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I for one welcome the "dryness" of mathematical writing. It feels clean, like reading a story without distracting ads. A beautiful advice that I received as a

I for one welcome the "dryness" of mathematical writing. It feels clean, like reading a story without distracting ads. A beautiful advice that I received as a student was to write mathematics as a series of definitions, propositions and proofs. No text is allowed to exist outside of these three. In practice it is difficult to enforce, but it is helpful to keep this as an aim.

taneq 5y ago
As a non-mathematician this seems to be lacking a little something. Would you not want your paper to provide some indication of what you're attempting to communicate and why? Or is that information (as the joke goes) possible, and therefore trivial, to infer?
- enriquto 5y ago
  A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions. I love the analogy with cooking recipes in another comment. Math papers are like recipe books, deliberately devoid of their social context. The same recipe may mean different things to different cooks, even contradictory! Having a clean, neutral description of the recipe allows both cooks to safely refer to the exact same recipe, without endorsing contexts that they may find odious. I agree that knowing the context in which a recipe was created, and the contexts where it has been used, is very useful. But it would be extremely annoying to have this explanation interleaved with the recipe description itself. This information is best kept separate. Then, at the beginning of the recipe you can have a list of links to cooks who have written different things about it.
  - abdullahkhalids 5y ago
    > A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions. Plenty of physics papers end up being useful in ways the authors never conceived. I don't think the authors writing down their best guess of the importance of their results, prevents others from using the results or techniques in alternate ways. In fact, I frequently see this happening. But the authors writing down their best guess of the importance of their results helps others judge the minimum importance of the results, and decide whether they want to read the paper in the first place.
    - enriquto 5y ago
      > But the authors writing down their best guess of the importance of their results helps others judge the minimum importance of the results, and decide whether they want to read the paper in the first place. In many cases it will be unlikely that the authors even know the importance of the results. Forcing them to come up with an explanation would be useless at best, and an unbearable burden at worst (e.g., when a student has proved a theorem proposed by his advisor). There are mathematical reviews, where mathematicians with a higher-level view on the field point to particular papers and explain their importance. Also, many journals have an editorial column that explains the papers on each issue.
  - hasmanean 5y ago
    While I was teaching c++ I would sometimes look at programs from a colleague and try to rewrite them in a more teachable form using c++14. The abstraction capabilities in modern c++ made it more fun and allowed me to condense programs in totally unexpected ways. More abstract code was simpler and sometimes easier to understand. But after a while (usually my version 4 or 5) if I abstracted it too much (Templatized it, made work for Unicode or char types) etc. the complexity would shoot up again. It became incomprehensible even to me even though I had written it. I knew it worked because it gave the same result but it was no longer anchored/rooted in anything real. There is a zone of usefulness that is between the completely abstract and the concrete. Physics lives in this zone…it uses mathematics but they don’t feel the need to generalize to N dimensions or to assume the constants of nature are variable—unless they have to. This is why all good descriptions of mathematics start by showing a concrete problem they want to solve. Gauss invented the FFT algorithm to simplify his calculations of the orbit of Ceres. He had the numbers in front of him and tried to reduce his computational workload by exploiting a repeated pattern in the computations. Teaching the FFT as a fait acompli and showing asteroid orbit as a sample application is ass backwards.
    - taneq 5y ago
      Reminds me of my general principle of the best solutions being "square" in the sense that the effort you expend on them is quite similar on any axis. By the time you're past abstraction's diminishing returns for the given problem, you're spending way more effort on that than anything else (which is understandable - it's fun!) and your efforts aren't "square" any more. (A toy example of the 'square' thing - imagine you have to make 10,000 widgets for 1 dollar each. Making them all by hand for $1 without optimizing the process would be inefficient. So would spending $9,999 to build a machine which could make widgets for $0.0001. But spending $100 to optimize production so you can make the widgets for $0.01 each is a massive win ($200 cost vs. $10,000 for the other alternatives.)
      - hasmanean 5y ago
        That is an interesting thought. What’s it an example of? Convex optimization?
  - 085335977350 5y ago
    @
- romwell 5y ago
  As a mathematician, let me assure you that the point of view in that comment is not popular amongst mathematicians. You are absolutely right, context is king.
ouid 5y ago
Trying to study a very abstract concept without any examples is almost useless. There is, perhaps, some philosophical sense in which we could only be writing down mathematics for god to read, but humans searching the proposition space necessarily move heuristically. Formal generalization is fine (and even great) for working with objects, but pretty terrible for recognizing them. For example, (and also meta-example), most people do not need to spend a lot of time convincing themselves that the pigeonhole principle is true, at least in the finite case, but show most people a problem that it solves, and they will be unable to do so.
romwell 5y ago
I'm sorry, but this is a horrible advice. It sacrifices communication for the sake of enforcing an arbitrary aesthetic. Until mid-20th century, mathematics had never been communicated in this austere manner. When you are a working mathematician, you never start with a definition. You start with a context in which your exploration begins. It might be a question which someone else asked that interests you (and there is a story as to why). It might be that you don't even have a question, but the objects of your study are not studied enough, so you hope you stumble into one. You can easily tell why this is an interesting thing to look at. You do some calculations, take a look at a few examples, see if you can make a stab at the chaos in front of your eyes and find a pattern, which we formally call a conjecture. Then you see if this pattern holds in other cases, and why. This shapes the backbone of the proof. Once you have a basic idea of a proof, you can formulate a theorem. In the formulation, you list all the conditions to which your proof applies. The pattern might be much more general, but your proof might work e.g. "only in cases when the order of the group is invertible in the base field", or some other. After all the work is effectively done, you decide that a concept that appears repeatedly in your reasoning deserves to have a name. Something convenient to call it by, so you don't have to repeat yourself. So you make a definition. You then decided to share your joy with this world, and write a paper. You listen to the "beautiful advice", and throw out anything that makes your paper interesting. Context goes out of the window, along with any hope for the reader to have any idea why your paper is worth looking at. At best, you'll advertise this in talks, or explain over beers. Side note: you will have to drink a lot of beers to make it in math. Then, you follow the advice again, and lay things out in the order exactly opposed to the one you were thinking in: definition - theorem - proof - conjectures - examples - context. Wait, you already scrapped context, and the examples you started with aren't illustrating the results you ended up proving. So you tidy up your paper, come up with more specific examples, and remove anything that wasn't on the direct path to your result. Having climbed to a place where you can see better, you pull the ladder up. Good luck to anyone outside the group of five people who are actively working in this niche! And finally, you write an abstract to your paper, where you mention the things you defined. The abstract doesn't make any self by itself, one needs to be in-the-know to get half of it, and read your paper to understand it. In practice, it acts as a "No Trespassing" sign for the outsiders (i.e. anyone not in direct contact with the five people you have beers with at the Annual Niche Field Conference). Satisfied, you lean back and post it to arXiv. It's been a beautiful day, you think. This practice is difficult to enforce, but you kept it as an aim, and got pretty close to perfection (as exemplified by a Bourbaki text, or anything by Serge Lang, but I repeat myself). Somewhere not too far away, a student in the class you're teaching cries. ---------------- I said "you", but as someone who's written a couple of math papers, that's really me too. We are all taught in a horrendously backwards (literally!) manner. This perversion of the beautiful art isn't a new observation. I can't write better about it than Vladimir Arnold[1] (a titan whose name is, I hope, familiar to you). It's worth a read to anyone who has ever studied mathematics: [1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
krisoft 5y ago
> A beautiful advice … Oh my goodness! So there are actual people who prefer that? I had to chew my way a fair share of books like that and I can’t stand them. Clearly the person who thought of these definitons, propositions and proofs had some reason to think of them. Sometimes they were trying to solve a problem, sometimes they were combining ideas, sometimes they were looking for structures with certain aesthetic properties. There was always a why behind why they thought about this and not something else. Sometimes there are multiple possible such reasons, that is fine. In that case the author can select whichever they fancy the most. Every time i had the misfortune to read a book like what you describe it felt like I was eating powdered milk without reconstituting it. There was a clear chain of thought between ideas and they choose to just hide it. I understand that math requires work. One needs to get a paper and a pencil and work out examples, check proofs, play with definitions. But why wouldn’t the author write down what made them care about the next item? > it is helpful to keep this as an aim. Why?

taneq 5y ago

As a non-mathematician this seems to be lacking a little something. Would you not want your paper to provide some indication of what you're attempting to communicate and why? Or is that information (as the joke goes) possible, and therefore trivial, to infer?

enriquto 5y ago
A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions. I love the analogy with cooking recipes in another comment. Math papers are like recipe books, deliberately devoid of their social context. The same recipe may mean different things to different cooks, even contradictory! Having a clean, neutral description of the recipe allows both cooks to safely refer to the exact same recipe, without endorsing contexts that they may find odious. I agree that knowing the context in which a recipe was created, and the contexts where it has been used, is very useful. But it would be extremely annoying to have this explanation interleaved with the recipe description itself. This information is best kept separate. Then, at the beginning of the recipe you can have a list of links to cooks who have written different things about it.
- abdullahkhalids 5y ago
  > A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions. Plenty of physics papers end up being useful in ways the authors never conceived. I don't think the authors writing down their best guess of the importance of their results, prevents others from using the results or techniques in alternate ways. In fact, I frequently see this happening. But the authors writing down their best guess of the importance of their results helps others judge the minimum importance of the results, and decide whether they want to read the paper in the first place.
  - enriquto 5y ago
    > But the authors writing down their best guess of the importance of their results helps others judge the minimum importance of the results, and decide whether they want to read the paper in the first place. In many cases it will be unlikely that the authors even know the importance of the results. Forcing them to come up with an explanation would be useless at best, and an unbearable burden at worst (e.g., when a student has proved a theorem proposed by his advisor). There are mathematical reviews, where mathematicians with a higher-level view on the field point to particular papers and explain their importance. Also, many journals have an editorial column that explains the papers on each issue.
- hasmanean 5y ago
  While I was teaching c++ I would sometimes look at programs from a colleague and try to rewrite them in a more teachable form using c++14. The abstraction capabilities in modern c++ made it more fun and allowed me to condense programs in totally unexpected ways. More abstract code was simpler and sometimes easier to understand. But after a while (usually my version 4 or 5) if I abstracted it too much (Templatized it, made work for Unicode or char types) etc. the complexity would shoot up again. It became incomprehensible even to me even though I had written it. I knew it worked because it gave the same result but it was no longer anchored/rooted in anything real. There is a zone of usefulness that is between the completely abstract and the concrete. Physics lives in this zone…it uses mathematics but they don’t feel the need to generalize to N dimensions or to assume the constants of nature are variable—unless they have to. This is why all good descriptions of mathematics start by showing a concrete problem they want to solve. Gauss invented the FFT algorithm to simplify his calculations of the orbit of Ceres. He had the numbers in front of him and tried to reduce his computational workload by exploiting a repeated pattern in the computations. Teaching the FFT as a fait acompli and showing asteroid orbit as a sample application is ass backwards.
  - taneq 5y ago
    Reminds me of my general principle of the best solutions being "square" in the sense that the effort you expend on them is quite similar on any axis. By the time you're past abstraction's diminishing returns for the given problem, you're spending way more effort on that than anything else (which is understandable - it's fun!) and your efforts aren't "square" any more. (A toy example of the 'square' thing - imagine you have to make 10,000 widgets for 1 dollar each. Making them all by hand for $1 without optimizing the process would be inefficient. So would spending $9,999 to build a machine which could make widgets for $0.0001. But spending $100 to optimize production so you can make the widgets for $0.01 each is a massive win ($200 cost vs. $10,000 for the other alternatives.)
    - hasmanean 5y ago
      That is an interesting thought. What’s it an example of? Convex optimization?
- 085335977350 5y ago
  @
romwell 5y ago
As a mathematician, let me assure you that the point of view in that comment is not popular amongst mathematicians. You are absolutely right, context is king.

ouid 5y ago

Trying to study a very abstract concept without any examples is almost useless. There is, perhaps, some philosophical sense in which we could only be writing down mathematics for god to read, but humans searching the proposition space necessarily move heuristically. Formal generalization is fine (and even great) for working with objects, but pretty terrible for recognizing them. For example, (and also meta-example), most people do not need to spend a lot of time convincing themselves that the pigeonhole principle is true, at least in the finite case, but show most people a problem that it solves, and they will be unable to do so.

romwell 5y ago

I'm sorry, but this is a horrible advice. It sacrifices communication for the sake of enforcing an arbitrary aesthetic. Until mid-20th century, mathematics had never been communicated in this austere manner. When you are a working mathematician, you never start with a definition. You start with a context in which your exploration begins. It might be a question which someone else asked that interests you (and there is a story as to why). It might be that you don't even have a question, but the objects of your study are not studied enough, so you hope you stumble into one. You can easily tell why this is an interesting thing to look at. You do some calculations, take a look at a few examples, see if you can make a stab at the chaos in front of your eyes and find a pattern, which we formally call a conjecture. Then you see if this pattern holds in other cases, and why. This shapes the backbone of the proof. Once you have a basic idea of a proof, you can formulate a theorem. In the formulation, you list all the conditions to which your proof applies. The pattern might be much more general, but your proof might work e.g. "only in cases when the order of the group is invertible in the base field", or some other. After all the work is effectively done, you decide that a concept that appears repeatedly in your reasoning deserves to have a name. Something convenient to call it by, so you don't have to repeat yourself. So you make a definition. You then decided to share your joy with this world, and write a paper. You listen to the "beautiful advice", and throw out anything that makes your paper interesting. Context goes out of the window, along with any hope for the reader to have any idea why your paper is worth looking at. At best, you'll advertise this in talks, or explain over beers. Side note: you will have to drink a lot of beers to make it in math. Then, you follow the advice again, and lay things out in the order exactly opposed to the one you were thinking in: definition - theorem - proof - conjectures - examples - context. Wait, you already scrapped context, and the examples you started with aren't illustrating the results you ended up proving. So you tidy up your paper, come up with more specific examples, and remove anything that wasn't on the direct path to your result. Having climbed to a place where you can see better, you pull the ladder up. Good luck to anyone outside the group of five people who are actively working in this niche! And finally, you write an abstract to your paper, where you mention the things you defined. The abstract doesn't make any self by itself, one needs to be in-the-know to get half of it, and read your paper to understand it. In practice, it acts as a "No Trespassing" sign for the outsiders (i.e. anyone not in direct contact with the five people you have beers with at the Annual Niche Field Conference). Satisfied, you lean back and post it to arXiv. It's been a beautiful day, you think. This practice is difficult to enforce, but you kept it as an aim, and got pretty close to perfection (as exemplified by a Bourbaki text, or anything by Serge Lang, but I repeat myself). Somewhere not too far away, a student in the class you're teaching cries. ---------------- I said "you", but as someone who's written a couple of math papers, that's really me too. We are all taught in a horrendously backwards (literally!) manner. This perversion of the beautiful art isn't a new observation. I can't write better about it than Vladimir Arnold[1] (a titan whose name is, I hope, familiar to you). It's worth a read to anyone who has ever studied mathematics: [1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

krisoft 5y ago

> A beautiful advice … Oh my goodness! So there are actual people who prefer that? I had to chew my way a fair share of books like that and I can’t stand them. Clearly the person who thought of these definitons, propositions and proofs had some reason to think of them. Sometimes they were trying to solve a problem, sometimes they were combining ideas, sometimes they were looking for structures with certain aesthetic properties. There was always a why behind why they thought about this and not something else. Sometimes there are multiple possible such reasons, that is fine. In that case the author can select whichever they fancy the most. Every time i had the misfortune to read a book like what you describe it felt like I was eating powdered milk without reconstituting it. There was a clear chain of thought between ideas and they choose to just hide it. I understand that math requires work. One needs to get a paper and a pencil and work out examples, check proofs, play with definitions. But why wouldn’t the author write down what made them care about the next item? > it is helpful to keep this as an aim. Why?