What is the recommended level of detail for published mathematical proofs?

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Suppose I am writing a Mathematics paper to a peer review reputed journal. I have to go from step 1 to step 2 in the paper which needs the knowledge of some well-established theorem X. Can I go from step 1 to step 2 without saying anything about X?



For example, this page contains some identities related to Fibonacci numbers, which (I assume that) may be well known to reviewers. So, instead of explicitly showing the identity, I want to use it implicitly from going one statement to another statement.



Does it give a poor impression on my paper? If I need to include all supporting identities, the paper may become too long!










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  • 2




    If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression.
    – GEdgar
    Aug 15 at 13:49







  • 2




    For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized.
    – user37208
    Aug 15 at 14:44






  • 2




    If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary.
    – Faheem Mitha
    Aug 15 at 17:24










  • I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience.
    – seteropere
    Aug 16 at 1:59














up vote
18
down vote

favorite
4












Suppose I am writing a Mathematics paper to a peer review reputed journal. I have to go from step 1 to step 2 in the paper which needs the knowledge of some well-established theorem X. Can I go from step 1 to step 2 without saying anything about X?



For example, this page contains some identities related to Fibonacci numbers, which (I assume that) may be well known to reviewers. So, instead of explicitly showing the identity, I want to use it implicitly from going one statement to another statement.



Does it give a poor impression on my paper? If I need to include all supporting identities, the paper may become too long!










share|improve this question



















  • 2




    If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression.
    – GEdgar
    Aug 15 at 13:49







  • 2




    For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized.
    – user37208
    Aug 15 at 14:44






  • 2




    If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary.
    – Faheem Mitha
    Aug 15 at 17:24










  • I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience.
    – seteropere
    Aug 16 at 1:59












up vote
18
down vote

favorite
4









up vote
18
down vote

favorite
4






4





Suppose I am writing a Mathematics paper to a peer review reputed journal. I have to go from step 1 to step 2 in the paper which needs the knowledge of some well-established theorem X. Can I go from step 1 to step 2 without saying anything about X?



For example, this page contains some identities related to Fibonacci numbers, which (I assume that) may be well known to reviewers. So, instead of explicitly showing the identity, I want to use it implicitly from going one statement to another statement.



Does it give a poor impression on my paper? If I need to include all supporting identities, the paper may become too long!










share|improve this question















Suppose I am writing a Mathematics paper to a peer review reputed journal. I have to go from step 1 to step 2 in the paper which needs the knowledge of some well-established theorem X. Can I go from step 1 to step 2 without saying anything about X?



For example, this page contains some identities related to Fibonacci numbers, which (I assume that) may be well known to reviewers. So, instead of explicitly showing the identity, I want to use it implicitly from going one statement to another statement.



Does it give a poor impression on my paper? If I need to include all supporting identities, the paper may become too long!







journals peer-review writing-style






share|improve this question















share|improve this question













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edited Aug 15 at 13:37









Bitwise

9,32213352




9,32213352










asked Aug 15 at 9:19









hind

633414




633414







  • 2




    If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression.
    – GEdgar
    Aug 15 at 13:49







  • 2




    For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized.
    – user37208
    Aug 15 at 14:44






  • 2




    If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary.
    – Faheem Mitha
    Aug 15 at 17:24










  • I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience.
    – seteropere
    Aug 16 at 1:59












  • 2




    If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression.
    – GEdgar
    Aug 15 at 13:49







  • 2




    For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized.
    – user37208
    Aug 15 at 14:44






  • 2




    If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary.
    – Faheem Mitha
    Aug 15 at 17:24










  • I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience.
    – seteropere
    Aug 16 at 1:59







2




2




If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression.
– GEdgar
Aug 15 at 13:49





If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression.
– GEdgar
Aug 15 at 13:49





2




2




For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized.
– user37208
Aug 15 at 14:44




For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized.
– user37208
Aug 15 at 14:44




2




2




If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary.
– Faheem Mitha
Aug 15 at 17:24




If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary.
– Faheem Mitha
Aug 15 at 17:24












I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience.
– seteropere
Aug 16 at 1:59




I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience.
– seteropere
Aug 16 at 1:59










4 Answers
4






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up vote
31
down vote













What is obvious to you may not be obvious to others.



Let me repeat a story I told in another answer here. I'll just repeat it, but also note that user Stella Biderman commented that the story may be apocryphal and not from KU at all.




Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes.



Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."




The conclusion is that you should make the connections yourself, and not rely on reviewers grokking it. If they don't make the connection they will comment on it and at least slow down acceptance. It is also possible, as the story above, indicates that you might actually be making too big a step or even an incorrect one.






share|improve this answer
















  • 2




    I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
    – Theoretical Economist
    Aug 15 at 16:25






  • 7




    I've heard multiple versions of this urban legend as well, but never about anyone in particular.
    – JeffE
    Aug 15 at 19:28






  • 3




    A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
    – CoffeeBliss
    Aug 15 at 20:55






  • 8




    @CoffeeBliss, Fermat did something similar as I recall ;-)
    – Buffy
    Aug 15 at 20:57






  • 5




    There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
    – Ray
    Aug 15 at 22:38


















up vote
20
down vote













You need to write your paper so that a reasonable reader is able to verify everything. (Here, "reasonable" reader depends on your audience. The level of detail is not the same if you write a course for high school, a paper for a top generalist journal, a specialized journal, a popularization magazine, slides for a talk, etc.)



You also need to keep in mind that the reader cannot read your mind, only what you have written. You are not writing for yourself; you are writing for your reader.



When you go from one step to the next in a computation, only you know what identity you have used if you do not write it. Even if the identity is well-known, it is not necessarily obvious for someone other than you which one it is. In some cases it's possible to determine easily what was used (e.g. if you write sum_nge0 1/2^n = 2 then hopefully everyone will be able to tell what's going on). But in other cases (especially if it's in the middle of a long computation), this may not be the case. So you need to seriously ask yourself whether someone who is not in your shoes can tell what's going on. If necessary, ask a colleague/friend to read it.



Because what will happen to your paper is even worse than giving a bad impression. It will be unconvincing. And an unconvincing math paper is basically worthless.






share|improve this answer


















  • 4




    +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
    – JeffE
    Aug 15 at 19:34











  • @JeffE For sure. I'll edit my answer.
    – Najib Idrissi
    Aug 16 at 8:11

















up vote
6
down vote













Look at what level of detail other papers in the area use.



Ask your advisor for advice.



With journals moving more and more online, length isn't really a problem, these days. Most people would probably prefer to read a 40-page paper that spells everything out than read a 20-page paper where they have to scribble down 20 more pages of calculations and 20 pages of failed attempts at calculations.



Finally, if it's just a matter of replacing "Therefore, Z" with "Z now follows from Theorem X", why not just do it?






share|improve this answer



























    up vote
    4
    down vote














    If I need to include all supporting identities, the paper may become too long!




    As others have noted, there's a middle ground here: you don't have to explain (let alone prove) every identity you use, but you can simply mention it ("by X, we have ...") to save the reader having to guess.



    Looking at it from an economic perspective, there's a small cost to you in mentioning X (it takes you that little bit closer to the journal's word/page limit, so you might have to tighten your wording elsewhere), but a potentially huge cost to the reader who has to work out how you got from step 1 to step 2 if you don't mention X at all (I'm sure as students we've all cursed authors who stole hours from our lives in this manner). And there are many readers vs one of you so that multiplies the cost even further.



    Thinking of this cost/benefit trade-off can help you determine the right level of detail. When you're wondering how explicit you need to be, you can ask yourself "what are the chances that a reader of this journal will know what I'm doing, and how much time will it cost them to work it out if they don't?"






    share|improve this answer
















    • 1




      Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
      – paul garrett
      Aug 16 at 0:04











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    4 Answers
    4






    active

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    4 Answers
    4






    active

    oldest

    votes









    active

    oldest

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    active

    oldest

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    up vote
    31
    down vote













    What is obvious to you may not be obvious to others.



    Let me repeat a story I told in another answer here. I'll just repeat it, but also note that user Stella Biderman commented that the story may be apocryphal and not from KU at all.




    Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes.



    Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."




    The conclusion is that you should make the connections yourself, and not rely on reviewers grokking it. If they don't make the connection they will comment on it and at least slow down acceptance. It is also possible, as the story above, indicates that you might actually be making too big a step or even an incorrect one.






    share|improve this answer
















    • 2




      I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
      – Theoretical Economist
      Aug 15 at 16:25






    • 7




      I've heard multiple versions of this urban legend as well, but never about anyone in particular.
      – JeffE
      Aug 15 at 19:28






    • 3




      A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
      – CoffeeBliss
      Aug 15 at 20:55






    • 8




      @CoffeeBliss, Fermat did something similar as I recall ;-)
      – Buffy
      Aug 15 at 20:57






    • 5




      There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
      – Ray
      Aug 15 at 22:38















    up vote
    31
    down vote













    What is obvious to you may not be obvious to others.



    Let me repeat a story I told in another answer here. I'll just repeat it, but also note that user Stella Biderman commented that the story may be apocryphal and not from KU at all.




    Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes.



    Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."




    The conclusion is that you should make the connections yourself, and not rely on reviewers grokking it. If they don't make the connection they will comment on it and at least slow down acceptance. It is also possible, as the story above, indicates that you might actually be making too big a step or even an incorrect one.






    share|improve this answer
















    • 2




      I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
      – Theoretical Economist
      Aug 15 at 16:25






    • 7




      I've heard multiple versions of this urban legend as well, but never about anyone in particular.
      – JeffE
      Aug 15 at 19:28






    • 3




      A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
      – CoffeeBliss
      Aug 15 at 20:55






    • 8




      @CoffeeBliss, Fermat did something similar as I recall ;-)
      – Buffy
      Aug 15 at 20:57






    • 5




      There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
      – Ray
      Aug 15 at 22:38













    up vote
    31
    down vote










    up vote
    31
    down vote









    What is obvious to you may not be obvious to others.



    Let me repeat a story I told in another answer here. I'll just repeat it, but also note that user Stella Biderman commented that the story may be apocryphal and not from KU at all.




    Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes.



    Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."




    The conclusion is that you should make the connections yourself, and not rely on reviewers grokking it. If they don't make the connection they will comment on it and at least slow down acceptance. It is also possible, as the story above, indicates that you might actually be making too big a step or even an incorrect one.






    share|improve this answer












    What is obvious to you may not be obvious to others.



    Let me repeat a story I told in another answer here. I'll just repeat it, but also note that user Stella Biderman commented that the story may be apocryphal and not from KU at all.




    Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes.



    Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."




    The conclusion is that you should make the connections yourself, and not rely on reviewers grokking it. If they don't make the connection they will comment on it and at least slow down acceptance. It is also possible, as the story above, indicates that you might actually be making too big a step or even an incorrect one.







    share|improve this answer












    share|improve this answer



    share|improve this answer










    answered Aug 15 at 12:55









    Buffy

    17.4k55497




    17.4k55497







    • 2




      I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
      – Theoretical Economist
      Aug 15 at 16:25






    • 7




      I've heard multiple versions of this urban legend as well, but never about anyone in particular.
      – JeffE
      Aug 15 at 19:28






    • 3




      A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
      – CoffeeBliss
      Aug 15 at 20:55






    • 8




      @CoffeeBliss, Fermat did something similar as I recall ;-)
      – Buffy
      Aug 15 at 20:57






    • 5




      There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
      – Ray
      Aug 15 at 22:38













    • 2




      I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
      – Theoretical Economist
      Aug 15 at 16:25






    • 7




      I've heard multiple versions of this urban legend as well, but never about anyone in particular.
      – JeffE
      Aug 15 at 19:28






    • 3




      A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
      – CoffeeBliss
      Aug 15 at 20:55






    • 8




      @CoffeeBliss, Fermat did something similar as I recall ;-)
      – Buffy
      Aug 15 at 20:57






    • 5




      There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
      – Ray
      Aug 15 at 22:38








    2




    2




    I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
    – Theoretical Economist
    Aug 15 at 16:25




    I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different.
    – Theoretical Economist
    Aug 15 at 16:25




    7




    7




    I've heard multiple versions of this urban legend as well, but never about anyone in particular.
    – JeffE
    Aug 15 at 19:28




    I've heard multiple versions of this urban legend as well, but never about anyone in particular.
    – JeffE
    Aug 15 at 19:28




    3




    3




    A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
    – CoffeeBliss
    Aug 15 at 20:55




    A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious!
    – CoffeeBliss
    Aug 15 at 20:55




    8




    8




    @CoffeeBliss, Fermat did something similar as I recall ;-)
    – Buffy
    Aug 15 at 20:57




    @CoffeeBliss, Fermat did something similar as I recall ;-)
    – Buffy
    Aug 15 at 20:57




    5




    5




    There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
    – Ray
    Aug 15 at 22:38





    There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved".
    – Ray
    Aug 15 at 22:38











    up vote
    20
    down vote













    You need to write your paper so that a reasonable reader is able to verify everything. (Here, "reasonable" reader depends on your audience. The level of detail is not the same if you write a course for high school, a paper for a top generalist journal, a specialized journal, a popularization magazine, slides for a talk, etc.)



    You also need to keep in mind that the reader cannot read your mind, only what you have written. You are not writing for yourself; you are writing for your reader.



    When you go from one step to the next in a computation, only you know what identity you have used if you do not write it. Even if the identity is well-known, it is not necessarily obvious for someone other than you which one it is. In some cases it's possible to determine easily what was used (e.g. if you write sum_nge0 1/2^n = 2 then hopefully everyone will be able to tell what's going on). But in other cases (especially if it's in the middle of a long computation), this may not be the case. So you need to seriously ask yourself whether someone who is not in your shoes can tell what's going on. If necessary, ask a colleague/friend to read it.



    Because what will happen to your paper is even worse than giving a bad impression. It will be unconvincing. And an unconvincing math paper is basically worthless.






    share|improve this answer


















    • 4




      +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
      – JeffE
      Aug 15 at 19:34











    • @JeffE For sure. I'll edit my answer.
      – Najib Idrissi
      Aug 16 at 8:11














    up vote
    20
    down vote













    You need to write your paper so that a reasonable reader is able to verify everything. (Here, "reasonable" reader depends on your audience. The level of detail is not the same if you write a course for high school, a paper for a top generalist journal, a specialized journal, a popularization magazine, slides for a talk, etc.)



    You also need to keep in mind that the reader cannot read your mind, only what you have written. You are not writing for yourself; you are writing for your reader.



    When you go from one step to the next in a computation, only you know what identity you have used if you do not write it. Even if the identity is well-known, it is not necessarily obvious for someone other than you which one it is. In some cases it's possible to determine easily what was used (e.g. if you write sum_nge0 1/2^n = 2 then hopefully everyone will be able to tell what's going on). But in other cases (especially if it's in the middle of a long computation), this may not be the case. So you need to seriously ask yourself whether someone who is not in your shoes can tell what's going on. If necessary, ask a colleague/friend to read it.



    Because what will happen to your paper is even worse than giving a bad impression. It will be unconvincing. And an unconvincing math paper is basically worthless.






    share|improve this answer


















    • 4




      +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
      – JeffE
      Aug 15 at 19:34











    • @JeffE For sure. I'll edit my answer.
      – Najib Idrissi
      Aug 16 at 8:11












    up vote
    20
    down vote










    up vote
    20
    down vote









    You need to write your paper so that a reasonable reader is able to verify everything. (Here, "reasonable" reader depends on your audience. The level of detail is not the same if you write a course for high school, a paper for a top generalist journal, a specialized journal, a popularization magazine, slides for a talk, etc.)



    You also need to keep in mind that the reader cannot read your mind, only what you have written. You are not writing for yourself; you are writing for your reader.



    When you go from one step to the next in a computation, only you know what identity you have used if you do not write it. Even if the identity is well-known, it is not necessarily obvious for someone other than you which one it is. In some cases it's possible to determine easily what was used (e.g. if you write sum_nge0 1/2^n = 2 then hopefully everyone will be able to tell what's going on). But in other cases (especially if it's in the middle of a long computation), this may not be the case. So you need to seriously ask yourself whether someone who is not in your shoes can tell what's going on. If necessary, ask a colleague/friend to read it.



    Because what will happen to your paper is even worse than giving a bad impression. It will be unconvincing. And an unconvincing math paper is basically worthless.






    share|improve this answer














    You need to write your paper so that a reasonable reader is able to verify everything. (Here, "reasonable" reader depends on your audience. The level of detail is not the same if you write a course for high school, a paper for a top generalist journal, a specialized journal, a popularization magazine, slides for a talk, etc.)



    You also need to keep in mind that the reader cannot read your mind, only what you have written. You are not writing for yourself; you are writing for your reader.



    When you go from one step to the next in a computation, only you know what identity you have used if you do not write it. Even if the identity is well-known, it is not necessarily obvious for someone other than you which one it is. In some cases it's possible to determine easily what was used (e.g. if you write sum_nge0 1/2^n = 2 then hopefully everyone will be able to tell what's going on). But in other cases (especially if it's in the middle of a long computation), this may not be the case. So you need to seriously ask yourself whether someone who is not in your shoes can tell what's going on. If necessary, ask a colleague/friend to read it.



    Because what will happen to your paper is even worse than giving a bad impression. It will be unconvincing. And an unconvincing math paper is basically worthless.







    share|improve this answer














    share|improve this answer



    share|improve this answer








    edited Aug 16 at 8:13

























    answered Aug 15 at 9:32









    Najib Idrissi

    8,94123455




    8,94123455







    • 4




      +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
      – JeffE
      Aug 15 at 19:34











    • @JeffE For sure. I'll edit my answer.
      – Najib Idrissi
      Aug 16 at 8:11












    • 4




      +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
      – JeffE
      Aug 15 at 19:34











    • @JeffE For sure. I'll edit my answer.
      – Najib Idrissi
      Aug 16 at 8:11







    4




    4




    +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
    – JeffE
    Aug 15 at 19:34





    +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly.
    – JeffE
    Aug 15 at 19:34













    @JeffE For sure. I'll edit my answer.
    – Najib Idrissi
    Aug 16 at 8:11




    @JeffE For sure. I'll edit my answer.
    – Najib Idrissi
    Aug 16 at 8:11










    up vote
    6
    down vote













    Look at what level of detail other papers in the area use.



    Ask your advisor for advice.



    With journals moving more and more online, length isn't really a problem, these days. Most people would probably prefer to read a 40-page paper that spells everything out than read a 20-page paper where they have to scribble down 20 more pages of calculations and 20 pages of failed attempts at calculations.



    Finally, if it's just a matter of replacing "Therefore, Z" with "Z now follows from Theorem X", why not just do it?






    share|improve this answer
























      up vote
      6
      down vote













      Look at what level of detail other papers in the area use.



      Ask your advisor for advice.



      With journals moving more and more online, length isn't really a problem, these days. Most people would probably prefer to read a 40-page paper that spells everything out than read a 20-page paper where they have to scribble down 20 more pages of calculations and 20 pages of failed attempts at calculations.



      Finally, if it's just a matter of replacing "Therefore, Z" with "Z now follows from Theorem X", why not just do it?






      share|improve this answer






















        up vote
        6
        down vote










        up vote
        6
        down vote









        Look at what level of detail other papers in the area use.



        Ask your advisor for advice.



        With journals moving more and more online, length isn't really a problem, these days. Most people would probably prefer to read a 40-page paper that spells everything out than read a 20-page paper where they have to scribble down 20 more pages of calculations and 20 pages of failed attempts at calculations.



        Finally, if it's just a matter of replacing "Therefore, Z" with "Z now follows from Theorem X", why not just do it?






        share|improve this answer












        Look at what level of detail other papers in the area use.



        Ask your advisor for advice.



        With journals moving more and more online, length isn't really a problem, these days. Most people would probably prefer to read a 40-page paper that spells everything out than read a 20-page paper where they have to scribble down 20 more pages of calculations and 20 pages of failed attempts at calculations.



        Finally, if it's just a matter of replacing "Therefore, Z" with "Z now follows from Theorem X", why not just do it?







        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered Aug 15 at 16:54









        David Richerby

        27k556113




        27k556113




















            up vote
            4
            down vote














            If I need to include all supporting identities, the paper may become too long!




            As others have noted, there's a middle ground here: you don't have to explain (let alone prove) every identity you use, but you can simply mention it ("by X, we have ...") to save the reader having to guess.



            Looking at it from an economic perspective, there's a small cost to you in mentioning X (it takes you that little bit closer to the journal's word/page limit, so you might have to tighten your wording elsewhere), but a potentially huge cost to the reader who has to work out how you got from step 1 to step 2 if you don't mention X at all (I'm sure as students we've all cursed authors who stole hours from our lives in this manner). And there are many readers vs one of you so that multiplies the cost even further.



            Thinking of this cost/benefit trade-off can help you determine the right level of detail. When you're wondering how explicit you need to be, you can ask yourself "what are the chances that a reader of this journal will know what I'm doing, and how much time will it cost them to work it out if they don't?"






            share|improve this answer
















            • 1




              Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
              – paul garrett
              Aug 16 at 0:04















            up vote
            4
            down vote














            If I need to include all supporting identities, the paper may become too long!




            As others have noted, there's a middle ground here: you don't have to explain (let alone prove) every identity you use, but you can simply mention it ("by X, we have ...") to save the reader having to guess.



            Looking at it from an economic perspective, there's a small cost to you in mentioning X (it takes you that little bit closer to the journal's word/page limit, so you might have to tighten your wording elsewhere), but a potentially huge cost to the reader who has to work out how you got from step 1 to step 2 if you don't mention X at all (I'm sure as students we've all cursed authors who stole hours from our lives in this manner). And there are many readers vs one of you so that multiplies the cost even further.



            Thinking of this cost/benefit trade-off can help you determine the right level of detail. When you're wondering how explicit you need to be, you can ask yourself "what are the chances that a reader of this journal will know what I'm doing, and how much time will it cost them to work it out if they don't?"






            share|improve this answer
















            • 1




              Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
              – paul garrett
              Aug 16 at 0:04













            up vote
            4
            down vote










            up vote
            4
            down vote










            If I need to include all supporting identities, the paper may become too long!




            As others have noted, there's a middle ground here: you don't have to explain (let alone prove) every identity you use, but you can simply mention it ("by X, we have ...") to save the reader having to guess.



            Looking at it from an economic perspective, there's a small cost to you in mentioning X (it takes you that little bit closer to the journal's word/page limit, so you might have to tighten your wording elsewhere), but a potentially huge cost to the reader who has to work out how you got from step 1 to step 2 if you don't mention X at all (I'm sure as students we've all cursed authors who stole hours from our lives in this manner). And there are many readers vs one of you so that multiplies the cost even further.



            Thinking of this cost/benefit trade-off can help you determine the right level of detail. When you're wondering how explicit you need to be, you can ask yourself "what are the chances that a reader of this journal will know what I'm doing, and how much time will it cost them to work it out if they don't?"






            share|improve this answer













            If I need to include all supporting identities, the paper may become too long!




            As others have noted, there's a middle ground here: you don't have to explain (let alone prove) every identity you use, but you can simply mention it ("by X, we have ...") to save the reader having to guess.



            Looking at it from an economic perspective, there's a small cost to you in mentioning X (it takes you that little bit closer to the journal's word/page limit, so you might have to tighten your wording elsewhere), but a potentially huge cost to the reader who has to work out how you got from step 1 to step 2 if you don't mention X at all (I'm sure as students we've all cursed authors who stole hours from our lives in this manner). And there are many readers vs one of you so that multiplies the cost even further.



            Thinking of this cost/benefit trade-off can help you determine the right level of detail. When you're wondering how explicit you need to be, you can ask yourself "what are the chances that a reader of this journal will know what I'm doing, and how much time will it cost them to work it out if they don't?"







            share|improve this answer












            share|improve this answer



            share|improve this answer










            answered Aug 15 at 22:41









            Simon Brady

            1411




            1411







            • 1




              Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
              – paul garrett
              Aug 16 at 0:04













            • 1




              Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
              – paul garrett
              Aug 16 at 0:04








            1




            1




            Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
            – paul garrett
            Aug 16 at 0:04





            Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted.
            – paul garrett
            Aug 16 at 0:04


















             

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