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There is a particular result from another author that I wanted to use in my research which is a central component to my research question (it’s an animal’s growth rate). However, deriving the results computationally requires an understanding of some areas of mathematics that neither I nor my supervisor have had training in. The nature of my work draws heavily on objects from mathematical biology, and I’ve recognized that this is a weakness in my education portfolio I intend to fill when I do my PhD.

I was making some progress in learning the math on my own, but I occasionally needed to reach out to the authors for clarification because I had nobody else I could speak with. In the end, the authors volunteered to give me their code so that I could produce their results and allow me to proceed with my research.

I find myself in the conundrum that I don’t know how to reconcile or could defend when pressed by a committee: conceptually I understand what the authors are doing, but I lack the training and time to do the computations entirely on my own. Yet I have the tool I need to continue on with my research.

I feel like this scenario is not entirely uncommon, particularly for graduate students, but I’m wondering if a reasonable defence can be formed for using another’s work (given the context above) if pressed by a committee. I only have an analogy that I would offer as a defence if pressed today. How can I approach this issue?

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    Reaching out to the authors is a good plan.
    – Buffy
    Jun 9, 2020 at 13:41
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    I’m definitely not shy when it comes to cold contacting, but once the initial relationship was established I didn’t want the email chain to converge to a point where I was always asking them questions - I don’t want to become their burden. I think the authors also recognized that, hence why they offered to just let me use the code they made.
    – GrayLiterature
    Jun 9, 2020 at 13:50
  • Ask them to point you to resources they know of that can help you get up to speed.
    – Buffy
    Jun 9, 2020 at 13:54
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    It's hard to say without knowing the details, but I would challenge the view that you understand the mathematical tools/concept if you cannot actually manipulate them (not having time/energy to re-derive them is another issue). That said, it's often fine in most field to not really understand some of the tools used.
    – fqq
    Jun 9, 2020 at 15:49

3 Answers 3

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When it comes to applying mathematical results, it is the norm that you do not understand all the nuts and bolts of what you are applying. This is nothing to be ashamed of. Instead, it’s the only way that modern science can work – it is too vast and intertwined to be understood in its entirety by one person. The crucial part is that there is a clear interface between theory and application where everything is communicated clearly.

To give an illustrative example: I recently wrote a paper where I derive some rules for building ecological models from basic principles. I did my best to describe said principles as well as the rules and their consequences so that those applying my work can use them. However, I expect that the vast majority of my readership cannot understand the mathematical proofs that go in between. I must expect that they treat this as a black box. On the other hand, in order to perform my proofs, I in turn relied on theorems whose proofs I did not understand. Mathematical theorems are great interfaces. They give you some statement of the form: “If X, then Y.” You only need to understand the statements X and Y, but you do not understand why one implies the other. That’s what mathematicians are for. On the other hand, the mathematicians who proved the theorems I use have no particular expertise in ecology as far as I know.

Now, there may not always be a clear interface. In that case, I suggest to create one yourself (and not only for yourself but also for your readers): Write a passage for your thesis that describes what you understand and need the tool to do. Then back up everything with references from the relevant publications and documentation. If necessary, ask the authors again.

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    (Good) Mathematicians themselves don't always understand the proofs of results they rely on. Some things are above one's weight class; a deep result relying on complex abstract nonsense and crossing across multiple fields may be very difficult for most people to grasp, but still relatively easy to utilize, as you describe.
    – zibadawa timmy
    Jun 10, 2020 at 6:11
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    @zibadawatimmy: I wouldn’t use the term nonsense here. If it’s nonsense, you obviously should not rely on it.
    – Wrzlprmft ♦
    Jun 10, 2020 at 7:00
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    @Wrzlprmft "Abstract nonsense" is a term in the mathematical community conveying the highest respect for difficulty and achievement, actually. Loosely speaking it refers to a high level of mastery/deployment of categorical concepts (often to achieve potent results in non-categorical fields/contexts). The proof can can be quick and concise (see the example on the link), but the density of the information content quite high, and it can be very difficult to see the "abstract nonsense" solution without having already been given it.
    – zibadawa timmy
    Jun 10, 2020 at 7:08
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    @zibadawatimmy: I see. Mathematicians really have a talent for unfortunate terminology.
    – Wrzlprmft ♦
    Jun 10, 2020 at 7:12
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    @Wrzlprmft It was originally used as a self-deprecating term, but quickly got repurposed as slang for categorical arguments. As category theory is the pinnacle of most of modern mathematics, both in complexity and utility, the utilization and mastery of such arguments is currently a highly valued skilled. The subject matter is indeed very abstract, as category theory generalizes huge swathes of mathematics and subsequently provides fertile ground for connections between seemingly disparate areas. It can thus seem like nonsense to those unaccustomed to it.
    – zibadawa timmy
    Jun 10, 2020 at 7:20
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Imho what matters is whether understanding the theoretical background is crucial for your own research: for instance if your research focuses on analyzing the outcome of this system in a new context, then you're just using this system as a tool in your research. Obviously it's always better to understand the tools you use very well, but you probably don't need to be able to re-implement their tool for using it correctly. On the other hand if your research goal is for instance to modify this system for a new application, then clearly you must acquire an in-depth understanding of it.

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If I understand correctly, you have a problem which is described in a mathematical model. You need to solve this model for some conditions and what you care about are the results.

If so, you do not need to understand how the calculations will be done, you can treat is as a black box. The same way as you do not need to understand assembler when using Python pandas.

There is however a very important thing you must understand: the conditions/limitations of the numerical recipe you will be using. You can have 20 methods to solve one problem (say, a differential equation) but some of them will be better, worse, unusable or plain wrong for that specific equation.

You can say: "I used method ABC to solve the equation in the model because it is a stiff equation and this and that. This allowed me to get a results with 10% precision in a reasonable time".

You do not need to explain how that method works (if you have no love for these methods, they are horribly complicated), but you must show you understand the limitations and therefore the implications on the result.

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