Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Mathematicians are chronically lost and confused

j2kun.svbtle.com

Post reply on HN
Print view View original

151–160 of 200 posts

Re: Mathematicians are chronically lost and confused

#151

Post by vargas84 »

vargas84 wrote: post #93
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…

Would anyone be interested in seeing a "Rap Genius for academic papers" to address the third paragraph of parent comment? To those who aren't familiar with Rap Genius, I basically just mean a place where academic papers can be annotated and explained in plain language by the community. Would you read/contribute?

I appreciate the quick feedback - I myself would love to use such a service. Just to clarify, I don't have the development experience to build this myself yet (I just switched to CSE from Finance this year) but if anyone else would be inclined to give it a shot, I'd love to help. Otherwise, free idea!

Re: Mathematicians are chronically lost and confused

#152

Post by ska »

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…

"and it pretty much just boils down to intuitive addition, multiplication, and division."

Or does it? After all, probability is one of the very first mathematical tools to be divised, but a rigorous theoretical underpinning for probability and statistics had to wait until measure theory, millenia later. And this was not for want of trying.

Part of what makes things "boil down" to the simple and intuitive is years of hard work. Reading Newtons original work on the calculus is painful and convoluted, it took many hands to polish it to the point you might have seen it in.

If we've done our jobs well as mathematicians, eventually the essence of an idea will be easy to understand an apply. If you really want to understand it though, you may have to dig into some much deeper work. And often, as with many thing is in mathematics, human intuition will just tend to be wrong about it (e.g. the Monty Hall problem).

Re: Mathematicians are chronically lost and confused

#153

Post by Crito »

johnbm wrote: post #138
logicallee wrote: post #85

Earlier quoted context omitted.

The issue is that a lot of intuitive stuff is wrong. When you formalize, you remove the simple, intuitive explanation - but you make it much harder for you to remain wrong, if you are wrong - or to become wrong, if you started off right. As a simple explanation, consider the difference between explaining the Monty Hall problem - which might seem to be philosophical, open to interpretation - and coding it up. The mome…

Nah, Monty Hall is trivial to demonstrate. Just do it with 100 doors instead of 3. Problem solved, intuition remains.

Such that the contestant chooses 1 door, the host then opens 98, and they are given the opportunity to switch to the last remaining door?

That's actually a pretty brilliant way of explaining it. With numbers like that the answer becomes much more intuitive.

Re: Mathematicians are chronically lost and confused

#154

Post by nnq »

alex-g wrote: post #130
nnq wrote: post #99

Earlier quoted context omitted.

> 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. Can't we do this in current mathematics?! I mean, no physicist or engineer ever thinks of numbers as sets, even if you are the kind of physicist that reads and understands mathematical proofs.

Right, this is how mathematics really works. But formalizations of mathematics may suffer from leaky abstractions. If we prove facts about numbers by compiling them into sets, and then using set-theoretic axioms, we might accidentally make it possible to prove things about numbers that are incorrect or meaningless.

Isn't this an abstraction problem that you solve by simply providing an "interface" or equivalent concept or access specifiers in oop like private/protected? All other modules that use the number module for applied math will just see an "interface" (let's call it GeneralNumber - as far as I know there are a few other alternate ways of defining numbers besides sets, right?), and the particular "implementation of numbers as sets".

For more abstract algebras or who knows what, the "numbers module" might also implement another more advanced interface that exposes more of the implementation, a "SetsNumber" interface. If you know have a proof that uses this interpretation of number that is tied to one particular "implementation", then there is nothing incorrect about it leading to weird or "meaningless" results, they would be correct for SetsNumber but not for GeneralNumber (or someone might need to take a good look and see if they can be made to work for GeneralNumber too).

(I know, the words are all wrong, it probably sounds either "all wrong" or like a gibberish to mathematicians that don't also happen to be programmers ...someone should figure out more appropriate terms :) )

And about leaky abstractions, I think they happen a lot in software because of the tradeoffs we make, like 'but we also need access to those low level stuff to tweak performance', 'but we need it done yesterday so it's no time to think it through and find the right mode' or 'our model has contradictions and inconsistencies but it's good enough at delivering usable tools to the end-user, so we'll leave "wrong" because we want to focus on something that brings more business value right now' etc. Also, there's a biggie: for some problems using no abstraction is not good enough (initial developing/prototyping speed is just to small), but if you figure out the right abstraction it will end up being understandable only by people with 'iq over n' or 'advanced knowledge of hairy theoretical topic x', and you can't hire just these kinds of people to maintain the product, so you knowingly choose something that's leaky but works and can be maintained by mediocre programmers, hopefully even outsourced :)

Re: Mathematicians are chronically lost and confused

#155

Post by acjohnson55 »

j2kun wrote: post #119
acjohnson55 wrote: post #101

Earlier quoted context omitted.

I just browsed your post, and it looks beautifully written! So you're saying there's nothing fundamental about the typical HS math sequence. I agree. But I also don't think there's that much of a compelling reason to change it, because there are going to be difficult portions no matter how you arrange it. But I think it's not exactly true that ellipses and congruent triangles have nothing to do with calculus. Graphin…

Ellipses aren't functions :) I think if everyone agreed that the goal is to teach critical thinking skills, and have the factual knowledge be a byproduct (and elementary facts are very easy to pick up if you have critical thinking skills), then it would make a world of difference. As to the motivations, after the students get going they don't need more real world motivation. They seem to be interested enough to ask t…

Ellipses can be functions, if plotted using polar coordinates :)

Re: Mathematicians are chronically lost and confused

#156

Post by [deleted] »

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…

[deleted]

Re: Mathematicians are chronically lost and confused

#157

Post by csours »

japhyr wrote: post #41

I currently teach math to at-risk students. I don't read all of these submissions about math education, but I skim the comments on most of them. The comments people make change the way I teach math. I have always done a decent job of teaching math. I focus on helping students understand concepts, even when they are focusing on mechanics. I use words like "shortcut" and "more efficient method" rather than "trick" when…

I love stories like these. I haven't gone much past calculus, what stories can I look for that will take me farther?

Re: Mathematicians are chronically lost and confused

#159

Post by logicallee »

Crito wrote: post #153
johnbm wrote: post #138

Earlier quoted context omitted.

Nah, Monty Hall is trivial to demonstrate. Just do it with 100 doors instead of 3. Problem solved, intuition remains.

Such that the contestant chooses 1 door, the host then opens 98, and they are given the opportunity to switch to the last remaining door? That's actually a pretty brilliant way of explaining it. With numbers like that the answer becomes much more intuitive.

Why 100? Why not just 5? Some people would 'get' it at five, some people at 100, and some people at a million. If you have to choose out of a million doors, and no matter what the host opens all but one of them, so that your prize is either behind the door you picked, or behind the other one -- then should you switch your choice?

Well, obviously, you should - with a million doors, it becomes obvious that you have just a 1 in 1,000,000 chance of having picked it.

But thing is - that "obvious" thing 'should' be just as obvious with 1000 doors, 100, 20, 5, or...3....

It's a matter of degree - not kind.

So appealing to a way of intuiting it that is a lot more 'obvious' - while in fact having the exact same format of question, just goes to underscore how fickle intuition can be.

That said, taking individual variables to ridiculous extremes is a great way to thought experiment and an awesome way to get intuition to work better.

Re: Mathematicians are chronically lost and confused

#160

Post by ColinWright »

johnbm wrote: post #138
logicallee wrote: post #85

Earlier quoted context omitted.

The issue is that a lot of intuitive stuff is wrong. When you formalize, you remove the simple, intuitive explanation - but you make it much harder for you to remain wrong, if you are wrong - or to become wrong, if you started off right. As a simple explanation, consider the difference between explaining the Monty Hall problem - which might seem to be philosophical, open to interpretation - and coding it up. The mome…

Nah, Monty Hall is trivial to demonstrate. Just do it with 100 doors instead of 3. Problem solved, intuition remains.

I never understood this assertion. Most people I explain it to in this way still think it's 50:50 because you only have two doors left.
Post reply on HN