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One of the techniques for proving statements in mathematics is "mathematical induction" (wikipedia entry). Very informally and not precisely speaking, when conducting a proof using this technique, (1) one proves that the statement is true in the simplest possible case, then (2) one assumes that the statement is true for a more complex case, and then (3) one proves that the assumption from step 2 implies the validity of the statement for even more complex case.

The assumption from step 2 is called "induction hypothesis". When executing step 3, one often reffers to this assumption. How should one refer to it? Is it

by induction hypothesis, ...

or

by the induction hypothesis, ...

?

In each of the wikipedia examples there is "the", which I believe to be correct, since we are referring to a specific, concrete assumption that has been made. I've seen mathematical textbooks that use only this variant, but also ones that mix both variants; the same goes for research articles.

abebebebahabe
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  • Always use 'the' here. – Mitch Oct 01 '19 at 15:23
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    Missing "the" is often a sign of second-language speakers of English who do not use definite articles in their first language. If the meaning is obvious (e.g. there is only one induction hypothesis at this point), it may not be picked up by editors or reviewers. – Henry Oct 01 '19 at 15:40
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    @Mitch By induction hypothesis, we can deduce. This is fine. The is optional. String theory tells us that .... – David M Oct 01 '19 at 16:13
  • @DavidM 1) that sounds awful. 2) I would think the rare instances of without the article would be be nonnative speakers (of non-arthrous languages). 3) String theory is not induction hypothesis. – Mitch Oct 01 '19 at 16:43
  • @Mitch So String Theory has special dispensation to drop it's article? I'm a native speaker like you, and while I'd probably use the article, I don't think it would offend me to hear it omitted. – David M Oct 01 '19 at 16:46
  • 'String Theory' has no special dispensation; it is one of innumerable non-count nouns. 'Hypothesis' is a count noun. – Mitch Oct 01 '19 at 16:53
  • @Henry so the reason for using "the" here is precisely because "induction hypothesis" is a countable noun and as such it requires an article? – abebebebahabe Oct 02 '19 at 07:31

1 Answers1

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The wording

by induction hypothesis

is a common solecism found in many mathematical texts. The natural way to say it in English is:

by the induction hypothesis.

The grammatical analysis is that one is referring to a particular hypothesis, which would require the definite article. One might say grammatically

by hypothesis X, ...

that is, X being the name of a given hypothesis, or

by hypothesis, ...

and in the following give an instantiation of the hypothesis with other things in order to make a deduction.

It is ungrammatical to say 'We use induction hypothesis' or 'Induction hypothesis implies...'; 'induction hypothesis' by itself is not a proper NP.

However 'by induction hypothesis' is a common wording, half as common as 'by the induction hypothesis', so it must be a partially accepted idiom patterned after 'by induction'.

Mitch
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