Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#141

Post by cma » Wed, Mar 05, 2014, 9:54 PM UTC

api wrote: post #128

j2kun wrote: post #125

Earlier quoted context omitted.

I agree wholeheartedly with how frustrating it is. I think part of the problem is that really great mathematicians are encouraged to stay as far away from teaching (and improving their teaching) as possible, and great teachers are often discouraged from pursuing more mathematics for a variety of reasons. And when I personally teach calculus I make sure to explain derivatives in the way you want in the very first day…

The problem is really very simple. First you teach the basics of the language. Then you teach how to express concepts in that language and what those concepts mean . Finally, you teach how to manipulate those concepts to build new higher-order forms. Mathematics is taught like this: First, students are shown how to manipulate symbols they do not understand. During this process, sometimes (if you're lucky) these symbo…

I don't think just doing everything from first principles is reasonable. Building up the natural numbers from set theory etc. would just be frustrating for kids.

Like trying to teach them their native language by grammar diagrams etc. instead of immersion

http://en.wikipedia.org/wiki/Language_immersion

Math is not exactly the same as natural language, but there are tradeoffs to doing it one way or the other and there needs to be balance.

Re: Mathematicians are chronically lost and confused

#142

Post by [deleted] » Wed, Mar 05, 2014, 10:01 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…

[deleted]

Re: Mathematicians are chronically lost and confused

#143

Post by TOMDM » Wed, Mar 05, 2014, 10:10 PM UTC

simias wrote: post #61

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…

For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions. Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand w…

This makes me realise the potential that a supposed "Math API" has. Mathematical functions described visually, with unambiguous terms, and even properly named variables so they can be implemented even if you don't understand how they function entirely.

I would love to see a wiki style website that aims to produce such a wealth of information.

Re: Mathematicians are chronically lost and confused

#144

Post by GFK_of_xmaspast » Wed, Mar 05, 2014, 10:14 PM UTC

graycat wrote: post #37

nbouscal wrote: post #27

Earlier quoted context omitted.

You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.

No, I'm correct: He set up an extreme straw man to knock it down. I clearly agreed that his extreme straw man is foolish. There is a common reason students fall for his straw man: They are concerned that if there is an exercise they can't work they are missing something important. My advice was, instead, for a very diligent student, to solve 90-99% of the exercises and just let go of the last few as illposed, stated…

Come on, dogg, re-read the article and see what he's actually arguing.

Re: Mathematicians are chronically lost and confused

#145

Post by CatMtKing » Wed, Mar 05, 2014, 10:16 PM UTC

cma wrote: post #139

simias wrote: post #61

Earlier quoted context omitted.

For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions. Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand w…

This helps so much: http://en.wikipedia.org/wiki/List_of_mathematical_symbols

Hmm, remind me to write a handwriting-recognition program to search through that.

Re: Mathematicians are chronically lost and confused

#146

Post by ska » Wed, Mar 05, 2014, 10:17 PM UTC

api wrote: post #128

j2kun wrote: post #125

Earlier quoted context omitted.

I agree wholeheartedly with how frustrating it is. I think part of the problem is that really great mathematicians are encouraged to stay as far away from teaching (and improving their teaching) as possible, and great teachers are often discouraged from pursuing more mathematics for a variety of reasons. And when I personally teach calculus I make sure to explain derivatives in the way you want in the very first day…

The problem is really very simple. First you teach the basics of the language. Then you teach how to express concepts in that language and what those concepts mean . Finally, you teach how to manipulate those concepts to build new higher-order forms. Mathematics is taught like this: First, students are shown how to manipulate symbols they do not understand. During this process, sometimes (if you're lucky) these symbo…

This may have been how you were taught, and that's unfortunate.

It is not, however, "how mathematics is taught".

Re: Mathematicians are chronically lost and confused

#147

Post by Figs » Wed, Mar 05, 2014, 10:27 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…

> you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.

"If I had more time, I would have written a shorter letter." -- attributed to Pascal.

Re: Mathematicians are chronically lost and confused

#148

Post by makmanalp » Wed, Mar 05, 2014, 10:27 PM UTC

DerpDerpDerp wrote: post #136

makmanalp wrote: post #132

Earlier quoted context omitted.

Hey - thanks for actually replying and taking me seriously. This is as far as this discussion has ever gotten between me and someone who seems to know what they are doing. I'm actually learning a lot. > It's because we have essentially picked the parts of mathematics we're going to force to be true about half way up the stack. If the axioms don't permit those theories, then we'll do away with the axioms and pick a di…

> It blows my mind that you can just pick any set of axioms you like. It feels like there should be a set of core axioms that is the fundamental truth. Think of it like this: it's not that Euclid set out a set of axioms and they just happened to make geometry we could use to talk about triangles, lines, squares, and matched up (to varying degrees) with behavior we see in the world; rather, it's that we saw those rela…

> rather, it's that we saw those relationships and went looking for the minimum set of rules from which they could all be derived. There are (of course) other choices we could have made, which are useful in different cases, such as talking about geometry on a sphere instead of geometry on a flat piece of paper.

OK, this makes sense now!

> It gave a corpus of different approaches, linked up the intuitive notions with formalisms, and led to several generalizations applied to different contexts.

And I guess this gives you multiple points from which to verify new ideas.

> If you need an analogy to what this would be like in a different field, we discovered (modern) formalism around the same time that biology discovered evolution, and formal (mechanical) computation around the same time they discovered DNA

This kinda reminds me of the Bohr atomic model from 1925: it went through many iterations since, a lot of which were kind of true but not quite. They discovered that, oops, the electron orbits are elliptical, not circular. And the more weird and odd particles and effects we discover, the more accurate the model gets, but the original wasn't that bad at modelling reality. Or the Newtonian physics vs modern stuff. Newtonian physics was 90% there, but not quite enough to explain things happening in extreme circumstances that are difficult to observe.

> you might want to check out Category Theory or Homotopy Type Theory. Both are very applicable to programming or science, if either is your field!

Category Theory is definitely on my list of things to learn. HTT seems quite new fangled, I remember seeing that the book made a big fuss on HN recently. I'll have to read more into it to see if it's worth it for me to delve deeper into.

I'd love to hear more about book recommendations! I'm a software engineer and I've had a CS education, so I've had a decent chunk of college-level math (including linalg, diffeqs...), automata theory, etc. So, some exposure to post-1700s math, some formal paper-reading, etc. You can find my e-mail in my HN profile if you like.

Thanks again. Good talk!

Re: Mathematicians are chronically lost and confused

#149

Post by ska » Wed, Mar 05, 2014, 10:32 PM UTC

pygy_ wrote: post #65

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…

> 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. Their intended audience is subject matter experts. Their purpose really is to let the authors show off how smart they are, impress their peers, and a…

I assume you mean the "reduction" process. The "redaction" process is an important part of the scientific process!

I know you were being sarcastic, but I think you are also being a bit unfair. Publication is necessary these days at perhaps an unfortunate rate, I agree.

However at least most research mathematicians publish largely to share ideas. After all, that's the best part of the job. Certainly this is true of the vast majority of them in my experience, and the ones weren't motivated by this tended to be very focused on teaching and very good at it.

Re: Mathematicians are chronically lost and confused

#150

Post by jpfr » Wed, Mar 05, 2014, 10:37 PM UTC

CatMtKing wrote: post #145

cma wrote: post #139

Earlier quoted context omitted.

This helps so much: http://en.wikipedia.org/wiki/List_of_mathematical_symbols

Hmm, remind me to write a handwriting-recognition program to search through that.

Someone already did exactly that. http://detexify.kirelabs.org/classify.html