name that some genius had to come up with and has complicated notation involving probabilities and logic…

Fixed link for the lazy: http://en.wikipedia.org/wiki/Bayes_theorem

Re: Mathematicians are chronically lost and confused

#60

Post by 11001 » Wed, Mar 05, 2014, 3:15 PM UTC

egdelwonk wrote: post #52

11001 wrote: post #42

Earlier quoted context omitted.

Textbooks and exercises. Avoid shortcuts such as online tutorials or anything with "for hackers" in the title.

I've been using Khan Academy to refresh my skills and it's been really helpful, but do you have any recommendations for textbooks? I'm 10 years out of school and haven't really needed to flex my math muscles in years.

Any particular "math" you're interested in?

If you just want to get your foot in with pure math I'd recommend studying basic abstract algebra and analysis at the same time. For abstract algebra look into Hungerford (Intro, not his grad text), for analysis, maybe Rudin, or Kolmogorov and Smirnov.