Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#121

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

makmanalp wrote: post #118

DerpDerpDerp wrote: post #109

Earlier quoted context omitted.

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 "s…

You seem to imply that mathematical notation should be standardized, but are omitting that even languages explicitly meant for computation are not.

Why do you think mathematics should be more standardized than programming? (Actually, I'd argue it's already more standardized than programming, and you're arguing for some kind of extreme position.)

Sorry, forgot to reply to part I had meant to:

> 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?

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 different set. (And perhaps explore why they failed to, and what is required in axioms to enable those theorems.)

As I mentioned before, mathematics already has a good way to relate high level theorems and such to these mid-level structures, and you see it employed all the time. The problem is that many of these structures aren't easy to compute with, so we're essentially having to work backwards to find a set of formalities that we can both do mechanistically and support the theorems/propositions we'd like to be true.

Mathematicians generally don't evaluate the truth of a new paper relative to the axioms, but relative to the already established results in a field. So you question about why libraries don't exist is essentially "Why have mathematicians not replicated hundreds or thousands of years of effort in to a format that's hard for them to personally use, but is good for these tools we've developed in the past couple decades?"

Well, people are working on it, but it's going to take some time. And the moment you pick a slightly different set of axioms, you need to rebuild large portions of the library you allude to, even if the results are still true.

(Derivations in terms of base steps are considerably longer than most mathematics proofs would be, which often omit some "standard" kinds of details. For an idea of what this is like, read portions of Principia Mathematica by Russell and Whitehead.)

Re: Mathematicians are chronically lost and confused

#122

Post by drivers99 » Wed, Mar 05, 2014, 7:26 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…

I don't think it's an extreme straw man, because I saw notebooks of my math teacher in high school where he decided to take classes and did exactly that, just to make sure he knew it really well. Just the other day I was wondering if I should try to do the same thing in order to get good at math.

Re: Mathematicians are chronically lost and confused

#123

Post by GFK_of_xmaspast » Wed, Mar 05, 2014, 7:31 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…

It's supposed to be read and written by people who have background and context, that's why. see also http://en.wikipedia.org/wiki/Polysemy

Re: Mathematicians are chronically lost and confused

#124

Post by enupten » Wed, Mar 05, 2014, 7:42 PM UTC

fidotron wrote: post #9

tokenadult wrote: post #8

Earlier quoted context omitted.

I'd be a lot more worried about the danger you mention if you could give even one example of that happening, ever. What ideas are mathematicians confusing each other into accepting that are basically nonsensical?

Cantor's conception of transfinite numbers is the one that I think has done most damage.

Is that you Henri ?

Re: Mathematicians are chronically lost and confused

#125

Post by j2kun » Wed, Mar 05, 2014, 7:43 PM UTC

api wrote: post #90

I completely agree about the power of math, and why programmers should learn it. There are two problems with math: (1) Math is IMHO the worst taught of all academic subjects. It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in th…

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 (before describing limits or anything else).

As to your second point, I think notation is a big problem, but it's a bit of a straw man. With very few exceptions that I doubt you would ever find yourself in, I have never met a professor or mathematician that would not explain notation if you asked (gladly stopping in the middle of a lecture or talk to clarify). There is still a lot of it, but every mathematician who is presenting the mathematics can explain the notation to any degree of precision you could ever want, and I have very few colleagues who have never stopped someone for this reason.

I think the bigger problem is trying to read mathematics by yourself, without the ability to ask questions. And even after understanding the notation, I feel programmers have bigger problems, which I've expanded more on in this post [1], the main difference between learning programming being there are simply more free and open resources for learning programming. This is probably because programmers invented the internet and filled it with their favorite content first.

But one point I make is that mathematical notation is inherently ad-hoc, and the only kinds of notation that stick around are the kinds that get used ad-hoc enough times to become standard. And even then people will make up their own notation for no other reason than that it's their favorite (Physicists are really good at this, and perhaps ironically it drives mathematicians crazy). Because of that (and because notation is introduced often to be rigorous, not to explain a concept) you're unlikely to ever find such a dictionary. Sorry :(

[1]: http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...

Re: Mathematicians are chronically lost and confused

#126

Post by j2kun » Wed, Mar 05, 2014, 7:45 PM UTC

trevorhartman wrote: post #94

Jeremy, I really appreciate this post and all the excellent content over at Math ∩ Programming. Thanks, and please keep it up!

I'm just so happy that I get to read everyone's interesting stories and thoughts in the HN comment threads! HN is really one of the highest-quality places for discussion on the web ^_^

Re: Mathematicians are chronically lost and confused

#127

Post by j2kun » Wed, Mar 05, 2014, 7:46 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 think that, for whatever reason, people tend to think mathematics is somehow different.

Re: Mathematicians are chronically lost and confused

#128

Post by api » Wed, Mar 05, 2014, 7:48 PM UTC

j2kun wrote: post #125

api wrote: post #90

I completely agree about the power of math, and why programmers should learn it. There are two problems with math: (1) Math is IMHO the worst taught of all academic subjects. It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in th…

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 symbols are explained in a piecemeal and oblique way. Sometimes conceptual meaning is discussed at the end to wrap things up (oh by the way this is what you'd use this for, now let's move on), but this is rare. Mostly you just get elaborate dances of symbols thrown at you with no explanation to tie what you're doing to any problem, reality, or conceptual meaning. In the end most students end up memorizing these meaningless opaque incantations and never understand why anyone would be interested in math.

Re: Mathematicians are chronically lost and confused

#129

Post by jk4930 » Wed, Mar 05, 2014, 7:53 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…

"a computer proof of a theorem can run in to the thousands."

For those interested in a small example, look at http://us.metamath.org/mpegif/mmset.html#trivia

Re: Mathematicians are chronically lost and confused

#130

Post by alex-g » Wed, Mar 05, 2014, 8:12 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.

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.