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I was reading the paper Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, where the author phrased something as this:

The analyses of these methods are ad hoc but illustrate interesting theoretical techniques.

The term "ad hoc" here for me means that the analyses in those methods show certain degrees of intuition and are somewhat qualitative(?). Then I googled about this latin word since I have seen it too many times in many mathematics literatures, and this is what the wiki entry said:

In science and philosophy, an ad hoc hypothesis is a hypothesis added to a theory in order to save it from being falsified.

Now I am confused for the term ad hoc in the first paper(it seems to me at least) does not exactly follow the definition in wiki. Also as title said, I have seen "ad hoc" sometimes being used similar to "a priori", using an example in error analysis for numerical PDE I am working on right now, in J.T.Oden's famous review article in my field, he said:

During the early 1980s the search for effective adaptive methods led to a wide variety of ad hoc error estimators. Many of these were based on a priori or interpolation estimates.

And basically "ad hoc" here means that we do not use any information after the numerical approximation to construct the error estimators, then it is somewhat similar to "a priori".

So my questions are:

(1) What does "ad hoc" exactly mean in mathematical paper? is it an interchangeable term with "a priori"?

(2) When should we use these kind of latin words, for example, "ad hoc","ad infinitum" in our paper? is there a rule? or mathematicians simply follow the convention in their field to use these latin phrases in the papers.

I am not sure this question suits here, mod please feel free to close it if this question is inappropriate.

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2 Answers

up vote 6 down vote accepted

Ad hoc never means anything much like a priori. An ad hoc method is one devised for the specific case at hand; as such it contrasts with a general method. It may be an impromptu method, one devised on the fly, though that’s less likely in for something being reported in writing.

A priori estimate has a specific technical meaning. In general, however, the term simply means something like based on theory rather than evidence or assumed without evidence, though the exact sense depends on context.

I’ve no problem with the use of common Latin expressions, and I don’t even think of a priori and ad hoc as particularly foreign: they’ve been rather thoroughly naturalized into the language. Phrases like ceteris paribus (‘all else being equal’) and mutatis mutandis (‘making the necessary changes’) are more problematic: they’re useful, but a significant fraction of most audiences probably won’t understand them.

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1) The intended meaning is the one from the first part of the Wikipedia article, not the second part:

Ad hoc is a Latin phrase meaning "for this". It generally signifies a solution designed for a specific problem or task, non-generalizable, and not intended to be able to be adapted to other purposes.

1.5) A priori has a more precise meaning.

2) These things are all social / stylistic conventions. I personally think that Latin words should be kept to an absolute minimum for the sake of readability by a wide audience (although plenty of non-mathematicians use "ad hoc" and it's not particularly uncommon).

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