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(A bit difficult to describe what I mean without sharing my definitions, but I am cautious of posting them since this is regarding my thesis.) I had to provide a definition for a term I came up with. Since it is a complex matter, I provided one definition and then wrote "alternatively put:" and gave a second definition that describes it a bit differently and more elaborate. My advisor just put a question mark. I thought of merging both definitions together, but they are more or less different "views" of the term, so it didn't make sense. Is there an academically adequate way for me to keep both definitions of the same term?

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    Is this math? Lots of answers seem to assume it is. Commented Dec 30, 2022 at 15:47
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    Shouldn't the general answer to "what to do if my adviser puts a question mark" be "then talk to your adviser!"? Commented Dec 30, 2022 at 23:10
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    Consider the seven definitions of a matroid. Commented Jan 1, 2023 at 16:58

6 Answers 6

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If they are "views" then they are hardly "definitions". A definition should be precise. If you want to give two statements, which might occur in math, you need to be able to prove equivalence.

If they merely overlap in meaning then there may be cases in which one applies and the other doesn't. In informal language that is probably fine, but in a dissertation that is unlikely to be the case. It is hard to "define" goodness for example.

Note that something like a metaphor or analogy isn't a definition. A definition needs to distinguish what is and what is not conformable.

I'll go along with your advisor and put a question mark.

You can choose one, perhaps, or unify them formally, but two is too many.

If the same term is used in your field in two distinct ways then it may be defined differently for different aspects. But in such a case you will need to be extremely clear when you use one definition or the other. What is it that distinguishes them?

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    If they are "views" then they are hardly "definitions". Counterpoint: en.wikipedia.org/wiki/NP_(complexity)#Formal_definition Though I do agree that the thesis should probably pick a single thing and call it the definition.
    – user541686
    Commented Dec 30, 2022 at 9:42
  • @user541686 the difference in that example is that all those different definitions are provably equivalent. Commented Dec 30, 2022 at 15:53
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    @JordiVermeulen That doesn't sound like a difference to me. If multiple definitions aren't provably equivalent, there's been a mistake.
    – wizzwizz4
    Commented Dec 30, 2022 at 21:14
  • @user541686 Well, the first one is the definition, and is why it's called NP. The second one is not so much a definition as a result saying that something is in NP if and only if it satisfies these conditions. And it's often more convenient to check NP-ness using the result than the definition. Commented Dec 31, 2022 at 8:25
  • @wizzwizz4 I just meant that in some mathematical settings, there are different "views" of a certain concept that are provably equivalent (e.g. NP-ness by Turing machine, or by certificates). Sometimes one view is more convenient for whatever it is you're doing, and that's fine, because they're all the same. It's different in some other sciences, where the same term may have multiple definitions that are not provably equivalent, but subtly different. Commented Dec 31, 2022 at 23:54
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Echoing @Buffy's good answer, and emphasizing a certain point: even if there is no truly "correct" order of logical development of the ideas (meaning what is a definition, versus consequence of definition), you probably need to pick one... Then perhaps prove that another "definition/characterization" is equivalent, or, possibly, is equivalent when both characterizations apply(?).

After proving such an equivalence, then we can forget the (possibly artificial) logical ordering that we used at the beginning (to avoid circular reasoning...)

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  • I often write definitions like "The following are all equivalent: def1, def2, def3. We call anything satisfying any one of these equivalent conditions a ____". Then prove the equivalence. Commented Dec 31, 2022 at 21:51
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Here is a familiar example of how to write equivalent definitions,

Lemma 1.3.4: For any set $A \subset X$ the following two conditions are equivalent:

(a) The complement $X-A$ is open in $X$.

(b) Any sequence $(x_n)$ in $X$ with limit $x$ that has all $x_i \in A$ also satisfies $x \in A$.

Proof: Yadda yadda yadda QED.

Definition 1.3.5: The set $A \subset X$ is said to be closed in $X$ to mean it satisfies one and therefore both of conditions (a) and (b) above.

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I guess you can give multiple definitions, as long as you then explain why there are multiple definitions. For example, I can imagine if you are talking about "fatigue" you could say "for engineers, fatigue means... But when sports scientists use the term, they mean... The difference is interesting because...."

Basically, you don't want your readers to be puzzled by what's going on. If your supervisor (who is an expert) is puzzled, then everyone else will be too. Have a bash at rewording it, and run it past your supervisor again.

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What field is this? If you're doing qualitative work, it wouldn't be out of line to provide an elaborated definition and a more concise (TL;DR) version of the same concept. The shorter definition may be shorthand for the former. The shorter version could also be a more abstract concept that would meet with wide agreement (and be easy to grasp, maybe even by non-academics), while the longer version is a detailed description of exactly what does and doesn't fit into the concept. Indeed, when doing quantitative analysis in the social sciences, it's very common to give a conceptual definition, followed by a definition that operationalizes the concept for the analysis in question.

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  • You aren't describing a "definition" but an explanation or, perhaps, an analogy. If the shorter "definition" really defines the concept, what use is the longer one?
    – Buffy
    Commented Dec 30, 2022 at 19:35
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    The definition of "definition", and the uses of definitions, vary by field. For example, I used to do research on democracy, and there are literally dozens of different definitions of it. Most everyone agrees that a "democracy" is a country ruled through elections, but there are all sorts of nuances, like who can vote, how free the press is, and how fair elections are. When there are no true bright lines between categories, you can't construct perfect definitions. OTOH, for quantitative analysis, I'd need a very specific definition, which might depend on what's feasible given imperfect data.
    – MTKnife
    Commented Dec 30, 2022 at 20:04
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A definition must be comprehensive and unambiguous.

  1. If you need a second definition, it can mean that your first "definition" is still ambiguous, that you still need to be more precise. In that case, it means that the first "definition" was actually not a definition. So it would not be okay to have two definitions. Work harder on the first definition.

  2. Another case is if you have readers from very distinct fields who have distinct jargons which makes that some of them who aren’t familiar with the jargon of the first definition (assuming it is a true unambiguous definition) don’t really understand it. In that case, a second definition can be interesting as long as it’s just a sound paraphrase. But the best solution remains to find a definition whose words are understandable by everyone.

TL;DR. It’s always a bad idea to have two definitions. However, it can be interesting to present equivalences. That is the fact that one term in a specific field corresponds to another word in another field. This can allow to jump from one field to another field and connect knowledges together.

If you want to help people better understand what your object is by explaining things about it, by stating some consequences of its existence, or by telling its origin, etc. it doesn’t stand in the "definition" part but in the rest of your text parts (etymology, origin, consequences, corollaries, theorems, etc. as it depends on your working field).

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