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I am writing a paper aimed at a physics journal which extends an old paper (not by me), which introduced a mathematical formalism for a physical problem in a rather handwaving way and without addressing most of its mathematical background. In addition to exentding this formalism, my paper elaborates on this mathematical background and relatedly introduces the formalism in a ”more rigorous” way. I consider the latter one of my paper’s key features for the following reasons:

  • One might consider the formalism and the methods derived from it not to be properly substantiated in the old paper.
  • Though the mathematical background has no other application so far, it is interesting for its own sake and perhaps from a philosophical point of view.
  • My approach to the formalism might be more accessible to some people and make them give a better understanding of what they are doing when applying the formalism and related methods.

As these points are rather opinion-based, I do not intend to elaborate on them in the paper – at most I might shortly mention the didactical aspects in the conclusion. (I mainly mentioned them here to give you some idea what I am talking about.) However, I do want to briefly mention in the abstract or the introduction that I “rigorified” the formalism, where briefly can be anything between one word and two sentences. For example, my abstract could look like:

[Old paper] introduced [old concept], which is useful for [application]. We extend this concept to [new concept] and also rigorify its mathematical background.

My problem is that rigorify its mathematical background is far from what I actually want to say. I am therefore looking for a way to say this without seeming arrogant or condescending on the old paper and in particular without implying that the old paper “did not do its mathematics properly”.

I intentionally do not give my best solutions for now, as I do not want to induce any bias on them, in case I may underestimate them.

I am also torn between Academia and English Language & Usage for posting this question.

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The word "physical" can be useful in situations like this (if the other paper was written by physicists). E.g. "this was done at a physical level of rigor by..." –  Noah Snyder Apr 25 at 20:57
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@NoahSnyder: This is exactly the kind of tone, I want to avoid. And that’s not even taking into account that I am a phyisicist and I am intending to submit the paper to a physics journal. –  Wrzlprmft Apr 25 at 22:29
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@Wrzlprmft: The fact that you are a physicist might be relevant information for the question. Also, if what you are adding is mostly mathematics, have you considered submitting to a math journal instead? (Or perhaps an interdisciplinary journal?) –  Pete L. Clark Apr 25 at 23:44
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For inspiration, here's an example of an existing paper that does something similar: synapse.cs.byu.edu/~dan/673/papers/wiggins.pdf . Quote: "I will attempt to make Boden’s descriptive hierarchy more precise. In doing so, I will suggest some additions to the theory, which may or may not be implicit in Boden’s account and show how some of the distinctions over which she has been challenged may perhaps be supported. The formalisation will also make it possible to identify desirable and undesirable properties of creative systems in abstract terms." –  Kaj_Sotala Apr 26 at 6:06
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I like Kaj_Sotala's comment. You can also add "tongue in cheek" remarks like, "At one level, my contribution can be seen as a mathematical paraphrase of the path-breaking intuitions of so and so; at another level, I like to think that I have built a small bridge to connect two distinct fields of inquiry" or something like that, make a joke about your ambition, to cancel out the possible perception that you're overstretching the importance of your contribution. Maybe. –  PatrickT Apr 26 at 9:07

4 Answers 4

up vote 14 down vote accepted

It sounds to me like you are viewing the older paper through the lens of the type of paper you would have written in the first place and are finding it wanting. It also sounds like you are viewing your own work as "fixing the flaws" of the older paper, almost as if you want to replace the older paper in your mind with your paper and pretend it had been written that way all along. More quantitatively you have your eye on the mark that the older paper should have hit, you are filling in the gap between the actual old paper and your eyeballed mark, and since you are measuring from the eyeballed mark you are giving the difference a minus sign. This framing seems to be behind most of your problems. Try recasting the entire thing more positively in your own mind.

1) Someone else published an inspirational paper way back when. This paper introduced some formalism and concentrated on its application to a physical problem.

2) Your paper gives much deeper attention to the mathematical aspects of the formalism, while also extending the formalism. Since 1) showed that the formalism is interesting and useful, your work has evident value.

There are no "minus signs" in the above description: 1) + 2) = your eyeball mark.

Unless there are actual mathematical errors in the older paper, you don't need to say that you are fixing or "rigorifying" (not really a word, by the way) the older paper. By the way, I don't know whether you've heard my rant about this use of the word "rigor" in mathematics and its applications. As an adult mathematician I have become increasingly skeptical of mathematical "rigor": the other paper either made mistakes (which you will need to correct), made claims which were unjustified or insufficiently justified (in which case you make clear that what you are contributing is the justification and not the claims themselves) or they weren't doing mathematics at all (which is fine: that's what you're adding). Adding rigor must mean one of the things above, right? It is not some generic ingredient that you can sprinkle more or less liberally over a piece of "unrigorous mathematics" and make it rigorous.

In summary: unless there are some clear mistakes, you don't need to bill your work as fixing their mathematics. You can bill it either as adding the math or adding more math. Both of these are good things added to other good things: no problem.

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Large parts of this answer capture my motivation for this asking my question really well: Adding more math is exactly what I consider myself to do, but I lack a word or phrase to describe it that does not sound like I am fixing some shortcoming of the old paper. I like your formulation “giving deeper attention to the mathematical aspects of the formalism“, which is at the very least a good starting point. — I also share your dislike of the word rigor, which is why I put in in quotation marks (except one omission). –  Wrzlprmft Apr 25 at 22:21

How about something like this:

WeakMath et al [foo] introduced a formalism for the problem we study. Their framework conveys valuable intuition* about [the problem] but is not precise enough for further development of the mathematical structures introduced. In this work, we elaborate on their ideas, placing them in a formal mathematical [something] that allows us to [do awesome things]

* if it does

You get the general idea: give them credit for intuiting the right ideas which will lessen the blow when you drop the hammer on them :)

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So far as I can see this may still be unnecessarily negative. The formalism of the older paper was precise enough for whatever their purpose was. The OP is adding new mathematical structure on something that has proven to have important applications. Adding to someone else's work is not a critique, is it? –  Pete L. Clark Apr 25 at 19:17
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It's possible. Hard to say without details. I've certainly been in a position where the prior work was NOT precise enough, because the paper somehow escaped reviewer attention and was intuitive. In such a case, I'd like to point out that my formalism adds a nonzero amount of value. –  Suresh Apr 25 at 19:52
    
In addition to Pete L. Clark’s comment, my “added mathematics” do not allow for anything new so far. –  Wrzlprmft Apr 26 at 9:20

You are taking a problem from another paper and providing additional perspective which will allow future research to solve or add to solving the problem. This does not make the original paper any less worthy.

To get into the mindset for writing in a non-dismissive fashion, you might adopt the following scenario: "Their paper is the greatest thing since sliced bread! My model and formalization will allow others to handle this and similar problems with a more formal perspective. How can it possibly be any more win-win?" . This may exaggerate the current situation, but it is not a lie when framed properly. With such an attitude, you can write glowingly about the inspiration provided by their paper, and the anticipated benefits your perspective and formalization will give. I don't see any need to "drop a hammer" on anyone. Putting myself in such a mindset, and making things general, I come up with:

"We take inspiration from Their paper [1], and provide additional mathematical perspective. Our model of the problem reflects the intuition in Their paper, and has among its benefits a framework which we feel can be carried to other situations. In particular, we believe it furthers formal and rigorous treatment of the problem."

Take reasonable precautions: whatever you write, have your mentor or colleagues review it. If someone in your department who is politically adept approves, you can try sending an advance copy to the authors of the inspiring article, to see what suggestions they may provide. DO NOT send them a copy without such outside wisdom and approval.

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I like “provide additional mathematical perspective”. “we believe it furthers formal and rigorous treatment of the problem” does not really apply however and also sounds as if I was claiming that the treatment in the old paper lacked something (formality and rigor), which is what I want to avoid. –  Wrzlprmft Apr 26 at 9:28
    
@Wrzlprmft, it may sound that way to you, and of course you have to be satisfied with your choice of words. I used "further" in an additive sense, not to claim the original paper lacked any, but that you found more to add. In any case, I hope you find something that works for you. –  Not Quite An Outsider Apr 28 at 18:48
    
Also, you may find English Language Usage.SE helpful in this regard. I don't know if the question is on topic, but the folks there may have useful opinions on the tone of a particular paragraph. –  Not Quite An Outsider Apr 28 at 18:50

In this paper, we re-examine the results of [OriginalReference], extending its ideas to [X and Y and Z], providing a new approach via [method] to the original which may better illuminate [aspect W].

Essentially, "okay, the original work was pretty cool but we can do more with it, so we're going to run with that to get to this new stuff, specifically by doing this thing here because it makes a point or points more accessible and obvious".

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