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I am 16 years old at the time of writing (so I have no supervisors to seek advice from) and I have written a mathematics research paper, which I plan on submitting to a journal for publication.

For a couple of the assertions that I make, I use proofs by induction. Now, in school we're encouraged to write proofs by induction in the following (rigid) format:

Base case:...

Assumption(s):...

Inductive step:...

Conclusion:...

I have noticed that no research articles that I have seen have written proofs by induction using this sort of format. The authors usually make it flow much more smoothly, eg 'For the base case, the result is trivial. Now assume the result holds for some $n=k$, so that... Now consider the expression for $n=k+1$...and by the inductive hypothesis this equals...hence the result is true by mathematical induction.'

So, is it good practice to write proofs by induction in the pretty rigid structure I first outlined or is it ok/better to write the proofs more naturally so that it flows better?

Thank you for your help.

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  • Who is your audience? Others like yourself or professionals?
    – Buffy
    Mar 18 '21 at 23:29
  • @Buffy I hope on the audience being professionals, but if a professional journal won't publish it then I'll probably put it on the arXiv. Mar 18 '21 at 23:31
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    You should get a mentor who can help you learn mathematical customs. Mar 18 '21 at 23:33
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    Let me urge restraint on the close votes. The age of the OP is not material to the question.
    – Buffy
    Mar 18 '21 at 23:40
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    This question is more suitable for math.stackexchange or mathoverflow.
    – Dan Romik
    Mar 19 '21 at 5:54
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As you probably have observed by now, by reading articles, people generally do not write steps down explicitly when those steps are considered "obvious." This trend is not special to proofs by induction.

In fact, this is true at any level, but what is considered "obvious" to an audience of people holding a PhD or with significant progress towards a PhD is different from what is "obvious" to someone taking first year calculus.

Think about it this way: people reading research articles know math quite well already. What they are interested in is what is new in your paper; demonstrating "obvious" facts is just a distraction. And, if you are at the stage where you will read several papers in a week, you generally want those papers to get to the point.

For people who are learning math, it's entirely different: you're probably only reading 1 textbook at a time, and the whole thing is new to you. So, you want everything explained in lots of detail because you've never seen it before. But again, if you're reading a calculus book, you probably don't need an explanation on solving a quadratic; you already know that. If you're learning induction, you want it spelled out; if you've seen proofs by induction ten thousand times, you don't need it spelled out for you.

Since you are a bit new to this, I would like to give some more examples about how proofs by induction are typically structured in upper undergraduate/graduate/research levels. Now, when it comes to proofs by induction, there are generally two categories: proofs where the induction step is "obvious" and proofs where the base-case is "obvious." Sometimes both are easy, but it's very rare in my experience to find a proof by induction where both the base case and the induction step require a lot of work to prove.

I suppose on the rare instance where both the base case and the induction step are non-obvious, I might split the proof into two lemmas or possibly I would call these "claims." In every other case, I write statements such as "the base case is trivial" or "the general case (the induction step) follows by induction."

I would also point out that there are even times where a proof will "prove" something by induction without the author ever writing that induction is being used. For instance:

Let f(n + 1) = 2f(n) and f(0) = 1. Then f(n) = 21 f(n - 1) = 22 f(n - 2) = . . . = 2k f(n - k) = . . . = 2n.

Induction is being used implicitly within the ellipses, but it hasn't been written explicitly so. I, personally, like to write "by induction" at the end of such a statement, but not everyone does.

I think your intuition of generally following how other people write mathematics is appropriate. You can deviate slightly, sure, but it's also good to understand why things are written the way they are and so it is great that you are asking this question.

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  • Thank you very much for your detailed and clear answer, I really appreciate it! Mar 19 '21 at 16:30
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If you are writing for professionals, then the paper, itself, should probably conform to what you have seen in print. That is, a more natural written style.

But, for your own work, an outline form may be best in the early stages so that you have more confidence that you don't miss something and that each piece of the puzzle is complete and correct.

I think the reasons for your teacher's suggestions is just that. Make sure you get it all and make sure you get it right. Then, the expository form for a paper can be extracted from your outline.


A note you might consider for the future. Once you write something technical it can be very difficult to proof read it. This is because you mind seems to read what you think you wrote, rather than what you did write. In other words, you "see" what you "expect to see". Thus, a formal, outline version can help you keep everything straight so that you can examine it more carefully, not "reading ahead" and thus missing something important.

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  • Thank you very much for your sound advice. I think you have answered every single one of my questions here on Academia.SE! Truly amazing. As a side point, I've been advised on Academia.SE more than once to get myself a mentor to help me; do you have any idea how I'd go about doing that? My Maths teachers have no useful contacts for me. Mar 19 '21 at 16:28
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    I don't have many ideas unless you are in a place close to a university. Much more complicated with the pandemic, but making a visit used to be possible for people trying to learn beyond their age level.
    – Buffy
    Mar 19 '21 at 18:50

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