[makmanalp]

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 lack of intuitive explanation) and low ambiguity (hence the seemingly backwards order of explaining things and the pages of "a is blah, b is foo").

It's an interesting discussion because from the mathematician's perspective, they don't see why they should cater to anyone who doesn't bother absorbing the lingua franca and the method of delivery. Countering that is the philosophical argument that information should be as available as possible. Countering that is how practical and useful that is, and whether the cost / benefit would be worth it.

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.

Then the explanation can reside alongside this unambiguous description, and can take whatever liberties it pleases.

[jorleif]

I think the process of peer review often causes papers to be optimized for being hard to critique rather than easy to understand. I have myself participated in writing papers where we decided to leave out some non-crucial but very useful detail just because it opens up too many questions and opportunities for critique.

[nly]

So "let's keep the source code closed so nobody can see how awful we are and just show off our wonderful working program"

[pseut]

The process of peer review also optimizes papers to be shorter than intended.

[j2kun]

Is this in mathematics or another field?

[DerpDerpDerp]

> 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 processing load when trying to manipulate several things at once.

[makmanalp]

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.

[DerpDerpDerp]

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.

[jk4930]

"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

[makmanalp]

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.

[DerpDerpDerp]

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.)

[makmanalp]

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 different set.

Is this related to the famous incompleteness theory? If you hit a proof that you can't prove with this set of axioms, you just try another one? 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. Maybe it's time that I read G.E.B and the Principia. Any recommendations for these kinds of questions?

> 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?"

Okay, so people are working on it. It seems like a miracle that the alternative of "we're doing it all in our heads" actually works:

> 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.

This is what I'm taking about. What if - somewhere in the middle - there was a mistake?

...

> 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.)

My main concern is - every year we get someone who thought they proved, for example, P != NP, only to find a few months and a billion man hours later that there is a minute error on page thirty-five where the author misunderstood and overlooked some very subtle thing. Wouldn't it me much more pleasant if this task was automated?

Wouldn't it be amazing if we could use machine learning methods or tree search methods to formulate a proof mechanically? Or automatically eliminate proof steps to make them less complicated! Compiler-style optimization if you will. Wouldn't it be amazing if thousands of existing proofs could be analyzed statistically? Wouldn't it be great if we could machine-translate standardized proofs into regular ol' mathematical notation or whatever language we want? When we have the ability in the future, we can go back and verify them all en masse! Am I deluded in thinking any of this could be valuable?

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

Well, at least programming languages without standards have an implementation that is the official implementation! So arguably, they are more standardized.

Anyway, thanks for replying, I haven't had such an interesting exchange on HN in a while. I'm eagerly awaiting your response (if you have the time)!

edit: Also it seems like learning a standardized mathematical notation would be only marginally harder that learning to typeset regular mathematical notation in latex! Heck, you could have it compile down to latex.

edit2: Another thought I had was, since mathematicians don't prove things all the way from the bottom axioms up, but from the closest accepted truth, couldn't we have verification systems do that instead? That seems a more easily reachable goal.

[cabinpark]

> 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.

[makmanalp]

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

[cabinpark]

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.

[eli_gottlieb]

In academia, there's some notion that eventually a teaching professor will consolidate many papers into one textbook, which will actually convey the material clearly for those without a graduate-level maths education in the specific subfield.

Occasionally, this even happens.