Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#101

Post by acjohnson55 » Wed, Mar 05, 2014, 6:00 PM UTC

j2kun wrote: post #69

acjohnson55 wrote: post #66

Earlier quoted context omitted.

I'm not sure how addressable that is. While math could be modeled as a DAG globally, I think it is inherently linear locally (no smooth function pun intended) and incremental. Sure you could jump around, but I think at the end of day, if a student is going to progress to advanced math, they can't dodge tricky concepts. But maybe I'm misinterpreting your point. Do you have links to what you've written?

I mean this about the typical subject matter of high school (which is what this branch of the comment thread concerns). Nobody needs to learn how to graph accurate ellipses and the various facts about congruent triangles before doing calculus. You also don't need excellence in algebra to do geometry. There are some fundamentals, like being able to work with fractions, but largely high school education is a lot of par…

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. Graphing ellipses is meant to help understand functional thinking, which is crucial to calculus. Those miscellaneous facts about triangles are as examples to motivate understanding of mathematical logic -- also crucial.

In other words, most of what we learn in math are really intended to illustrate underlying mathematical concepts with some level of concreteness. Otherwise, we'd just start with category theory in kindergarten and derive all other math from that :)

I can certainly understand the perception that these things are often taught solely as ends in and of themselves. I think that part of the challenge is that there is a tradeoff between taking the time to provide a concrete motivation for every math concept upfront versus saving time by dealing with math concepts in their own world to cover more ground. For instance, the seven bridges problem serves as a great motivator for graph theory (and is used very often for this purpose), but can we really afford to find a similar motivating problem for every single graph theoretical concept?

Re: Mathematicians are chronically lost and confused

#102

Post by cabinpark » Wed, Mar 05, 2014, 6:16 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 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

We do. It's called mathematical notation.

Re: Mathematicians are chronically lost and confused

#103

Post by coldpie » Wed, Mar 05, 2014, 6:17 PM UTC

applesinjuice wrote: post #59

coldpie wrote: post #25

Earlier quoted context omitted.

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

Thanks. I reported this bug nearly two years ago :)

https://news.ycombinator.com/item?id=4112327

Re: Mathematicians are chronically lost and confused

#104

Post by GIFtheory » Wed, Mar 05, 2014, 6:18 PM UTC

Reminds me of this great quotation, which Oksendal places before the preface to his stochastic differential equations book:

We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things. Posted outside the mathematics reading room, Tromsø University

Re: Mathematicians are chronically lost and confused

#105

Post by makmanalp » Wed, Mar 05, 2014, 6:18 PM UTC

DerpDerpDerp wrote: post #92

makmanalp wrote: post #55

Earlier quoted context omitted.

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 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. Math papers are written with high compression using standard tables of translations to reduce the proce…

But why present these in English, where you have to manually apply those tables of translations, knowing full well that humans are error prone?

Why not use a computer readable and standardized language like coq / gallina (http://en.wikipedia.org/wiki/Coq), where you can verify the proof unquestionably and immediately AND you can use a compiler to translate the theorem into latex / english / whatever form you want immediately?

If the "standard tables of translations" really exist and are really as standard as you claim, this method is clearly superior and that table can be utilized to translate to "mathematician lingo" if people still desire to stick to that.

Re: Mathematicians are chronically lost and confused

#106

Post by makmanalp » Wed, Mar 05, 2014, 6:24 PM UTC

cabinpark wrote: post #102

makmanalp wrote: post #55

Earlier quoted context omitted.

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 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 We do. It's called mathematical notation.

Mathematical notation is not easily computer parseable / checkable. And I'm not sure it's really standardized (no standards body that I could find) or even unambiguous for that matter. You could maybe call it a de facto standard.

Please read my other comment: https://news.ycombinator.com/item?id=7348666

Re: Mathematicians are chronically lost and confused

#107

Post by charlieflowers » Wed, Mar 05, 2014, 6:32 PM UTC

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 think that would be fantastic. I have envisioned that kind of thing often. Something like "autodidact.stackexchange.com" (which doesn't exist), for example. The best explanations could be voted up. For bigger topics, it would not be explanations you're voting on, but rather "learning roadmaps" -- a recommended set of materials in a recommended order that is designed to help one grok the topic.

For academic papers, perhaps people could vote on two dimensions: (1) papers you'd like to see explanations for, and (2) particular explanations that were effective for you.

And yes, I would read the hell out of it, and yes, if there were areas I could contribute, I would do so.

Re: Mathematicians are chronically lost and confused

#108

Post by pbhjpbhj » Wed, Mar 05, 2014, 6:32 PM UTC

acjohnson55 wrote: post #66

j2kun wrote: post #46

Earlier quoted context omitted.

I think part of the problem is that the math track is way too linear. It doesn't need to be that way, as I've written about a few times in the past.

I'm not sure how addressable that is. While math could be modeled as a DAG globally, I think it is inherently linear locally (no smooth function pun intended) and incremental. Sure you could jump around, but I think at the end of day, if a student is going to progress to advanced math, they can't dodge tricky concepts. But maybe I'm misinterpreting your point. Do you have links to what you've written?

If you look at the Khan Academy lessons they have a map of the lesson structure that's not linear - https://www.khanacademy.org/exercisedashboard.

Re: Mathematicians are chronically lost and confused

#109

Post by DerpDerpDerp » Wed, Mar 05, 2014, 6:37 PM UTC

makmanalp wrote: post #105

DerpDerpDerp wrote: post #92

Earlier quoted context omitted.

> 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. Math papers are written with high compression using standard tables of translations to reduce the proce…

But why present these in English, where you have to manually apply those tables of translations, knowing full well that humans are error prone? Why not use a computer readable and standardized language like coq / gallina ( http://en.wikipedia.org/wiki/Coq ), where you can verify the proof unquestionably and immediately AND you can use a compiler to translate the theorem into latex / english / whatever form you want i…

They're generally published in symbols, which largely have a direct translation in to more formal methods, with the English being included to comment on the motivations, things which might not be formalized in the theory, etc.

The primary purpose of mathematics papers is for distributing information between mathematicians in a form which it's easy for them to integrate in to reasoning about new theorems. To reason about new theorems, you really want as many ideas/concepts to be able to be present in the mathematician's head free of context (ie, with the structure represented, but ignoring the traditional intuition about what it is or used for, which English naming can bias towards).

Part of the problem of Coq is that you'd have to formalize a lot of the metatheory in to something computable, and we've been struggling to find a good approach to that since the 40s/50s. We have a lot of trouble with trying to formalize the axioms in a way that you can compute results from them. (Axiom of choice + axiom of excluded middle can cause problems in Coq, for instance.)

A secondary concern is that the translations from full sequences of axiomatic steps to constructs which are higher level (ie, built on the idea that some axiomatic steps must exist in this instance, but we haven't found them explicitly) can be really complicated, as well as much bulkier.

A math paper is usually no more than tens of pages, while a computer proof of a theorem can run in to the thousands.

Re: Mathematicians are chronically lost and confused

#110

Post by kalid » Wed, Mar 05, 2014, 6:43 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…

Exactly, many lessons are written to be defended from assault, not to be inviting. A castle, while romantic, is not as comfortable as a hotel.

(My personal mission is to find/share the aha! moments that actually make the details click. Why do we force everyone to laboriously discover them for themselves? Can't want talk about the underlying insights directly?)