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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. H
by makmanalp 13y ago
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 13y agoI 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.
- DerpDerpDerp 13y ago> 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 13y agoBut 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 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 13y agoThey'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.
- cabinpark 13y ago> 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 13y agoMathematical 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 https://news.ycombinator.com/item?id=7348666
- cabinpark 13y agoSo? 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 13y agoIn 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.