Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#111

Post by cabinpark » Wed, Mar 05, 2014, 6:47 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…

> 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.

You have the benefit of hindsight. Everything is obvious in hindsight. All the research I've done is obvious and straight forward, if only I had known what I know now and would be able to draw a few simple diagrams.

That is, until you realise it's not. In those days and weeks spent deciphering mathematics you are actually learning a lot. I cannot count how many times I've been reading mathematics and struggled for weeks on a concept. Then one day it clicks and it all makes sense. Then I re-read the description again and the answer is clear as day. The answer was always there, I just hadn't learnt enough to appreciate it.

>So many ideas seem closed to mortals because of the nature of mathematics.

I disagree with this statement 100%. No ideas are closed because of mathematics. The ideas are only closed if you are not willing to put in the time.

Re: Mathematicians are chronically lost and confused

#112

Post by countrybama24 » Wed, Mar 05, 2014, 6:51 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?

Yes. Although the field of "academic papers" is significantly broader with a smaller user demographic than hip-hop, but if you could pull it off I would be an avid user.

Re: Mathematicians are chronically lost and confused

#113

Post by cabinpark » Wed, Mar 05, 2014, 6:52 PM UTC

makmanalp wrote: post #106

cabinpark wrote: post #102

Earlier quoted context omitted.

> 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

So? Why does it have to be computer parseable and checkable? I've never heard anyone complain about that before.

As for your other comment, DerpDerpDerp provides a good answer.

Re: Mathematicians are chronically lost and confused

#114

Post by nbouscal » Wed, Mar 05, 2014, 7:05 PM UTC

nnq wrote: post #95

nbouscal wrote: post #24

Earlier quoted context omitted.

I think it can in theory, but the reason it hasn't been done in practice is basically that mathematics is too damn big. Look at Whitehead and Russell's original attempt to do basically exactly what you're asking for: it took them a few hundred pages to build up enough machinery to prove that 1+1=2.

I never got to why this kind of work was stopped. I only heard the same argument (this is the phrasing from Wikipedia but everybody seems to say something along these lines): >"However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there wou…

It wasn't stopped, it's just way, way harder than one would naively expect. Russell continued to work on it by developing type theory, which was later carried on by people like Alonzo Church and Per Martin-Löf. This led not just to types in programming, but also to the use of types in proof checkers like Coq and Agda, which are very much the kind of thing you're talking about. Efforts are ongoing to continue working towards the vision of automated theorem proving, but again, it is an extremely difficult problem.

Re: Mathematicians are chronically lost and confused

#115

Post by baby » Wed, Mar 05, 2014, 7:07 PM UTC

That's what I tell people around me. Studying math is hard because it makes you feel stupid. You always feel lost, you always feel like you missed so many things when you're starting to learn a new thing, you always feel like your questions are stupid (until you get that the rest of the class is pointless as well).

Especially with talented professors (Lyon 1, France, the professors there are not really good educators, but they are geniuses), they make you feel bad for not understanding things that seem so simple to them.

Studying math is depressive if you take it too seriously.

Re: Mathematicians are chronically lost and confused

#116

Post by nbouscal » Wed, Mar 05, 2014, 7:07 PM UTC

nnq wrote: post #99

alex-g wrote: post #53

Earlier quoted context omitted.

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., yo…

> 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.

The vast majority of working mathematicians work in set-theoretic foundations, in which numbers are sets.

Re: Mathematicians are chronically lost and confused

#117

Post by countrybama24 » Wed, Mar 05, 2014, 7:16 PM UTC

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…

No, thank you for teaching math to at-risk kids. Seriously, bravo.

Re: Mathematicians are chronically lost and confused

#118

Post by makmanalp » Wed, Mar 05, 2014, 7:18 PM UTC

DerpDerpDerp wrote: post #109

makmanalp wrote: post #105

Earlier quoted context omitted.

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 ab…

Your reply is fascinating - the part about size difference and the difficulty in expressing certain things in theorem provers.

About the size difference, I don't understand why it takes so much longer. What is so fundamentally different about Coq (or E or whatever) that it takes so much more space that just specifying it with mathematical notation?

Is it because you have to start from scratch? Has no one created a "standard library of existing theorems / proofs" that one can depend on?

Or is it because mathematical notation is just that much more expressive?

So if verifying proofs is that difficult, what the heck are mathematicians doing when they read a paper? Not really actually verifying the proof? It's hard to believe that verification is one of those things that humans are just better at. (As opposed to, say, formulating new ideas, which is fundamentally difficult for computers)

What is it that is so fundamentally difficult about verifying proofs formally, when a mathematician can just read a paper and call it done?

If provable theorems are too hard, why don't we simply just start with a parseable mathematical notation standard? It's be pretty similar conceptually, but would be standardized. Surely that's not too difficult? That could still serve as the portion of the paper that is the "transmitting new findings formally" part, whereas the remaining part can be devoted to explanation.

Re: Mathematicians are chronically lost and confused

#119

Post by j2kun » Wed, Mar 05, 2014, 7:21 PM UTC

acjohnson55 wrote: post #101

j2kun wrote: post #69

Earlier quoted context omitted.

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. 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 their own questions about graphs, try to answer them, or come up with their own relations to the real world. This is where I think a lot of the critical thinking happens, not in learning facts about graphs. The facts (what degree means, what planar graph means, etc) come as a byproduct of following these paths of thought.

Re: Mathematicians are chronically lost and confused

#120

Post by SilasX » Wed, Mar 05, 2014, 7:24 PM UTC

logicallee wrote: post #85

jseliger wrote: post #81

Earlier quoted context omitted.

The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal. In other words, the purpose is primarily tenure and advancement. This may be a problem.

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…

That would be more convincing if the scientific papers were written in a way that make the point as clearly as the coded-up version of Monty Hall problem. In practice, it's more like they publish the assembly code and when you ask why they didn't do it in Python or something, they lecture you about the need for formal rigor.